Computerized Tomography: Proceedings of the Fourth International Symposium Novosibirsk, Russia [Reprint 2020 ed.] 9783112314067, 9783112302798

Citation preview

Computerized Tomography

COMPUTERIZED TOMOGRAPHY Proceedings of the Fourth International Symposium Novosibirsk, Russia

Editor-in-Chief: M.M. Lavrent'ev

///VSP/// Utrecht, The Netherlands, 1995

VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

© VSP BV 1995 First published in 1995 ISBN 90-6764-187-1

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

CIP-DATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG Computerized Computerized tomography : proceedings of the fourth international symposium held in Novosibirsk (Russia) on 10-14 August, 1993 / ed.-in-chief M.M. Lavrent'ev. Utrecht: VSP ISBN 906764-187-1 bound NUGI812 Subject headings: computerized tomography.

Printed in The Netherlands by Koninklijke Wöhrmann BV, Zutphen.

C O N T E N T S Agafonov M.I., Stankevich K.S. Image reconstruction of celestial objects from a few strip-integrated projections by lunar occultations

1

Alekseev G.V. Numerical study of a non-linear sound radiating inverse problem of underwater acoustics

7

A l p a t o v V . V . , Likhachov A . V . , Pickalov V . V . , R o m a n o v s k y Yu.A. Optical tomography of natural and artificial optical disturbances in the near terrestrial space

12

Anikonov D . S . , K o v t a n y u k A.E., Prokhorov I.V. Some numerical experiments in tomography of scattering medium

25

Anikonov Yu.E. Integral geometry

31

A p a r t s y n A. Inversion formulae for one class of multidimensional integral Volterra equations of the first kind

49

Arbuzov E. V . and B u k h g e i m A. L. The Cauchy problem for /1-harmonic functions

54

A v d e e v A. Simultaneous determination of sounding signal and wave propagation velocity in a vertically-inhomogeneous medium

70

Balandin A., Fuchs G., Pikalov V . , Rapp J., Sostwisch H. Vector tomography of plasmas using Faraday rotation

78

Baranov V . A . A variational approach to non-linear backprojection

82

Boulfelfel D . , Rangayyan R . M . , Hahn L.J., Kloiber R. Three-dimensional restoration of signal photon emission computed tomography images using the Kalman-Metz filter

98

D a x A. An extended Kaczmarz's method for l p minimum norm solutions

106

Denisjuk A.S. On explicit formula« for inverting the X-ray transform

117

D e s b a t L. Algebraic reconstructions from efficient sampling schemes in tomography

124

D i t m a r P.G. An approach to 2-D ray tracing intended for non-linear seismic tomography

138

11

Falaleev M . V . Asymptotic expansions of continuous solutions of system of Volterra integral equations of the first kind

155

Fokin M . V . , Skazka V . V . On new methods for determining static time corrections from phase velocities of surface waves

158

Fuchs G . , M i u r a Y . , M o r i M . Soft A'-ray tomography on Tokamaks using flux coordinates

165

Goldin V.A. Ray reflection tomography: review and comments

169

Golov V.I. Tomographic methods for information processing in seismilogy. Nonlinear version

188

G r y a z i n a G . Y u . , K a b a n i k h i n S.I. Penalty method with internal prox-regularization for an inverse source problem.

. . . 194

G u l l b e r g G . T . , Z e n g G.L., Clack R . Inverse problems in single photon emission computed tomography

201

H a t t o r i K . , Y a m a g u c h i M . , I w a m a N . , Hayakawa M . Application of GCV-aided Phillips-Tikhonov regularization to direction finding of magnetospheric V L F / E L F radio waves

218

Isanov R . S h . A problem of emission tomography for a domain of small diameter

229

Isanov R . S h . , Y a k h n o V . G . Direct and inverse problems of thermoelasticity

232

I w a m a N . , Y o s i d a H . , T a k i m o t o H . , Teranishi M . GCV-Aided regularizations in sparse-data computed tomography with an application to plasma imaging

235

Izen S . H . Frames on the range of the Radon transform

240

K a b a n i k h i n S.I., B a k a n o v G . B . On the stability estimation of finite-difference and differential-difference analogues of a two-dimensional integral geometry problem

246

K a n e k o A . , Takiguchi T. Hyperfunctions and Radon transform

259

Kazantsev I.G. Inversion of some projection matrices

264

K a z a r o v a A . Y u . , L y u b a v i n L.Ya., N e c h a e v A . G . Interference tomography of the ocean bottom

270

K e r g a k o v B . V . , Fokin V . N . , Fokina M . S . Determination of sea bottom characteristics using acoustic field spatial dependence

