Neural Disjunctive Normal Forms for Learning Exact Rank Filters

Diego Marcondes, Wonder A. L. Alves, Ronaldo F. Hashimoto

Gradient-based learning of morphological operators is hindered by locally sparse derivatives with respect to the structuring elements. We develop a method for learning rank filters that retains exact operator evaluation while distributing the derivatives across the window. Using threshold decomposition, we construct a neural disjunctive normal form representation and establish a sufficient condition for exactness with integer-valued images and structuring elements. We then derive a smooth surrogate from the unique root of a smooth threshold equation. Implicit differentiation gives closed-form derivatives with normalised local weights, whose distribution is controlled by a smoothing parameter. These derivatives are used in a straight-through procedure: the forward pass evaluates the exact rank filter, while the backward pass uses the surrogate. For fixed ranks, the procedure learns real-valued structuring elements of any rank filter, including the median, and extends to compositions, infima, and suprema of rank filters. Experiments on image denoising using MNIST and Fashion-MNIST and retinal vessel segmentation show that, starting from a generic random initialisation, the proposed…