Hierarchical pixel clustering for image segmentation

The paper focuses on the domain of image segmentation by optimal approximations that minimally differ from the image of N pixels in the standard deviation  or total squared error 2  N E  . Although related approaches, namely Otsu methods [1, 2], K-means method [3] and Mumford-Shah model [4–7] have a long history, the opportunities of minimizing of the total squared error E are far from being exhausted, especially, in the task of multiple optimization for each number of pixel clusters or, in particular, connected image segments. In this task Otsu's multi-thresholding [2] provides an accurate but incomplete solution for clustering of pixels. Mumford-Shah model [4–7] provides a complete sequence of image partitions into each number of segments, but minimizing effect is poor. K-means method for image segmentation is too heuristic to provide any of mentioned two requirements, but it can be advanced for application in conjunction with Otsu method and Mumford-Shah model [8]. To solve the task of multiple optimization without any difficulties we use a special data structure of Sleator-Tarjan dynamic trees [9] that essentially optimizes the computing, but does not affect the obvious meaning of algorithms. Therefore, to avoid the cumbersome details of implementation, here we address rather to motivation of solutions and do not dwell on the software that supports the fast generation, storing in the available RAM and effective transformations of pixel clusters in a computer memory. To substantiate the study of segmentation results without appealing to the subsequent detection of a priori specified objects, we have calculated the optimal and nearly optimal approximations for the simplest examples of real images [10]. These proved important for the formulation of the problem caused by two challenges.