Geometric Tomography

By: Richard J. Gardner

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Paperback | 11 September 2006 | Edition Number 2

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Paperback
516 Pages

Edition Type
Revised

Dimensions(cm)
23.39 x 15.6 x 2.92

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Geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on planes or cross-sections by planes. It is a geometric relative of computerized tomography, which reconstructs an image from X-rays of a human patient. The subject overlaps with convex geometry and employs many tools from that area, including some formulas from integral geometry. It also has connections to discrete tomography, geometric probing in robotics and to stereology. This comprehensive study provides a rigorous treatment of the subject. Although primarily meant for researchers and graduate students in geometry and tomography, brief introductions, suitable for advanced undergraduates, are provided to the basic concepts. More than 70 illustrations are used to clarify the text. The book also presents 66 unsolved problems. Each chapter ends with extensive notes, historical remarks, and some biographies. This new edition includes numerous updates and improvements, with some 300 new references bringing the total to over 800.

Industry Reviews

praise for the first edition ... 'The work is written in a very clear style, is well organized, and is amply provided with helpful illustrations and diagrams.' Wm. J. Firey, SIAM Review 'This is an extremely useful monograph, collecting together all the currently available material on geometric tomography ... The book is well organized and easy to read, with many illustrations, and copious notes to each chapter ... I recommend the book highly as a valuable introduction and continuing reference.' Peter McMullen, Bulletin of the London Mathematical Society '... A valuable reference source. Libraries should have this book.' A. G. Ramm, Inverse Problems Newsletter 'This well-written book describes the recent theory ... it includes copious notes and a comprehensive bibliography.' Mathematika 'The book is self contained, readable and can be warmly recommended not only for specialists.' European Mathematical Society Newsletter

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Non-Fiction Mathematics Geometry Differential & Riemannian Geometry Calculus & Mathematical Analysis Medicine Other Branches of Medicine Medical Imaging Tomography

