Morse Operators for Digital Planar Surfaces and Their Application to Image Segmentation

Abstract

This paper introduces the concept of digital planar surfaces and corresponding Morse operators. These operators offer a novel and powerful method for construction and de-construction of such surfaces in a way that global topological control of the resulting object is always maintained. In that respect, this paper offers a complete pixel characterization tool. Image handling is a natural application for such approach. We present a novel fast algorithm for image segmentation using Morse operators for digital planar surfaces. It classifies as a region growing technique with added topological control and is extremely useful for applications that need proper object description. Results from real data are stimulating, and show that the segmentation algorithm compares very well with other methods. The topological approach also forms a base for future expansion to applications such as volume segmentation.

Alternative formulations to compute the binary shape Euler number

“…Let m′ be the new number of tetra-pixels when a new tetra-pixel is found again by analysing S c . We have to prove (18) …”

“…As shown in [18], Morse operators form a powerful topological tool to handle object classification. Both the 4-and 8-connected cases are considered in this reference.…”

“…Many methods have been developed to obtain the Euler number of a digital binary image. Refer, for example, to [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. For instance, the algorithm outlined in [5] was certainly one of the first reported in literature.…”

Topological triangle characterization with application to object detection from images

“…An important issue in this context is the topological characterization of the basic image elements, i.e., pixels. Solutions to the previous problems, and to many others, either explicitly or implicitly employ such a characterization [18,14].…”

Circulation and Topological Control in Image Segmentation

“…For example, the segmentation of an axial image of the brain with Euler number equal to 1 could produce a single contour of the skull (assuming this is the most external anatomical structure present). But, if Euler number is set to values smaller than 0, other internal structures (along with the skull itself) would appear as the result of the segmentation [11].…”