By Peter W. Hawkes (Ed.)
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Extra info for Advances in Electronics and Electron Physics, Vol. 75
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. (73) By taking the 2D 28 M. BERTERO Fourier transform of both sides of this equation, we have (Natterer, 1986) (r) %r).
0 < x N s b: n = 1 , . , N . (145) When we have a differentiablesolution f ( x )of this problem, then a solution of the corresponding problem of numerical derivation is just f ’ ( x ) . It must be pointed out, however, that the problem of numerical derivation can be formulated independently of the corresponding problem of interpolation. If gn = f ( x , ) ; 46 M. , N (146) which is already in the form (115), at least when X = LZ(a,b). , a Hilbert space of continuous functions such that all the evaluation functionals are continuous.
It must be pointed out, however, that the problem of numerical derivation can be formulated independently of the corresponding problem of interpolation. If gn = f ( x , ) ; 46 M. , N (146) which is already in the form (115), at least when X = LZ(a,b). , a Hilbert space of continuous functions such that all the evaluation functionals are continuous. In fact, iff E X then, from the Riesz representation theorem (Balakrishnan, 1976),it follows that for given x E [u, b ] , there exists a function Q , E X such that f ( x ) = ( f ,Q J x (147) which shows that the problem (145) has the form (1 15), the functions r#+, being the functions Q , associated with the points x,.