By Victor Palamodov

ISBN-10: 3034879415

ISBN-13: 9783034879415

ISBN-10: 3034896298

ISBN-13: 9783034896290

One hundred years in the past (1904) Hermann Minkowski [58] posed an issue: to re 2 build a fair functionality I at the sphere eight from wisdom of the integrals MI (C) = fc Ids over huge circles C. Paul Funk discovered an specific reconstruction formulation for I from facts of huge circle integrals. Johann Radon studied an identical challenge for the ecu clidean airplane and house. The curiosity in reconstruction difficulties like Minkowski Funk's and Radon's has grown greatly within the final 4 many years, prompted by means of the spectrum of latest modalities of picture reconstruction. those are X-ray, MRI, gamma and positron radiography, ultrasound, seismic tomography, electron mi croscopy, artificial radar imaging and others. The actual ideas of those equipment are very various, besides the fact that their mathematical types and resolution meth ods have a great deal in universal. The umbrella identify reconstructive indispensable geom etryl is used to specify the range of those difficulties and techniques. the target of this ebook is to provide in a uniform approach the scope of good recognized and up to date effects and techniques within the reconstructive quintessential geometry. we don't contact right here the issues coming up in model of analytic how to numerical reconstruction algorithms. We discuss with the books [61], [62] that are taken with those difficulties. a number of facets of interaction of essential geometry and differential equations are mentioned in Chapters 7 and eight. the implications offered listed below are partly new.

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**Extra resources for Reconstructive Integral Geometry**

**Sample text**

V k ] where v j -- Xl-2 (0 ,U - l ,XlU2 j j - j X2Ul"" ,XlUjn - xnujl ) ,J. -- 1, ... , k . It is easy to check that {L (X), v I , ... l U~ Y2 1 0 U~ 0 U~ ... .. Y~) un . U~ = xII {L(x) ,L (u l ) , ... ,L (uk)}. 46 Chapter 3. The Funk Transform Therefore dV (L (x) , L (A)) dV (x,A) [L(x),vl, ... ,V k ] [X,U 1, ... 4) since Xl [L(x),L(u l ) , ... ,L(u k )] [X,U 1 , ... ,U k] = Po (L (e (X))). ~ Example 4. Let S be the unit sphere in a Euclidean space E with the center in the origin, E\ {O} ---+ S be the central projection.

It is the projective subspace of S of dimension n - 1. l) is defined in S. Take the point y = (1,0) ESC E n + 1 and consider the family of spheres {z; (z,y) = cos¢},O:::; ¢:::; n/2 in sn. l)dS. 4 If n is even, then any sufficiently smooth even function 9 on sn is reconstructed by (-1t/ 2 ("/2 ( 8 )n-l d¢ g(y) = 2n-2nn/2f(n/2) Jo cos¢ 8¢ cos¢ G(¢) sin¢· ... 8) G(¢,y) = sec¢F(y,tan¢) hyperplane integrals MEf (H), f = {d(H) where F(r) is the spherical mean of (1 + Ix12) -n/2 9 over the sphere = r = tan¢}.

V k ] where v j -- Xl-2 (0 ,U - l ,XlU2 j j - j X2Ul"" ,XlUjn - xnujl ) ,J. -- 1, ... , k . It is easy to check that {L (X), v I , ... l U~ Y2 1 0 U~ 0 U~ ... .. Y~) un . U~ = xII {L(x) ,L (u l ) , ... ,L (uk)}. 46 Chapter 3. The Funk Transform Therefore dV (L (x) , L (A)) dV (x,A) [L(x),vl, ... ,V k ] [X,U 1, ... 4) since Xl [L(x),L(u l ) , ... ,L(u k )] [X,U 1 , ... ,U k] = Po (L (e (X))). ~ Example 4. Let S be the unit sphere in a Euclidean space E with the center in the origin, E\ {O} ---+ S be the central projection.

### Reconstructive Integral Geometry by Victor Palamodov

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