Non-homothetic granulometric mixing theory with application to blood cell counting

A granulometry is a family of morphological openings by scaled structuring elements. As the scale increases, increasing image area is removed. Normalizing removed area by the total area yields the pattern spectrum of the image. The pattern spectrum is a probability distribution function and its moments are known as granulometric moments. Modeling the image as a random set, the pattern spectrum is a random function and its moments are random variables. The original granulometric mixing theory provides closed-form representation of the granulometric moments, shows that the distributions of the moments are asymptotically normal, and gives asymptotic expressions for the means and variances of the granulometric moments. The theory applies to random-set models formed as disjoint unions of randomly scaled image primitives (grains). The scaled grains are known as homothetics. The theory can be used in a method-of-moments fashion to estimate the parameters governing the random scaling factors and mixture proportions. Application is limited by the homothetic assumption. This paper drops the homothetic requirement and provides a mixing theory for a disjoint union of fully randomized primitives. Whereas the original asymptotic theory gives expressions for the moments themselves and is applied by taking expectations afterwards, the non-homothetic theory involves the granulometric size density, which is the mean of the original size distribution prior to normalization. Hence, the representation concerns expectations, not random variables. Nonetheless, a similar method-of-moments approach can be used to estimate mixture proportions. A large part of the paper is devoted to estimate blood cell proportions that correspond to cell-age categories. Each cell class is represented by a random grain, and the problem is to estimate the proportions of cells occurring in the various age categories. The random behavior of the cells in each category makes the non-homothetic theory appropriate. Because the estimation strategy leads to a system of nonlinear equations whose solution presents computational di$culties, the estimation is accomplished via a divide-and-conquer strategy in which the full mixture problem is partitioned into smaller problems, and the solutions of these problems are joined to solve the full problem. 2001 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.