Generation of Optimal Approximations of Color Images

Using color images as an example, the problem of optimal clustering of multidimensional vectors in the form of three-dimensional pixels is posed and solved, that is, the problem of the maximum possible improvement of the image approximation quality based on the total square error for each number of colors from a given range. The solution is achieved by dividing the image into independent parts processed by Ward’s method, where the part is understood as any subset of pixels. As a new result, this paper proposes an elementary algorithm that allows transforming a set of approximation hierarchies for parts computed by Ward’s method into a single approximation hierarchy for the entire image, described by a convex sequence of total squared deviation values. Processing the image in parts solves the problem of excessive computational complexity that is typical when applying Ward’s method to images with a large number of pixels. And the generation of a series of hierarchical sequences of approximations for a color image provides an estimate of the optimal values of the total squared deviation and the selection of appropriate approximations for the values of the number of colors in the entire specified range.