Preface to the second edition	p. xv
Preface	p. xvii
Background material	p. 1
Basic concepts and terminology	p. 1
Transformations	p. 3
Basic convexity	p. 6
The Hausdorff metric	p. 9
Measure and integration	p. 10
The support function	p. 16
Star sets and the radial function	p. 18
Polar duality	p. 20
Differentiability properties	p. 22
Parallel X-rays of planar convex bodies	p. 28
What is an X-ray?	p. 28
X-rays and Steiner symmetrals of planar convex bodies	p. 29
Open problems	p. 52
Notes	p. 52
Computerized tomography	p. 52
Parallel X-rays and Steiner symmetrals of convex bodies	p. 54
Exact reconstruction	p. 55
Well-posedness and stability	p. 56
Reconstruction of convex bodies from possibly noisy data	p. 57
Geometric probing	p. 58
Jakob Steiner (1796-1863)	p. 59
Parallel X-rays in n dimensions	p. 60
Parallel X-rays and k-symmetrals	p. 61
X-rays of convex bodies in E[superscript n]	p. 65
X-rays of bounded measurable sets	p. 69
Open problems	p. 86
Notes	p. 87
Parallel X-rays and k-symmetrals of convex bodies	p. 87
Switching components and discrete tomography	p. 88
Parallel X-rays and k-symmetrals of measurable sets	p. 91
Blaschke shaking	p. 92
Reconstruction of polygons and polyhedra from possibly noisy X-rays	p. 93
Ridge functions and the additivity conjecture	p. 94
X-rays of bounded density functions	p. 94
Johann Radon (1887-1956)	p. 95
Projections and projection functions	p. 97
Homothetic and similar projections	p. 98
The width function and central symmetral	p. 106
Projection functions and the Blaschke body	p. 110
Open problems	p. 125
Notes	p. 126
Homothetic and similar projections	p. 126
Bodies with congruent or affinely equivalent projections	p. 128
Sets of constant width and brightness	p. 129
Blaschke bodies and Blaschke sums	p. 130
Determination by one projection function	p. 131
Determination by more than one projection function	p. 132
Determination by directed projection functions, etc	p. 134
Reconstruction	p. 135
Mean projection bodies	p. 137
Projections of convex polytopes	p. 137
Critical projections	p. 138
Almost-spherical or almost-ellipsoidal projections, and related results	p. 138
Aleksander Danilovich Aleksandrov (1912-1999)	p. 139
Projection bodies and volume inequalities	p. 141
Projection bodies and related concepts	p. 142
Smaller bodies with larger projections	p. 154
Stability	p. 164
Reconstruction from brightness functions	p. 171
Open problems	p. 180
Notes	p. 180
Projection bodies and zonoids	p. 180
The Fourier transform approach I: The brightness function and projection bodies	p. 182
The Minkowski map and Minkowski linear combinations of projection bodies	p. 183
Generalized zonoids	p. 184
Bodies whose projections are zonoids	p. 185
Projection bodies of order i	p. 185
The L[superscript p]-Brunn-Minkowski theory and L[superscript p]-projection bodies	p. 186
Characterizations in terms of mixed volumes	p. 186
Results related to Aleksandrov's projection theorem	p. 187
Smaller bodies with larger projections	p. 187
Stability results	p. 189
Reconstruction from brightness functions	p. 190
Hermann Minkowski (1864-1909)	p. 192
Point X-rays	p. 194
Point X-rays and chordal symmetrals	p. 195
The X-ray of order i	p. 201
Point X-rays of planar convex bodies	p. 206
X-rays in the projective plane	p. 221
Open problems	p. 225
Notes	p. 226
Point X-rays and chordal symmetrals	p. 226
Point X-rays of planar convex bodies	p. 226
Reconstruction	p. 227
Well-posedness	p. 228
Point X-rays in higher dimensions	p. 228
Discrete point X-rays	p. 228
Point projections	p. 229
Wilhelm Suss (1895-1958) and the Japanese school	p. 229
Chord functions and equichordal problems	p. 232
i-chord functions and i-chordal symmetrals	p. 233
Chord functions of star sets	p. 237
Equichordal problems	p. 254
Open problems	p. 264
Notes	p. 264
Chord functions, i-chordal symmetrals, and ith radial sums	p. 264
Chord functions of star sets	p. 265
Equichordal problems	p. 265
Wilhelm Blaschke (1885-1962)	p. 267
Sections, section functions, and point X-rays	p. 269
Homothetic and similar sections	p. 270
Section functions and point X-rays	p. 276
Point X-rays of measurable sets	p. 286
Open problems	p. 288
Notes	p. 289
Homothetic and similar sections	p. 289
Bodies with congruent or affinely equivalent sections	p. 290
Sets of constant section	p. 290
Determination by section functions	p. 290
Determination by half-volumes	p. 291
Point X-rays of measurable sets	p. 293
Sections of convex polytopes	p. 294
Critical sections	p. 295
Almost-spherical or almost-ellipsoidal sections	p. 296
A characterization of star-shaped sets	p. 297
Sections by other sets of planes	p. 297
Integral geometry	p. 299
Mean section bodies	p. 300
Stereology	p. 301
Local stereology	p. 301
Paul Funk (1886-1969)	p. 303
Intersection bodies and volume inequalities	p. 304
Intersection bodies of star bodies	p. 305
Larger bodies with smaller sections	p. 324
Cross-section bodies	p. 334
Open problems	p. 336
Notes	p. 337
Intersection bodies	p. 337
The Fourier transform approach II: The section function and intersection bodies	p. 340
The map I	p. 340
Generalized intersection bodies	p. 340
Bodies whose central sections are intersection bodies	p. 341
Intersection bodies of order i	p. 341
k-intersection bodies and related notions	p. 341
Characterizations in terms of dual mixed volumes	p. 342
Larger bodies with smaller sections I: The Busemann-Petty problem	p. 343
Larger bodies with smaller sections II: Generalizations and variants of the Busemann-Petty problem	p. 345
Stability results	p. 346
Cross-section bodies	p. 346
Problems involving both projections and sections	p. 348
Herbert Busemann (1905-1994)	p. 348
Estimates from projection and section functions	p. 350
Centroid bodies	p. 351
Some affine isoperimetric inequalities	p. 355
Volume estimates from projection functions	p. 362
Volume estimates from section functions	p. 370
Open problems	p. 375
Notes	p. 376
Centroid bodies and polar projection bodies	p. 376
The floating body problem	p. 376
Affine surface area, the covariogram, and convolution and sectional bodies	p. 377
Affine isoperimetric inequalities	p. 379
The L[superscript p]-Brunn-Minkowski theory: centroid bodies, ellipsoids, and inequalities	p. 380
Volume estimates from projection functions	p. 382
Volume estimates from section functions	p. 384
The slicing problem	p. 385
Central limit theorems for convex bodies	p. 387
Estimates concerning both projections and sections	p. 388
Estimates for inradius and circumradius	p. 389
Hugo Hadwiger (1908-1981)	p. 389
Appendixes
Mixed volumes and dual mixed volumes	p. 391
An example	p. 392
Area measures	p. 395
Mixed volumes and mixed area measures	p. 397
Reconstruction from surface area measures	p. 401
Quermassintegrals and intrinsic volumes	p. 403
Projection formulas	p. 406
Dual mixed volumes	p. 409
Inequalities	p. 413
Inequalities involving means and sums	p. 413
The Brunn-Minkowski inequality	p. 415
The Aleksandrov-Fenchel inequality	p. 419
The dual Aleksandrov-Fenchel inequality	p. 420
Other inequalities	p. 422
Integral transforms	p. 424
X-ray transforms	p. 424
The cosine and spherical Radon transforms	p. 427
Open problem	p. 436
References	p. 437
Notation	p. 471
Author index	p. 476
Subject index	p. 482
Table of Contents provided by Ingram. All Rights Reserved.

Geometric Tomography

At a Glance

Paperback

Industry Reviews

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