274

iii Klimenko O.A. A n a l g o r i t h m for solving a p r o b l e m in e l a s t o - e l e c t r o d y n a m i c s K u l a k o v I.Yu., K e s e l m a n S.I. T h r e e - d i m e n s i o n a l s t r u c t u r e of l a t e r a l h e t e r o g e n e i t i e s in p-velocities m a n t l e of t h e s o u t h e r n m a r g i n of S i b e r i a f r o m teleseismic d a t a

283 in t h e u p p e r 288

Lavrent'ev M.M.(jr), Priimenko V.I. S i m u l t a n e o u s d e t e r m i n a t i o n of elastic a n d e l e c t r o m a g n e t i c m e d i u m p a r a m e t e r s .

. . . 302

Likhachov A.V., Pickalov V . V . A m o d i f i c a t i o n of A R T m e t h o d for c o n e - b e a m t o m o g r a p h y of high s p a c e resolution

309

Likhachov A.V., Pickalov V . V . A new a p p r o a c h t o t h e p r o b l e m of 3D i n t e r p o l a t i o n f r o m an a r b i t r a r y set of p o i n t s

318

Lyadina E.S., Tanzi C.P., D a Cruz D.F., Ingesson L.C., D o n n e A.J.H. A s p a c e - t i m e t o m o g r a p h y a l g o r i t h m for t h e five-camera soft X - r a y d i a g n o s t i c a t the RTP Tokamak 324 M e r a z h o v I.Z., Y a k h n o V . G . Direct a n d inverse p r o b l e m s for s y s t e m of e l e c t r o m a g n e t o - e l a s t i c i t y e q u a t i o n s

332

Morozov A.K., Snegovoy A.A., Tishchenko S.A. A M e t h o d of t h e n u m e r i c a l ray t r a c i n g for a 3 D - i n h o m o g e n e o u s isotropic m e d i u m .

336

N a t t e r e r F. S a m p l i n g a n d r e s o l u t i o n in C T

343

Schwab A.A. S o m e p r o b l e m s of c o m p u t e r i z e d t o m o g r a p h y in t h e s t a t i c p o t e n t i a l s

fields

355

Selishchev S.V., Tereshchenko S.A., Masloboev Yu.P. L u m i n e s c e n t t o m o g r a p h y of s e m i c o n d u c t o r m a t e r i a l s

360

T a n a b e S. O n a r e c o n s t r u c t i o n f o r m u l a for ray t r a n s f o r m a f t e r d i s c r e t e i n c o m p l e t e d a t a given on a line

363

Tushko T.A. C o n s t r a c t i o n of a velocity m o d e l f r o m seismological d a t a

373

Voronin A.F. Solution of c o n v o l u t i o n i n t e g r a l e q u a t i o n s of t h e first k i n d on a s e g m e n t

377

Zhao Shuang-Ren, Horst Hailing I m a g e r e c o n s t r u c t i o n for f a n b e a m X - r a y t o m o g r a p h y using a n e w i n t e g r a l transform pair

384

Preface This volume contains the Proceedings of the Fourth International Symposium on Computerized Tomography (CT-93) which was held in Novosibirsk (Russia) on 10-14 August, 1993. The CT-93 was managed and sponsored by • Siberian Branch of the Russian Academy of Sciences; • Institute of Mathematics, Novosibirsk; • Institute of Theoretical and Applied Mechanics, Novosibirsk; • Institute of Automatics and Electrometry, Novosibirsk; • Computing Center, Novosibirsk; • Institute of Applied Physics, Novosibirsk; • Novosibirsk State University. The Ct-93 was held under protection and in cooperation with the following associations: • Russian Society "Tomography Corporation", Moscow; • Tomography Society, Novosibirsk; • Russian Chapter of the International Society for Optical Engineering (SPIE/RUSS), Moscow. A great interest to theoretical and numerical analysis of tomography problems and their applications leads to sufficiently large amount of participants from Australia, Belorussia, Canada, Estonia, France, Germany, Israel, Italy, Japan, Kazakhstan, Kirgizstan, Russia, The Netherlands, Ukraina, USA, Uzbekistan. The symposium had three plenary sessions concerned with • mathematical problems of computerized tomography; • algorithms of computerized tomography; • tomography applications in physics, geophysics, industry, and medicine. Some lectures and talks presented to the CT-93 are included to this book. Regretfully not all plenary lecturers prepared their lectures for publication and some limitations did not allow all authors of communications to take part in this volume. The Editorial Board refrained from changing the authors' terminology due to the wide range of computerized tomography applications.

