Computer Analysis of Images and Patterns: Proceedings of the International Conference on Automatic Image Processing Held in Wismar (GDR), September 2–4, 1987 [Reprint 2021 ed.] 9783112473184, 9783112473177

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Mathematica/ Research Computer Analysis of Images and Patterns edited by L.P. YaroslavskiT- A.Rosenfeld W. Wilhelmi

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Computer Analysis of Images and Patterns

Mathematical Research Wissenschaftliche Beiträge herausgegeben von der Akademie der Wissenschaften der D D R Karl-Weierstraß-Institut für Mathematik

Band 40 C o m p u t e r Analysis of Images a n d Patterns

Mathematische Forschung

Computer Analysis of Images and Patterns Proceedings of the II. International Conference CAIP '87 on Automatic Image Processing held in Wismar (GDR), September 2 - 4 , 1 9 8 7 edited by Leonid P. Yaroslavsku Azriel Rosenfeld, Wolfgang Wilhelmi

Akademie-Verlag Berlin 1987

Herausgeber: P r o f . Dr. sc.techn. Leonid P. Yaroslavskil Akademie der Wissenschaften der UdSSR Institut

für Probleme der Informationsübertragung

(IPPI)

P r o f . Dr. phil. A z r i e l Rosenfeld University of Maryland Center for Automation

Research

P r o f . D r . sc.techn. Wolfgang

Wilhelm!

Akademie der Wissenschaften der DDR Zentralinstitut

für Kybernetik und

Informationsprozesse

Die Titel dieser Schriftenreihe werden vom Originalmanuskript Autoren

der

reproduziert

ISBN 3-05-500451-5 ISSN 0138-3019 Erschienen im Akademie-Verlag Berlin.DDR-1086 Berlin, Leipziger Str.3-4 ( ^ Akademie-Verlag Berlin

1987

Lizenznummer: 202 • 100/410/87 Printed in the German Democratic

Republic

Gesamtherstellung: VEB K o n g r e ß - und Werbedruck, 9273 Oberlungwitz Lektor: Dipl^-Phys. Gisela Lagowitz LSV 1085 Bestellnummer: 763 790 6 (2182/40) 03800

F o r e wo r d

volume

This

is

a collection

International

Conference

held

in W i s m a r ,

The

success

Automatic

GDR

from

achieved

Image

encouraged

of

on

Automatic

September by

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Processing

the

selected

2

papers Image

to

I.

4,

Processing

Processing for

Control

a

conference

doing so The

so

far

we

hope

played

image

of

machines

new

hand,

CAIP

intelligence

methods

reader

not

will

balanced The

quality

dependent invited

of

on

was

in

cooperation

WGMA.

"87

in

and

the

of

by

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the

and

years

importance

on

Berlin)

Automatic later.

and

great

its

influence

recognition

In role

progress real

image

towards world

the

on

generations

scenes. of

On

the

artificial

field.

breakthroughs

the

principles

new

penetration

analysis

reached

methods

infor m a t i o n - p r o c e s s i n g the

the

for

two

members was

Executive

showing and

volume.-

given

by

Technik

containing

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of

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pages

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of

highest to

complete

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taken

topicality

this

is

publication

refereeing.

decision

presentations (KdT)

of

accepted

final

approach

publication

'87

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but

rather

a

well

advance.

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innovative

image

spectacular

quality

with

its

the

demonstrated

into

evaluated The

shown

understanding

papers

the

Committee.

and

observe

continuous

have

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of

in

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increase

II.

circles.

kinds

parallel capable

second

will

expert

processing

(especially

other

CAIP

contributions

introduction

of

that

among

presented

by

organize

CAIP

Group

Society

to

-

the

Conference

17-18,1985

Scientific-Techno1ogical (WGMA)

at

1987.

International

(October

Image

delivered

by

paper

except

for

those

Programm

W.Wilhelmi

R.Klette

Image score CAIP

on

and

Processing in

by

Group

scientific

were

overview

abstract

essentially

International

publication of

was

the

every

level,

selected

including

in

for

poster

Kammer

der

contribution.

The of

editors the

would

program

like

to e x p r e s s

S.Fuchs

(Kiev)

(Wismar)

F.Sloboda

(Bratislava)

K.Voss

(Oena)

competent and

and

chairman

contributors.

A great

Dr.

R.Klette

during

are

very

obliged

of

help

to

sponsoring

made

in

Informatics

work

to h o s t

of t h e s e

Last,

not

a successful

Looking

the

editors and

this

and was

decision

V.Kempe,

and

due

of

this

conference and

the

to

CAIP'87

to

all

other

the

support

of

making

process.

We

Director

Information

of

the

Central

Processes,

Institute,

and

for

in c o n n n e c t i o n

to

their

with

the

WGMA.

Prof.Dr. in W i s m a r

providing

we

and M r . and

forward

would

D.Schmidt and

excellent

like

K.-D.Muller

fruitful

to C A I P

Azriel Rosenfeld Leonid Yaroslavski? Wolfgang Wilhelml

6

for

to

for

to M r s .

his

G.Lagowitz

conditions

for

the

Proceedings.

least,

H.Rustenbach,

ensure

sessions,

the G D R

are

publication

Mrs.

of

Akademie-Verlag

but

the

Director

thanks

organizational the

of

Cybernetics

efforts

from

cooperation,

Prof.Dr.

Deputy

special

reliable

the e v a l u a t i o n

Prof.Dr. H.Fuchs,

Our

invited

(Hannover)

O.Schmidt

of

the

(Berlin)

M.I. Schlesinger

Society

members

(Berlin)

C.-E.Liedtke

Institute

other

(Linkoping)

V.A.Kovalevsky

much

the

(Dresden)

R.Klette

speakers

to

(Budapest)

B.Gudmundsson

their

thanks

committee

D .Chetverikov

for

their

CAIP

'89

for '87.

to their

thank

Mrs.

untiring

E.Bahn,

efforts

to

Table

of

I. S I G N A L

PROCESSING

Oackel , Koschan , Neumann : Design of Operators for B e z i e r - A p p r o x i m a 1 1 on 3ahn

:

Edge

Detection

Vitkus

on

the

, Yaroslavsky

Recursive

Basis

Content

Image

Processing

Based

on 11

of

Piecewise

Linear Models

23

:

Algorithms

for

Local

Adaptive

Linear

Filtration

Matej : Fast Transform Methods of Image Reconstruction P a r a l l e l I m p l e m e n t a t i o n on S I M D - T y p e C o m p u t e r Morhac : Error-Free Dec o n v o l u t i o n and its Application Processing Krasnov , Glotov , Starodubtsev : I n t e g r a l I n v a r i a n t D e s c r i p t i o n of G r e y - S c a l e II.

34

and

Their 40

in

Signal

Images

48 55

ALGORITHMS

HegedOs Fast

Geometric

Correction

Lucka , Creutzburg Parallel Sommer The

SIMD

Design

Vilser

of

Optimal

Matrix

, Vajtersic

Using

the

Gaussian

DOLP

, Grundmann

Multiplication

I I I . AI Klette

: Systems-

on

A Modern Software

, Kleifi , K l e t t e

Knowledge

Based

62 : Number

Transform

70

Program

Edge

Detectors

82

:

an A r r a y - L o g i c a l

R et si eo n nt ae ld Te rp ar di

Processor

by a n e nL ti s t Compo

Ds ae td a Ba

90

101 95

TECHNIQUES

Liedtke . Ender : U t i l i z a t i o n of K n o w l e d g e C o n t e n t s of an I n d u s t r i a l V i s i o n S y s t e m Hesse

Fermat

, Gössel

L ul t eh s. N .: : Kö AI fm fa i rc a en s sf io nrgm a tA il og no sr i ton g e sa gn ee PT ro h m sI m aon Se tl rl up cr to uc r e s sor C

Expert

Images

:

, Creutzburg

Parallel

, Grundmann

Convolution

, Meinel

of D i g i t a l

Technology

for

the

107

Automated

Adaptation

121

: Construction

for

Sagerer , Schroder , Niemann : An A s s o c i a t i v e N e t w o r k as S y s t e m S h e l l Understanding

Image

for

Analysis

Knowledge

Based

133

Image

141

7

IV. Klette The

,

Voss

Three

MODELLING

:

Basic

Kovalevsky

IMAGE

Formulas

of

Neighborhood

Structures

149

:

The Topology of Cellular Complexes as Applied to Image Processing 162 Hub 1 e r : A n A x i o m a t i c A p p r o a c h to D i s c r e t e G e o m e t r y a n d I t s R e l a t i o n s to U s u a l D i g i t a l G e o m e t r y for I m a g e P r o c e s s i n g 174 K i e s e w e t t er

:

T w o di m e n s i on a l Chetverikov On

Some

Toward

Concepts

, Stiehl a Testbed

Vos s : Discrete

Orders

and

Picture

Processing

187

:

Basic

Neumann

Topologie

of

Texture

Analysis

196

: for

Integral

Evaluation

of

Early

Visual

Processes

202

Geometry V.

209 APPLICATIONS

Fritzsch : V i s i o n for I n d u s t r y

213

Danz , Elter , Mangoldt , Möhlmann , Rubbert , Weidlich I m a g e P r o c e s s i n g for t h e V E G A - M i s s i o n to C o m e t H a l l e y Höfling , Osten Image Modelling Interferometry Pietrzyk The

: and

Kühnert

in

Holographic

and

Speckle

, Michel

Based ,

Fringe

Henniger

,

Pattern Hussack

Processing

the

, Schwarze

Programmable

235

: Patterns

in 242

IMPLEMENTATION

A6472

Image

Processing

System

by

Use

of 247

Unger , Winter , Feist , Marzok , Zedier : P r o g r a m P a c k a g e for P r o c e s s i n g L i n e Flow Like A6472 Using A d a p t i v e and A n i s o t r o p i c M e t h o d s

Images

by

BVS 255

:

Hardware-Structure

Owczarczyk : D e s i g n of a H i g h - P e r f o r m a n c e

8

System

Aided Processing of SEM-Interference M e c h a n i c s and M i e r o m e c h a n i c s

Luth.W. . Gössel : L i n e a r F i l t e r i n g on Residue Arithmetics

The

Pattern

:

VI.

Kutschke

222

227

Microcomputer

Computer Fracture

Analysis

:

Binary

GIPP Image

and

Its

Application

Processor

266 275

AUTHOR

INDEX

Chetverikov

196

Creutzburg

70.

90

Danz

222

Elter Ender

222 121

Feist Fritzsch

255 213

Glotov Gössel Grundmann

90, 70,

Hegedüs Henniger Hesse Höfling Hübler Hussack

62 242 133 227 174 242

Jack el 3ahn Kiesewetter Klein Klette Köles Koschan Kovalevsky Krasnov Kühnert Kutschke Liedtke Lucka Luth, N Luth, W Mangoldt Marzok Mat e j Mei nel Michel Möhlmann Morhac

55 247 90

11 23

107,

133,

187 133 149 101 11 162 55 242 266 121 70 95 247 222 255 40 82 242 '2 2 2 48

Neumann Niemann

11 141

Osten Owczarczyk

227 275

9

Pietrzyk

235

Rubbert

222

Sagerer Schröder Schwarze Sommer Starodubtsev Stiehl

141 141 266 82 55 202

Unger

255

. . ."

Vajtersic Vilser Vitkus Voss

149,

70 90 34 209

Weidlich

222

Winter

255

Yaroslavsky Zedier

10

34 255

DESIGN OF OPERATORS FOR IMAGE PROCESSING BASED ON THE BÉZIER-APPROXIMATION D. Jackél, A. Koschan, H. Neumann

»

ABSTRACT The function values of a two-dimensional image function can be considered as a threedimensional surface. The often degraded image function should be preprocessed with a smoothing operator prior to a differentiation step to detect significant changes In the intensity surface. In consideration of the B£zler surface approximation technique such operators can be generated

on a quadratic segment with an odd number of control

points. We will call them 'B6zier' operators. The Bernstein polynomials Involved are to be understood as weighting functions of the control points. The first and second derivatives of a 'B6zler' operator can be developed easily.

An

efficiency

evaluation

of

the

'Bezier'

operators

compared

to

'Gaussian'

operators suggested by Harr and Hildreth has been made. As a result, the substantially easier generation

of

'B6zler'-convolution

operators

has to

be stressed.

Furthermore,

empiric studies done on some test Images have shown that 'B6zler' operators of smaller size

are

reacting

less

strong

with

regard to

discretization

errors

than

'Gaussian'

operators. 1. INTRODUCTION For the detection of discontinuities in the Image function which can be caused by different

physical

rotationally

phenomena,

symmetrical

Harr/Hlldreth

operators

based

on

16] suggest the

a

Laplacian

set of

of a

variable

sized

two-dimensional

Gaussian:

F*G(x,y,o)

-1 n-o«

l

-

x> + y s 2Bj , • ( v ) i=C j=0 with

r i j = [ xi j , y i j , Zi j

]T

,

u,v

e [ 0.0

1.0 ] ,

Bi,n(u) - (n!/(i!•(n-i)!))>u'•(1-u)»"1 , Bj , • ( v ) = ( m ! / ( j ! - ( m - j ) ! ) ) - v J • ( l - v ) - - J and n,m = Order of function (surface with ( n + l ) x ( m + l ) control points r i j ) The functions Bi,n(u) and Bj,»(v), often referred to as 'blending functions', are Bernstein polynomials. The Cartesian components of these vector functions can be simplified because the control points are equidistant. Thus, the functions x(u,v) and y(u,v) are defined by the equations x(u,v) = n-u and y(u,v) = m-v. With this the vector function r(u,v) can be considerably reduced. We receive:

r(u,v)

n-u m-v

n m Z ZI j -Bt , o ( u ) -Bj , • ( v ) I i=0 j=0

(1)

With regard to the definition of rotationally symmetrical operators and the local approximation of the image intensity function by a Monge patch r = u-ei + v-ea + /(u,v)-e3 equation (1) gives

r(u,v)

12

= n

u v z(u,v)/n

1/n

n I i=0

u v

m I zi j -Bi , n ( u ) - B j , D ( v ) j-0

Based on the equation

(1) the first and second partial derivatives of r(u,v) can be

calculated easily. Differentiating the vector function only Its z-components z(u,v)

=

n I i=0

m I 21 j - B i , > ( u ) - B j , » ( v ) j-0

have to be considered. If the Bernstein polynomials are broken into their components Bi,.(u)

= C(n,i)*F(u)

with C ( n , i ) the

partial

-

(2)

n ! / ( n - i ) ! and F ( u )

differentiation

of first

« u'-(l-u)»-' and

second

,

order

of z(u,v)

is

reduced

to

the

differentiation of the function F(u). We receive for the first derivative F ' (u)

« i-u'"1 •(1-u)»-'

-

(n-i)-u'•(1-u)»-'

and for the second derivative F"(u)

-

i - [ ( i - 1 ) - u ' - *1 • ( 1 - u ) ' - 1 1 - ( n - i ) • u ' - ' • ( 1 - u ) " - ' " 1 ] (n-i)•[i-u'" •(1-u)»-'" - (n-i-1)-u'•(1-u)»-'-»]

(3)

For the disassembly and the differentiation of the Bernstein polynomial as a

function

of the parameter v the corresponding equations result analogously. 2.2 GENERATION OF DISCRETE TWO-DIMENSIONAL CONVOLUTION KERNELS Depending on the number of given control points a corresponding number of Bernstein polynomials are established Bezier surface

as 'blending functions'.

Inside a defined interval of the

these define an unambiguous function value

for a random point

(u,v)

depending on the number of control points. The influence of every control point value when calculating a specific function value at a random location (u,v) is a direct result of

the

functional

values

of

the

corresponding

Bernstein

polynomial

for

u

or

v,

respectively. Thus, these have to be understood as weighting functions for the- control points. For

the

development

mentioned principle

of

rotationally

symmetrical

can be applied. Such

convolution

operator can

kernels

be generated

the

on a

above

quadratic

segment with an odd number of control points. If it is assumed that this set of picture elements for the specifically viewed local neighborhood of a point Pij control

points

for

a

Bezier approximation,

this point

is

located

is the set of

Inside

the

viewed

interval at the location (0.5,0.5). For the calculation of the functional value of the Bezier surface at the location (u,v) = (0.5,0.5) the weighting of the individual control point from the Bernstein polynomials at the

location

u=0.6

or

v=0.5,

respectively,

is

neighborhood of the central picture element. Fig.

received

depending

on

the

1(a) shows by way of example

local the

shape of the Bernstein polynomials with the order n=14.

13

Fig. 1: Bernstein polynomials with 16 control points. The broken line in the diagrams indicates the weighting of the function values based on the values of the polynomials at u = 0.6. (a) Original function, (b) first derivative. For the approximation of differentiation operators from Bernstein polynomials the partial derivatives of the equation (2) can be calculated with the support of equation (3). As a result we receive the slope and/or the curvature of the polynomials for each point Inside the interval. Fig. 1(b) represents the shape of the first derivative of Bernstein polynomials with

n + 1 = 15 control points.

The

operators

convolution

for

smoothing

and

partial

differentiation of

the

image

function can be generated from Bernstein polynomials and their derivatives at location (u,v) = (0.5,0.5). With regard to the definition of the discrete convolution (see e.g. [1]) the convolution kernels of different operators are defined as follows: B(i,j)

= (1/n)-Bt , . (0.5)-Bj , • (0.5) ,

B. ( i , j )

= ( 1 / n ) - B ' i , . ( 0 . 5 ) - B j . . (0.5) ,

B,(i,j)

= (1/n)-Bi , . ( 0 . 5 ) - B ' j , n (0.5) ,

Bu i ( i , j ) = (1/n« ) - B " i , • ( 0 . 5 ) - B j , . ( 0 . 5 ) .. Bw(i,j)

= (1/n» ) - B i , n ( 0 . 5 ) - B 1 ' j , o ( 0 . 5 ) , a n d

B.»(i,j)

-

( l / n 2 ) « B ' i , o (0 . 5) • B ' j , • ( 0 . 5 ) .

For a more complete development, see [3]. Fig. 2 shows convolution operators and their partial derivatives generated by applying the 'B6zier' approximation. Fig. 3 displays by way of example the corresponding power spectra In frequency domain for B(iJ) and its Laplaclan.

