Resolution in Computerized Tomography

Abstract

In computerized tomography one reconstructs a function in ℝ² from a finite set of line integrals. The arrangement of the lines is referred to as scanning geometry. In the present paper we shall investigate the possible resolution of various scanning geometries. We say that a scanning geometry has resolution d if functions containing no details of size ≦ d can be recovered reliably from the integrals over the lines making up the scanning geometry. By a function containing no detail of size ≦ d we mean a function which is (essentially) band-limited with band-width $\rm b=2{\pi\over \rm d}$, i.e. a function f whose Fourier transform $\rm {\hat f}(\xi)=(2\pi)^{-n/2}\int\limits_{R^n} e^{-ix\cdot\xi}f(x)dx$ is negligible for | ξ | ≧ b. Here, n denotes the dimension (in our case n=2) and x · ξ is the dot product. This definition of resolution is quite common in image processing, see e.g. Pratt (78). Note that in the engineering literature a factor of 2π shows up in the exponent in the Fourier integral. This means that our results have to be modified correspondingly when compared to others.