V

Acknowledgement We wish to acknowledge with thanks all contributors and participants, who made the CT-93 a success. It is our pleasure to express our gratitude to the group at Institute of Mathematics that has prepared these proceedings for publication. Of those people whose contribution to this work was of particular importance we would like to mention T . V . Bugueva, I. G. Kabanikhina, S. V . Martakov, D. V. Nechaev.

Professor

M. M. Lavrent 'ev, Editor-in-Chief

Professor S. I.

Kabanikhin, Editor.

Image Reconstruction of Celestial Objects from a Few Strip-Integrated Projections by Lunar Occultations M. I. AGAFONOV and K. S. STANKEVICH Radiophysical Research Institute B. Pecherskaya st. 25/14, Nizhny Novgorod 603600, Russia

ABSTRACT In this paper we consider the technique of image reconstruction based on iterative algorithms and perform a numerical analysis of the solution convergence. We represent the results of numerical experiments. The brightness images of celestial objects are very important for studying physical processes in astrophysics. In certain cases, usually for low frequencies, the angular resolution of up-to-date radiotelescopes is not enough to get sufficiently accurate images. Nevertheless, observations of the objects in their Lunar occultations allow us to obtain integral brightness profiles with a high angular resolution. One-dimensional (1 — D) projections of several occultations may be used for the 2 — D image reconstruction. The Lunar limb is approximated by a plane screen moving at different position angles. A similar problem of image reconstruction arises in observing radiosources with the knife diagram. The stretched beam of the plane reflector of RATAN-600 radiotelescope can change its position in a limited range of position angles. The irregular distribution of position angles such that a sector of about 100 degrees is not filled is typical both for the case of Lunar occultations of radiosources and for the observations by the knife-beam radiotelescopes. We present here: 1. the technique of image reconstruction from a limited number of projections using iterative algorithms with nonlinear constrains (Agafonov and Podvoiskaya, 1989, 1990); 2. the results of numerical experiments using the known iterative algorithms such as the standard CLEAN (Hogbom, 1974) and the Trim Contour CLEAN (TC-CLEAN) (Steer et al., 1984); the study of the solution convergence by minimizing the error of initial and control profiles with variations of the loop gain (g) and the trim contour level (TC); 3. the experimental results of the Crab Nebula images reconstruction on frequencies of 178 and 750 MHz with a high angular resolution from four one-dimensional projections of Lunar occultations which were obtained using the 70-meter radiotelescope (RT-70) (Agafonov et al., 1984).

2

M. I. Agafonov

and K. S.

Stankevich

The Deconvolution Problem The problem is reduced to the classical deconvolution equation G = H * F(+noise), where F(x, y) is the brightness distribution of the observed object on the celestial sphere, H(x,y) is the fan beam (the instrument point response function or dirty beam), G(x,y) is the dirty map or summary image. The functions G(x,y) and H(x,y) are calculated by summing the corresponding values of one-dimensional profiles. The classical case by Bracewell (Bracewell and Riddle, 1967) with the function H(x,y) filling completely the region of spatial frequencies of UV-plane up to the radius with the boundary frequency / needs the number of projections N to be not less than 3,14 D f , where D is the diameter of the considered region. To achieve the desired angular resolution 1 / / from a limited number of projections, it is required to extrapolate the solution and to use nonlinear processing methods. Iterative algorithms with nonlinear constraints (see, for example, Vasilenko and Taratorin, 1986) are the most effective in the image reconstruction from a few strip-integrated projections. The paper by Agafonov and Podvoiskaya (1989) is devoted to solving the reconstruction problems by the iterative algorithms used in radioastronomy. The study of reconstruction possibilities was accomplished by the numerical experiments based on the following scheme of calculations. One-dimensional profiles and the dirty map were obtained from a 2 — D object model. To clean the dirty map, we applied independently both of the iterative algorithms CLEAN (Hogbom, 1974) and TC-CLEAN (Steer et al., 1984), with the use of a fan beam. After the scanning of the obtained images, the error of control and initial projections (ERROR) was calculated. The solution convergence was investigated by the ERROR minimization over the variable parameters g and TC. The stand CLEAN algorithm (Hogbom, 1974) is the realization iterative algorithm which has been used extensively in radioastronomy since 1974. It needs only one component of the image net at every iterative step. The low computational rate and probable distortions - stripes and ridges which may appear in the case of extended areas - are the shortages of the method. The algorithm does not provide with a clear criterion for the choice of the parameter g (loop gain), but this choice was found to influence the result in a complicated manner. The Trim Contour CLEAN algorithm (Steer et al., 1984) differs essentially from the former. The Trim contour method is used here for selecting components at each iteration. Its advantage is particularly seen when processing extended areas. A great number of components exceeding the fixed TC level are considered simultaneously at each iteration thus preventing from the appearance of defects. The Trim Contour CLEAN inspired a hope for improving the image quality on extended areas, but it needs a study in different practical cases (Cornwell, 1988). We studied the solution convergence by the ERROR minimization over the variable parameters g (loop gain) and TC (trim contour level). The known maps of radioastronomical objects were taken as initial models. The numerical experiments were carried out on the net of 64 x 64 pixels. The BESM-6 computer was used . It was shown (Agafonov and Podvoiskaya, 1989) that the search for the solution Fopt{x, y) using the above algorithms is to be made by the optimization of the parameters of iterative processes: the loop gain g (in both of the algorithms) and the TC level (in the TC-CLEAN case only). The parameters are chosen in such a way to minimize the error of initial and control profiles. This corresponds to a discrepancy solution. The process was illustrated in details in the paper by Agafonov and Podvoiskaya (1989). It was shown also that the T C - C L E A N is more effective than the former algorithm and, moreover, it