14

Fig. 2: Convolution operator of order n=32 generated by t h e method based on t h e B6zler approximation. (a) Operator for noise suppression B, (b) f i r s t partial derivative B«, (c) second partial derivative Buu, (d) operator for generation of negative Laplacian - v * B = -(B..(1J)+Btv(1J))

Fig. 3: Corresponding power spectra for (a) operator for noise suppression B, (b) negative Laplacian operator. 3. IMPLEMENTATION AND RESULTS The

'Bezler'

operators

of

parametric

size

as

developed

In

Chapter

2 have

been

implemented for a floating point arithmetic. For the evaluation of the processing results from convolutions

of operators

with digital

image matrices,

operators

from a

two-

dimensional Gaussian and their partial derivatives of equal size have been applied to t h e same images. In the following, the applied operators will be presented and compared with each other. Subsequently,

the

image data

and the

processing purposes will

be specified and

a

representation of the processing results will be given.

15

3.1 'BEZIER' OPERATORS IN COMPARISON WITH GAUSSIAN OPERATORS For the application of 'Bezier' operators for digital image matrices and the evaluation of the processing results Gaussian operators with different generated.

standard deviations were

d=5.08-o as suggested by Grimson |2) served as criterion for the definition

of the kernel size. For the present implementation the specific operator size was defined to be d =L5.08-o| + 4 (I.e. functional values in the periphery are only 1% of the maximum magnitude),

where the

absolute value of the

diameter has

to be

an odd

number. Dependent on the parametrlzatlon of the Gaussian operators the operator sizes defined this way are serving as predetermination for the number of control points for generating corresponding 'Bezier' operators. Table 1 shows the operator parameters and their size in a survey.

Gauss o

Diameter

0 75 1 41 3 18 Table 1:

Bézier d

Order

7 11 19

n

Control points

6 10 18

Parametrization

(n+1)

7 11 19

and s i z e o f a p p l i e d

operators

Numerical differentiation of Images is an ill-posed problem which can be regularized by filtering the data prior to differentiation [8], Small sized 'B6zier' operators for example with a diameter of 7 control points show better low pass characteristics than Gaussian operators of equal size (standard deviation o = 0.75). Real-valued kernels and corresponding

frequency

spectra

for

equal

sized

(same

diameter)

Gaussian

and

'Bezier'

convolution kernels are presented in Table 2/Fig.4 and Table 3/Fig.5. Besides the operator for the suppression of noise there have been generated specific convolution kernels

for each partial derivative

up to second order. With this it

is

possible to evaluate the magnitudes and the directions of the gradients as well as zero crossings of the second derivative apart from image smoothing.

0.00000 3.00000 ). 00003 0.00000 >.00023 1.00332

0.00003 1.00332 1.04782

0.00009 0.00808 0.11632

0.00003 0.00000 1.00000 0.00332 0.00023 1.00000 0.04782 0.00332 1.00003

0.00009 0.00808 0.11632

0.28294

0.11632 0.00808 1.00009

0.00003 1.00332 1.04782 0.00000 1.00023 1.00332 0.00000 1.00000 .00003

0.11632 0.00808 0.00009

0.04782 0.00332 1.00003 0.00332 0.00023 1.00000 0.00003 0.00000 1.00000

Table 2: Gaussian kernel 7x7 with a = 0.75.

16

Fig.4: Plot of corresponding freqency spectra.

0.00024 0.80146 0.00366 0.00146 0.00878 0.02197 0.00366 0.02197 0.05493

0.00488 0.02929 0.07324

0.00366 0.00146 0.00024 0.02197 0.00878 0.00146 0.05493 0.02197 0.00366

0.00488 0.02929 0.07324

0.09765

0.07324 0.02929 0.00488

0.00366 0.02197 0.05493 0.00146 0.00878 0.02197 0.00024 0.00146 0.00366

0.07324 0.02929 0.00488

0.05493 0.02197 0.00366 0.02197 0.00878 0.00146 0.00366 0.00146 0.00024

Table 3 :

' B 6 z i e r * k e r n e l with 7x7 c o n t r o l p o i n t s . F i g . 5 : P l o t of corresponding frequency s p e c t r a .

3.2 IMAGE DATA AND PROCESSING PURPOSES For t e s t i n g t h e 'B6zier' operators Image matrices a s described

in the following

chapter

were applied: CIRCRAMP_0 Is a s y n t h e t i c a l l y generated

image matrix where different

ratios

can

and

slopes

of an

Image function

be simulated

examplarily.

local

contrast

The

function

contains a ramp of c o n s t a n t slope, a circular region of c o n s t a n t I n t e n s i t y being located in the image c e n t e r (Fig. 6(a)). Thereby the ramp is approximated by a step function of two pixels width with an increase of two i n t e n s i t y units each (Fig. 6(b)).

Fig. 6: Structure of the Image function CIRCRAMP_0: (a) surface representation of t h e i n t e n s i t y function, (b) approximation of t h e ramp of c o n s t a n t slope. With t h i s image function a closed contour of c o n s t a n t c u r v a t u r e with uniformly changing local contrast ratio is simulated. The accuracy

of t h e

and direction)

detection

can

be tested

as

well

as t h e

gradient calculation of

discontinuities.

pattern can be used to evaluate the behaviour of the operator response in

(magnitude This

test

dependence

of discretization errors and weakly contrasted images.

17

LSPHERE60 shows a synthetically generated image function of a luminance distribution for a Lambertlan surface of a sphere model. The radius of

calculated using a computer-graphics illumination

the sphere is 50 pixels

(equals 101 pixels In diameter). The

simulated light source is in the viewer position. Viewer and light source are assumed to be at infinity, so in this case orthographic projection and parallel light rays can be assumed. For the illuminated surface of the sphere a continuous intensity surface of negative curvature is generated, bounded by a tangential occluding contour. Vlth

the

help

of

this

image

function

the

susceptibility

to

disturbances

of

the

convolution kernels of small size with regard to discretization errors with intensity distributions (r.g. with shaded surfaces) can be evaluated. 3.3 PROCESSING RESULTS Fig. 7(a) and (b) show the processing results of Gaussian operators with a = 1.41 and 'B6zler' operators with n = 10 on CIRCRAMP_0, respectively. The processing results for the visualization

in Fig. 7 are strongly dependent on the selection of

an Interval

( - £ , + i ) for the definition of a real-valued zero. This arrangement is arbitrary. An 'a priori' calculation based on the image data and the convolution kernels remains an unsolved problem. The empirical determination for e =

10-"

showed consistently good

results. The absolute values of Intensity gradients vary according to the local contrast ratio continuously along the contour of the embedded region. The directions are orthogonal to the tangent of the circular contour. In regions of constant intensity the operators yield the value zero; the constant slope of the ramp causes gradients of constant norm In the direction of the steepest descend. The second derivative of (Laplacian)

and

by

this

considerably from v8B"I

also

the

number

of

the

detected

to paG"I. It is recognizable that the

the image function

zero V2B

crossings

differ

operator for the

digital 6tep function of constant slope Indicates a curvature value with the absolute value of zero (linear approximation) while opposed to this the V-2G operator specifically shows across a picture element alternatively positive and negative curvatures (waving approximation). This filter response therefore leads for

y 2 G'I

to the detection of zero

crossings in the 'subpixel' area over the whole ramp.

Fig. 8(a)

and (b) show processing results on LSPHERE60 for the calculation

of the

second derivative of curved Intensity surfaces for operators of small size. For o = 0.75 the y*G operator shows non-significant changes In curvature (zero crossings) inside the intensity surface that has a uniform curvature. The corresponding

V'B-Operator with

n = 6 shows only one alteration of curvature for the enclosing occluding contour.

18

Fig. 7: Convolution of CIRCRAMP_0 with (a) G'l, P 2 G'I = (Gxx + Gyy)"I, | vG'I | = / ( ( G * ' I ) , + (Gy"I) J ), ^?»=tan" 1 (Gy"I/Gi"I) with o = 1.41, (b) B"I, = (B.a+B» v )'I, | pB"I I =V((Bu"I) a +(Bv'I) J ), V«=tan-»(Bv'I/Bu - I) with n = 10. For the representation of a r e a l - v a l u e d zero an i n t e r v a l ( - £ , + £ ) with t = 10-® has been chosen, i.e. all values with a magnitude |x| < 10-® have been mapped to zero.

19

•

ttfij «ft -»ith B „ ' i * r • •

o Fig. 8: C o n v o l u t i o n s of LSPHERE60 w i t h (a) G'l, V 2 G = ( G « + G , y ) , l with (b) B'l, y2B=(Baii+Bvv)'I with

20

o = 0.75 n = 6

4. SUMMARY Based on a given number af control points a surface can be sufficiently approximated with

the

help

of

polynomials.

elements as a set

We

looked

upon

the

of control points representing

two-dimensional

set

of

quantized intensities. The

picture

plecewlse

approximation of this set of control points by a Bernsteln/B6zler surface segment causes a smoothing of the discrete functional shape.

The Bernstein polynomials used for the approximation are to be understood as weighting functions of

the specific

discrete

discrete control point interval

control points. Choosing

the

central

element

of a Bezler surface of even order parametric

of

a

operators

for image processing can be generated.

The generation

of

'Bäzler'

convolution

kernels

is

easily

practiced:

Mainly, only

the

calculations of integer-valued factorials and of the powers of 0.5 are necessary. Thus, the

generation

of

parametric

Bfezier'

convolution

kernels

and

their

derivatives

is

numerically more stable than the generation of Gaussian kernels and their derivatives. The

definition

boundary

for

of

the

size

of

the functional

Gaussian

value

operators

(minimum

depends

value),

on

a randomly

contrarlly

the

size

selected

of

'B6zier'

operators is uniquely defined by the number of given control points.

The processing results In Chapter 3 show that Gaussian operators of the smallest size react very strongly to discretization errors. Corresponding 'Bezler'

operators,

however,

do not perform this way. Neurophyslological examinations of the human retina showed a sensibility profile for the receptive field which can be described In good approximation by a Gaussian distribution or a difference of two Gaussians (see f.e. Korn [4]). It has to

be

checked

If

maybe the

use

of

Bernstein

polynomials

could

yield

to

a

better

modeling of these profiles.

Furthermore, the less strong reaction of 'B6zler' operators with regard to discretization errors

could

analysis

in

maybe "lower

result scales"

in

a

(see

further Investigations. Finally

more

accurate

Pentland

operator

17]). These

response

aspects

will

for be

local a part

shading of

our

the advantages of 'B6zier' operators In comparison with

Gaussian operators should be summarized:

-

substantially easier generation of 'Bezler' operators,

-

more numerical stability,

-

less strong reaction with regard to discretization errors.

This work was carried out within SFB 203 at the Technische Universität Berlin and was funded by the Deutsche Forschungsgemeinschaft (DFG).

21

REFERENCES (1| |2| [3] [4) (6) [6] [71 [8]

22

Gonzalez, R.C., P. Wintz: Digital Image Processing. Reading: Addlson-Vesley 1977. Grlmson, W.E.L.: From Images to Surfaces: A Computational Study of the Human Early Visual System. The MIT Press, Cambridge 1981. Jackél, D., A. Koschan, H. Neumann: Entwurf lokaler Blldvorverarbeitungsoperatoren auf der Grundlage der Bézler- Approximation. Techn. Bericht 86-11 , Institut für Techn. Informatik, TU Berlin 1986. Korn, A.: Das visuelle System als Merkmalsfilter, Fachberichte Messen, Steuern, Regeln, Band 13: Aspekte der Informationsverarbeitung, Springer Berlin 1985. Marr, D., E. Hildreth.: Theory of Edge Detection. Proc. of the Royal Society of London, Vol.207 (1980), Series B, pp.187-217. Newman, W.M., R.F. Sproull.: Principle of Interactive Computer Graphics. McGrawHill Tokyo. 2nd Edition 1979. Pentland, A.P.: Local Shading Analysis. Tech. Note 272, AI Center, SRI International 1982. Torre, V., T. Poggio: On Edge Detection. MIT, AI Laboratory: AI Memo 768 1984.

EDGE DETECTION ON THE BASIS OF P1ECEWISE LINEAR MODELS Herbert Jahn

'

1. Introduction The detection of edges in images is one of the most important steps towards an automatical analysis of scenes. Because of the 3D - origin of 2D - images there may exist objects in the images which are not limited by closed contours (one may think about the objekt "nose" in the image of a human face). Therefore a primary forming of regions is not always the right first step and the recognition of objects primarily should be oriented at (not always closed) contours. The generation of a raw primal sketch in the sense of Marr and Hildreth /1/ therefore should be the first step in object recognition in natural scenes. This does not mean that the use of region forming methods generally fails; in simple (artificial) scenes they may work sufficiently but in complex (natural) scenes the edge detection should be primary. Towards a simple description of scenes by edges one first has to find edge elements (edgels). In a second step these edgels must be linked to get edge fragments. In this process besides the strength of the edgels their direction is needed too. The method for the generation of edgels which is described in this paper therefore provides both the strength and direction of edge elements. According to their different origin in the 3D - scene there exist edges of different types. The most important types are the step edge and the roof edge /2/ (Fig. 1).

Fig. 1 There exist many methods (e.g. /1/ - /4/) for the detection of edges and especially step edges. Most important are the methods basing on the gradient operator, the Hueckel operator, template matching and the zero crossings of the 2n

=

k = ( M ^ ) " 1 - 4 T ( 1 /2-k)-H .

(13)

(12) i s a f i n i t e impulse response ( F I R ) f i l t e r f o r the c a l c u l a t i o n of the l e f t - hand l i m i t s y^(i+i/2) and y ' L ( i + y 2 ) which i s easy to implement on a d i g i t a l computer. The estimation q u a l i t y of the V t t l s t a t e component z^(i+i/2) ( i s given by the mean quadratic estimation e r r o r o ^ ( i

+

y

2

)

=

y 5 i t follows from f i g . 4- that *.» there i s a d i s c o n t i n u i t y i f the i n e q u a l i t y

SiL,v(i+1/2) = K , v ( i + 1 / 2 ) ] V 2

y L - K=$L > / + or y - ;T > K'(«T + S ) ^ T

^ p

T

-

p

i s f u l f i l l e d . Here K i s a c e r t a i n parameter.

Fig. 4 In the ; e n e r a l case the i n e q u a l i t y - y1! > K - ( ? r + ? L ) or ( f o r both s t a t e components) f«E + L 1 O =1,2) P.* - ' < „ 1 > must be f u l f i l l e d . One obtains d i s c o n t i n u i t i e s of o ^ order i f st v a l i d f o r v = 1 and d i s c o n t i n u i t i e s of 1 order for v = 2. If there i s a d i s c o n t i n u i t y of order at x=i+1/2 but not one order, then i t corresponds to a step edge ( f i g . 5 a ) . Reversely, 26

(18) (13) i s of , 1 s t a roof

edge corresponds to a discontinuity of 1 s t order (fig. 5b). If there are discontinuities of Otl1 and 1 s t order then one has a combined edge type (fig. 5c). 1 ! I._J J * i r^-»-"— i ^ I I I I LLI I I L_L L. _J I I LLI I I I

JS.

a)

b)

1

"i-4-*. 1 "*i J-* i-_J 1—J LLI 1 1 1— c)

Fig. 5

Now it must be discussed how to choose the numbers nJu T and n,-, which deK fine the extent of the left - hand and right - hand estimation interval respectively. Obviously n (n = n L ,n R ) must be grater than or equal two in order to make possible the estimation of two unknowns z^ and Zg. A reduction of noise results for n> 3 only. But an unlimited increase of n in general is iiot possible because then the polynomial model becomes invalid. According to /7/ one gets an optimal value n L ^ of n L (and analogously of n^) if one looks for the minimum of the estimation error f ^ v (i+1/2) with respect to n L (for n L > n L rnin » 3). In order to do that the quantities must be calculated for'all possible values ¡ujjj» This can be done recursively if one makes use of the equations f

nL+1^ =f n

^

m:

=

mJ

L

^

+

4T (-niT1/2)'H "yi-nL •

+ (f,T(-nL-y2)-H'HT. (-nL-i/2) ,

which immediately follow from (10) and (11) (see /7/). But much computer time is needed if one uses this method, and therefore it is only justified if very weak edges have to be detected. In most cases it is better to choose fixed values n^, n^ in dependence of the class of pictures which are to be processed. Now the advantages and disadvantages of the method have to be discussed. First of all, as already has been said, the method enables one to detect edges of a general type with step and roof edges as special cases. In large regions with smoothly increasing or decreasing grey values no edges will be found. In this respect the method is superior to the well - known gradient methods. The possibility of detection of an edge according to (18) depends on the dissimilarities in the vicinity of the edge. The less the deviations of the grey values from P L linear polynomials are, that is the less the quantities ? " and f ^T? fc f n^jp zA T/ are, the less may be the difference |zn ~ n v ! which represents the strength of the edgel. So very weak edges may be detected if the data values left - hand and right - hand of the discontinuity fit the polynomials exactly. To avoid the detection of too weak edges one may change the decision criterion (18) slightly to get

27

IznR,» — zn. ,»l

>

v

+ÌÌ v ]

.

(19)

Here the quantities A y (y = 1,2) are certain thresholds which define lower limits for the detection of discontinuities of O^*1 and order respectively. According to (19) one can define a quntity •¿R sL nn nT S„ ; , (20) i> = MIX which may be associated to an edgel as a feature with ls„ ) being the strength of the edgel of order V-1. If one has found a discontinuity in row i at position j with a strength |S^ 0W j and another one in column 0 at position i with a strength then it is possible to defind an edgel at (i,j) which is characterized by the feature vector (S* ow , S ° o 1 ) giving the strength and direction of the edgel. Another good property of the method follows from the possibility of an optimal choice of the numbers n R and n^. At a given noise strength the capability of detection of a discontinuity increases with increasing n^ and n^ as long as the used polynomial model is not violated. This is a property which the visual system of humans has too: The perception of an edge increases with the extends of the adjacent homogeneous regions. Another advantage is the possibility of a straightforward extension of the method to the detection of discontinuities of higher order using polynomials of order V -1 (v2 3)• The main deficiency of the method follows from the fact that the used model does not represent the reality sufficiently good. Sharp discontinuities usually do not oo°ur in real images because the modulation transfer function of the receiving camera provides a blur. Therefore in most cases "soft" edges will be observed, and in the next step the convolution of the polynomial pieces with the line spread function of the used camera must be provided. A further deficiency of the method results from the used one - dimensional half - neighborhoods. If n L and / or n^ are small then the reduction of noise may be not sufficient resulting in false edgels. Therefore a generalization of the method to two - dimensional half neighborhoods would be desirable. 3. Results First of all the method has been tested using simulated data. Fig. 6 shows a step - like grey value profile corrupted by uncorrelated Gaussian noise with different dispersions = 0.1 for is 4-0, o* = 0.2 for iS 41).

