Determination of compactly supported functions in shift-invariant space by single-angle Radon samples

Abstract

While traditionally the computerized tomography of a function f∈L2(R2) depends on the samples of its Radon transform at multiple angles, the real-time imaging sometimes requires the reconstruction of f by the samples of its Radon transform Rpf at a single angle θ, where p=(cos⁡θ,sin⁡θ) is the direction vector. This naturally leads to the question of identifying those functions that can be determined by their Radon samples at a single angle θ. The shift-invariant space V(φ,Z2) generated by φ is a type of function space that has been widely considered in many fields including wavelet analysis and signal processing. In this paper we examine the single-angle reconstruction problem for compactly supported functions f∈V(φ,Z2). The central issue for the problem is to identify the eligible p and sampling set Xp⊆R such that f can be determined by its single-angle Radon (w.r.t. p) samples at Xp. For the general generator φ, we address the eligible p for the two cases: (1) φ being nonvanishing (∫R2φ(x)dx≠0) and (2) being vanishing (∫R2φ(x)dx=0). We prove that eligible Xp exists for general φ. In particular, Xp can be explicitly constructed if φ∈C1(R2). Positive definite functions form an important class of functions that have been widely applied in scattered data interpolation. For the case that φ is positive definite, the corresponding single-angle problem in SIS V(φ,Z2) is addressed such that Xp can be constructed easily. Besides using the samples of the single-angle Radon transform, another common feature for our recovery results is that the number of the required samples is minimum.