Mathematical morphology: The Hamilton-Jacobi connection

In this paper we complement the standard algebraic view of mathematical morphology with a geometric, di erential 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 su cient to consider how the boundary is modi ed; 3) the modi cations of the boundary are such that we can move each point 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 di erential deformation of the boundary along the normal, governed by a Hamilton-Jacobi partial di erential equation (PDE). A second theme of this paper is to show that mathematical morphology operations can be numerically implemented in a highly accurate fashion as the solution of these PDEs. Introduction In this paper we complement the standard algebraic view of mathematical morphology with a geometric di erential view. Classically, mathematical morphology treats binary images as sets and grayscale images as functions and operates on them in the spatial domain via morphological transformations, using structuring elements [17, 27, 5, 28]. While there is a growing interest in applying mathematical morphology to abstract spaces such as lattices [28, 6], graphs [29] or manifolds [20], our interest lies in considering geometric interpretations of morphological transformations; see also [2, 4, 25]. We restrain ourselves to the Euclidean plane and consider the morphological operators : IR ! IR transforming a set (shape) into another set. In this framework, we formulate basic morphological operators as partial di erential equations governing the geometric evolution of the shape. To illustrate our approach consider the dilation transformation of a shape with a disk structuring element. The transformed shape is the union of all disks centered on points of the original shape. The boundary of the transformed shape is a curve parallel to the boundary of the original shape with a distance equal to the radius of the disk. This is precisely Huygens' principle for wavefront propagation [7, 10], relating operations on algebraic constructs, i.e., sets of points and operations on geometric entities, i.e., curves representing the boundary. We will now show that this is true for all convex structuring elements, by studying morphological operations in a shape evolution framework. The geometric evolution of shapes was extensively studied by Kimia et.al. [9, 13]. Their \shape from deformation" framework captures the essence of shape through the deformations of it which were shown to be modeled geometrically as follows: Consider a shape represented by the curve C0(s) = (x0(s); y0(s)) undergoing a deformation, where s is the parameter along the curve (not necessarily the arclength), x0 and y0 are the Cartesian coordinates and the subscript 0 denotes the initial curve prior to deformation. Now, let each point of this curve move by some arbitrary amount in some arbitrary direction; see Figure 1. It can be shown that all deformations reduce to an equivalent deformation along the normal described by @C @t = (s; t) ~ N C(s; 0) = C0(s); (1) Observe that when (s; t) = 1; the evolution describes unit movement along the normal which, through an application of Huygens' principle, is equivalent to the morphological operation of dilation with a circular structuring element. Similarly, (s; t) = 1; corresponds to erosion with a circular structuring element. It is therefore reasonable that alternate forms of , should correspond to interesting morphology operations, as we will show in this paper. For a review of mathematical morphology see [1], and [17, 27, 5, 28]. Mathematical Morphology and Geometric Evolution The geometric view of the algebraic operations of mathematical morphology, is founded on three observations. First, observe that for a class of convex structuring elements B̂, n repeated dilations (or erosions) with a structuring element B 2 B̂, is equivalent to a single dilation with a structuring element of the same shape, but which is \scaled up" n times. Let us denote a structuring element scaled by as B( ). For example, 10 dilations with a circle of radius of 1, B(1), is exactly a single dilation with a