23

Fig. 6 Various combinations of the thresholds K„ and d* has been checked. In the case of fig. 6 (and in other cases too) one obtains good results for K„ = 3...5 (* = 1,2), A, = 0...0.3 and \ = n^ = n R = 5 and K^ = 3, Kg =

A, = 0.3,

0...0.03. Choosing

= 0.03 one finds disconti-

nuities of 0 t h order between 20 and 21, 40 and

and discontinuities

of 1 s t order between 17 and 18, 18 and 19, 22 and 2 3 . The latter disappear if one makes use of the algorithm with optimal choice of n^ and n^ (njj,n-g>5). Then discontinuities of 0^'a order between 19 and 20, 20 and 21, 38 and 39, 40 and 41, 41 and 42 will be found ( with the greatest strength between 20 and 21, 40 and 41). st Fig. 7 shows a discontinuity of 1 order, that is y^

(i = 1..40),

y ± = - 2 + 0.05-i +«-• ^ ( i = 41...80), 45 with a sheet of paper each the visual system of humans does not find the discontinuity too. But the algorithm with the optimal choice of n^ and

(n^.n-^ > 5) finds dis-

continuities in the vicinity of i = 40, and the one with the greatest strength is situated between 39 and 40. At this discontinuity the optimal values of n L and n R are 41 and 39 respectively. This means that all data have been used for the estimation.

29

Ill the upper part of fig. 8 a simulated grey value step (on the left grey value 190, on the right - 210) corrupted by white Gaussian noise with the dispersion (i = 5 is shown. The lower part of fig. 8 shows the overlay of the result of the application of the O tl1 order edge operator with the parameters K^ = 3, A, = 0.3. The optimal values of the numbers n L and rig automatically have been chosen within the interval [5,20].

Fig. 8

The upper part of fig. 9 represents another simulated scene. In the columns 1 - 4 4 and 85 - 128 there is a constant intensity with mean grey value 100. Prom column 45 to column 64 the grey value linearly increases up to the top level 200. After that it linearly decreases, reaching the value 100 at column 84. This spike as well as the other parts of the image again are corrupted by white Gaussian noise with order at tf = 8. It is difficult to recognize the discontinuity of column 64 in the photographic picture which means that it is not a very strong one. The lower part of the picture displays the strength st of the found discontinuities of 1 order (maximum brightness corresponds to a strength of 15» ¿i = 0.02, 5 - n0pi; - 20). In almost all image rows the existing discontinuities have been found but the strong noise influence is evident. Fig. 10 shows parts of the Baltic Sea islands Hiddensee and Rugia. The essential edges are found by the edge operator of 0 tl1 order with the parameters \ = 0.3, 5 - n0p-(; - 20 (maximum brightness corresponds to edge strength 8), as is shown in the lower part of fig. 10. Edges which are bounds of very narrow regions are not detected satisfactorily beoause n m i r i = 5 was chosen. With respect to this better results will be obtained if one chooses n m i n = 3« But then many additional noise edgels arise.

30

The a e r i a l photograph of an agricultural f i e l d ( f i g . 11) shows s t r i p e l i k e structures which very much resemble the simulated spike of f i g . 9. Applying the O^*1 order edge operator one does not obtain useful results st but the 1 order edge strength displayed in the lower part of the picture s a t i s f a c t o r i l y reproduce the structure ( A* = 0.02, 5 i n ^f 10). I t f o l l o w s that in natural scenes roof edges may be essential which means that the use of the usual edge operators f o r the detection of step edges i s not s u f f i c i e n t . The last picture ( f i g . 12) has been generated by computer graphics. I t contains regions with gradually changing grey values as w e l l as edges of d i f f e r e n t type and strength. The lower part of the picture shows discontinuities of 0 ^ order. Maximum brightness corresponds to an edge strength of 10 ( A„ = 0.3. 5 i n ^ i 20). Weak edges which cannot be r e produced by the photographic picture ( e . g . in the right upper p a r t ) 31

precisely have been found and the bright central point within the circle on the left of the picture too.

Fig. 11

Pig. 12

REFERENCES /1/ D. Marr, E. Hildreth; Theory of edge detection; Proc. R. Soc. Lond. B 207, 1980,p. 187 - 217 /2/ R. M. Haralick; Digital step edges from zero crossing of second directional derivatives; IEEE Trans., PAMI-6, 1984-, p. 58 - 68 /3/ G. B. Shaw; Local and regional edge detectors: some comparisons; Computer Graphics and Image Processing 9» 1979, P. 135 - 149

/4•/ R. Nevatia; Machine Perception; Prentice - Hall, Inc., Englewood Cliffs, N. J., 1982 /5/ I. Leclerc, S. W. Zucker; The local structure of image discontinuities in one t dimension;

Proc. of the 7

32

Int. Conf. on Pattern Recogn., Montreal, 1984

/ 6 / L. A. Zadeh, C. Ä. Deeoer; Linear System Theory; Mo Graw - H i l l , New York, 1963 / 7 / H. Jahn; Zustandsschätzung mit adaptivem Gedächtnis MESSEN STEUERN HEGELN 24, 1981, S . 146 - 149

RECURSIVE ALGORITHMS FOR LOCAL A D A P T I V E LINEAR FILTRATION R. Vitkus, L. Yaroslavsky

Introduction The class of adaptive linear filters for picture processing as d e scribed in [ 1 , 2 ] features frequency response adjustable to the observed spectrum of the picture to be processed. These filters can be adaptive locally or globally. With global adaptation, the filter parameters are adjusted to the spectrum estimate of all the picture w h i c h is processed as a whole. W i t h local adaptation, the filter parameters are adjusted to the spectra of individual image fragments, and the image is processed on the fragment-by-fragment

basis. Finally, w i t h sliding processing

where the passage from one fragment to another is done by shifting the fragment center by one sample, at each step only the central sample is estimated and the adjustment is performed over all the fragment.

This

implies that in order to perform filtration one must estimate the spectrum of each picture fragment, generate the filter frequency response

in

terms of the spectrum, multiply it by the fragment spectrum and perform the inverse spectral transform in a single point, the central

fragment

sample. The technique of adaptive filter parameter adjustment is independent of the chosen orthogonal basis

^ 1 , Zj

w i t h respect to w h i c h pic-

ture fragment spectrum is estimated. Filtration effectiveness,

there-

fore, is defined by the effectiveness of the algorithms for computation through picture fragment

spectra. The discrete Fourier transform

(DFT) is one of the most common orthogonal transforms used in picture processing. The two-dimensional DFT is known (see, for example, £ 2 J ) to be computable recursively w h i c h dramatically reduces the amount of computations in sliding processing. However, the DFT-based

estimation

of Fourier spectra has disadvantages due to the pronounced influence of the boundary effects on the spectrum. Special spectral windows provide better spectral estimation. The discrete cosine transform

(DCT) is an

alternative to spectral windows. It makes better use of signal

samples

for spectrum estimation. Recursive spectral analysis algorithms based on DFT w i t h spectral windows or o n DCT have n e v e r been described in the literature. This paper is devoted to the study of the class of linear orthogonal two-dimensional transforms for w h i c h recursive spectral a n o The authors are w i t h the Institute of Information Transmission Problems, at the USSR Academy of Sciences, 19, Ermolovoyst. Moscow, 101447, USSR

34

lysis algorithms can be constructed and to their description for DPT with cosine spectral window and for DCT.

The class of base functions and recursive algorithms of local spectral analysis. We confine our consideration to two-dimensional linear transforms that are performed over rectangular picture fragments and are separable. All the widely used picture-independent two-dimensional orthogonal transforms meet this constraint by construction. Starting with the property of separability, the two-dimensional recursive algorithms may be constructed by means of one-dimensional transforms along picture rows and columns. Therefore, let us consider at first the algorithms of onedimensional recursive transforms. Let (

{(X(/7)J be sequence of, generally, complex numbers, and k

e

a set of base functions of a transform that are ortho-

gonal over an interval of length mple vector

M

. The components of special sa-

Oti, •••, M~i J

the fragment

, as computed through

{a (f1-M+i),a(n-M+2),...,Cl(n)] are defined by

M-i (")=2- a(n-M+k) k=o

(k) ,

..., M-i

(1)

Write this expression as

M-i a(n-k)

,

1=0,1,M-i

(2>

k'O where h^(k) = (f^ (M~i~ k) . it describes M linear filters with finite pulse response that compute spectrum samples when the processed fragment slides along the sequence. Let us determine the form of filter pulse responses allowing recursive determination of the coefficients (n) with respect to n . At the sample-by-sample shifting of the processed fragment, one end sample leaves it and another appears on the opposite side. Therefore, the requirement that the spectral samples be computed as a linear combination of (a) input and output of the initial sequence at shifting the fragment and (b) the value of the same spectral samples in the previous position of the processed fragment

d*(n) = is Ot

(n-i)

a(n) +d?a(n-M)

,

^

M-i ,

where { C ^ j , and fct*i j are arbitrary numbers, is one of the simplest and natural recursiveness conditions. According to (3), the spectral samples j are computed independently of each other, and at most three multiplications and two

35

additions of, generally, complex numbers are required for one sample. The following expressions may be written for the pulse responses of linear systems described by the difference equations (3): C

f

L

£

k = 0 i

'

''

'

Since we are interested in the final pulse responses of length M

, the

condition

6i

= - di

must be obeyed. Finally, obtain the following relations

and the

coefficients of (3):

(k) = Si c* where

and

; d7 =-Sz c?

C^

are arbitrary numbers. Hence, the following con-

straints on ^cfjf(k)^

result:

(0) cik

t = -(fi (o) Cn Si

=

one substitutes

^f-i(k)

k,? = 0,1,..., M-i

,

are arbitrary real numbers. If the

i s orthonormal,

{ifijz(k)J

that

R

K

=

- ¿Afpf

•

The

choice of the

defines a particular base. If for t/ftf1

and

of shifted DPT (SDFT) with parameters

K

, obtain the known base

and R

defining the shift

in the space and in the spectral domain, respectively, ( [ 2] ). In this case, the parameters =

(*/{M)

C*

f^P-tfO)^

anc

*

from (4) are

exp[i2rft*R)K/M]

= ex/7 l-L2ir( ? + R)]

, (5)

.

By substituting them into (4) obtain after appropriate transformations the following recursive algo ithm for computation of SDFT spectrum -

W

36

1

) =

) ^(n-i)

+e*plizïï(>t +R) K/M J

x

7

(exp(iZTTR) a(n) -a(n-M))}

e%p[-i2ïï(t+R)/M]

(6,

Notably, the amount of arithmetic operations for computation of mz ] =

R

i L , i 2 E i ( n i» n 2 )] R i : L ,i 2 C x ( n i ' n 2 0 ( m o d u >

il»i 2 = 0,1,...,N-1 and 11 is the Fermat number defined as ,t M = F ^ = 2

+1

t=

0,1,2,...

(14)

The properties of the solution aire : - it does not require the calculation of inverse matrix - there do not arise rounding-off errors due to the integer operations executed in modular arithmetic 3. Algorithm of two dimensional deconvolution using Fermat transform Two dimensional Fermat transform pair can be expressed as > 1 ?=J ,alnl ,a2n2 I x(a T ,a P ) = (mod M )

(15)

respectively x(nlfn2) =

_2

H=1 aj=0

where n^, n 2 > a^, a 2 ¿jCj1 (mod if) = 1

fcl oC/

1

1

/ ~a2n; oL 2 2 X ^ . a ^

(mod it)

(iß)

= 0,1,...,N-l; N is a power of 2 and there holds (17)

Similarly the same transform pair can be written in matrix notation as X = [T. X.T] (mod M ) resp.

50

(18)

x = (t_1. X'T_1J (mod M)

(19)

where T, T-^" are Fermat resp.inverse Fermat transform matrices. The error-free one dimensional algorithm was derived and published in

.

According to this derivation the two-dimensional error-free deconvolutioia algorithm can be described as follows 1. Calculate the matrix of the transformed values of the two-dimensional response function h H = T.h.T (mod M )

(20)

2. Calculate the matrix H ^ such that for all their elements H ^i^jig) there holds H (i1,i2)- H~ 1 |i 1 ,i 2 ) (mod M) = 1

i-^ig = 0,1,..., N-l

(2l)

Using the matrix H from. eq.(20) we can write for the determinant of system (7 ) N-l

D(mod M) = DJJ

3. Let

N-l

flo

(mod M )

i i

y n i Q ^ j n g ) = yfn^iig)

(22)

n-^,^ = 0,1,...,N-1

4. Calculate the transform of ym^ Y M 0 = [T.ym 0 .T](mod M ) 5. For i 1 »i2

=

(23)

0,1,...,N-1 calculate

X0(ilfi2) = [ H - ^ i ^ i ^ . Y M ^ i ^ i ^ m o d 6. The inverse transform of X« is xM

p. M x M (mod m ) = — XJJ(mod m )

D = [ V ^ X 0 «T _ 1 ](mod m ) = —

7. Let x(i l f i 2 l

- XQII^IJ

= DJJ X ^ i ^ ^ ij.^

8. According to y? =

%

(mod M )

=

0

(25) (26)

N"1

'2?)

*0

7 (

(24)

(6) ,(7) calculate the vector

form the matrix =

tO

y

v

yQ and then D

-yo)

yn^ = y-^mod M )

calculate matrices

j + i = yj+ifmod

M

(36)

)

15. If for a l l i 1 » i 2 = 0 , 1 , . . . , » - ! y ^ i 1 ! » 1 ^ ! = °> t h e n f i n i s h the calculation. If not, increment j and repeat f r o m point 10 on. 16.

= x ^ t ^ /

D

ii»^

=

0,1,...,N-1

(37)

Well known Euclidian algorithm can be used for calculation of needed inverse number

(see eq. 21} in the framework of modular arithmetic. One

version of such; algorithm is described in

.

The error-free algorithm also requires the value of determinant in (28) ,(37) . In; £ t h e

algorithm for precise calculation of the deter-

minant o f one-dimensional convolution, system was introduced. The algorithm to calculate the determinant of matrix for two-dimensional. case (see eq. (6)and (7)) was found and described in ^ 7 ^ •

4. Applications The described method of deconvolution was used to the resolution enhancement in processing of spectroscopic data. The algorithms were applied to data given in fig.l and 3. Fig. 1 shows spectrum consisting of 4 peaks located very closely to each other (positions

86, 89, 93, 97 } heights 150, 100, 50, 200 ) .

Looking at this figure we cannot state the existence of second and third peak in the middle of the spectrum. Fig. 2 shows the spectrum after deconvolution.

52

A

t ^ X Fig. 1

—

X

Fig. 2

Similarly the two-dimensional deconvolution algorithm described in part 3 of this paper was verified. Fig. 3 shows the spectrum before deconvolution (positions x 1 = 7 = 9, x 2 = 10, y 2 = 8, x^ = 8, y^ =5» heights 200, 150, 50) and fig. 4 shows deconvoluted spectrum with evidently better resolution«

Fig. 3

Fig. 4

The algorithms were implemented for a TPA-70 minicomputer(Hungarian production ) . The modulus M = F^ = 65537 and coefficient oL= 3 were chosen in order to permit the length of the transformed vector or matrix up to 65536.

5. Conclusion The method described in the paper shows that Fermat transform can be successfully used also for the two-dimensional deconvolution. The main point of the usage of modular arithmetic is an elimination of the rounding-off errors. On the other hand commonly used Fourier transform gives a solution with oscillations. Unfortunately, the algoritlm is rather time-consuming since the

53

calculation process must be repeated several times. But there is a possibility to eliminate this disadvantage using specialized processor of the fast Fermat transform.

6. Literature [l] f2j [3^

£4] [5J [6] [7J [ßj [9j

54

Ahmed,N., Rao, K.R. : Orthogonal Transforms for Digital Signal Processing. Springer-Verlag, Berlin 1975« Goutte, R., Prost, R., Georges, A. : Déconvolution numérique avec prolongement spectral. Applications aux signaux et aux images. Analusis £ (1980) 1, 6-15. Grabaric, B.S., o'Halloran, R.J., Smith, D.E. : Resolution Enhancement of A.C. Polarographics Peaks by Déconvolution Using Fast Fourier Transform. Analytica Chimica Acta ^¿¿(1981), 349- 358. Kennet, T.J., Prestwich, W.V.,: Incremental Déconvolution I. Algorithm Development and Assessement. Nuclear Instruments and Methods 201 (1982), 317-327. Lacoste, L.J.B, s Déconvolution by Succesive Approximations. Geophysics ¿X (1982) 12, 1724-1730. MorhéS, M. : Precise Déconvolution Using the Fermat Number Transform. Computers and Math, with Appls. 12A (1986) 3, 319329. Morhäö, M.: System Identification and Déconvolution Using the Fourier and the Fermat transforms . In Slovak . Dissertation, Bratislava 1983. Nussbaumer, H.J. : Fast Fourier Transform and Convolution Algorithms. Springer-Verlag, Berlin 1981. Ore, 0. : Number Theory and Its History. McGraw-Hill, New York 1948.

