Reconstructive Integral Geometry

Couverture
Springer Science & Business Media, 20 août 2004 - 164 pages
One hundred years ago (1904) Hermann Minkowski [58] posed a problem: to re 2 construct an even function I on the sphere 8 from knowledge of the integrals MI (C) = fc Ids over big circles C. Paul Funk found an explicit reconstruction formula for I from data of big circle integrals. Johann Radon studied a similar problem for the Eu clidean plane and space. The interest in reconstruction problems like Minkowski Funk's and Radon's has grown tremendously in the last four decades, stimulated by the spectrum of new modalities of image reconstruction. These are X-ray, MRI, gamma and positron radiography, ultrasound, seismic tomography, electron mi croscopy, synthetic radar imaging and others. The physical principles of these methods are very different, however their mathematical models and solution meth ods have very much in common. The umbrella name reconstructive integral geom etryl is used to specify the variety of these problems and methods. The objective of this book is to present in a uniform way the scope of well known and recent results and methods in the reconstructive integral geometry. We do not touch here the problems arising in adaptation of analytic methods to numerical reconstruction algorithms. We refer to the books [61], [62] which are focused on these problems. Various aspects of interplay of integral geometry and differential equations are discussed in Chapters 7 and 8. The results presented here are partially new.
 

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Pages sélectionnées

Table des matières

Distributions and Fourier Transform
1
13 Tempered distributions
5
14 Homogeneous distributions
9
15 Manifolds and differential forms
14
16 Push down and pull back
17
17 More on the Fourier transform
20
18 Bandlimited functions and interpolation
26
Radon Transform
29
52 Odddimensioned subspaces
94
53 Even dimension
98
54 Range of the flat transform
99
55 Duality for the Funk transform
101
56 Duality in Euclidean space
102
Incomplete Data Problems
105
62 Radon transform of Gabor functions
106
63 Reconstruction from limited angle data
107

22 Inversion formulae
31
23 Alternative formulae
34
24 Range conditions
37
25 Frequency analysis
38
26 Radon transform of differential forms
41
The Funk Transform
43
32 Spaces of constant curvature
47
33 Inversion of the Funk transform
49
34 Radons inversion via Funks inversion
50
35 Unified form
51
36 FunkRadon transform and wave fronts
53
37 Integral transform of boundary discontinuities
55
38 Nonlinear artifacts
61
39 Pizetti formula for arbitrary signature
62
Reconstruction from Line Integrals
65
42 Sources at infinity
68
43 Reduction to the Radon transform
71
44 Rays tangent to a surface
73
45 Sources on a proper curve
74
46 Reconstruction from plane integrals of sources
77
47 Line integrals of differential forms
78
48 Exponential ray transform
83
49 Attenuated ray transform
86
410 Inversion formulae
87
411 Range conditions
89
Flat Integral Transform
93
64 Exterior problem
108
65 The parametrix method
111
Spherical Transform and Inversion
115
73 Hemispherical integrals in space
119
74 Incomplete data
124
75 Spheres centred on a sphere
125
76 Spheres tangent to a manifold
127
77 Characteristic Cauchy problem
130
78 Fundamental solution for the adjoint operator
133
Algebraic Integral Transform
135
82 Special cases
136
83 Multiplicative differential equations
139
84 Funk transform of Leray forms
141
85 Differential equations for hypersurface integrals
142
86 Howards equations
144
87 Range of differential operators
146
88 Decreasing solutions of Maxwells system
147
89 Symmetric differential forms
149
Notes
153
Chapter 4
154
Chapter 5
155
Chapter 7
156
Bibliography
157
Index
Droits d'auteur

Autres éditions - Tout afficher

Expressions et termes fréquents

Fréquemment cités

Page 157 - Beylkin, G., 1984. The inversion problem and applications of the generalized Radon transform, Comm. Pure Appl. Math., 27, 579-599.
Page 157 - The Radon transform on a family of curves in the plane
Page 157 - An example of non-uniqueness for a generalized Radon transform, J. Anal. Math. 61 (1993), 395-401.
Page 157 - On the determination of a function from spherical averages SIAM J. Math. Anal. 19 214-232 2.

Informations bibliographiques