Digital topology

Boundary tracing

Boundary tracing, also known as contour tracing, of a binary digital region can be thought of as a segmentation technique that identifies the boundary pixels of the digital region. Boundary tracing is an important first step in the analysis of that region.Boundary is a topological notion. However, a digital image is no topological space. Therefore, it is impossible to define the notion of a boundary in a digital image mathematically exactly. Most publications about tracing the boundary of a subset S of a digital image I describe algorithms which find a set of pixels belonging to S and having in their direct neighborhood pixels belonging both to S and to its complement I - S. According to this definition the boundary of a subset S is different from the boundary of the complement I – S which is a topological paradox.To define the boundary correctly it is necessary to introduce a topological space corresponding to the given digital image. Such space can be a two-dimensional abstract cell complex. It contains cells of three dimensions: the two-dimensional cells corresponding to pixels of the digital image, the one-dimensional cells or “cracks” representing short lines lying between two adjacent pixels, and the zero-dimensional cells or “points” corresponding to the corners of pixels. The boundary of a subset S is then a sequence of cracks and points while the neighborhoods of these cracks and points intersect both the subset S and its complement I – S. The boundary defined in this way corresponds exactly to the topological definition and corresponds also to our intuitive imagination of a boundary because the boundary of S should contain neither elements of S nor those of its complement. It should contain only elements lying between S and the complement. This are exactly the cracks and points of the complex.This method of tracing boundaries is described in the book of Vladimir A. Kovalevsky and in the web site. (Wikipedia).

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Moore neighborhood | Digital image | Pathfinding | Optimization problem | Pixel connectivity | Curve sketching