Integral Invariant Description of Grey-Scale Images A.E.Krasnov, A.K.Glotov, V.F.Starodubtsev

Introduction In certain applikations of the automatic monitoring one has to generate image description that are independent of unknown initial conditions of the object in question. A most complicated and most common is the case where all the information on object state is contained in the degrees of grey of its image. In this connection, the problem of constructing an integral invariant image description cannot be regarded as the new. vThen one comes across an individual object or a group of objects making up a complex scene, the problem may by of interest by itself or be the basic element of a more general, e.g. structural, description. Mathematically, construction of the integral description involves calculation of various functionals of two-dimensional functions which may be exemplified by the well known methods of moment, spectral or statistical invariants [1,2,3J. The ever increasing amount of methods for separation of image features attests the importance of the problem and, at the same time, points to the lack of a unique approach to it[4]. The lack of generality is the common disadvantage of these methods because they depend on the nature of image and disregard the physical properties of the image as the subject of inquire. As the result, degenerate descriptions are obtained where similar descriptions correspond to structurally different images, and the causes of degeneracy and the ways to its elimination are not clear. This paper aims at an integral invariant method of image description that is free of the above demerits and emphasizes a description invariant to translation-and-scale image transformations. Image is primarily regarded here not as a function but as a physical object or the result of interaction of certain physical processes such as spatial propagation of electromagnetic oscillations or waves and detection of physical v a lues related to them. Owing to the common nature of the dynamic processes of electromagnetic wave propagation, the method suggested here may be used for image description in the optical, radio-frequency or X-ray bands. In each of the bands there are, however, specific features of wave detection. In this connection, the paper considers the promising digital, analogue and optical hardware for implementation of the proposed method in different applications.

^ The authors are with the Institute of Control Sciences, 65, Profsoyuznaya St., Moscow, 117342, USSR.

55

Qualitative description of image. Phase portrait of the image spatial structure. Let us consider a scalar electromagnetic signal corresponding to the intensity of electromagnetic field propagating from the object under consideration to the detector. The signal is described in mathematical terms as a real function E(x,y,z,t) of space (x,y,z) and time (t) coordinates. The interaction of the field and detector including operations of measurement and computation results in the image I(x,y,t), a positive definite function giving mathematically adequate spatial structure of the electromagnetic signal observed at the aperture of the detector. It goes without saying that in the process of image generation the physical properties of both signal and detection are taken into account in order to render as much information contained in the signal as possible. In the final image, however, the usefui qualitative properties of the signal, interrelations between its individual components are reduced. Description of their relations is the integral image description. This interrelation may be established by taking into account the space-time relations between the components of electromagnetic signal E or the physical process corresponding this signal. This space-time relation is known to be expressible according to the laws of the classical electrodynamics as the wave equation c^ V^E = d t^ that is satisfied by field E in the free space [5}. In practice, at image generation one observes quasi-stationary electromagnetic fields E(x,y,z,t) = ft(x,y,z) cos(cot) having the variation speed of envelope f much smaller then the mean cyclic frequency 0> of the electromagnetic signal carrier. The en2 2 velope or the "video" signal f t satisfies the wave equation v f^ + k f^ = 0 where k is the wave number corresponding to co and light speed c in vacuum (k = CJ c). Superposition of flat waves propagating from the object to detector is the real part of the equation solution having a profound physical sense [6] ft(x,y,z) = £

j ^ L ^ L

k z

R k x , k y > k z ( t ) exp (-i (kxx+kyy+kzz)),

p O O p where k^+k^+k^ x y z = k and the amplitudes iRj of flat waves depend on the boundary conditions as defined by the object and the distance to the detector aperture. Let us group the terms of the solution so as to explicitly separate the harmonic dependence of lines and omit the coordinate z=Z of the flat detector aperture: f t (x,y) = ft(x,y,Z) =

L

k x

R kx (y.t) exp(-ikxx)

Thus, the video signal of any line y in the detector aperture plane is the superposition of harmonic processes q w ( x ) with different spectral kx p 2 2 weights R ^ satisfying each the second-order equation d The geometric correction can also be defined by points whose as well as their points or

OUTPUT

INPUT

positions

positions are known. These are called reference

Fiducial marks, and their positions can be experimentally measured.

This is not an exact way of generating geometric corrections,as,due to measuring precision, the exact positions of the reference points are not known. The positions of all other points are not known at all, but must be interpolated or extrapolated from corrections found at the reference points. However the theoretical and experimental approaches can be combined, i.e.,the correction obtained theoretically can be parametrized by calculating the positions of a few reference points. In section 2, concerning the computational problems, we suggest an efficient solution of the general geometric correction problem. In sections 3 and 4 we discuss how to obtain the correction theoretically and experimentally. Section 5 shows the results of our implementation. A brief bibliography is presented in section 6.

62

2. Discussion of computational problems a) Without loss of generality we may assume that if a correction is restricted. to an appropriately small domain it may be considered as a linear function e.g.

in the case vhen the dlstorsion to be corrected is not too

large. Linear and affine correction of digital pictures are well developed disciplines. There are several fast algorithms published

[2,5] which are

based on the fundamental rules of digital straight line representation [lj . It is useful to use these fast algorithms, with appropriate adjusment even in the case of globally nonlinear corrections. One might use a point-by-point modification but with a smooth correction function one may adjust at much less frequent materials. Theoretically a good solution is to adjust less frequently where actual correction changes slowly and more frequently at other places. Although from the point of view of the number of adjustments needed it Is only suboptlmal, In our case the following method seems to be promising: -Let us consider the picture domain

e.g.

in system

A. We can cover the pic-

ture with a regular square lattice such that in each square the correction may be considered to be linear. A single square in this lattice will be referred to as a lattice element. The more linear the correction is, the more points of the picture can be Included in a single lattice element. -Obtain the position of each vertex of each lattice element in system

B .

These are called control data. -Knowing the positions of the vertices in system

B , one can perform the cor-

rection as a four-point interpolation at each lattice element of the picture to be correct. If the correcting relation is invertable, then the transformed lattice elements are disjoint and form a perfect tiling In system

B .

This method can be justified as follows: -To control an adjustment whose density changes, one has to perform extra computations

i.e.

gradient computation. In the case of a regular lattice it

can be omitted. -Although in this method one has to obtain the control data more often than it would be necessary, since its frequency is determined by the intervals with the fastest change, the amount of extra computation is not considerable since the number of control points is generally much less than the total number of points in the picture. -If one handles the lattice elements of a row in the system

A , then one can

raster scan the image. This is compatible with the data structures of many of the common secondary storage systems.

63

b) In geometric transformations one obtains the position of the vertices of regular lattice defined in one system in another system. If the regular lattice is defined in the

IHPUT

is defined in the

OUTPUT system, the one obtains a n inverse transformation.

system, then one obtains direct transformation. If it

If the correcting relation is invertable, direct and inverse methods will theoretically yield the same result. From several points of view

[4] it is more useful to choose the inverse

method, for example i n this case every position of the picture in the corrected system is defined

i.e. there is no need for initialization.

c) To assign the original and the corrected lattice to each other one needs an interpolating function

R . If a spot is homogeneous in the picture, one

cannot see the difference between them. However i n the case of a picture with outlines there occur considerable differences. The nearest neighbour technique, which is often used i n fast algorithms, cannot be used here since it would produce a false outline. Hore precise results can be obtained by bilinear or spline interpolating functions. However these need much more computation. It is fruitful to let the picture adjust the choice of the interpolating function. However, this requires the fast detection of changes in the picture. One can detect edges using fixed point arithmetic by convolving the picture function with a quickly decaying function that stresses the local change in the picture. Thus, when transforming the lattice elements -one can vise a fa3t algorithm based o n a fixed point arithmetic in the homogeneous parts of the picture. I n this case the interpolation is the same as in the nearest neighbour technique. - I n addition one can use a n additional correction of the position in the less homogeneous parts of the picture, and one should use a more precise interpolating formulae i n the neighbourhood of this position. Thus it is possible to arrange that only the minimal amount of computation Is performed Jo achieve the requested accuracy.

3.

A theoretical way of performing geometric corrections

If one knows the correcting relation

f: A — B

, then the steps of

correction are as follows. a) Choose the lattice spacing coefficient according to the smoothness of the correcting relation. It can be done by investigating the extremal values of the gradient function or the coefficient can be determined from prior knowledge. b) Perform the correction at the vertices of the lattice, and so obtain the control data.

64

c) Using the control data, perform the row-to-row corrections -perform tha general four-point transformation of the elemental squares using fixed point arithmetic -process simultaneously the elemental squares including the row which is currently being processed -use the picture controlled interpolation (incorporating a fast edge detecting technique together with the correction).

4« An experimental way of performing geometric correction The geometric correction can be defined by obtaining the positions of the points (the so called reference points) which are the same in both the and in the

OUTPUT

INPUT

pictures. Even if corresponding points can be identified

unambiguously quantisation of the position limits the accuracy. In the view of the above points, one should find a correcting relation such that: -the error of assigning the reference points to one another should be minimal, -the mapping to be found should be as smooth as possible ever the whole plane of the picture, -one should be able to obtain a more precise correction by using additional reference points. Requirements for the algorithm for obtaining the correcting relation are: -it should require no other information than the positions of the reference points, but if there are such additional pieces of information, then there should be a way to utilize them (such as the parametrization of the correction obtained theoretically by the reference points). -it should contain both simple and more complex approximations, and one should be able to prescribe the necessary complexity. -it should converge fast enough even in the case of a large set of reference points. -it should be possible to control the errors. Consider a set of reference points that consists of fine a one-to-one correspondence between systems

A

N

and

points and that deB :

P s= ' W W

V

I

8=1

y

*ì

D(5) s=ls

,

D(Ì) =

(f

m - S i-1

_ 1

2 1

(6)

denotes the actual value of the approximated function at points F g (s=l,..,H)

fjdenotes the weights. The weights

should be choosen according to the distribution of points P g .

^fj should be smaller where the density of points If f

m « N

, the second constraint

Fg

is greater.

(smoothness) can be fairly since,function

has relatively few parameters and so when the functions g. are smooth it has

less chance to differ from f . The smoothness can also be achieved by appropriate regularization. In this case, instead of (^(a), one can use ^ ( S , j ) when obtaining coefficients by minimization:

66

a^

(7) Functional \|/(a)

must express the smoothness of the function« so it can

be,for example, an approximation of the integral (8) where K is the plane of the picture,

pCK.

The choice of parameter J determines the balance between whether the approximation is more precise at the points given, or whether the approximating function is smoother. Unfortunately, both the computation of the functional (computation of the gradient in the whole domain) and the redefinition of it, into a form with maximum gradient using general basis functions, need a lot of computation. In our case another method seems to be suitable: a) perform the approximation with $ »0

i.e. without any constraint of

smoothness, b) try to achieve a smooth approximation by distributing tha identification points

P s uniformly and by choosing smooth functions

g^ .

c) choose the faster solution rather than optimizing one after performing such global investigations as: -perform several approximations with different complexities (see later) -check their smoothness subsequently,by a tilt, visual technique, and leave the choices that need trade-offs to human decision. Note that this method can yield almost the same result as using formula (7), but it is much faster and it can be coupled with the computation of the data directly controlling the correction. The computation needed by this method depends crucially on approximating function case of large

m

m , one must invert an

m. If one uses

m by m matrix. Thus, in the

it is difficult to change the basis functions

g^ , and to

alter the number of functions used in the approximation. Zherefore these computations are slow. This difficulty can be avoided by finding a recursive relation between the approximations with

m

and

m+1

(but suboptimal)

terns. Moreover the

program should choose the functions used in the ra -term approximation from a so called basis function library

containing

M ^ m

functions, such that the

error caused by the approximation should be minimal. In this way we do not have to determine the number of terns in an approximation in advance and we use the basis functions that yield the best results at every stage We also require that the algorithm should utilize the results of the m-teim approximation when it computes the

m+1 -tera, i.e., the magnitude of computa-

tion should not increase. Obviously, the number of terms one can use is limited since in the computer a number can be stored with finite accuracy. The algo-

67

rithm is also required to control the errors. The recursive algorithm worked out by us yields

m

approximating functions

different from one another and considering the requirements of accuracy and smoothness, we have to select the best one. One gains information about the satisfaction of the requirements of accuracy from the messages of the program implementing the algorithm. It is useful to couple the check of whether the requirements of smoothness are satisfied with the computation of the control data correction i.e., it is useful that, when perforating what is described in Article

b) of section

3,

the program enables the user to check the result visually. This can be realized by displaying the corrected lattice on a

TV-monitor,

The correction can actually be done as it is described in Article

c) of

section 3. 5. In the

MIP

system

Implementation (Modular Image Processing, made by SzKI)

there is a

subsystem for solving the tasks in connections with the geometric corrections. The approximating method allows us to approximate with a rational fractional function of degree 5, and so the basis function library consists of

21+20

elements. The recursive algorithm uses the elements of a subset of it. Both the subset and the number of the functions used in the algorithm

(max. 15)

can be specified by the user. The maximum number of reference points is 255. These points can be defined numerically or interactively by defining a controllable position in the picture displayed. If the number of reference points is ten then the approximation takes about one minute. The best approximation is selected by displaying the corrected lattice, producing simultaneously the data directly controlling the correction. Thus the time to investigate an approximation is one minute. In the case of

256x256 point pictures, the actual correction takes 0.5-2

minutes. This timo depends on the need for secondary storage and on the threshold value of the change of interpolation mode. The time needed is (practically) independent of the complexity of the correction approximated and although the approximation is several magnitudes faster than the precise correction they yield almost identical results.

68

6.

[1]

References

B.W.Jordan,W.J.Lemon,B.D.Holm,"An Improved Algorithm for the Generation of Nonparametric Curves", IEEE Transaction on Computers, C-22, 1973,1052-1060.

[2]

C.Braccini,G.Marino,"Fast Geometrical Manipulations of Digital Pictures", Computer Graphics and Image Processing, 13.,1980,127-141.

[3]

Carl F.R.V/eiman, "Highly Parallel Digitized Geometric Transformations Without Matrix Multiplications",Proceedings of the 1976. International Conference on Parallel Proceedings, Detroit, Michigan,

1-10. [4]

D.Pertly,M.Gangnet,Ph.Coueignoux,"Perspective Mapping of Planar Textures", Computer Graphics, 16.,1982.,72-89.

[5]

H.L.Ingram,R.Hoolcer, "The Selection of Approximating Functions for Tabulated Numerical",Technical Report, X-64658,George C.Marshall Space Flight Center, Alabama.

Acknowledgement The author would like to thank Dr. G6za Alio (Computer Research and Innova® tion Center,HUNGABY) and Dr. John Illingworth (Rutherford Appleton Laboratory, ENGLA1TD) for their corrections and useful pieces of advice.

Dr.Gy.Csaba

Hegediis

deputy head of department Computer Research and Innovation Center H-1015 Budapest I.

, Donati u. 35-45.

HUNGARY

69

PARALLEL SIMO CONVOLUTION USING THE FERMAT NUMBER TRANSFORM M&riS LuckS

Reiner Creutzburg

++

, Hans-Jorg Grundmann

and Marian Vajter¥iC

+++

,

+

1. Introduction With the rapid advances in large scale integration, a growing number of digital signal processing operations becomes attractive. The convolution property of certain transforms can be used to compute the cyclic convolution of two discrete signals. One such transform is the discrete Fourier transform (DFT) in the modified version of the fast Fourier transform (FFT) / if- / which operates on signals in the complex number field. The FFT of length N requires (N/2) log 2 (N/2) complex multiplications / k=0

h(n-k) x(k)

.

formulae

n = 0.1,

...,N-l.

(8)

In order to obtain the same result in (7) and (8) it is necessary choose a unique representation of the m elements of the residue

to

class

71

ring of integers modulo m . Often the symmetric representation of the m elements —

m-1 ^ , t«*r

,

1 Q 1 o t "J.» Wi J. «

• ••»

m-1 ^

(9)

J

if m is an odd integer, is used. Note that m has to be an odd integer in order to get nontrivial transform lengths N in If

max

| , m < m-1 I y(n)l =

(4). (10)

.

then the modulus m is large enough and no overflow modulo m in (7) occurs. Then (9) is called the dynamic range of the N T T . To obtain (10) from (8)

follows

|y(n)| t max |y(n)| i N max [x(n)[ max|h(n)l If

maxlx(n)l

=

'

(11)

= A, max|h(n)| = A, then (11) yields

1*1 -I/tS-^' •

With (12) a simple bound for"the signals x and h is given to prevent overflow in aonvolution and correlation computations. Better bounds can be obtained using the -ip-norm of the signal x / j / . The convolution

(7)

can be interpreted as a filtering of the signal x with the filter h . For such filtering applications often the following bound is useful |y(n)j

= maxlx(n)|

h (n)[ = ¡¡¿i . n=0 because in most cases the filter h and the maximal amplitude

max|x(n)[

of the signal x are known. From the numerical point of view one wishes to choose a NTT with parameters N,o£, ii, that have the following

properties:

- the transform length N has to be large enough and highly

factorizable

(preferably a power of 2), - the primitive N-th root of unity ai modulo m has a simple

binary

arithmetic, - the modulus m has to be large enough to avoid overflow and should have a simple binary a r i t h m e t i c . For most practical applications

and N are given and a convenient

dulus m has to be found. In / £ , ? / a useful way is shown to solve problem by studying cyclotomic polynomials and primitive

factors of the term

o£

N

mo-

this

$fj( x ) f° r integer valuesol

- 1. The following theorem

is

given here without proof. For a detailed describtion the reader is referred to / t*-/. T h e o r e m : Let

oCe. Z

(| X(k) 2 " n k mod F r k=0

b = 2

N = 2 d + 1 , oL - 2,

with =

_ 2 2 2

g)

M k l < - S H F D M k l (2°)

)

LOC (FXN)^ (Mkl) k -1

h)

i

i)

Fk

j)

go to e )

then go to step 4 . else

continue

i+j LOC

(FXN)^(Mkl)

4 . j « - j/2 5 . Fl«- F 1 . 2 F k 6. F 2 < - Fl2k-1 |2k 7 . Create the mask Mkl^*-(I . , a 8 . F1 9.

F1

k.....I

^

k.

2

0

k)

2

(Mkl s )

F1F1+F2

10. k

k+1

if k i logN

then go to 3 .

11. Decoding the elements of F1 from D-code into the B - c o d e . tie note that the multiplication in step 5 . can be performed as shown in (22). In step 3., each computational stage for k=l,2, ...,logN into Fk exponents located in the field FXN on positions j.l, 2k_1

delivers j=0,l,...,

- 1, l = N / 2 k .

