This work analyses several approaches for determining optimal sparse data sets for image reconstructions by means of linear homogeneous diffusion. Two optimisation strategies for finding optimal data locations are presented. The first one impresses through its simplicity and is based on results from spline interpolation theory. However, this approach can only be applied to one dimensional strictly convex and differentiable functions. Due to these restrictions we derive an alternative approach which uses findings from optimal control theory. This new algorithm can be applied on arbitrary signals. Both approaches are analysed for their convergence behaviour. Further, we discuss the problem of selecting good data values for fixed data positions. This problem can be analysed as a least squares problem. An important relationship between the optimal data locations and the data values is derived and we present efficient numerical schemes to obtain these values. Finally, we present a image compression approach based on the findings from this work. Experiments show that is possible to outperform popular compression algorithms.