Notes on image processing with partial differential equations

We discuss the underlying physics of transport coefficients in the PDE image processing approach and present a more principled method for their construction. Transport coefficients in PDEs play important roles, e.g., the diffusion coefficient c in the Perona-Malik equation ∂u/∂t = ∇^T (c∇u). Hence different models for c have been proposed in the literature. We present a physics-based method for constructing c and compare its performance with existing models. We also clarify an issue that arises in implementing the PDE approach. It is well-known that the product rule for derivatives in continuous spaces, namely, ∂(fg) = g∂f + f∂g, does not hold in discrete spaces. Thus computing such quantities as ∇^T (c∇u) in spatially discretized images becomes ambiguous depending on which of the two alternatives we use, namely, whether to treat c∇u as a single function or treat c and ∇u as two functions and use the product rule. We examine this ambiguity and show which alternative is better in the sense of yielding approximations that are closer to the continuous space results.