As a consiuding step in the inverse FNT algorithm, one has to divide the result by the value N = 2 n . Since N

can be expressed from (18) as

N - 1 = _ 2 n " l ° 9 N ; (nultiplication by this value can be computed by means of formula (23), using shift and negation operations 6 . Fermat Number Transforms in convolution

only.

computing

As mentioned in the chapter 2, the F N T saves the convolution It means that the convolution computed by where

iFNT

„

I F N T

property.

of two periodic signals x, h_ can be (24)

£ ( F N T * „ FNT hj]

denotes the inverse Fermat Number

Transform.

Hence, to evaluate the convolution one needs to perform 2 direct one inverse FNT and the points-wise multiplication of the

FNT's,

transformed

79

signals. This multiplication the f o r m u l a The of

following formula

is c o m p u t e d in the D - c o d e , a c c o r d i n g

(22) g i v e n in the p r e v i o u s table summarizes e x p e c t e d

times

(24) on PPS S I M D in d e p e n d e n c e

N . For c o m p a r i s o n

purposes,

the t a b l e

for p a r a l l e l

brings

also the t i m e s

256. Analyzing

the

the l o s t of e f f i c i e n c y l e n g t h of d a t a n e e d e d

run on t h i s

for l a r g e N is c a u s e d by the i n c r e a s i n g

FFT

32

0,0057

0,0268

64

0,0141

0,0376

128

0,0360

0,0559

256

0,09500

0,0811

it is to n o t e

rence

on the p a r a l l e l

that

word-

using

the t i m e s g i v e n are o b t a i n e d

c o s t s in c o r r e s p o n d i n g

time we i n t e n d

that

FNT.

FNT

the i n s t r u c t i o n

for

for N = 2 5 6 , it is to e x p e c t

T i m e s / i n s e c . / for c y c l i c c o n v o l u t i o n

Finally,

computing computer

g a i n s in f a v o u r of the F N T are t r a n s p a r e n t

time e t t i m a t i o n s in

N

parameter for

bits.

A s s e e n , the e f f e c t i v i t y N
**n-

(19)

Some simple production rules demonstrate the power of this operation f o r k = 1,2,3, . . . 86

E

E

k{Qk)**Ok

2 k + l } * * A 2k+1

{ A

k+1

=Qk*

(E2k-i{A3} **n

=

A2(k+1)2-1

- A2k+1-i

•

In f i g u r e 1 the impulse response of the DOLP edge d e t e c t o r —DOLP = C9 ~ C13

r

1 3

aid those of the lowpass f i l t e r s

cn3)**2**

-

(n5)**2

are shown. The convolution of f with Qg i s r e a l i z e d in three p a s s e s by using quadrant symmetry of A 3 with 9 s h i f t s and 24 additions per p i x e l . The convolution with C 13 weeds f o u r passes with a t o t a l of 4 m u l t i p l i c a t i o n s and 16 additions per p i x e l . Figure 2 shows the t r a n s f e r functions H(u) of those f i l t e r s and the modulations t r a n s f e r function MTF of the imaging system, an Anger s c i n t i l l a t i o n camera, which i s used in Nuclear Medicine. The upper 2 boundary frequency of the MTF i s indicated as u^. Because u^ » UJJ the influence of a l i a s i n g onto true image c o n t r a s t can be n e g l e c t e d . 5. Design of a Gaussian DOLP edge detector The design of Gaussian DOLP f i l t e r s by synthesis on the base of cascaded and expanded b o x f i l t e r s means the m u l t i p l i c a t i o n of t r a n s f e r functions

with

H (u)

= 7 7 n ± (u)

Hj (u) i

= Hn nk

(u)

=

Bin(k(w). and Note, and in

- classical

At that point,

it is possible to define features of knowledge which are

essential for specifying the structure of concrete expert systems. features, as incompleteness, inconsistency, non-monotonity, relativity,

uncertainty,

Such

inaccurracy,

imprecision, informality, significance,

rele-

vance etc. are considered in (2), e.g., and these features are characterizing

at some extent the new quality by going from data

knowledge

Progress

logic,

such features of knowledge as listed will have an essential

and

in

to

processing.

studying these "non-classical extensions"

impact on the futher development of expert system

of

technology.

5. Conclusions In rating the present situation in the field of expert systems, ally

in

Computer

Vision,

some critical remarks and important

futher research may be listed.

for

expert systems,

and

connected with

years

ago

decreasing

interest

be

that

AI in general have gained during the last years it Computer

Vision were highly

at many places (and now,

underestimation),

topics

Despite of the high attention

may be said that this field is still at its beginning. ties

especi-

in

it seems,

As the difficul-

underestimated

we do have

a

this field what is the normal result

some

time of

of this

now the same is true for expert systems. There should

much more concrete projects in this field for stimulating real

pro-

gress, and less papers "just talking about it". Research

in the field of expert systems will make contributions to

knowledge

about KNOWLEDGE,

i.e.

stems which may be used for certain types of reasoning as well as features deal types ving the

of knowledge as mentioned at the end of Section 4,

with knowledge which is characterized by these features.

expert

systems

in

our

about the structur e of inference sy-

Computer Vision have

demonstrated

that

about

and how to So

far,

different

of inference systems (several agents) have to cooperate for complex problems (as illustrated in Fig.6 for three agents) iteration

essential.

solwhere

between low-level and high -level processes seems to

Only close to the configuration or program synthesizing

there is some hope that operational systems may be possible in the

be area near

future. General research issues of expert systems are, for example: (i)

Learning for simplifying the knowledge acquisition processes,

(ii)

explanations

of

"why" this solution path

was selected, and not

"in what sequence of steps" the solution was generated so far, (iii) generalisations of specifi ed expert system approaches, (iv)

inclusion of

causal knowledge

of the given domain into the infe-

rence system, (v)

more comfort at the user surface (Natural language Interface, spe-

(vi)

judgement of its own competence, or the development of own strate-

cialized Hardware/Software components etc.), and gies which may be (partially) in conflict to the user's requests.

119

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13. 14. 15. 16. 17.

18. 19. 20. 21. 22.

120

D . H . B a l l a r d , C . M . B r o w n , Computer V i s i o n . E n g l e w o o d C l i f f s , N . J . , P r e n t i c e - H a l l , 1982. W . B i b e l , P h . J o r r a n d (eds.), Fundamentals of A r t i f i c i a l I n t e l l i gence . LNCS 232, Springer, Berlin H e i d e l b e r g New York London Paris Tokyo, 1985. T.O.Binford, Survey of m o d e l - b a s e d image a n a l y s i s s y s t e m s . Int. J. of R o b o t i c s R e s e a r c h 1(1982), 18-64. W . C l o c k s i n , C . M e l l i s h , P r o g r a m m i n g in Prolog. S p r i n g e r , Berlin H e i d e l b e r g New York , 1981. P . R . C o h e n , E . A . F e i g e n b a u m , The Handbook of A r t i f i c i a l I n t e l l i g e n c e . Heuristech Press, Stanford, C.A., 1982. J.Mc D e r m o t t , R1 r e v i s i t e d : f o u r years in the t r e n c h e s . AI M a g a z i n 5 ( 1 9 8 4 ) 3 , 21-32. M . E n d e r , C . - E . L i e d t k e , R e p r ä s e n t a t i o n der r e l e v a n t e n W i s s e n s i n h a l t e in einem s e l b s t a d a p t i e r t e n B i l d d e u t u n g s s y s t e m . IFB 125 (8.DAGM S y m p . ) , S p r i n g e r - V e r l a g 1986, 2 1 9 - 2 2 3 . A . G r z e g o r c z y k , An O u t l i n e of M a t h e m a t i c a l Logic • D . R e i d e l Publ. Company D o r d r e c h t - H o l l a n d 7 B o s t o n - USA, 1974. B . H a y e s - R o t h , A b l a c k b o a r d a r c h i t e c t u r e for c o n t r o l . A r t i f i c i a l Int e l l i g e n c e 26 (1985), No. 3. R . H e s s e , R . K l e i n , R . K l e t t e , XAMBA - A knowledge-based programming e n v i r o n m e n t for a u t o m a t e d picture a n a l y s i s . P r o c . I n t . C o n f . " S y s t e m s of k n o w l e d g e and picture p r o c e s s i n g " , S m o l e n i c e , C z e c h o s l o v a k i a , 24 - 2 8 N o v . 1 9 8 6 , 105-109. M.D.Levin, A.B.Nazif, Rule-based image segmentation. Computer Vision, G r a p h i c s , and Image P r o c e s s i n g 32 (1985), 104-126. R . K l e t t e , R . W i e h a g e n , R e s e a r c h in the theory of i n d u c t i v e i n f e r e n c e by GDR m a t h e m a t i c i a n s - a survey. Information Sciences, 22(1980), 149-169. T . M a t s u y a m a , K n o w l e d g e o r g a n i z a t i o n and control s t r u c t u r e for image u n d e r s t a n d i n g . P r o c . 7 t h I n t . C o n f . o n P a t t e r n R e c o g n i t i o n , July 30 August 2, 1984, M o n t r e a l , IEEE, 1118-1127. T.Matsuyama, M . O z u k i , LLVE-: An E x p e r t system for top-down image s e g m e n t a t i o n . Journal of IPS Japan 2 7 ( 1 9 8 6 ) , 191-204 (in J a p a n e s e ) . M . M i n s k y , A framework for r e p r e s e n t i n g k n o w l e d g e , in: P s y c h o l o g y of Computer V i s i o n (P.H. W i n s t o n , ed.), M c - G r a w Hill, New York, 1975. N . N i l s s o n , P r i n c i p l e s of A r t i f i c i a l I n t e l l i g e n c e . S p r i n g e r , Berlin H e i d e l b e r g New York, 1980. E.M.Riseman, A.R. Hanson, A m e t h o d o l o g y for the d e v e l o p m e n t of general k n o w l e d g e - b a s e d v i s i o n systems, in: Wissensbasierte Systeme (W. Brauer, B.Radig, Hrsgb.), I n f o r m a t i k - F a c h b e r i c h t 112, S p r i n g e r Verlag Berlin H e i d e l b e r g New York Tokyo, 1985, 257 - 288. A. R o s e n f e l d , Image a n a l y s i s : problems, progress and prospects. P a t t e r n R e c o g n i t i o n 17 (1984), 3-12. G . S a g e r e r , D a r s t e l l u n g und N u t z u n g von E x p e r t e n w i s s e n für ein B i l d a n a l y s e s y s t e m . IFB 104, S p r i n g e r - V e r l a g 1985. K.Sakane, RTTamura, A u t o m a t i c g e n e r a t i o n of image p r o c e s s i n g p r o g r a m s . P r o c . of CVPR, 1985, 189-192. M.Stefik, J.Aikins, R.Balzer, J.Benoit, L.Birnbaum, F.Hayes-Roth, E . S a c e r d o t i , The o r g a n i z a t i o n of expert systems, a T u t o r i a l . A r t i f i cial I n t e l l i g e n c e 18(1982), 135. P.H.Winston, B.K.P.Horn, LISP . A d d i s o n - W e s l e y P u b l i s h . C o m p . , R e a ding , M a s s . , 1981.

U T I L I Z A T I O N O F K N O W L E D G E C O N T E N T S FOR THE ADAPTATION OF AN INDUSTRIAL VISION C.-E. L i e d t k e , M.

AUTOMATED

SYSTEM

Ender

I n s t i t u t für T h e o r e t i s c h e N a c h r i c h t e n t e c h n i k Informationsverarbeitung, Callinstr.

1.

problem

becomes

the

in the

use of v i s i o n

adaptation

properties,

changes

of

in

these

area

of

image

are

highly

processing

expensive

question

arises

entering

a

itself,

if

part

so

that

Since

we

the

cannot to

recognition

a

of

and

systems

pattern

system

can

this

very

specific

and

relevant is

ourselves

contained

vertices,

in

the f o l l o w i n g concept

developed

the

circles,

for

a

fast

the

to be a n a l y z e d

(b)

The

Fig.l. During

more

one

This

adaptation

hand and

is

and

quite

support

by

flexible

all

the

situation vision the

generality,

by

system

user

we

application, position

industrial

adapt

of

line call

from

and

line

a

the

single

f u r t h e r , we where

the

orientation

structures these

limited

namely

scenes,

type, p o s i t i o n ,

image

analysis

considerations:

needed

(a)

less

like

edges,

structures

system

Many

available.

The

frequently, concept

p h a s e a fast a l g o r i t h m by hand

is

some

contents

on the other

the

experts

limited

this the

their

of

object

algorithm phase

knowledge

these

has

similar

in

about

is

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illustrated

generated

mechanism,

object

domain

knowledge contents about

fast,

therefore

is used for the

automatically

inference

and

been images

the a n a l y s i s p r o c e d u r e m u s t b e e x t r e m e l y is

time

camera in

cues".

and

a processing

interpretation.

interest

and

object

the

to s i m p l i f y this p r o b l e m

relation

following

adaptation

is into

of

and e l l i p s e s . We will

have

preceding

spatial

in

area

class the

"scene d o m a i n

under

considerably

a

about

and

of

Since

overcome

little

problem

3D-objects

to

information

objects

the h e l p of e x p e r t s

number

to

automation

contents.

s t a n d a r d TV c a m e r a image. In order restricted

new

changes

knowledge

with

solve

2D-

their

to new s c e n e

to

industrial

recognition.

possible expert

in

and

requires

since

is

this

itself automatically

ourselves

and

it

of

systems

illumination,

p o s i t i o n . Up to now the a d a p t a t i o n

on

Hannover

1, FRG

INTRODUCTION

A major

A

Universität

32, D 3000 Hannover

und

which of image

in

image in

a

employs

immediate analysis

algor ithms.

121

2. C O N C E P T OF THE A U T O M A T E D A D A P T A T I O N The

concept

knowledge offered

of

contents to

the

corresponding system The

can

the of

about

object

in

the

image.

the

The

ideal

processing

path.

criterion

Parts the

of

has been

the

the

is e s s e n t i a l

for

have

been

a s p e c t of

described

the

knowledge

o b j e c t can b e expressed

Measure tances

the center

the

image 3.1 In

to

knowledge

to

contrast

be

is

From

representations contents

122

have

the

this

frequently

the

shape

by the

a

in d e t a i l .

of

procedure.

a

particular

procedure:

If all

the disin(3.2)

both

examples

apparent

difference knowledge

in

are

the

refers

to

content.

If

an

attribute

"round"

in

to m e e t

the

adapt

use

concluded, in

the

modified.

object

the new

that

the

procedure declarative where

the

in

in an

declaration situation. 3.2

3.2 h a s c o m p l e t e l y

applications,

same,

of

flexibility

"angular" to

the

use

by

be

to be

In

which

object.

s h a p e . The p r o c e d u r e

can

CI - 33.

coding,

by a d e c l a r a t i o n or a

tolerance

in

difficult

find

and

the

replaced

quite

will

One

of

d e s c r i p t i o n of an angular rewritten.

in

termination

literature

is round."

angular,

to be

it

repeated

pregiven

is d i s c u s s e d

any p o i n t of

become

schemes.

modification only

a

the

be expressed

contents

differences

appears

needs

processed

modifications

(3.1)

a pregiven

the o b j e c t

representational respect

in

about

of g r a v i t y .

remain within

significant

actual in

representation

of g r a v i t y of the

the center

t e r v a l , then

Though

is round"

the d i s t a n c e b e t w e e n

o b j e c t and

until

results.

processing

declaration:

k n o w l e d g e c o n t e n t could

"Calculate

the

a the

CONTENTS

content

by the

"The o b j e c t

The same

are

and

by

resulting

the s o l u t i o n of the p r o b l e m

Any k n o w l e d g e c o n t e n t can b e e x p r e s s e d example

processing

obtained

is p e r i o d i c a l l y

cycle

knowledge

3. C O D I N G M E T H O D S FOR K N O W L E D G E

For

image

fulfilled.

system

following

are

between

procedure

This

The

the r e f e r e n c e d e s c r i p t i o n

evaluated

processing-evaluation-modification

Fig.2. interest

reference

intermediate

differences

in

immediate

a

results

r e s u l t s are

shown

of

of

From

ideal

processing

is

domain

form

automatically

intermediate

and

the

procedure

reference description.

reference the

adaptation

system

derive

actual

results

the

SYSTEM

to

the

to b e

knowledge knowledg'e

A second

difference

contents. that

it

The

concerns

procedural

contains

a

content.

The

that

validity

of

the

gives

no

In

Therefore

used.

In

and

where

Knowledge for

than

those

in

and

the the

knowledge

contents

"explicite". has

the

and

control

The

term

The

advantage,

of

combinations

of

most

different

efficient They

languages

like

caused

mainly

for

the

of

the

knowledge

prefered.

which

a

properties This

form the

of

situation

systems

more

human

a different

in

is

is

In c a s e a

in

easily

the

similar

called

representation can

are

processed procedural

representation".

to

procedural mainly

the

coding

to

be

machines

understand

and

supported

by

technical

C,

RULE,

and

are

classical

of

indicated

There

today's

for

SEMANTIC

and

up

of

algorithmic etc..

vs.

methods

manipulation

LISP,

properties

implicite

coding

methods

the

formulation

PASCAL,

and

four

PARAMETER,

represent

strongly

FORTRAN, by

be

in

work.

very

methods

to b e

contents,

technical

ALGORITHM,

PARAMETER

are

have

representation

vs.

led

properties

tools

contents.

have

namely

and

they

operator

declarative

contents,

ALGORITHM

the

importance,

knowledge

human

checked

of

different

the such,

systems.

knowledge The

the

in

knowledge

the

data

of

representation

inefficient

represented

which

be

therefore

knowledge

explicite

representation

Fig.3.

the

that

technical

in

validity

can

knowledge

to

represented

them,

use

The

will

usually

"implicite

explicite

NET.

are

of

knowledge advantage,

formulated

validity

minor

systems' d u e

are

use

of

the

the the

knowledge

usually

amounts

of

has

mostly

content

contents

environments

by

humans

is

use

check

been

the

and

abstract.

technical

to

has

check

large

purposes

described

as

are

reasoning

systems

to

where

the

declarative

general

knowledge

contents

uses

how

knowledge

flexibility for

of

the

how

more

applications,

representations

form

the

contrast,

in

representation

description

indication,

content.

efficiency

description

knowledge

efficiently.

the

knowledge

in

to

now

knowledge

programming wide

use

existing

is

computer

systems .