Image Reconstruction

of Celestial

Objects

3

is stable and prevents from image defects in the case of extended areas. The TC-CLEAN gains an advantage in computational time. The reconstruction of a complicated object is performed by the TC-CLEAN in 13-18 iterations but needs 400-600 iterations using the standard CLEAN. Moreover, the parameters are chosen more easily in the TC-CLEAN algorithm. The minimum error of initial and control one-dimensional profiles for stable solutions (in the CLEAN case it needs sometimes additional efforts (Agafonov and Podvoiskaya, 1989)) may be approximately the same for both of the algorithms. Then we prefer the result reflecting the physical peculiarities of the object to a great extend or assume the existence of the class of probable solutions ranged from "obtuse" (TC-CLEAN) to "sharp" (CLEAN) variants. As is known, the CLEAN summarizes peaks in the solution, which explains the result trend towards the sharpest variant (as far as it is possible for the fixed constraints). The TC-CLEAN, on the contrary, accumulates the result by involving the whole areas into the domain of the given constraints and beginning, as a rule, with the most extended components. It is natural that in this case a more smoothed solution should be expected. Thus, the two methods are proposed to study the solution existence in complicated cases and, in particular, for insufficient preliminary information. For simple objects, the domain of the solution uncertainty narrows. The results obtained by both of the algorithms coincide. In the paper by Agafonov and Podvoiskaya (1990), the examples of typical different structures are considered. The trivial object consisting of peaks may be successfully restored by the standard CLEAN. The results obtained by the two methods practically coincide for the simple object consisting of independent components of finite dimensions. But the TC-CLEAN is more effective. A difference of the results is observed when considering a more complicated object having strong components on an extended area. The image reconstructed by the standard CLEAN has a grooved structure on extended areas. For smoothed 1 — D profiles with small hillocks it was possible to get a solution both from isolated, contrasting components (the CLEAN under stability conditions) and from more smoothed components (the TC-CLEAN). Since the errors of initial and control profiles had the same value, the preference was not given to one of these variants. The standard CLEAN increased the contrast of small components, but the quality of the extended background deteriorated considerably because of grooves. Thus, the pair of solutions obtained in the complicated case by the two methods allows us to estimate the range of possible change of the structure. As a result, the conclusions about the components may be drawn. At the same time, the availability of preliminary information on the object may indicate the solution in accordance to the physical peculiarities of the object. The questions as how to improve the reconstruction quality from a limited number of projections are of great importance both in radioastronomy and tomography (Vishnyakov et al., 1985). We hope that the presented technique based on the two above algorithms will be useful for researchers.

Images Reconstruction A detailed study of the Crab Nebula structure on the frequencies less than 1 GHz is of great interest since good synthesis telescope maps are absent. The brightness profiles of the Lunar occultations were obtained by means of the radiotelescope RT-70 (West

4

M. I. Agafonov

and K. S.

