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Download Image Reconstruction from Projections: Implementation and by G. T. Herman PDF

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By G. T. Herman

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Example text

A two-dimensional p2-point discrete function is a mapping from 772 into tF. f,x g] (m)=f(m) x g(m). I'M x,q, is the M-point discrete function defined by Ef~-~ a] (m)=f(m) x g(m). The convolution product of discrete functions. )= ~ f(") a('"-"), n ~ -- 03 provided that the sum exists for all m. f(n),q(m-n). nEZ7M Theorem. I f f and 9 are two discrete functions and ,q is such that for some K >0 g(k)=0 whenever kK H, then for every M>2K and every m~77K [f*g] (m)= If'M**9'](m) where f ' and g' are the M-point discrete functions satisfying f(m)=,f'(m) and g(m)--9'(m) for m~77M.

By the commutative and associative properties of discrete convolution, we could equivalently average the data, 9,, but this adds to the computing time (we only convolve H and qR once). This property is easily shown by computing the Fourier series coefficients of the Hamming window. [ [ct + (1 - a) cos (27r,,Yd)] exp ( - 2r~imXd)dX. Xd then H(m)= ~-n ~ o~ i (cosmU-isinmU)dU 2~ _~ d + ~ = [~+(1 - a ) c o s U] exp(-imU)dU ,f cos U (cos m U - i sin m U)d U a (1 - ~)/2 0 if m = 0 if Iml = 1 otherwise.

11a--d. Standard deviation of the noise as a function of position ill a reconstruction of pure noise for 30 projection angles. a Hidden line three-dimensional display - sampling method 1. b Value alo,lg x = 0 - sampling method 1 b . (ILl i] I 51 ,qt,lC E b Let u =x cosnA +ysinnA and u = - x s i n nA + y c o s h A . /(,/a-o 1'),1~.

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