On Computing Gradient of Products in Discretized Spaces and Its Effects in PDE Image Processing

Abstract

Many digital signal and image processing methods involve computing the gradient of products of functions. However, the product rule for derivatives in continuous spaces, $\partial (fg)=g\partial f + f \partial g$ , does not generally hold in discretized spaces. Hence, computing the gradient of products becomes ambiguous as the results depend on whether to treat the product $fg$ as a single function or treat $f$ and $g$ as two separate functions and use the product rule. The two alternatives lead to different results, particularly for iterative solutions of differential equations since these small differences compound. Although this ambiguity is well-known, thus far a resolution has not been proposed in the literature. In this letter, we propose a mathematically rigorous procedure that selects the better alternative in the sense of yielding approximations that are closer to the continuous space results. As an example, we discuss the Perona-Malik anisotropic diffusion equation used in image processing.