Stankevich

Crimea). Two images with a high angular resolution in this range were obtained on the basis of these observations. Maloney and Gottesman (1979) succeeded in restoring the Crab Nebula image on a frequency of 114 MHz from four projections by the standard algorithm CLEAN. The reconstruction of the structure on a frequency of 750 MHz with the angular resolution of 20 x 35 arcsec was also made using the standard CLEAN in Agafonov et al. (1986). It should be noted that the Nebula diameter is equal to about 6 arcmin. The images presented in Maloney and Gottesman (1979) and Agafonov et al. (1986) have some defects on extended areas which are typical for the standard CLEAN. The results of numerical experiments (Agafonov and Podvoiskaya, 1989, 1990) which have been briefly described above clearly demonstrate the effectiveness of the T C - C L E A N (Steer et al., 1984) as compared with the CLEAN. Using the T C - C L E A N yields highquality images of extended areas which are typical for the Nebula. Images on frequencies of 178 and 750 MHz were reconstructed with the angular resolutions of 20 x 35 and 45 x 65 arcsec, respectively (Agafonov et al., 1984). Four profiles obtained in the Lunar occultations of 1982 and 1983 years were used on every frequency. The numerical experiments using the well-known image of the Nebula on a frequency of 1,4 GHz (Agafonov and Podvoiskaya, 1989) allowed us to estimate beforehand possibilities of the object structure restoration by a low UV-plane filling. The earlier restoration (Agafonov et al., 1986) by the standard CLEAN sharpened the contrast of small components. This was useful for establishing the correlation with the position of filaments in optics. However, the numerical experiments (Agafonov and Podvoiskaya, 1989, 1990) showed that the image obtained by the T C - C L E A N reflect the physical features of the Nebula to a large extend. The described technique may be used for the image reconstruction from the Lunar occultations profiles and may be adapted for observations by the radiotelescope RATAN600 and for solution of tomographic problems. REFERENCES Agafonov M. I., Aslanyan A. M., Gulyan A. G., Ivanov V. P., et al.(1986), Pis'ma

v

Astronom. zhurnal, vol. 12, 275. Agafonov M. I., Ivanov V. P., Podvoiskaya 0 . A. (1984), Astronomicheskil zhurnal, vol. 137, 159. Agafonov M. I., Podvoiskaya 0 . A. (1989), Izvestiya Vyssh. Uchebn. Zaved., Radiofizika, vol. 32, 742. Agafonov M. I., Podvoiskaya 0 . A. (1990), Izvestiya vol. 33, 1185. Bracewell R. N., Riddle A. C. (1967), Astrophys. Cornwell T. J. (1988), Astron.

Astrophys.,

Hogbom J. A. (1964), Astron.

Astrophys.

Vyssh.

Uchebn. Zaved., Radiofizika,

J., vol. 150, 427.

vol. 202, 316. Suppl. Ser., vol. 15, 417.

Maloney F. P., Gottesman S. T. (1979), Astrophys. Steer D. G, Dewdney P. E, Ito M. R. (1984), Astron.

J., vol. 234, 485. Astrophys.,

vol. 137, 159.

Vasilenko G. I., Taratorin A. M. (1986), Image Restoration, "Radio i svyaz",

Moscow,

(in Russian) Vishnyakov G. N., Gil'man G. A., Levin G. G. (1985), Opt. Spectroscopy,

vol. 58, 406.

Image Reconstruction

of Celestial

Objects

5

1. The example of numerical modeling: a) TEST OBJECT (Crab Nebula map 1 . 4 GHz, VLA); b) DIRTY MAP; c) standard CLEAN reconstruction; d) TC-CLEAN reconstruction (TC-0.79, g-0.26, 16 iterations). FIGURE

-2*

O'

2'

b)

F I G U R E 2. The obtained results of the Crab Nebula reconstruction on a frequency of 750 MHz from four profiles of Lunar occultations: a) CLEAN and optic image; b) TC-CLEAN.

Numerical Study of a Nonlinear Sound Radiating Inverse P r o b l e m of Underwater Acoustics G. V. ALEKSEEV Institute of Applied Mathematics, Far Eastern Branch of Russian Academy of Sciences 690041, Vladivostok, Russia

ABSTRACT In this paper we consider a nonlinear inverse extremal problem of sound radiating. This problem is to minimize the power of the sound field,which is radiated by a source into the far zone of a regular acoustic waveguide, by locating a number of secondary point sources. The near field technique is used to calculate the power output for an arbitrary number of primary and secondary point sources. This power output is a function nonconvex in coordinates of the secondary point sources. For given complex strengths of sources this function reaches its (nonunique) minimum for an optimal location of the secondary sources. Results are presented for the case when the sound field is emitted by a single primary point source and by a number of secondary point sources located in an optimal way. In particular, it is shown that a substantial reduction in power output can be achieved even for a few secondary sources.

Let D = Dh = { x = (a;, z) : 0 < z < H, —oo < x < 00} be a regular plane acoustic waveguide of the depth H with parameters w > 0 ,

p ( z ) , c ( z ) G L°°(0,H),

p > p o > 0 ,

c { z ) > c o > 0 .

(1)

Introduce a 2 x N - matrix (array) _

/

Xi

X2

...

xN

\

^

Z\

Z2

...

Zjv

/

and denote by (Z, Q), q = (