On the detection of simple points in higher dimensions using cubical homology.
Simple point detection is an important task for several problems in discrete geometry, such as topology preserving thinning in image processing to compute discrete skeletons. In this paper, the approach to simple point detection is based on techniques from cubical homology, a framework ideally suited for problems in image processing. A (d-dimensional) unitary cube (for a d-dimensional digital image) is associated with every discrete picture element, instead of a point in epsilon(d) (the d-dimensional Euclidean space) as has been done previously. A simple point in this setting then refers to the removal of a unitary cube without changing the topology of the cubical complex induced by the digital image. The main result is a characterization of a simple point p (i.e., simple unitary cube) in terms of the homology groups of the (3d - 1) neighborhood of p for arbitrary, finite dimensions
Niethammer
Marc
M
School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250, USA. marcn@ece.gatech.edu
Kalies
William D
WD
Mischaikow
Konstantin
K
Tannenbaum
Allen
A
D000465:Algorithms; D001185:Artificial Intelligence; D003196:Computer Graphics; D007089:Image Enhancement / Q000379:methods*; D007090:Image Interpretation, Computer-Assisted / Q000379:methods*; D021621:Imaging, Three-Dimensional / Q000379:methods*; D016247:Information Storage and Retrieval / Q000379:methods; D009716:Numerical Analysis, Computer-Assisted; D010363:Pattern Recognition, Automated / Q000379:methods*; D012815:Signal Processing, Computer-Assisted