Abstract

Consider the function f(x,y) to belong to the set of all integrable functions with compact support. The projections of f(x,y) may generally be written as where the h_i are strip-like response functions corresponding to each of the N available projection measurements. The objective of computed tomography (CT) is to reconstruct the source function f(x,y) from these N measurements. Clearly a limited number of such measurements cannot completely specify an arbitrary f(x,y). Since Eq. 1 may be viewed as an inner product between h_i and f in the Hilbert space of all acceptable functions, each measurement consists of a projection of the unknown vector f onto the basis vector h_i. The available measurements can only provide Information about those components off that lie in the subspace spanned by the response functions called the measurement subspace. The components off that lie in the orthogonal (null) subspace do not contribute to the measurements and, hence, cannot be determined from the measurements alone. Without prior information about f(x,y) it is at least necessary to restrict the solution to the measurement space in order to make it unique, i.e., have minimum norm. The null-space components of such a solution are obviously zero. It is known that this leads to identifiable, objectionable artifacts when the projections span a limited range of angles.^1,2 In its generality, Eq. 1 is representative of any discretely sampled, linear-imaging process. Thus, the above statements and the approach that follows are applicable to many other problems such as restoration of blurred images and coded-aperture imaging.

© 1983 Optical Society of America