The

coding

methods

RULE

representation

of

interpretation

mechanism.

supported

The

a

level

of

of

image

symbolic

processing of

extraction

a

Explicite

analysis methods:

NET

For

permit

their

knowledge

first

can (1)

description,

includes

of

SEMANTIC

contents.

use

the they

representation

explicite require is

an

mostly

systems.

processing

first

extraction the

expert

methods

groups of

by

and

knowledge

and

(3)

preprocessing

symbolic

scene

usually

domain

be

subdivided

Low-level-processing,

cues.

The

into

three

extraction

high-level-processing. and

description

(2)

image

refers

in

processing

Low-

segmentation. our

application

methods

(1)

and

The to (2)

123

have

ir common, that

they work on the data

structure of pixel

arrays,

which contain usually a large amount of data. In this group mainly high efficiency and only to a lower degree flexibility is required

to handle

the

knowledge

large

data

representation processing,

volumes. schemes

where

Therefore

are

prefered.

hypotheses

inference

mechanisms

explicite

knowledge

implicite

are

are

In

set

applied,

the up,

the

representation

and

procedural

group

(3) of

plans

are

request

becomes

for

high-level-

generated

and

flexibility

and

dominant.

The

processing

methods work mostly on reduced data sets of symbolic data, which employ more general and flexible data structures like trees or graphs. 4. CODING OF KNOWLEDGE CONTENTS IN THE AUTOMATED ADAPTATION SYSTEM The

solution

requires

of

the

the

adaptation

utilization

of

problem, which

knowledge

has been

contents

stated

(KC) , in the

above,

following

ca tegor ies: (a) KC about the object domain Prior

to

with

a

the

adaptation

relevant

recognized

during

phase

the operator

description

of

the

has to provide

workpiece,

which

the has

system to

the production phase, and a reference description

be of

the reference scene, to which the system will be adapted. (b) KC concerning processing Image is

interpretation

enhanced

processing

step

In

step

The

KC

are contained

connection

preprocessing, first

is a multistep by

methods.

information

symbolic

with

methods

the

by

process.

applying

about

The relevant

a

the

sequence

enhancement

information

of

of

elementary

the

relevant

in the set of elementary processing

given

application

segmentation,

the

description,

and

this

extraction about

includes

of

scene

grouping

KC

methods. about

domain

methods

cues

for

the as

object

recogni tion. (c) KC about evaluation methods The evaluation

is based on the calculation of quality measures and

interpretation of the values of these measures. Since the evaluation applied

sequentially

optimization Consistency

requires

different the

use

is obtained by stating

measures on a higher

124

to

level.

levels of

of

abstraction,

consistent

quality

a

the is

global

measures.

the common properties of the quality

Essential

KC are c o n t a i n e d

in

- the p r o p e r t i e s of the q u a l i t y -

the

(d) K C a b o u t the a d a p t a t i o n The

measures

i n t e r p r e t a t i o n of the v a l u e s of the q u a l i t y

adaptation

adaptation,

strategy

measures

strategy

employs

heuristic

KC a b o u t

the m e t h o d o l o g y

the use of the e v a l u a t i o n m e t h o d s and KC a b o u t

the

of

context

for w h i c h a d a p t a t i o n r u l e s are v a l i d . (e) KC a b o u t the s y s t e m In order

to p r o c e s s

control

the d a t a

the a c t u a l p r o c e s s i n g

KC are

needed, which describe

- the f u n c t i o n i n g of the e x p e r t s y s t e m -

the c o m m u n i c a t i o n p a t h s of the

- the m e c h a n i s m s of r u l e -

the e l e m e n t a r y p r o c e s s i n g procedures

- the i n d i v i d u a l

rules

All

KC,

procedures

some

which

have

been mentioned

so

a r e a of a p p l i c a t i o n , or

general

"edges",

and the e v a l u a t i o n

KC and

are

needed,

logic,

"vertices",

KC

like

about

"circles",

far

(g) KC a b o u t the p r o c e s s i n g

specific

KC

the

about meaning

"ellipses",

for

system.

the

laws

of

general

and

KC

the In

and

object

addition rules

terms,

about

of like

general

etc..

algorithm

about

- the o p t i m a l -

are

the p r o c e s s i n g

p r o c e d u r e s like m i n i m a t i o n , m a x i m a t i o n ,

The KC

system

KC

the

mathematics

shell

subsystems

- the a c t i v a t i o n o f the p r o c e s s i n g

domain,

run

selection

- the e v a l u a t i o n

(f) G e n e r a l

h o w to

s y s t e m . These i n c l u d e KC a b o u t

s e q u e n c e of e l e m e n t a r y p r o c e d u r e s

the o p t i m a l p a r a m e t e r

represent obtained

the by

results

the

of

inference

w h i c h have b e e n m e n t i o n e d

and

values the

adaptation

mechanism

of

process. the

system

These from

KC the

have

been

other

KC

above.

125

The coding

methods

which are e i t h e r which

are

used

for

the

related

KC are

the

quality

explicite

-

in

to the processing

frequently

are coded

scheme ALGORITHM. H e u r i s t i c of

listed

measures,

procedural

for

about

methods

table

of

Fig.4.

KC

of l a r g e volumes o f data or reasons of

KC ahout the

representation

the

e f f i c i e n c y by

interpretation

of

the

and methodologies and

are

coded

require

by

the

the

values an

scheme

RULE. KC, which have f r e q u e n t l y to be exchanged, l i k e

the KC about the

object

or

domain,

which

processing

methods,

frequently

updated

explicite

-

changes the

with

each

evaluation

methods

and m o d i f i e d during

declarative

application and

the

KC about

rules,

which

the are

the adaptation process need an

representation.

They

have

been

coded

by

the

scheme SEMANTIC NET. Examples

for

ALGORTITHM,

the

RULE,

different and

representation

SEMANTIC

NET

as

they

and have

coding been

methods

used

in

the

proposed system are shown in F i g . 5 - 7. 5. RESULTS The d e s c r i b e d 11/780.

knowledge based v i s i o n system has been simulated on a VAX

The symbolic data base c o n t a i n s p r e s e n t l y about

40 nodes of a

semantic net and about 60 production r u l e s . An example f o r the

stepwise

adaptation

an angle

iron of by

in connection

i s shown in F i g . 8 .

with

the r e c o g n i t i o n

of

vertices

of

The adaptation process needed about 20 minutes

CPU-time, where about 10% were used by the e x p e r t system, about 90% the

image

processing

system

and

less

than

1% by

the

internal

communication between the e x p e r t system and the image processing

part.

6. REFERENCES 1. L i e d t k e ,

C.-E.;

Ender,

wissensbasierten Objekten,

7.

M.; Heuser, M.: Komponenten e i n e s

Bildverarbeitungssystems

DAGM

Symposium

zur

adaptiven

Lageerkennung

"Mustererkennung

1985",

von

Erlangen,

1985. 2. Ender,

M. ;

Processing

Liedtke, Quality

C.-E.: in

a

Evaluation Fully

Criteria

Automatic

for

Knowledge

Assessment Based

of

Vision

System, EUSIPCO-86, The Hague, 1986. 3. L i e d t k e , Automated

C.-E.;

Ender, M.: A Knowledge Based V i s i o n System f o r

Adaptation

to

New Scene

Contents,

Eighth

Conference on P a t t e r n R e c o g n i t i o n , P a r i s , October

126

the

International

1986.

Fig.1. Concept of a fast and flexible image analysis system (KC = knowledge contents)

result Fig.2. Concept of automated adaptation

127

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129

Quality: Q = 1 + (K-G)/(C*N) + Diff/N with

Diff - - J ]

MIN

m «M Notation

C d(m,r)/D

, 1 3

r tR

used:

R

-

Processed

M

-

Ideal set of scene domain cues

set of scene domain cues

N

-

Number of elements of M

G

-

Number of elements of R

K

-

Number of correctly extracted scene domain cues

D

-

Tolerance bound

C

-

See text below

d(,) -

Weighted Minkowsky Distance

Knowledge contents about properties of the evaluation 1. Increasing

similarity between

the ideal and

criterion:

the processed

set of

scene domain cues increases the numerical value of Q. 2. If

n

scene

domain

cues

are

found

exactly

in

addition

to

a

previous state the value of Q is increased by n/N. 3. A tolerance bound of Q does not

is defined

increase

for each scene domain cue. The value

if an additionally detected

scene

domain

cue is on the tolerance bound. 4. The value of Q does not change scene

domain

cue

scene

domain

cues

which have

if for each additionally

has been been

found

found

in

its exact

outside

the

detected

position,

tolerance

C

bound

additionally, too.

Fig.5. Example

for

the

implicite

and

procedural

representation

of

knowledge contents about the properties of an evaluation measure. The

evaluation

adaptation

measure

process

scene domain cues.

130

and

is

used

to

assess

is

based

on

the

the

quality

comparison

of

of sets

the of

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e

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goal

or

action

is

slots:

not),

program

starting

refinement

program

the

during

the

the

potential

processing,

rule

of

the

refinement

algo-

rithm ,

(5) father the

(6) a

and

children

refinement

fragment

action

to

of

the

achieve

(7) arguments

frame

lames

according

to t h e

position

within

hierarchy,

which

program's

the

are

goal

data

source

(only

in

code

case

structures

and

which

represents

of t e r m i n a l

connect

the

goals),

action

the

with

others.

The

whole

generated called cut-out graphs. tional by

program during

program of The

net

an

stage

the

being

example

rectangular

n o d e s ) of

is r e p r e s e n t e d

past

refinement a set

successor—relations

of r e l a t e d

program nodes

different

by t h e

steps.

net

in

represent

program

form

of

type

program

list

list

of t h e

represents

graphs. two

potential

organisational

(organisational

full

This

Fig.

program

structure

a

program

actions

split)

so-

1 shows

connected

(action,

frames a

(func-

connected

or

135

I ••« w•«» * 5

1 L (ù

$ O u_

Pig. 1:

136

Program net representation of a refinement situatic

c i r c u l a r n o d e s r e p r e s e n t d a t a w h i c h are

connected

w i t h f u n c t i o n a l n o d e s by i n p u t - o u t p u t - r e l a t i o n s to e n s u r e the

control

flow).

necessary

interaction functions

The

b e t w e e n the f u n c t i o n a l

child-relation graph)

(means

generates

different also

completely independent program

(informational program structure that

or d a t a f l o w ) .

The

the f a t h e r node w a s r e f i n e d b y

a h i e r a r c h y of p r o g r a m g r a p h s w h i c h

l e v e l s of s p e c i f i c a t i o n d e t a i l of the

e n s u r e s the correct m a n a g e m e n t

the

layers

program

sub-

fatherchild

describe

function.

of g l o b a l a n d local d a t a by

an i n h i e r i t a g e m e c h a n i s m p a s s i n g the r e l a t e d d a t a of the f a t h e r node the c o r r e s p o n d i n g f u n c t i o n a l n o d e s of the c h i l d The program

program

construction

process starts w i t h a

representing .the fully unspecified overall program consider the

trivial

initial

function.

only the c o n n e c t e d d a t a n o d e s a n d can be p r o c e s s e d

reto

independent-

The c o m p l e t e

refinement

p r o g r a m w i l l be a c h i e v e d by r e f i n i n g r e p e a t e d l y a l l goal

until t h e r e are only t e r m i n a l nodes i n the l o w e s t l e v e l s of the graph

nodes

The

of one goal f u n c t i o n node of a c e r t a i n p r o g r a m g r a p h n e e d s

ly of all other n o d e s of the w h o l e p r o g r a m n e t . of

to

graph.

g r a p h c o n s i s t i n g of only one f u n c t i o n a l n o d e and no d a t a

finement

It

causing

nodes program

hierarchy.

3. K n o w l e d g e b a s e d The

Refinement

b a s i c a l g o r i t h m of goal n o d e d e c o m p o s i t i o n a n d a c t i o n

t i o n is as

genera-

follows:

(1) c h a r a c t e r i z a t i o n

of the goal n o d e f u n c t i o n to be d e c o m p o s e d

user-system-interaction

during

and s t o r i n g the o b t a i n e d p r e d i c a t e

values

in the d a t a b a s e , (2) c l a s s i f i c a t i o n of

the

of the goal node f u n c t i o n a p p l y i n g p r o d u c t i o n

k n o w l e d g e base w h i c h m a t c h the g i v e n goal

characteristics (3) a l l o c a t i o n

of

of

data

a p r e d e f i n e d subgoal

of

structures

structure

the subgoal

algorithm

belonging

to

the

of

its

node),

s t r u c t u r e by s p e c i f i c a t i o n

in order to fit it a c c o r d i n g to the real

t i o n s of the goal node to be This

rules

function

(1),

i n f e r e d goal node class ( c h i l d g r a p h of the goal (4) s p e c i f i c a t i o n

node

condi-

refined.

is a p p l i e d r e p e a t e d l y to e a c h goal node until t h e r e

are

only t e r m i n a l n o d e s w i t h i n the frame l i s t . T h e n a p r o g r a m g e n e r a t o r

will

produce

base

contains

the source code of the p r o g r a m . two d i f f e r e n t kinds

T h e r e f o r e the

of k n o w l e d g e u n i t s :

knowledge

production rules

and

137

predefined subgoal The

units

patterns.

structures.

of

the k n o w l e d g e base are a d d r e s s e d by

B e c a u s e of the fact t h a t slot (4) of the frame

formal

string

representation

of f u n c t i o n a l n o d e s m a y c o n t a i n s u c h an a d d r e s s p a t t e r n as a p o i n t e r of the n o d e ,

of

generalisation-specialisa-

the

k n o w l e d g e b a s e are o r d e r e d w i t h i n a

tion-hierarchy. determine

the

the

to

the rule f i r s t to a p p l y in case of r e f i n e m e n t

elements

T h i s h i e r a r c h y , the i n t r o d u c e d s u b s t r u c t u r e s , and d o m a i n of a p p l i c a t i o n of the s y s t e m .

All

rules

knowledge

d e c l a r e d w i t h the help of a f r a m e - l i k e d e s c r i p t i o n l a n g u a g e and c a n

is easy

be c h a n g e d or m o d i f i e d . To p r e p a r e the s y s t e m for a special a p p l i c a t i o n d o m a i n the

knowledge

can be i n s e r t e d w i t h the h e l p of a c o m f o r t a b l e g r a p h i c - s u p p o r t e d ledge a q u i s i t i o n

know-

module.

The s e p a r a t e p r o g r a m g e n e r a t o r a l l o w s e a s i l y to a d a p t the s y s t e m to d i f f e r e n t goal p r o g r a m m i n g l a n g u a g e s . system

4. A p p l i c a t i o n to Image The ting

2 shows a flow chart of the

Processing

s y s t e m w a s i m p l e m e n t e d i n P A S C A L a n d runs u n d e r M S - D O S

system

on I B M - P C - c o m p a t i b l e

m e m o r y and 6 4 0 K b y t e o p e r a t i o n a l Firstly

it

was

- repeatedly processing

(2) s u b s t r u c t u r e s - picture - picture

of single

of series of pictures

input preprocessing s e a r c h and object

- object

classification

- result

for the

for

for

object class output

operaexternal

memory.

- object -

disk

analysis

slides. The k n o w l e d g e b a s e c o n t a i n s a b o u t

(1) basic p r o g r a m s t r u c t u r e s

- processing

16-Bit-PC's with hard

a p p l i e d to d e v e l o p p r o g r a m s

medical microscopical

138

Fig.

kernel.

evaluation

isolation

pictures

of

hundred

I Fig.

I

Mnou ledge engineer

2: Flow chart of the system's

domain oriented user

kernel

139

(3) data structures for - olass membership oriterions - olase features - output formats - parameters of functions, and (4)

simple tasks

menue driven rules for specification and

ohoosing functional

of

substructures

program from

the

knowledge base. The

generated

language

IAHBA

picture

processing programs

are

written

in

the

(Robotron,1985).

During the test phase the following results oould be stated: - The maximum time-out of the system between two user requests is some seconds.

The

development of a typical program takes about half

an

hour. - The generated programs are well structured and commented w - The

efficiency of the automatically generated programs doesn't sic-

nifioantly differ from that of hand-made programs. - Additionally inserted verbal or graphic help and overview make

the

handling

of the system very comfortable to

functions

the

domain-

oriented user. Up to now the system was tested in analysis of microscopical

slides

in pathology.

5. References Cohen, P. R., Feigenbaum, E. A., 1982, The Handbook of Artificial

Intel-

ligence, Heuristeoh Press, Stanford, C. A. Robotron

Vertrieb

Berlin

(VEB),

1985,

Dialog-und-Programmier-System

AMBA/R, Berlin Sacerdoti,

E.

D., 1977, A Structure for Plans and Behavior, Artificial

Intelligence Series, Elsevier North-Holland Inc., New York Voss,

K.,

Hufnagl,

P., Klette, R., 1985, Interactive Software Systems

for Computer Vision, Progress in Pattern Recognition 2 (Kanal, L. N., Rosenfeld, A., eds.), North Holland, 57:78 Stefik, M., 1981, Planning with constraints (MOLGEN: Part 1 and Part 2), AI-Journal 16 (1981), pp. 111-170

140

AN A S S O C I A T I V E NETWORK AS SYSTEM SHELL f'up KNOWLEDGF BASED

' MAGE UNDERSTANDING «

G . Sagerer, S. Schroder, H. Niemann Lehrstuhl fUr Informatik 5 (Mustererkennung) Martensstr. 3 D-8520 Erlangen Federal Republic of Germany Abstract: For the analysis of complex images, or image sequences explicit knowledge about the task domain is required, in order to extract a symbolic d e s c r i p t i o n out of the input data automatically. In this p a p e r , a shell for such image analysis systems is described. It combines a powerful knowledge representation scheme with a control a l g o r i t h m , which m a y be refined for special tasks. Tools for knowledge acquisition c o m p l e t e the shell. It does not d e p e n d on a special field of problems to be attacked. It was successfully u s e d for the analysis of scintigraphic image sequences of the heart. Applications for the analysis of industrial scenes, and speech understanding are in progress. 1.

