Wavelet Analysis with Applications to Image Processing [1 ed.] 9780849331695, 9780367811310, 9781000721980, 9781000718072, 9781000714166

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ANALYSIS with Applications to

IMAGE PROCESSING L. Prasad

dh

=xF o

±

I

E

nF

R R+ C N , N+ Z, Z+ Q , Q+ lim max min sup

inf supp

InI

iff w . r .t . w .l.g. dim codim

r

1

Phi , null set , empty set less than or equal to greater than or equal to is not equal to / not equal to is identically equal to plus or minus minus or plus cross product / Cartesian product , cross composition implies is implied by implies and is implied by goes to / is mapped onto is perpendicular to integral of summation of product of any field of numbers; usually R or C the field of real numbers the field of positive real numbers the field of complex numbers the set of natural numbers the ring of integers the field of rational numbers limit of maximum of minimum of supremum of infimum of support of

absolute value of if and only if with respect to without loss of generality dimension of codimension of f inverse

Introduction and Mathematical Prelimaries

1.2

3

Basic Set Operations

In this section, some elementary aspects of set theory used in this book will be introduced. The notion of a set has always eluded formal definition and , therefore, is taken to be a primitive notion in axiomatic set theory. However, in practice, one can easily work with sets without a formal definition because of their frequent occurrence in various branches of the mathematical sciences. It suffices, therefore, to say that the term “set” refers to a collection of objects. In actual mathematical usage this loose description becomes tighter once the context of the set and the category of objects in it is clear. For instance, all real numbers form a set . Likewise all integers, rationals between 1 and 3.7, etc., form sets. The term “set” is used synonymously with the equivalent terms “class,” “family,” “collection,” and “system.” The objects contained in a set are called its “elements” or “members.” If x , y , z , u , v , w belong to a set 5, then this is written: S = { x , y , z , u , v , w }. i. e., a set is represented by writing its elements within curly brackets.

To indicate that an object a is an element of a set A, the membership relation “ ” is used as follows: a G A (read “a belongs to A” or “a is a member of A” or “a is contained in A” ). To indicate that an object a is not an element of a set A, the symbol is used as follows: a £ A ( read “a does not belong to A” or “a is not a member of A” or “a is not contained in A” ) .

^

“ ”

If A and B are two sets such that every element of A is also an element of B , then A is called a subset of H , written A C B or B D A ( read “ B contains A” ). A C B allows for the possibility that A and B coincide (i.e., they have identical set of elements in them ) . This is written A=B ( read “A is equal to J5” ). Clearly A C B and B C A implies A=B .

If Ac B but A / H, then A is said to be a proper subset of B . Consider the set of all odd integers divisible by two. Of course since there are no such odd integers, this “set” contains no elements. Such sets are called empty Thus ( read “A is the null set” ) implies sets or null sets, denoted by that A has no elements. For any set A, $ C A. Given two sets A and B , one can construct a set C which contains all elements belonging to at least one of the sets A and B. The set C is called the union of the sets A and B , written A \J B ( read “A union Bv ).

Wavelet Analysis with Applications to Image Processing

4

Given two sets A and B , One can construct a set C which contains all elements belonging to both the sets A and B . The set C is called the intersection of the sets A and B , written An B ( read “ A intersection .£?” ) . If A fl B = 4> then A and B are said to be mutually disjoint.

If Aa are sets, where the index a runs through an arbitrary set I called the indexing set ( I could be the set of natural numbers, a finite set , or the set of real numbers. It is not necessary for a to take integer values) , then one can define the union of the sets Aa , a G / as the set consisting of all elements which belong to at least one of the sets Aa . This set is written:

(J Aa

or just

|J Aa oc

if the indexing set is clear from the context , or is left unspecified . Similarly, one can define the intersection of the sets Aa , a G / as the set consisting of all elements which belong to every one of the sets Aa This set is written:

Pl Aa

or just

a /

Pl Aa a

if the indexing set is clear from the context , or is left unspecified .

The difference A\B of the set A from the set B is the subset of A consisting of all those elements in A not belonging to B. It is also called the relative complement of B in A. The following identities are easy to verify: 1. ( A\B ) D ( B\A )

=$

2. ( A\B ) U { A fl B )

=A

3. ( A\B ) U { A fl B ) U { B\A )

= AUB

4. ( A u B )\A = B

5. A\B

= A \ (A n B )

The set ( A U B \ ) { A fl B ) = ( A\B ) U ( B\A) is called the symmetric difference of the sets A and B , written AAB . The operations U , H and A are symmetric, i.e., AUB = BuA, AnB = BnA , and AAB = BAA, while ’\ ’ is not ; i .e., A\B B\A; in fact , as already noted, ( A\B ) C\ { B\A) = 4> .

