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

By G. T. Herman

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5 to see that it is the sampling of the data which is responsible for the varying standard deviation ; the standard deviation of the reconstruction is greatest where the sampling lines are most dense because the least averaging occurs there. A special case of the interpolating function worth noting is the nearest point interpolation. This choice results in a uniform level of noise in the reconstruction. When f = 11, we have that for every x and y 11((x cos nA + y sin nA )/d - I) × 11 ((cos nA + y sin nA)/d - l') I if I=1' and - - L < ( x c o s n A + v s i n n A ) / d - - I < ~ otherwise.

A desirable window, 14, should then have the following properties: 1) 14"(0)= 1 and [@W-I(0)=0 so that the behavior o f a b s near the origin is retained. 35, 36) remain equivalent specifications of the discrete filter. 3) IW(X)I <1 for IXl_-_ l/2d. Violation of this condition would result in an overemphasis of those frequencies for which {W(X){> 1. ] 4) W(X) is real and even so that the resulting convolving function, q, is real and even. In Sect. 4, we will examine the effects of using the following windows: Band limiting window Low pass, band limiting windows Generalized Hammin 0 low pass window 14~n(X)= [ct +(1 -cQcos(2r~d)]ll(Xd) O

Y-'~ ~,~,;,:: ~:T~' ~ r,;,,:~ e:i~:: =. ~',';ty,;r S; ' t : ' , ~ . , . ~,. ,,'*',~i,. ~,~%~,~, - • ::" 7 ~ - , = ~ % ~ -,z,,'~ ~ Fig. 5a, b. ach line is g i v e n by the e q u a t i o n x c o s n A + y s i n n A + I/2 = k Computer Implementationof hnage Reconstruction Formulas 43 reconstruction problem. In spite of the unique nature of the origin, the properties of the point response functions at the origin will be shown to be related to the effects of using a particular filter and interpolation method in reconstructing a complicated object.

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