Theory and Algorithms for Constructing Discrete Morse Complexes from Grayscale Digital Images

• Published in: IEEE transactions on pattern analysis and machine intelligence Vol. 33; no. 8; pp. 1646 - 1658
• Main Authors: Robins, V, Wood, P J, Sheppard, A P
• Format: Journal Article
• Language: English
• Published: Los Alamitos, CA IEEE 01-08-2011; IEEE Computer Society; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Artificial intelligence; computational topology; Computer aided design; Computer science; control theory; systems; Construction; Critical point; Differential geometry; Digital; Digital images; digital topology; Discrete Morse theory; Exact sciences and technology; Geometry; Gray-scale; Joining processes; Level set; Manifolds; Mathematical analysis; Mathematics; Pattern analysis; Pattern recognition. Digital image processing. Computational geometry; persistent homology; Sciences and techniques of general use; Software; Three dimensional displays; Topology; Differential topology; Contour line; Homology; Critical point; Digital image; Discrete function; Discrete Morse theory; Gray scale; Homotopy; Voxel; computational topology; persistent homology; Tridimensional image; digital topology; Algebraic topology
• ISSN: 0162-8828; 1939-3539
• DOI: 10.1109/TPAMI.2011.95

We present an algorithm for determining the Morse complex of a two or three-dimensional grayscale digital image. Each cell in the Morse complex corresponds to a topological change in the level sets (i.e., a critical point) of the grayscale image. Since more than one critical point may be associated with a single image voxel, we model digital images by cubical complexes. A new homotopic algorithm is used to construct a discrete Morse function on the cubical complex that agrees with the digital image and has exactly the number and type of critical cells necessary to characterize the topological changes in the level sets. We make use of discrete Morse theory and simple homotopy theory to prove correctness of this algorithm. The resulting Morse complex is considerably simpler than the cubical complex originally used to represent the image and may be used to compute persistent homology.