^

If { Aa } a £i is a family of sets, and B is any other set , then the following distributive laws hold :

( ( jA,) ns = ( ( AanB)

J

5

Introduction and Mathematical Prelimaries

( i.e., intersection distributes over union ) . Indeed ,

x

J

([ Aa ) H B => a: G a

[ J Aa and x 6 B a

x E Aa fl B for some a , and i.ex E Aa for some a and x G B , x e \Ja ( Aa fl B ) . Conversely, one can see that xG

U (4

a

n B ) =$

xE

Aa n B

a

x G |J Aa and x G B ,

for some a.

x G ( (Ja Aa ) fl B

Similarly, it is easy to verify:

1.

(

f|Aa ) U S = f|(A“ UB) a

a

( i .e ., union distributes over intersection ) , 2.

(

U A„ ) \ B = U (AAS) a

a

{ i .e., difference distributes over union ) . If in a particular context every set referred to is a subset of a bigger set U , then U is called the universal set for all sets in that context. Then , for any set A C U , the set U\A is called the “complement of A” or “ A complement ,” and represented by the symbol Ac . The following identities are easily verified ; c 1. ( Ac ) = A ,

2. Uc =

3. $c

= U,

= ACDBC , { A Z given by p{ n ) = ( l ) n [n / 2] , where [ x ] is equal to the largest integer less than x . p is thus a one-to-one correspondence between N and

—

•

Z. Two finite sets have the same cardinality if they have an equal number of elements. It is clear that:

# ( A ) = # { B ) and# { B ) = # { C )implies# { A ) = # (C) . ( b) If A = LU Aa B = Ua where the indexing set is the same in both the cases, and # ( Aa ) = # ( Pa ) Va , then # ( A ) = # (P) . ( a)

—

Definition 1.2 A set A is said to be countable if it has the same cardinality as the set of natural numbers N , i.e., if there is a one to one correspondence between the elements of A and those of N .

Introduction and Mathematical Prelimaries

7

In other words, countable sets are those infinite sets whose elements can be labeled uniquely and exhaustively using all natural numbers. The cardinality of countable sets is denoted by the symbol No ( read “aleph nought ” ) .

-

Taken together , the following two theorems show that countable sets are infinite sets of “smallest ” cardinality:

Theorem 1.3 Any infinite set A has a countable subset. Proof: Pick an element from A and label it a\\ pick another and label it L’ given by h { ax + /3 y ) = ah ( x ) -f /3h{ y ) . Va , /3 G F , Vx, y E Ly then L is said to be a homomorphism of L onto L\ }

•

Remark: If h:L ~> L’ is a homomorphism , then /i (0) and 0 G L' . Indeed , /i (0) = h { 0 + 0) = 2 h ( 0).

= 0 where 0 G

L

14

Wavelet Analysis with Applications to Image Processing

Definition 2.4 If h: L-+ L’ is a homomorphism such that h( x ) 0, then h is called an isomorphism.

= 0 L’ is an isomorphism , then \/ x , y G L , h { x ) x = y.

= h( y ) &

If /i:L-> L’ is an isomorphism , the correspondence h( x ) «-> x between elements of L and L’ is one-to-one and onto: h{ x ) x , h{ y ) y => h{ x + y ) x + y , h { ax ) a:c. Isomorphic linear spaces may be thought as different realizations or representations of the same linear space. •

2.1 . 1

Subspaces

If L is a linear space over a field F , and L/ is a nonempty subset of L such that V x , y Gl/ , Va , /? G F , ax + f ) y Gl/ , then L' is said to be a subspace of L , written L ' Cs L.

A subspace L' of L is a linear space in its own right. The trivial space consisting of the zero element only and the whole space L are subspaces of L. A proper subspace of L is a subspace of L which is not the whole space L and contains at least one nonzero element of L. Definition 2.5 If x i , x 2 , . .. , xn are elements of a linear space L such that there exist scalars a i , 0:2 , . . . , an GF not all zero , satisfying OL\ XI -f «2 2 + ^ . • . + OLnxn = 0 , then x i , X 2 , . . . , £n are said to be linearly dependent . If there exists no such set of scalars , then x i , X 2 , . . . , xn are said to be linearly independent . Definition 2.6 A linear space L is said to be n- dimensional if n linearly independent elements can be found in L , but every set of ( n+1 ) elements is linearly dependent . If , however, for every n there exist n linearly independent elements in L , then L is said to be infinite dimensional. Remark ; Any set of n linearly independent elements of an n-dimensional linear space L is said to be a basis for L , and every element of L can be expressed uniquely as a linear combination of elements of the basis.

