A new and interesting approach to image segmentation is proposed; it is both mathematically motivated and efficient. The main idea is applying symmetries.
The book consists of six chapters:
(1) Introduction to Image Segmentation,
(2) Towards a Theory of Image Segmentation,
(3) Gray Level Segmentation,
(4) Feature Selection for Texture Description,
(5) Texture Segmentation Using a Multi-dimensional Feature Space, and
(6) Image Segmentation--A Problem Solved?
Chapter 1 surveys existing approaches, and chapter 2 provides a general formalism. On one hand, the segmentation must divide black and white points (or, more precisely, points with different brightnesses). On the other hand, the boundary between the segments must be smooth (or at least not too irregular). In the ideal case with no noise these demands are consistent, but in real cases where noise is present they are somewhat contradictory, because the noise makes the boundary between black and white points random and hence irregular. The traditional approach to such a problem is to postulate some aim function that expresses the quality of the image segmentation and to minimize this aim function. The author’s approach is based on symmetry ideas, namely, on scale invariance: because the formulation of the problem contains no fixed unit of length, the restoration algorithm must lead to the same restored image when the scale changes. In particular, doubling the unit of length produces a new image in which each new pixel corresponds to four old ones. It is natural to assign this new pixel a brightness equal to the arithmetic mean of the four old brightnesses. This procedure diminishes the number of pixels. Repeating this procedure again and again leads to a new image with few pixels for which segmentation is easy, i.e., boundaries are easily constructed. Due to the invariance demand, the original segmentation must have the same boundaries. However, the boundaries for an averaged image determine the boundaries of the original segmentation only approximately. Therefore, the authors propose to analyze all the intermediate images in order, from the simplest image to the original image, and to determine the boundary more precisely at each step--this can be done using a statistical criterion. This method turns out to be efficient in practice; both theoretical and experimental estimates of its quality are given.
A similar scaling-motivated algorithm is proposed for a realistic situation in which the segments can correspond to different textures. Experiments show that what we call texture can be expressed by characteristics of the image’s Fourier transform (spectrum). Therefore texture segmentation is, crudely speaking, segmentation in the frequency domain. Because projections on regions in frequency domains and projections on regions in spatial domains do not commute, it is impossible to obtain strict segmentation both in texture and in brightness (something like the Heisenberg inequality applies). Here again a scaling approach turns out to be efficient.