Introduction

The purpose of image a n a l y s i s is the automatic investigation of a symbolic description of an image, or an image sequence. The content of the symbolic d e s c r i p t i o n must c o r r e s pond to the requirements of a concrete application. E.g., for a medical application like X-ray images, diagnostic descriptions are to be extracted out of an image; for industrial scenes descriptions of the quality of built objects or special tasks for a robot a r e required. To achieve this, complex applications need the explicit representation of extensiv knowledge about objects to be recognized, their relations to each other a n d their task specific domain. Furthermore there is a need for algorithms to use the stored knowledge during the analysis process. The efficiency of an image analysis system therefore depends on both the knowledge base a n d the control a l g o r i t h m . Besides this two modules, knowledge base a n d control, a database for the results a n d methods for preprocessing a n d segmentation of images are necessary/1/. In many approaches the system is c o m p l e t e d by an explanation module which allows the user to check system activities a n d to evaluate final and intermediate results after an analysis process has been finished. Up to this point the structure of an image analysis system, c o m p a r e Fig.l, is quite similar to a so called e x p e r t system/2/. The differences are mainly the input data and therefore also the data transformation processes. While expert systems transform symbolic d a t a into other symbolic data, image analysis systems has to extract symbolic descriptions out of image matrices, additionally. This y i e l d in a s e c o n d distinction. In expert systems input d a t a can be viewed as to be correct, while images are noisy, segmentation errors may occur, and so on. That is the reason why image analysis systems have to take into account certainty values of data they manipulate. Viewing the knowledge base of a system, different approaches for formalizing knowledge exist. If a knowledge acquisition is done by hand and the obtained special approach is fixed, knowledge is coded into the desired approach. First investigations are done in the a u t o m a t i c acquisition mainly in rule basjd approaches. Up to now, a rough knowledge is also necessary for training processes and training is only possible for a fixed representation scheme. The a c q u i s i t i o n modules, basic knowledge and training processes c o m pletes the system structure in Fig. 1. Depending on the scheme used for knowledge representation, image a n a l y s i s or more g e n e ral pattern analysis systems can be divided into the following groups: - syntactical approaches based on formal grammars - relational database systems - rule b a s e d systems - logic systems based on first order predicate calculus - a s s o c i a t i v e network systems Overviews a n d descriptions of different systems are given for example in /1,3,4/. Control algorithms mainly used are meta rules, depth first or best first search, a n d the A * - A l g o r i t h m / 9 / . To handle certainties heuristic functions, probabilities, or fuzzy sets are used. In this paper a system shell for pattern analysis systems will be described which In this shell, an depends on an associative network formalism a n d the A*-Algorithm.

The work reported in this paper is partially supported by SIEMENS A G , Erlangen

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active hierarchical knowledge base incorperates with an control algorithm, which only knows the syntax and semantics of the network but not it's content. This allows, to create homogenuous knowledge based systems for various applications. Therefore, theshell itself is independend of a special application. Besides the knowledge base structure and the control module which may be refined, the shell actually subsumes also tools for allows interactive acquisition, for creating a n d updating the database for results and difierent types of certainty-functions, only restricted by the A*-Algorithm. Tools for at most a u t o m a t i c knowledge acquisition are in development. The shell was successfully used in a system which achieves automatically an d i a g n o s t i c interpretation of scintigraphic image sequences of the human heart. Two further systems, a speaker independent continuous speech understanding system and for the interpretation of industrial scenes are in progress. 2. Knowledge Representation by an Associative Network 2.1 General

Approach

Remarks

Associative (or semantic) networks are special d i r e c t e d graphs, with labeled nodes and edges. First developed to represent the semantics of natural language, they were used as knowledge representation scheme for different applications. The basic idea is quite simple: Information about conceptions (objects, relations, events) are r e p r e s e n t e d by nodes, interrelationships between nodes are represented by labeled edges, to mark also the kind of the relationship. Additional, a special kind of edge, mostly named " I S _ A M , is used to connect a conception to another one which represent a class. Along such a is hierarchie of more general to more special nodes an inheritance of the properties defined. The theoretical background of such networks was studied with large effort by many groups/5/. It was c a r r i e d out, that associative network formalisms has to be epistemological adequate. That means on the one hand that there is a need for a clear distingtion, what kind of information has to be represented by nodes and what kind by edges/6/. On the other hand the syntax of a network must be well defined a n d also the semantics of the different types of nodes a n d edges must be fixed/7/. This leads to formalisms which are independend from the special content of knowledge they are used for. Further investigations were done to expand the prior only declarative content of such networks due to procedural knowledge. Starting with procedural attachment at some slots of a node this y i e l d to the powerful integration of procedures into "procedural associative networks", e.g./8/. By our opinion, such well structured networks with the capability to handle also procedural knowledge are, at the moment, the adequate way to built up knowledge bases for image (and speech) analysis systems. Our approach which will be described in the next sections is mainly influenced by KL-ONE/7/ and PSN/8/.

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3.2 7Vpeg of Nodes thi**e different types of nodes exist in the network. Fii st of all, a CONCEPT gives the nterisional description of an object, an event, or classes of objects or events. The intensional description subsumes all properties of the conception to be modeled. Examples of such concepts are general ones like "3D-object", "movement", or "disease", or more special ones like "car", "run", or "enlarged left ventricle". The second type, called INSTANCE, represent a hypotheses that a real world object or event satisfies the description of a concept. If, for example, some region of an image is interpreted to be a "car" during the analysis process, an instance of car is created referring this region, may it be true or not. Therefore one real object may be referred to by more than one instance, but each instance is associated with exactly one concept. The concepts form the knowledge base, while their extensional sets, the instances, are the intermediate and final results. Viewed as data structures, they are both identical, exept that instances hold results while concepts subsume descriptions. The third type of node, the hUTA-OONCUfl , build up the knowledge base for the acquisition process, which results in a knowledge base for the image analysis system. Concepts can be looked at to be created by meta-concepts and data. This third kind of node will be described in section 4. The features of concepts and their interrelation will be discussed in the following. Fig. 2 shows the syntactical structure of a concept. Hatched blocks are necessary for the knowledge acquisition process and are explaned in section 4. 2.3 Types of Edges Relationships between concepts (and also between instances) can be built up by three different types of edges. They are denoted by generalization, semantic-part, and necessary-part. Each of these edges is described in a concept, see Fig.2. While a concept can only have an unique generalization which is another concept, it may consist of arbatrary number of semantic-, and/or necessary-parts. All the edges define a hierarchie in the network. Both part relationships are dscribed by a complex slot, see Fig.2. If, e.g., a concept A has concept B as semantic-part, B is the domain inside a slot semantic-part in A. For all the three relationships also the inverse is defined by specialization, semantic-part-of, and necessary-part-of, respectively. In the following, first the semantics of the relationships will be discussed, and after that the different facets inside the both "part"-slots. Generalization combines a concept to another one which represent a superclass compared to the concept itself, e.g. vehicle would be the generalization of car and object of vehicle. Like in KL-ONE all properties, that are parts, attributes, and structural_relations defined in a concept A are inherited to all concepts which are specializations of A, but they may be explicitely modified. Such modifications are to be described in special facets of the different slots. The restrictions and the rules for such modifications are comparable to those defined in KL-OHE. Contrary to other approaches of semantic networks we distinguish two different "part" relationships. Semant ic-part s of a concept must hold the restriction that they are parts in a natural sense. E.g., semantic-parts of a car are wheels, semantic-parts of a special movement can only be other special movements, not an object. Therefore we can denote this by: a concept and its semantic-parts must have the same degree of abstraction. By the relationship necessary-part concepts of different abstraction levels can be combined. If a concept A has necessary-par. B, this fact can be expressed by: to talk about A requires B. Therefore a object can be necessary-part-of a movement, or a line of a car. All slots in a concept are referred uniquely defined by their role, which describes the functional role of the slot inside the concept. E.g., the fact that one car passes another one has to be modeled in the following way: a concept "pass" has two necessaryparts. Both have the domain "car". One of the part-slots has the role "passes", the other one "is passed". The facet modified explain modifications of an inherited slot. The value can be NO, if the slot is not defined in a more general concept, YES, if a slot with identical role has been modified compared to the more general concept, and therefore been overlayed by a new definition. If also the rqje has changed, the old one has to referred to by this facet. Finally, the facet computation, which is only defined for semantic-parts (and attributes) gives information about the context dependency of the referred part. That is, the conception defined by this part depends on a larger context, i.e., the concept referring the part. E.g., the conception "(chair-)leg" depend on the context "chair". It is not possible to describe a concept "chairleg" without the knowledge that it is a part of a "chair". Contrary, a wheel can be modeled without knowing if it is part of a car, or a bike and so on. Different semantic-parts or different necessary-parts can be combined to sets of modality. Each such set describe which combinations of such parts are sufficient for the concept (obligatory set), and for each set the optional and inherent parts which may complete the optional set.

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2.4 Attributes. Structures. Certainty Attribute* characterize properties of a concept which a r e essential with respect to the task domain. In most cases such attributes are physical properties, like the volume or the color of an object, or start or end time of a movement. But also, e.g., textual explanations of diagnostic interpretations may be attributes of concepts. But they must not be used to handle properties which describe relationships to another attribute or special relationships between properties of parts. Identical to parts, attributes are uniquely defined by their role. Additional facets give the domain, select ions for the domain and default (or initial) values. As mentioned a b o v e , attributes are inherited from a more general to a more special concept. But it is also possible to modify an attribute in the same way as a modification can be done for parts. Substantial for the definition of a n attribute is the procedure how a value for an instance can be calculated (computation). Such a procedure can be a simple multiplication, but also a parser combined with a grammar, or, even more complex, a complete rule based system. There is no necessity for such procedures to give an unique result. Based on the same input data they may build up different competing results. Each is stored in one instance, and therefore competing instances are created, each having one unique value for all its attributes. The results of the computation can be restricted due to the conception they are used for. The procedure referred to by restrictions do this job. The arguments of the computation procedure are referred in a further facet of the slot. The arguments are a subset of those attributes which are d e f i n e d in parts of the concept, or if the concept depend on a larger concept also in the concept describing this context, compare section 3. In order to support control algorithms with mixed bottom-up and topdown strategies, also a procedure for an inverse computation is referred. Properties which must hold between different attributes and/or attributes of parts, are to be tested by structures. The semantics of these slots, which are also uniquely defined by their role, is: a relation tests the arguments due to the fact substantial for the concept. E.g., in a concept "car" a structure "above" may b e defined, which is able to test, whether the "bodywork" is above the "wheels". The results of structural tests are essential for an instance to be more or less valid for the concept. Other informations which may help to judge the quality of an instance are the restrictions facets in the different slots and the quality of the instances referred to by part relationships. Out of these facts the CF-arguments can be chosen to check the quality of an instance by the CF-procedure.

3. Use of Knowledge. Creating Instances The main idea to use knowledge stored in an associative network as described above is concentrated in the basic rule for instantiation. This rule completes the definition of the network scheme by a third step. Besides the syntax a n d the semantics of the formalism, the use of stored knowledge and therefore the pragmatic is defined. This can also be done without respect to a task domain. The rule can be expressed as follows: of the modality of A, IF for a concept A with respect to one obligatory-set instances for those concepts exist, which are referred to by the following slots in A or slots inherited to A without modificat ions: necessary-parts AMD semant ic_part s with facet computation equal to A O AND one of context if this slot is not equal A O THEN build u p instances ivp(A) of A as follows: - create empty instances of A, - connect these instances to those instances, which are mentioned in the condition of the rule, - calculate the attribut«-values for all ivp(A), with respect to the argument s - test the restrictions - estimate the quality of ivp(A), by CF-procedure with respect to the CFarguments referring yet known instances a n d restrict ions By this rule partial instances are built up with calculated attribute values and tested restrictions, a n d an estimated quality. As mentioned in section 2.4, for structural tests attributes out of all parts may be used. Therefore a n additional rule, the rule for completing an instance, is required: IF an instance ivp(A) of a concept A exists AND instances for all those concepts exist, which are referred to by the semantic-part slots in A, with respect to the obligatory set of the modality used for building up ivp(A)

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THEN

build up iv(A) out of ivp(A) as follows: - connect iv(fl) to the instances referred to by the condition of this rule and not y e t c o n n e c t e d to ivp(A) - test the structures d e f i n e d in A with respect to the arguments - calculate the quality of iv(A), by CF-procedure with respect to the CFarguments If a goal concept for an analysis process is known, fecursive application of these b o t h rules results in a search path for the g o a l concept. By c o m p e t i n g instances, generated for a concept, this path is splitted into a search tree. Based on the q u a l i t i e s of instances qualities for concepts r e s t r i c t e d to one path of the tree can be estimated. This yield to quality functions for the nodes of the search tree, w h i c h are used for the A - A l g o r i t h m /10/. Up to this point only obligatory sets of the modality of the concepts were used. An a d d i t i o n a l rule, the rule for extending an instance, a l s o handles the optional and inherent parts. IF an instance iv(A) of a concept A exists AND instances of concepts e x i s t , which are optional or inherent due to the modality used for building up iv(A) THEN e x t e n d iv(A) to ive(A) as follows: - connect the instances m e n t i o n e d in the c o n d i t i o n to ive(A) - c a l c u l a t e attributes with r e s t e c t to the arguments - test structures with respect to the arguments - c a l c u l a t e the CF of the new ive(A) If the goal concept is k n o w n , this three rules together w i t h the A*-Algorithm form the skeleton for d i f f e r e n t c o n t r o l strategies. But they are also one complete strategie. F u r t h e r m o r e , it is a l s o possible to e x t r a c t automatically potential g o a l concepts /15/. 4. Knowledge

Acauisistion

4.1 General Remarks The ability of knowledge acquisition (learning) is very important for m o d e r n image analysis systems. A l t h o u g h the analysis system is designed modularly, the a d a p t i o n to another field of interest is a time spending process. The bottleneck of this process is of course the c o n s t r u c t i o n of the new k n o w l e d g e base. There are basically two approaches to knowledge a c q u i s i t i o n : leave it to the d e s i g n e r , that is, the manual a p p r o a c h , or leave it to the machine, that is, the (automatic) learning a p p r o a c h . And there is an a r b i t r a r y number of r e l a t e d approaches resulting from some mixture of the m a n u a l a n d the learning approach /l/. Examples of knowledge a c q u i s i tion for image a n a l y s i s systems are g i v e n in /13,14/. Until now we followed the manual a p p r o a c h . The system designer was able to b u i l d up the declarative part of the knowledge base using a special network editor. This editor is described in /12/. The goal of our knowledge a c q u i s i t i o n system will be a n almost automatically c o n s t r u c t i o n of the d e c l a r a t i v e part of the knowledge baSe. This learning will be done by e x a m p l e s . Our knowledge acquisition system will enable persons, w h o are not well v e r s e d in a c q u i s i t i o n networks, to build up a new knowledge base. This is an important p r e c o n d i t i o n for a n easy a d a p t i o n of the analysis system to different fields of interest. 4.2 Features used by the Acquisition Process The hatched blocks in Fig. 2 show the features which are used by the a c q u i s i t i o n process. If a concept is g e n e r a t e d by the a c q u i s i t i o n process using a special m e t a - c o n c e p t , these two c o n c e p t s are c o n n e c t e d by an edge named meta-concept from the generated concept to the m e t a - c o n c e p t . The inverse edge is named mode I-concept. A concept of the knowledge-base of the analysis process can only have one meta-concept while a metaconcept can have'several m o d e l - c o n c e p t s . There is no inheritance from a m e t a - c o n c e p t to its m o d e l - c o n c e p t s . Instead each m e t a - c o n c e p t contains a description about h o w to construct such a m o d e l - c o n c e p t . During the a c q u i s i t i o n process the e x i s t i n g parts, structures, a n d attributes of the meta-concept may be split. For instance the m e t a - c o n c e p t 3D-0BJECT may have a part surface. Of course the g e n e r a t e d concepts of real objects w i l l have several surfaces. Therefore the process of splitting, named differentiation, is c o n t r o l e d by a given interval. This interval defines the a l l o w e d number of differentiations. The slot differentiation contains the bounds of this interval. The slot computation of differentiation refers to the procedure that c o m p u t e s the real number of differentiations. The same p r o b l e m o c c u r e s in the case of dimension. Different model-concepts of the same meta-concept may have d i f f e r e n t d i m e n s i o n s of their parts, structures a n d attributes. T h e problem is s o l v e d in the same way as the differentiation. The slot dimension con-

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- i'I i 11 b tut bounds of the allowed dimension-interva; while t li 1 • • imputation of dimension refers tc the procedure which computes the real numbec. The slot rnnodlfied is used if a part, structure or attribute of the meta-concept has to be splitted. In this case the slot of the model-concept contains the role of the part, structure or attribute of the meta-concept. Otherwise rnnodlfied is set to NO. The slot argument-test always refers to a procedure that checks the existance of arguments of procedures that are used by the analysis process. The slot computation of default refers to a procedure that computes the default-values during the acquisition process. 4.3 System Structure The system structure of the acquisition system is shown in Fig. 1. It is similar to the the structure of the analysis system. The modul METHODS is equal to the same modul of analysis system. The content of the modul META-KNOWLEDGE will be explained later. During the acquisition process the database for the results contains concepts instead of instances in the case of analysis. The control of the acquisition process is done by a special algorithm which only knows the syntax a n d semantic of the assosiative network but not it's content. Of course this algorithm is not as sophisticated as the control algorithm of the analysis process. The acquisition process does not have to classify the input data. The goal is to determine a symbolic description of the input data of a given object or a given class of objects that can be used by the analysis process. In our system the symbolic description is stored in an associative network. The two steps of the acquisition algorithm are: creating one concept using a single observation and creating a new concept by comparison of existing concepts. The input data is a sample. But this sample may not only consist of images. For example the process for acquisition of workpieces may use images a n d CAD data of these workpieces as input data. 4.4 Meta-Knowledqe The knowledge base of the acquisition process contains the meta-knowledge. It is separated into a declarative and a procedural part. The declarative part is the meta-model which is formed by the meta-concepts. The meta-model is based on the associative network formalism, which was explained in chapter 2. The procedural part of the meta-knowledge consists of the procedures referred to by the slots argument-test , computation of differentiation, computation of dimension, and computation of default a n d of the rules used to compare intermediate results. The meta-knowledge incorperates with the control algorithm of the acquisition process. But of course they are independent in the way that the same algorithm will work with different meta-knowledge each for one field of interest. 4.5 The Acquisition Process The acquisition process is divided into two steps. The first one is to create one concept using a single observation. This observation may be for instance an image of an object or CAD-data. The algorithm has two parameters. The first parameter is the name of the meta-concept that leads the algorithm. The second one is the name of the concept to be created. During the first phase of step one the acquisition algorithm will b u i l d up a concept shell. Concept - shell means that the new concept, it's parts, attributes a n d structures, and the edges between the parts and the concept are created, but the slots which will later contain special values are empty. This means during this phase computation of differentiation and computation of dimension takes place. The parts, attributes and structures of the meta-concept may be splitted. The goal of the second phase is to test the existance of the arguments of the analysis procedures. This is done by calling the routines referred to by the slots argument-test . A part, attribute or structure of the meta-concept may no more occur in the modelconcept because of differentiation. If a part, attribute or structure of the metaconcept is splitted into several parts, attributes or structures, then they will have new a n d different roles and they can not have the role of the part, attribute or structure from the meta-concept. In this case the names of the arguments which cure composed of the roles have to be determined again. During the third phase, the default values are computed by calling the computation procedure of each attribute. The analysis process may be improved by restriction of the domain of attributes. But if, for example, the analysis is supposed to recpgnize an object independent from its position and the computation procedure for a special 3dpoint computes the position values, it would be fatal to restrict the domain of this attribute. On the other hand it may be useful to restrict the domain of an attribute to

