Decomposition of Gray-Scale Morphological Templates Using the Rank Method

Convolutions are a fundamental tool in image processing. Classical examples of two dimensional linear convolutions include image correlation, the mean filter, the discrete Fourier transform, and a multitude of edge mask filters. Nonlinear convolutions are used in such operations as the median filter, the medial axis transform, and erosion and dilation as defined in mathematical morphology. For large convolution masks or structuring elements, the computation cost resulting from implementation can be prohibitive. However, in many instances, this cost can be significantly reduced by decomposing the templates representing the masks or structuring elements into a sequence of smaller templates. In addition, such decomposition can often be made architecture specific and, thus, resulting in optimal transform performance. In this paper we provide methods for decomposing morphological templates which are analogous to decomposition methods used in the linear domain. Specifically, we define the notion of the rank of a morphological template which categorizes separable morphological templates as templates of rank one. We establish a necessary and sufficient condition for the decomposability of rank one templates into 3 ¥ 3 templates. We then use the invariance of the template rank under certain transformations in order to develop template decomposition techniques for templates of rank two.