Segmentation into Three Classes Using Gradients

Abstract
Consider a three-dimensional "scene" in which a density f(x, y, z) is assigned to every point (x, y, z). In a discretized version of the scene the density D(i, j, k) assigned to the (i, j, k) th volume element (voxel) is the average value of f(x, y, z) over the voxel. Suppose that the points in the original scene can be meaningfully segmented into classes 1, 2, and 3 separated by two threshold values l and u. Partial volume artifact is the phenomenon that a voxel (i, j, k) which is at the interface of class 1 and class 3 (and thus contains only points with low and high densities) usually has a density D(i, j, k) between l and u, and so cannot be distinguished by density alone from a voxel which contains only points in class 2. We describe how a two-dimensional (gradient, density) feature space can be used for the segmentation of such discrete scenes into three classes in a meaningful way. We illustrate the method using examples from medical imaging.

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