Mathematical Morphology: The Hamilton-Jacobi Connection

Abstract

The authors complement the standard algebraic view of mathematical morphology with a geometric, differential view. Three observations underlie this approach. (1) Certain structuring elements (convex) are scalable in that a sequence of repeated operations is equivalent to a single operation, but with a larger structuring element of the same shape. (2) To determine the outcome of the operation, it is sufficient to consider how the boundary is modified. (3) The modifications of the boundary are such that each point can be moved along the normal by a certain amount, which is dependent on the structuring element. Taken together, these observations, when the size of the structuring element shrinks to zero, assert that mathematical morphology operations with a convex structuring element are captured by a differential deformation of the boundary along the normal, governed by a Hamilton-Jacobi partial differential equation (PDE). A second theme is to show that mathematical morphology operations can be numerically implemented in a highly accurate fashion as the solution of these PDEs.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>