Reconstruction Of Image From Projections

Reconstructive Integral Geometry V.P.Palamodov Tel Aviv University Preface One hundred years ago Hermann Minkowski (1904) has started the problem: to reconstruct an even function f on the sphere S 2 from knowledge of the integrals M f (C) = Z f ds C over big circles C. Paul Funk (1916) has found an explicit reconstruction formula for f from data of big circle integrals. Johann Radon studied the similar problem for the Euclidean plane and space. The interest to reconstruction problems like Minkowski-Funk’s and Radon’s ones grew tremendously in the last four decades, stimulated by the spectrum of new problems and methods of image reconstruction. These are X-ray, MRI, gamma and positron radiography, ultrasound, thermoacoustic, seismic tomography, electron microscopy, synthetic radar imaging and others. Analytic methods of reconstruction in two and three dimensions from plane, ray or spherical averages are now in the focus of studies, being motivated by applications. The objective of the Chapters 2-5 and 7 of this book is to represent the scope of recent results and new methods in the reconstructive integral geometry 1 in a uniform way. Keeping in mind the applications to real problems, the problems with incomplete data are studied in Chapter 6. The phase space analysis is applied to show the limits of stable reconstruction. We do not touch here the problems arising in adaptation of analytic methods to numerical reconstruction algorithms. We refer to the books [63],[64] which are focused on these problems. Various aspects of relations between integral geometry and differential equations are discussed in Chapter 8. The results presented here are partially new. Necessary information from the harmonic analysis and the distribution theory is collected in Chapter 1. The book is an extended version of the lecture course which was read for students of Tel Aviv Uni