The Radon transform and local tomography:
Radon Transform and Local Tomography presents new theories and computational methods that cannot be found in any other book. New material, aimed at solving important problems in tomographic imaging and image processing in general, as well as detailed descriptions of the new algorithms and the result...
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Pr.
1996
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | Radon Transform and Local Tomography presents new theories and computational methods that cannot be found in any other book. New material, aimed at solving important problems in tomographic imaging and image processing in general, as well as detailed descriptions of the new algorithms and the results of their testing, are expertly covered The theory described in this book solves the important problem of finding discontinuities of function from its tomographic data. A detailed theoretical analysis and three different solutions to this problem are given, as well as algorithms for practical solutions and examples of applications for both simulated and real-life data |
| Physical Description: | XVIII, 485 S. Ill., graph. Darst. |
| ISBN: | 0849394929 |
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MARC
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| 245 | 1 | 0 | |a The Radon transform and local tomography |c A. G. Ramm ; A. I. Katsevich |
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Pr. |c 1996 | |
| 300 | |a XVIII, 485 S. |b Ill., graph. Darst. | ||
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| 520 | 3 | |a Radon Transform and Local Tomography presents new theories and computational methods that cannot be found in any other book. New material, aimed at solving important problems in tomographic imaging and image processing in general, as well as detailed descriptions of the new algorithms and the results of their testing, are expertly covered | |
| 520 | |a The theory described in this book solves the important problem of finding discontinuities of function from its tomographic data. A detailed theoretical analysis and three different solutions to this problem are given, as well as algorithms for practical solutions and examples of applications for both simulated and real-life data | ||
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Record in the Search Index
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| adam_text | TABLE OP CONTENTS
Preface vii
Chapter 1. Introduction 1
1.1. Brief description of new results and the aims
of the book 1
1.2. Review of some applications of the Radon
transform 6
1.2.1. Applications in medicine and
non destructive evaluation 6
1.2.2. Applications in geophysics 8
Chapter 2. Properties of the Radon
transform and inversion formulas 11
2.1. Definitions and properties of the Radon
transform and related transforms 11
2.1.1. Definition of the Radon transform 11
2.1.2. Some generalizations 13
2.1.3. Simple properties of the Radon
transform 13
2.1.4. Radon transform of a convolution 14
2.1.5. The Fourier slice theorem 15
2.1.6. The adjoint operator R* 16
2.1.7. Formulas for R*R and RR* 17
2.1.8. Formula for (R*g) * / 20
2.1.9. The Parseval and Plancherel
equalities 20
2.1.10. Integrals over a domain 23
2.1.11. Consistency and moment conditions 23
2.1.12. The Radon transform of spherically
symmetric functions 25
2.1.13. Concluding remarks 26
2.2. Inversion formulas for R 26
2.2.1. The first method 26
2.2.2. The second method 28
xii TABLE OF CONTENTS
2.2.3. Inversion in two and three
dimensional spaces 31
2.2.4. Radon s original inversion formula 32
2.2.5. Inversion via the spherical harmonics
series 33
2.2.6. Inversion in the spherically
symmetric case 35
2.3. Singular value decomposition of the Radon
transform 35
2.4. Estimates in Sobolev spaces 41
2.5. Inversion formulas for the backprojection
operator 45
2.5.1. Motivation and problem formulation 45
2.5.2. Inversion formulas 45
2.6. Inversion formulas for X ray transform 48
2.6.1. Definition of X* and a formula for
X*X 48
2.6.2. Inversion formula for X ray transform 50
2.7. Uniqueness theorems for the Radon and X ray
transforms 52
2.7.1. Uniqueness theorems for the Radon
transform 52
