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 spaceH^α of order α > 1/2 is to be reconstructed from n line integrals with a root mean square error ε, then the root mean square error of the reconstruction is of the order (ε α /(α + 1/2) + n- α / 2 )\| y \|H^α at least. We give a reconstruction method which achieves this optimal error bound.