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an »«tervfel e l t o -the coayutetf default value. "J^is way the- analysis w l l oiJy iflarch u>r segments of the image that satisfy the given restriction. The conclusion from these examples is: Restriction of the domain is not always useful and if a restriction takes place, some inaccuracy should be taken into account. The concepts created by the first step are the intermediate results of the acquisition process that are stored in the database. The second step will create new concepts by rulebased comparison of these intermediate results. The comparison may be done to combine several observations of the same object. Especially if you want to build up a complete 3d-model of an object you have to examine several images from different views. On the other hand the comparison of concepts may lead to a description of a given class of objects (generalization). This can be done by specifying positive and negative examples. After these two steps have been processed several times the final results of the acquisition process, the created concepts, have to be transmitted to the knowledge base of the analysis process. 5. Applications and Results The shell described was used for the automatic diagnostic interpretation of scintigraphic image sequences of the heart. This system is discussed in detail in /10/. The knowledge base consists of about 180 concepts with more than 1000 edges. Procedures to calculate attributes are, beside others, also a parser with formal grammars, dynamic programming routines, and fuzzy rules. All the tests and qualities are realized by fuzzy-membership-functions. The system was tested with 21 image sequences. 18 were interpreted correctly, while 3 had result "undefined". A speech understanding system using this shell is in progress. A rapid prototyted version with about 200 concepts is tested. The Strategie of this system is mixed bottom-up and top-down due to the basic rule of instantiation. Therefore also "inverted" attribute calculation is used. Datadriven the knowledge base is restricted to created instances. These variable knowledge bases are build up by NDDIFIED-OONCEFTS. The quality of instances is based on probabilistic features. The main point of the third application, interpretation of industrial scenes, will be automatic knowledge acquisition. The meta-knowledge base of this system consists at the moment of about 20 meta-concepts. The complete shell, including further investigations, is, respectivelly will be realized in C under UNIX. The reason is the high efficiency. Other versions with less functionality were realized in FORTRAH /10/ and LISP /15/. References; /I/ Niemann, H.: Pattern Analysis. Berlin, Heidelberg, New York: Springer 1981 /2/ Hayes-Roth, F., Waterman, D.A., Lenat, D.B. (eds): Building Expert Systems. McGraw Hill 1982 /3/ Ballard, D.H., Brown, C.M.: Computer Vision. Englewood Cliffs: Prentice-Hall 1982 /4/ Hanson, A.R., Riseman, E.M. (eds): Computer Vision Systems. New York: Academic Press 1978 /5/ Findler, N.V. (ed): Associative Networks. New York: Academic Press 1979 /6/ Woods, W.: What's in a Link: Foundations for Semantic Networks. In Bobrow, D., Collins, A. (eds): Representation and Understanding. New York: Academic Press 1975 111 Brachman, R.J.: On the Epistemological Status of Semantic Networks. In /5/, 3-50. /8/ Levesque, H., Mylopoulos, J.; A Procedurel Semantics for Semantic Networks. In /5/, 93-121. /9/ Nilsson, N.J.: Principles of Artificial Intelligence. Berlin, Heidelberg, New York: Springer 1982 /10/ Sagerer, G.: Darstellung und Nutzung von Expertemwissen fUr ein Bildanalsesystem. Berlin, Heidelberg, New York: Springer 1985 /11/ Niemann, H., Sagerer, G: Semantische Netze als Ansatz zur Repräsentation und Nutzung von Wissen fUr die automatische Bildanalyse. In Robotersysteme 1, 1985, 139-150. /12/ Hofmann, I., Gamlich, R., Niemann, H.: A Human Interface for Control of an Image Processing System. Proc. 8th ICPR, Paris, 1986, 1256-1258. /13/ Perkins, W.A.: INSPECTOR: A Computer Vision System that Learns to Inspect Parts. IEEE PAMI, Vol.5, No.6, 1986, 584-592. 1141 Yashida, M., Isuji, S.: A Variable Machine Vision System for Complex Industrial Parts. IEEE Trans, on Comp., Vol.C-26, No.9, 1977, 882-894. /15/ Eichhorn, W., Niemann, H.: A Bidirectional Control Strategy in a Hierarchical Knowledge Structure. Proc. 8th ICPR, Paris, 1986, 181-183.

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the

the

impor-

features

areas".

approach

The

and

neighborhood

of

of

practical

measures

presented

way

of

Mesh

facts

and

to

considered

the

2)

characterized

algorithms,

importace

defined

the

fundamental discipline

of

notion

character-

structure.

point

The

neighborhoods.

by

the

During eral

the

the

digital

last

a

series and

brief

of

of

theory

is

connected

has

to

be

well-developed

it-

the

digital

geometry;

"low

neighborhood

"image this

of

a

Image to

topics

vision"

characterized

relations

in

theory".

papers

will

as

be

-

just

given 1985

of

have

use

in

cite

pub-

of

irre-

sets

most

presented

to

essentially

point

the

self-

represen-

we

bj> t h e

discrete Some

sev-

a

connectivity,

closed

is

Since

formu-

Processing

[ 1 , ..., 10 ]

[llj.

for

formulate

relatively

level

is

necessary

recorded

covering

sets

processing

series

Digital

be

[12]

point

[ 1 3 ] which

tools

compare

In

to

random

of

may

approach

papers.

related

of

papers

an in

and

field

attempts

the

as

the

important

this

paper

way. of

images of

digital

generated by

other

hand,

ables

of

image

a given

specifying

acterized

or

symmetric

subjects

were

in

theory

contained

lected

decades

successful

mostly

the

facts

crete

available.

topics

objects

The

be

books

of

flexive

a

scientific

results)

to

related

basic

in

a

Mathematics

have

two

less

topology,

upon

lished

of

of

theoretic-mathematical

of

based

methods,

branches

theory

or

contained

tation

(notions,

different

more

few

foundation

preconditions,

and

lating

a

is

which

central

Voss

will

deducing

STRPCTURES

1 ) and Klaus

Vertex

formulas

theoretic-mathematical two

self, -

by

the

the

NEIGHBORHOOD

Introduction

to -

of

complete

orientation

that

processing, sets

is

Klette

upon

three

importance

point

a

shown

based

These

image

a mesh

be

be

Theorem.

topological

of

will

may

OF

by

the

field

processing of

assumptions

on

abstraction, so-called

"4-,

such

models

may

be

a given

theory

which

are

application digital there

6-,

or

should

be

certain

topology

and

obtained

images. a

are

some

or

For

a

has

geometry.

"favorite

8-neighborhood", by

dealing

model

generalization

These

example.

of of

be

models"

for

interpretation

with to

some all

consemodels charOn

the

vari-

the

con-

1 ") Zentralinstitut fUr Berlin, DDR-1086

Kybernetik

2 ) Friedrich-Schiller-Universitat, haus,

Jena,

und

Informationsprozesse,

Sektion

Technfllogie,

KurstraBe

33,

Universitatshoch-

DDR-6900

149

crete m o d e l s relevant to image p r o c e s s i n g . At this level of v a r i a b l e s of the t h e o r y are lation tion

points,

a certain

abstraction,

n e i g h b o r h o o d

d e f i n e d on the set of these points, and a c e r t a i n

d e f i n e d by cyclic o r d e r i n g of all the finite n e i g h b o r h o o d s

By i n t e r p r e t i n g these v a r i a b l e s ,

re-

o r i e n t a of

for example points by points of an

points. ortho-

gonal, or by cells of a c e l l u l a r space, or by r e g i o n s of images etc., crete m o d e l s may b e o b t a i n e d . At this level of a f u n d a m e n t a l t h e o r y , e s t a b l i s h e d branches of M a t h e m a t i c s may be c h a r a c t e r i z e d to be the tools, as Set Theory, Graph Theory, Theory of Algebraic Complexes Grid T h e o r y

con-

some methodic

[ 14 ] , or

[ 15 1. But sets, g r a p h s , c o m p l e x e s , or grids are not the

ry subjects of a tlteory of digital

prima-

image p r o c e s s i n g . This point of v i e w

may be compared with the I n v e s t i g a t i o n of spatial m o t i o n s in P h y s i c s , example: The subject is a model of the moving mass point, and the of vector spaces and d i f f e r e n t i a l

e q u a t i o n s are p r o v i d i n g the

tools.

So far we see the following four m a i n topics of a t h e o r y of image sing w h i c h may be studied in the same m e t h o d i c a l - the t o p o l o g i c a l t h e o r y of n e i g h b o r h o o d as support structures of digital

n e i g h b o r h o o d structures - the theory of discrete

s t r u c t u r e s w i t h specific

structures,

in r e g u l a r grids based u p o n mappings' of

into the real space,

and

image functions d e f i n e d on r e g u l a r grids

In the present paper we shall restrict ourself-s to the most

is a finite set of

s t r uct u r e points,

N we obtain concrete

is defined to be a tuple

(P, N) where P

and N - P i Pis an i r r e f l e x i v e

r e l a t i o n called n e i g h b o r h o o d models

relation.

of n e i g h b o r h o o d

of point pi P, and

& (p,q)fcNjis

Let £ = card (P) be the n u m b e r of points, and let structure

graph theory is g i v e n by the V e r t e x

symmetric

structures. A neighborhood

^ ( p ) = card (N(p)) denotes

of edges cf a g i v e n n e i g h b o r h o o d

and

By i n t e r p r e t a t i o n of P and

(P, N) is i s o m o r p h i c to an u n d i r e c t e d g r a p h without

loops and m u l t i p l e edges. The set N ( p ) = ^ q | q £ P

P6P

fundamental

structures.

Structures

A n e i g h b o r h o o d

borhood

charac-

continuosity.

theory, the t o p o l o g i c a l t h e o r y of n e i g h b o r h o o d

structure

models

images,

t e r i z e d by a principal e x c l u s i o n of

2. N e i g h b o r h o o d

proces-

line,

- the geometric theory of digital figures on n e i g h b o r h o o d - the metric theory of figures

for

theories

elementary the

neigh-

its v a l e n c e .

card (N)/2 be the

number

(P, N). Then, a basic f o r m u l a of

Theorem

*(p) = 2 St.

(1 )

Then, ^ = 2at/£ roay be called the a v e r a g e d

valence

of the

given

n e i g h b o r h o o d structure. By g r a p h theory some common notations are g i v e n : A way 9

150

(q 1 , . . . ,

in

a

subset

Q satisfying q i € N ( q ± _ 1 ) ,

Q S P is an ordered sequence

for i = 2, . . . , n . A set QS.P is

of

points

c o n n e c t e d

if a n d o n l y

if

(iff) there

exists

po i n t s p, qft Q. A r e g i o n of a s e t

Q £ P denotes

3. O r i e n t e d By c y c l i c

ordering the

3pecific

realization a

.... Q j ^ Let

H

C be t h e

(P, N, C) is t h e s t r u c t u r e A

(directed)

ed) edge

(p, q ) u n i q u e l y

successor

within

edges,

the

defines

structure

g e n e r a t e s

an

sequence

finite,

any generated way

(p, q, r, s, ...)

ment

of o u r a p p r o a c h w i t h the

c ( p ) = ^ r , t, q ^

the

ordering

predecessor any

its

any edge

(direct-

direct

(p, q )i N

(q,r),(r,s),...

way

of p o i n t s

Because

the

set

P of

Figure

1. The

of

edges,

e. g.

or c ( q ) = ^ r , s, t, p ^ . T h e n u m b e r

points is

period

by m ( p , q).

by

of

passed

A complete

, and denoted

is i l l u s t r a t e d

clockwise

edge

q. T h u s ,

(q, r ) as

(p,q),

P =

n e i g b o r h o o d

(directed)

is c y c l i c .

a g e n e r a t e d w a y w ( p , q ) is c a l l e d a m e s h coincide

a

edge

a c y c l e

p C P. T h e n ,

p is a d i r e c t

of p o i n t

of e d g e s .

is

sequel.

In t h i s w a y ,

by this g e n e r a t e d

notion

by

sequence

a n d w ( p , q ) = (p, q, r, s, ...) d e n o t e s

cycles

c(p),

in the

(directed)

through

basic

cycles

c(q)

(P, N, C).

infinite

an For

ordering { l ^ t Qg'

of a n o r i e n t e d

cycle

an

processing

A cyclic

, i . e . point

the n e i g h b o r h o o d

N(p),

structure.

a n e i g h b o r h o o d

f o l l o w e d

— > —

1 i

0

0

1

2

3

4

5

6

7

8

9

10

11

12

13

H

15

Fig. 2. A structure P ( p o i n t s o, », edges — , • » ) and a structure (points •, e d g e s « » ) , where points p.. are identified by 1J row number i and column number j. For example, in Fig. 2 cycle

c(p

cycle o Q ( p 5 8 ) = < p 6 8 P/.n> P > , -48' *57 r nQ. for p, qfcQ and (p, q ) £ N (i)a

core

P , too

(for

(ii) or a

mesh example,

border

3 8

^

in Figure

p m

Q

68' p59' p48' Q^p' ^

= m

Then, m n

is

and

57 me

^

is restricted

®h

=

p

P

P

Q ^ 2 2 ' 32' 31 ' mesh of Tq iff

P

in Figure 2), 21^ n{p, q), i. the edge

from m^ (for example,

m^

structure

(p, q ) geu-

p < PAR48 ' Prb> ^58 'PC 57' •

2).

of Q , ( p , q ) is an (ordered) b o r d e r

( p , q ) are a t t a c h e d

to

P and cycle c ^ ( q ) =

(undirected) b o r d e r

pair

of Q, and both [ p , q j

the border mesh m Q ( B , q ) = m g ( q , r ) ,

cycle c (q) =(s, s^, ..., s^, p, p^, ..., p-^, r,

of point q in

of point q in substructure

the border edges { p

p

p

p

Euler t _ » « .

example are at... in

by successively

adding a single point

In this way it may be proved

or a

structure)

by induction

that

Theorem +

v

= X $ 2

holds for connected c h a r a c t e r i s t i c

152

P

(connecting two points already existing within the

to a given substructure. the

p

Theorem

Structures may be constructed single edge

p

for

structure

P . For

56' ^57 Ì l 5 6 ' 6 6 $ ' i 55' 5 6 } ' Ì 4 5 ' 5 5 Ì tached to the border mesh m^ip^g, P 5 7 ) m (p Q 5 7 ' p 6 7 } = m Q ( p 6 7 ' p 66' Figure 2 . 4. Euler's

to

substructure

called

For qtQ and pfcP - Q, if q t N ( p ) then £ p , q j is an edge

p a

of PQ iff m^ = m(p , q ) , i . . m^ is a mesh of

erates a mesh in P different P

58 . Let

16

F_

(3)

(!) structures. X

Also

it follows that the

of a given oonnected

Euler

structure may be equal to an

even n u m b e r +2, 0, -2, -4, ... only. In g e n e r a l , = (P, N, C ) consists

of

wise building up a structure striction

(deletion

we have

iff X = X ( P ) =

exists

planar sible

because

C on this graph

X = 2 would of

* ( P )

= 0. For example,

r

In toroidal nets it holds that 2 \ + 2 V

-t. - * a . = o,

thus this equation may be

transformed to where g is called the

total

curvature

of the single border mesh

of P R . The total curvature may be equal to +1 in case of planar

regions,

or equal to -1 in case if toroidal regions, the border mesh of the region is called an

outer

border

mesh,

or an

inner

border

mesh

respectively. Now assume that region R has r border meshes m^ , m^, . . . , m^. Let

156

be the

length mesh

of

m..

mesh

m^ , and

Then,

let

similarty

n^

to

be

(5)

the we

number

of

border

edges

attached

to

obtain

r Vf

-

2 H n i =1

*(u-r)+

+

* for

the

2 *

,

(7.1)

- 1. = 1=1 1

2 *

, and

(7.2)

*

(7.3)

R

r

P _ = (R, N„ , C D ) . n H H

=

J. *T

"

the

global

calculated

logical

*R

^

-

topological

by

7

Because

All

- ^ 1 and

total

Pick R

n. a n d l

1 = ^ - 1 . . . 1

It

1=1

n

and

1

which

Furthermore, of

mesh

exactly

other

characterized

by

the

value

of

r

may

*

based

on

curvature

one

border

may

outer

meshes

be

equational may

border

of

R

obtained

be

local

system

topo-

(7)

the

proved:

mesh

are

by

inner

of

total

ones

curvature

with

total

cur-

n

may

be

process. or

calculated

signalizes Finally,

e x t e r i o r

during

the

the

type

of

.equation

relatively

computation

border

(6) to

may the

mesh be

of

at

used

a

the to

computed

border

mesh,

end

this

of

define

the

border

mesh.

border

meshes.

Formulas

be

For ^

.

•

curvature

i n t e r i o r

Let

1=1

structure

values

R has

the

=

computational

8.

the

region

+1.

the

2—