2.7.2. Uniqueness theorems for X ray
transform 54
2.7.3. Example of the lack of injectivity 55
2.8. Attenuated and exponential Radon transforms 57
2.8.1. Simplest properties 57
2.8.2. Inversion formulas 59
2.8.3. Generalized Radon transform 63
2.9. Convergence properties of the inversion
formulas on various classes of functions 64
Chapter 3. Range Theorems and
reconstruction algorithms 67
3.1. Range theorems for R on smooth functions 67
3.1.1. The classical range theorem 67
3.1.2. What happens if the moment
conditions are violated? 70
3.2. Range theorem for R on the Sobolev spaces 76
3.2.1. Introduction 76
3.2.2. Proof of Theorem 3.2.1 77
3.2.3. The range theorem in terms of the
Fourier coefficients 79
3.3. Range theorems for R* 81
TABLE OF CONTENTS xiii
3.4. Range theorem for X ray transform 84
3.5. Numerical solution of the equation Rf = g
with noisy data 86
3.5.1. Introduction 86
3.5.2. Regularization 1 87
3.5.3. Regularization 2 89
3.5.4. Regularization 3 90
3.6. Filtered backprojection algorithm 91
3.6.1. Derivation of the algorithm 91
3.6.2. The parallel beam protocol 92
3.6.3. The fan beam protocol 92
3.7. Other reconstruction algorithms 95
3.7.1. Fourier algorithm 95
3.7.2. Algebraic reconstruction algorithms 96
Chapter 4. Singularities of the Radon
transform 98
4.1. Introduction 98
4.2. Singular support of the Radon transform 99
4.3. The relation between S and S 100
4.4. The envelopes and the duality law 103
4.5. Asymptotics of Rf near S 104
4.6. Singularities of the Radon transform: an
alternative approach 112
4.7. Asymptotics of the Fourier transform 116
4.7.1. Introduction 116
4.7.2. Statement and proof of the result 116
4.8. Wave front sets 120
4.9. Singularities of X ray transform 121
4.9.1 Introduction 121
4.9.2. Description of the procedure 121
4.10. Stable calculation of the Legendre transform 124
4.10.1. Introduction 124
4.10.2. The Legendre transform 124
4.10.3. Calculation of the generalized
Legendre transform 129
4.10.4. A sufficient condition for (4.10.2) 133
Chapter 5. Local Tomography 134
5.1. Introduction 134
5.2. A family of local tomography functions 135
5.2.1. Definition of a family. Basic property 135
5.2.2. An elementary proof of the relation
WF(f) = WF{4 ) 138
5.3. Optimization of noise stability 139
xiv TABLE OF CONTENTS
5.4. Algorithm for finding values of jumps of a
function using local tomography 142
5.4.1. Derivation of the algorithm. Basic
result 142
5.4.2. Proof of Theorem 5.4.1 in case of the
locally flat S 144
5.4.3. Proof of Theorem 5.4.1 in case of the
convex S 149
5.5. Numerical implementation 153
5.5.1. The first numerical scheme for
computing values of jumps 153
5.5.2. The second numerical scheme for
computing values of jumps 157
5.6. Local tomography for the exponential Radon
transform 161
5.7. Local tomography for the generalized Radon
transform 165
5.7.1. The first approach 165
5.7.2. The second approach 168
5.7.3. Remarks on numerical
implementation 169
5.8. Local tomography for the limited angle data 173
5.9. Asymptotics of pseudodifferential operators,
acting on a piecewise smooth function /, near
the singular support of / 177
5.9.1. The case of a convex boundary 177
5.9.2. The case of a flat boundary 184
5.9.3. Further generalizations 187
5.9.4. Asymptotics of PDO, symbols of
which have discontinuities on a
conical surface 191
5.9.5. Proof of the auxiliary results 199
Chapter 6. Pseudolocal Tomography 206
6.1. Introduction 206
6.2. Definition of a pseudolocal tomography
function. Basic property 207
6.3. Investigation of the convergence fp(x) — f(x)
as p 0 209
6.4. More results on functions /|, fp, and on
convergence fp tf 217
6.5. A family of pseudolocal tomography functions 221
6.5.1. Definition of a family. Basic property 221
6.5.2. Relation between pseudolocal and
local tomography functions 223
TABLE OF CONTENTS xv
6.5.3. Proof of auxiliary results 226
6.6. Numerical implementation of pseudolocal
tomography 228
6.7. Pseudolocal tomography for the exponential
Radon transform 231
6.7.1. Definitions. Basic property 231
6.7.2. Some auxiliary results 235
6.7.3. Investigation of the convergence
fcp{x) f(x) as p * 0 238
6.7.4. Remarks on numerical
implementation 243
6.7.5. Proofs of Lemmas 6.7.2 6.7.4 244