Linear Spaces , Metric Spaces, and Hilbert Spaces

15

Factor spaces ( quotient spaces )

2.1 . 2

If L' Cs L then x , y G L belong to the same residue class generated by L/ if x- y G L/ . The set of all such residue classes is a linear space and is called the factor space of L relative to L' , written L / L' . If £ , 77 G L / L/ are two residue classes , then choose x G £ , y G 77 . Then the class £ -f 77 is the class to which x -fy belongs , and L / L' is a linear space. If L' CSL then the dimension of L / L' is called the codimension of L' in L.

Theorem 2.7 If L1 Cs L then V has finite codimension n iff 3 linearly independent elements X \ , . . . , xn G L and every x G L has a unique repre sentation of the form x = a\X\ + . . . + anxn + y where , . . . , an G F and yeL' . Proof: If £ G L / L' and Xk G £& , then £ = £iai -f . . . + £n «n V£ G L / L' . Thus £1 , . . . , £n is a basis for L / L' and dim L / L' = n . If codim L' = n , 3 basis of L / L' = < £1 , . . . , £n > . If x G L and x G £ G L / L' , then £ = ai £i + . . . + an £n , i .e. , x = aiXi + . . . + anxn -f y for fixed Xi , . . . , xn G +V £1 , . . . , £n respectively, and yGL'. Indeed , if x — a [ x1 -f . . . -f - y' = 0 , i .e . , ( a - - ai ) £* = 0 , a contradiction and hence then (a - ai ) xi the theorem . •

2.1 . 3

Linear functionals

Definition 2.8 A mapping f : L F is a functional on L . f is additive if f { x -\- y ) = f { x ) + f { y ) and homogeneous if f [ a x ) = ocf { x ) . If f is defined on a complex linear space then f is conjugate homogeneous if f { a x ) = a f { x ) . Accordingly a functional with the above two properties is a linear / conjugate linear functional . Examples: 1 . If ( a i , . . . , a n ) G Rn or Cn is a fixed n- tuple , then

/ : Rn -> R given by (x i , . . . , xn ) \—> a i X\ + a^ x 2 . . . + anxn or / : Cn -> C given by { x \ , . . . , xn ) 1—> a i X i + a2 x 2 . . . + anxn are

real or complex linear functionals

x { t ) i p { t ) d t ( p { t ) G C [ a , b\ fixed and x { t ) G C [ a , b ] , then I is 2 . If I { x ) = a linear functional on C[a , b ] where C d , b] is the space of all continuous functions on the closed interval [ a , b

Wavelet Analysis with Applications to Image Processing

16

2.1.4 If

Null space (kernel ) of a functional - hyperplanes

/ : L -> F is a linear functional, consider the set L f = {x | / (x) = 0}. •

Since / (0) = 0 and / (ax + py ) = a / (x) + P f { y ) , f { x ) = 0, f ( y ) ( ax / + P y ) = 0. Hence L/ is a subspace of L, written L/ C L.

= 0^

Theorem 2.9 Let Xo he any fixed element of L \ L f . Then every element x G L has a unique representation of the form u = axo + y , where y L f . Proof: / (xo ) 0 and xo 0 (obvious) . Without a loss of generality, assume / (xo ) = 1 (else replace XQ by xo / f { x o ) ) - Let x G L. Let y = x — ax0 , where a = f { x ) . Now y L f . : . x = ax0 This representation is unique. Indeed, x = a' x0 + y' => (a a1 ) x0 = y — y' and a = oc' => y = y 1 . , a « 4 x o 6 L / , a contradiction and hence the theorem.

^

^

—

^

Corollary 2.9.1 a?i , X2 are in the same residue class in L with respect to L / (» .c., Xi - x2 e L f ) iff f ( xi ) = / (x2 ) . Corollary 2.9.2 L f has Codim= l . Corollary 2.9. 3 Two nontrivial linear functionals space are proportional .

Indeed, let XQ L \ L f 3 / ( XQ ) y ( y L / ) g { x ) = f ( x ) g { x0 ) . g { x0 ) f { x )

= 1, then