Optimizing Data for Interpolation and Approximation with Homogeneous Diffusion

Simon Setzer
Saarland University

A major task in many applications is to reconstruct a function, e.g. a signal or an image from given values at a few distinct points. A successful way to do this is to ?ll in the unknown data by a PDE-based diffusion process. In applications like image compression, we are free to extract suitable data from our original image for transmission and reconstruction. The approach presented here is to choose a simple diffusion method – homogeneous diffusion – but to optimize the spatial location and tonal value of the data points. We present strategies in 1D and 2D on how to ?nd this data and show that this approach can greatly improve the reconstruction results.

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