Chapter 7. Geometrical tomography 248
7.1. Basic idea 248
7.2. Description of the algorithm and numerical
experiments 250
Chapter 8. Inversion of incomplete
tomographic data 259
8.1. Inversion of incomplete Fourier transform
data 259
8.1.1. The basic result 259
8.1.2. Numerical aspects 264
8.2. Filtered backprojection method for inversion
of the limited angle tomographic data 265
8.3. The extrapolation problem 267
8.3.1. Formulation of the problem 267
8.3.2. The first method of solution 268
8.3.3. The second method of solution 269
8.3.4. The third method of solution 270
8.4. The Davison Grunbaum algorithm 273
Chapter 9. Inversion of cone beam data 276
9.1. Inversion of the complete cone beam data 276
9.2. Inversion of incomplete cone beam data 280
9.3. An exact algorithm for the cone beam circle
geometry 284
9.3.1. Reconstruction algorithm 284
9.3.2. Geometry of the fan beam data 287
9.4. 7 ray tomography 289
9.4.1. Brief description of three different
protocols 289
9.4.2. Uniqueness results and inversion
formulas for Problems 9.4.1 and 9.4.2 292
9.4.3. Investigation of Problem 9.4.3 293
xvi TABLE OF CONTENTS
9.4.4. Sufficient condition for a linear
operator to be a convolution 301
Chapter 10. Radon transform of distributions 303
10.1. Main definitions 303
10.2. Properties of the test function spaces 307
10.3. Examples 309
10.4. Range theorem for the Radon transform on £ 313
10.5. A definition based on spherical harmonics
expansion 315
10.6. When does the Radon transform on
distributions coincide with the classical Radon
transform? 318
10.7. The dual Radon transform on distributions 319
10.7.1. Definition of R* on certain classes of
distributions 319
10.7.2. Singularities and singular support of
the solution to the equation R*fi = h
321
Chapter 11. Abel type integral equation 325
11.1. The classical Abel equation 325
11.2. Abel type equations 326
11.3. Reduction of Equation (2.2.42) to a more
stable one 328
11.4. Finding locations and values of jumps of the
solution to the Abel equation 330
Chapter 12. Multidimensional algorithm for
finding discontinuities of signals from
noisy discrete data 335
12.1. Introduction 335
12.2. Edge detection algorithm 337
12.3. Thin line detection algorithm 338
12.4. Generalization of the algorithms 341
12.5. Justification of the edge detection algorithm 343
12.6. Justification of the algorithm for thin line
detection 349
12.7. Justification of the general scheme 351
12.8. Numerical experiments 352
12.9. Proof of auxiliary results 355
Chapter 13. Test of randomness and its
applications 361
13.1. Introduction 361
13.2. Consistency of rank test against change points
(change surfaces) alternative 363
TABLE OF CONTENTS xvii
13.2.1. One dimensional case, m = 2 363
13.2.2. One dimensional case, m 2 368
13.2.3. Multidimensional case, fixed design
model 371
13.2.4. Random design model 373
13.2.5. Numerical experiments 377
13.3. Consistency of rank test against trend in
location 377
13.3.1. One dimensional case, equispaced
design model 377
13.3.2. Multidimensional case, regular design
model 386
13.3.3. Random design model 387
Chapter 14. Auxiliary Results 390
14.1. Abstract and functional spaces 390
14.1.1. Abstract spaces 390
14.1.2. Lebesgue and Sobolev spaces 391
14.2. Distribution theory 394
14.2.1. Spaces of test functions and
distributions 394
14.2.2. Fourier transform of distributions 397
14.2.3. Wave front of a distribution 400
14.3. Pseudodifferential and Fourier integral
operators 400
14.3.1. Oscillatory integrals 400
14.3.2. Fourier integral operators 401
14.3.3. Pseudodifferential operators 402
14.4. Special functions 405
14.4.1. Gamma and beta functions 405
14.4.2. Bessel functions 406
14.4.3. Orthogonal polynomials 408
14.4.4. Integration over spheres 411
14.4.5. Spherical harmonics 412
14.4.6. The Hankel transform and the
Fourier transform 414
14.5. Asymptotic expansions 415
14.5.1. Definitions 415
14.5.2. Laplace s method 417
14.5.3. The stationary phase method 419
14.5.4. The Morse lemma 422
14.6. Linear equations in Banach spaces 423
14.6.1. Closedness. Normal Solvability 423
14.6.2. Conditions for surjectivity 424
14.6.3. Compact operators 425
xviii TABLE OF CONTENTS
14.6.4. Resolution of the identity 427
14.7. Ill posed problems 428
