Ebook: Reconstructive Integral Geometry

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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This book covers facts and methods for the reconstruction of a function in a real affine or projective space from data of integrals, particularly over  lines, planes, and spheres. Recent results are collected stressing explicit analytic methods. Another focus consists of the relations between algebraic integral geometry and partial differential equations.
A concise basic course in harmonic analysis and distribution theory is given in the first chapter. The first half of the book includes the ray, the spherical mean transforms in the plane or in 3-space, and inversion from incomplete data. It will be of particular interest to application oriented readers. Further chapters are devoted to the Funk-Radon transform on algebraic varieties of arbitrary dimension.
The material appeals to graduates and researchers in pure and applied mathematics who are interested in image reconstruction, inverse problems or functional analysis.

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This book covers facts and methods for the reconstruction of a function in a real affine or projective space from data of integrals, particularly over  lines, planes, and spheres. Recent results are collected stressing explicit analytic methods. Another focus consists of the relations between algebraic integral geometry and partial differential equations.
A concise basic course in harmonic analysis and distribution theory is given in the first chapter. The first half of the book includes the ray, the spherical mean transforms in the plane or in 3-space, and inversion from incomplete data. It will be of particular interest to application oriented readers. Further chapters are devoted to the Funk-Radon transform on algebraic varieties of arbitrary dimension.
The material appeals to graduates and researchers in pure and applied mathematics who are interested in image reconstruction, inverse problems or functional analysis.

Content:
Front Matter....Pages i-xii
Distributions and Fourier Transform....Pages 1-28
Radon Transform....Pages 29-41
The Funk Transform....Pages 43-64
Reconstruction from Line Integrals....Pages 65-91
Flat Integral Transform....Pages 93-104
Incomplete Data Problems....Pages 105-113
Spherical Transform and Inversion....Pages 115-134
Algebraic Integral Transform....Pages 135-152
Notes....Pages 153-156
Back Matter....Pages 157-164

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This book covers facts and methods for the reconstruction of a function in a real affine or projective space from data of integrals, particularly over  lines, planes, and spheres. Recent results are collected stressing explicit analytic methods. Another focus consists of the relations between algebraic integral geometry and partial differential equations.
A concise basic course in harmonic analysis and distribution theory is given in the first chapter. The first half of the book includes the ray, the spherical mean transforms in the plane or in 3-space, and inversion from incomplete data. It will be of particular interest to application oriented readers. Further chapters are devoted to the Funk-Radon transform on algebraic varieties of arbitrary dimension.
The material appeals to graduates and researchers in pure and applied mathematics who are interested in image reconstruction, inverse problems or functional analysis.

Content:
Front Matter....Pages i-xii
Distributions and Fourier Transform....Pages 1-28
Radon Transform....Pages 29-41
The Funk Transform....Pages 43-64
Reconstruction from Line Integrals....Pages 65-91
Flat Integral Transform....Pages 93-104
Incomplete Data Problems....Pages 105-113
Spherical Transform and Inversion....Pages 115-134
Algebraic Integral Transform....Pages 135-152
Notes....Pages 153-156
Back Matter....Pages 157-164
....