  The traditional approaches to computerized tomography (CT) depend on the
samples of Radon transform at multiple angles. In optics, the real time imaging
requires the reconstruction of an object by the samples of Radon transform at a
single angle (SA). Driven by this and motivated by the connection between Bin
Han's construction of wavelet frames (e.g [13]) and Radon transform, in
refinable shift-invariant spaces (SISs) we investigate the SA-Radon sample
based reconstruction problem. We have two main theorems. The fist main theorem
states that, any compactly supported function in a SIS generated by a general
refinable function can be determined by its Radon samples at an appropriate
angle. Motivated by the extensive application of positive definite (PD)
functions to interpolation of scattered data, we also investigate the SA
reconstruction problem in a class of (refinable) box-spline generated SISs.
Thanks to the PD property of the Radon transform of such spline, our second
main theorem states that, the reconstruction of compactly supported functions
in these spline generated SISs can be achieved by the samples of Radon
transform at almost every angle. Numerical simulation is conducted to check the
result.