14.7.1. Basic definitions 428
14.7.2. Examples of ill posed problems 429
14.7.3. Methods for stable solution of
ill posed problems 430
14.7.4. Asymptotics of singular values of the
Radon transform 434
14.8. Examples of regularization of ill posed
problems 437
14.8.1. Stable differentiation 437
14.8.2. Stable summation of the Fourier
series 440
14.9. Radon transform and PDE 441
14.9.1. Fundamental solutions of elliptic
equations 441
14.9.2. Fundamental solution of the Cauchy
problem 444
14.9.3. Proof of identity (14.9.2) 444
14.10. Statistics 445
14.10.1. Random variables and some of their
basic properties 445
14.10.2. Modes of convergence and limit
theorems 447
14.10.3. Hypothesis testing 449
14.10.4. Randomness and deviations from it,
ranks, rank tests, order statistics 450
14.10.5. The Monte Carlo method 453
14.10.6. Image processing 454
Research Problems 456
Bibliographical notes 458
References 464
List of notations 476
Index 480
|
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| id | DE-604.BV010881532 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:00:29Z |
| institution | BVB |
| isbn | 0849394929 |
| language | English |
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| physical | XVIII, 485 S. Ill., graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
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| publisher | CRC Pr. |
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| spelling | Ramm, Alexander G. Verfasser aut The Radon transform and local tomography A. G. Ramm ; A. I. Katsevich Boca Raton [u.a.] CRC Pr. 1996 XVIII, 485 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Radon Transform and Local Tomography presents new theories and computational methods that cannot be found in any other book. New material, aimed at solving important problems in tomographic imaging and image processing in general, as well as detailed descriptions of the new algorithms and the results of their testing, are expertly covered The theory described in this book solves the important problem of finding discontinuities of function from its tomographic data. A detailed theoretical analysis and three different solutions to this problem are given, as well as algorithms for practical solutions and examples of applications for both simulated and real-life data Radontransformatie gtt Tomografia larpcal Transformada de radon larpcal Mathematik Radon transforms Tomography Mathematics Tomografie (DE-588)4078351-0 gnd rswk-swf Radon-Transformierte (DE-588)4195862-7 gnd rswk-swf Tomografie (DE-588)4078351-0 s Radon-Transformierte (DE-588)4195862-7 s DE-604 Kacevič, Aleksandr I. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007275675&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ramm, Alexander G. Kacevič, Aleksandr I. The Radon transform and local tomography Radontransformatie gtt Tomografia larpcal Transformada de radon larpcal Mathematik Radon transforms Tomography Mathematics Tomografie (DE-588)4078351-0 gnd Radon-Transformierte (DE-588)4195862-7 gnd |
| subject_GND | (DE-588)4078351-0 (DE-588)4195862-7 |
| title | The Radon transform and local tomography |
| title_auth | The Radon transform and local tomography |
| title_exact_search | The Radon transform and local tomography |
| title_full | The Radon transform and local tomography A. G. Ramm ; A. I. Katsevich |
| title_fullStr | The Radon transform and local tomography A. G. Ramm ; A. I. Katsevich |
| title_full_unstemmed | The Radon transform and local tomography A. G. Ramm ; A. I. Katsevich |
| title_short | The Radon transform and local tomography |
| title_sort | the radon transform and local tomography |
| topic | Radontransformatie gtt Tomografia larpcal Transformada de radon larpcal Mathematik Radon transforms Tomography Mathematics Tomografie (DE-588)4078351-0 gnd Radon-Transformierte (DE-588)4195862-7 gnd |
| topic_facet | Radontransformatie Tomografia Transformada de radon Mathematik Radon transforms Tomography Mathematics Tomografie Radon-Transformierte |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007275675&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT rammalexanderg theradontransformandlocaltomography AT kacevicaleksandri theradontransformandlocaltomography |