Peter W. Hawkes (Ed.)'s Advances in Electronics and Electron Physics, Vol. 75 PDF

By Peter W. Hawkes (Ed.)

ISBN-10: 0120146754

ISBN-13: 9780120146758

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N . , the scalar product is defined by Eq. (122) with W,, = h,,, then the operator given by Eq. (128) coincides with the Gram matrix (which is symmetric) and the latter is given by e G,, = 2c sinc[c(x, - x,)/n]. ( 134) The singular values 0, of the problem are the square roots of the eigenvalues of G and the singular vectors v k are the corresponding eigenvectors of G . , N , (136) where d is the sampling distance. When d = n/c, the Fourier transform is sampled at the Nyquist rate. In this case, the functions +,(y) = exp( - ix,y) are orthogonal and the Gram matrix is 43 LINEAR INVERSE AND ILL-POSED PROBLEMS a multiple of the unit matrix, G,, = 2c d,,,,.

BERTERO harmonics a,= 4n( (39) from Eq. (36) we obtain +a, c (L*g)(r) = l = O 1 in-1 ol( Js2g(sr)xrn(s’)*dsj)ulm(r) (40) In conclusion, the subspace of visible sources is the closed subspace spanned by the functions ulm(r)of Eq. (38). Moreover, the set of triples {ol;ulm,ulm} is just the singular system of the compact operator L , as follows from the relations Lutm = olulm, L*vIm = OlUlm, (41) which are easily obtained from Eqs. ( 3 9 , (36), and (37) using the orthogonality properties of the spherical harmonics.

We have again the problem of restoring a function f from limited values of its Fourier transform. An explicit inversion formula for the transform (69) was already obtained by Radon (1917), as we recalled in the Introduction. Here we sketch an approach which is the basis of the algorithms currently used in practical applications. If we introduce the formal adjoint R# of the Radon transform, also called the back projection operator, (R#g)(x)= then the equation g = Rf lS, s(e7 (x7 0)) dB7 can be replaced by R'g = R'Rf.

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Advances in Electronics and Electron Physics, Vol. 75 by Peter W. Hawkes (Ed.)

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