A Sobolev Space Analysis of Picture Reconstruction

The problem of recovering the density function of a plane picture from its line integrals is considered. If a picture of finite extent whose density function belongs to the Sobolev space$H^\alpha $ of order $\alpha > \frac{1} {2}$ is to be reconstructed from n line integrals with a root mean square error $\varepsilon $, then the root mean square error of the reconstruction is of the order $(\varepsilon ^{\alpha /(\alpha + 1/2)} + n^{ - \alpha / 2} )\| y \|_{H^\alpha }$ at least. We give a reconstruction method which achieves this optimal error bound.