Klyachin V.A. Using the Laplace Operator Spectrum in Contour Analysis

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https://doi.org/10.15688/mpcm.jvolsu.2026.1.5

Vladimir A. Klyachin
Doctor of Sciences (Physics and Mathematics), Head of the Department of Computer Sciences and Experimental Mathematics, Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation

Abstract. Calculating the contours of objects in articles/images is often used in computer vision systems to detect and identify objects. In particular, contour analysis allows one to determine the shape of an object in an image. Using the spectrum of the Laplace operator for a region bounded by a contour for this purpose is based on the solution to M. Katz’s well-known problem in geometric analysis: “Can you hear the shape of a drum?” This problem is known to have a negative answer: there are non-isometric regions with the same spectrum of the Laplace operator. However, in 2020, S.A. Titarenko proved that almost all regions are uniquely defined by their spectrum. This provides grounds for applying this approach to contour analysis. This article describes an attempt to practically use the spectrum of the discrete Laplace operator to identify objects in articles/images.

Key words: Laplace operator, image contour, contour region graph, image processing, feature extraction.

Klyachin V.A. Using the Laplace Operator Spectrum in Contour Analysis is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Mathematical Physics and Computer Simulation. Vol. 29 No. 1 2026, pp. 58-65

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