Abstract
The problem of curve matching appears in many application domains, like time series analysis, shape matching, speech recognition, and signature verification, among others. Curve matching has been studied extensively by computational geometers, and many measures of similarity have been examined, among them being the Fréchet distance (sometimes referred in folklore as the “dog-man” distance).
A measure that is very closely related to the Fréchet distance but has never been studied in a geometric context is the Dynamic Time Warping measure (DTW), first used in the context of speech recognition. This measure is ubiquitous across different domains, a surprising fact because notions of similarity usually vary significantly depending on the application. However, this measure suffers from some drawbacks, most importantly the fact that it is defined between sequences of points rather than curves. Thus, the way in which a curve is sampled to yield such a sequence can dramatically affect the quality of the result. Some attempts have been made to generalize the DTW to continuous domains, but the resulting algorithms have exponential complexity.
In this paper we propose similarity measures that attempt to capture the “spirit” of dynamic time warping while being defined over continuous domains, and present efficient algorithms for computing them. Our formulation leads to a very interesting connection with finding short paths in a combinatorial manifold defined on the input chains, and in a deeper sense relates to the way light travels in a medium of variable refractivity.
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Alon Efrat earned his Bachelor in Applied Mathematics from the Technion, Israel’s Institute of Technology in 1991, earned his Master in Computer Science from the Technion in 1993, and his Ph.D. in Computer Science from Tel-Aviv University in 1998. During the years 1998–2000 he was a Post Doctorate Research Assistant at the Computer Science Department at Stanford University, and at IBM Almaden Research Center. Since 2000, he is an assistant Professor at the Computer Science Department at the University of Arizona. His main research areas are Computational Geometry, and its applications.
Quanfu Fan is a Ph.D. student in the department of Computer Science at the University of Arizona. His research interests include object recognition and image understanding, information retrieval, and geometric algorithms.
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Efrat, A., Fan, Q. & Venkatasubramanian, S. Curve Matching, Time Warping, and Light Fields: New Algorithms for Computing Similarity between Curves. J Math Imaging Vis 27, 203–216 (2007). https://doi.org/10.1007/s10851-006-0647-0
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DOI: https://doi.org/10.1007/s10851-006-0647-0