ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 70, 1 (2001)
pp. 33-50

APPLICATIONS OF NONLINEAR DIFFUSION IN IMAGE PROCESSING AND COMPUTER VISION
J. WEICKERT

Abstract. Nonlinear diffusion processes can be found in many recent methods for image processing and computer vision. In this article, four applications are surveyed: nonlinear diffusion filtering, variational image regularization, optic flow estimation, and geodesic active contours. For each of these techniques we explain the main ideas, discuss theoretical properties and present an appropriate numerical scheme. The numerical schemes are based on additive operator splittings (AOS). In contrast to traditional multiplicative splittings such as ADI, LOD or D'yakonov splittings, all axes are treated in the same manner, and additional possibilities for efficient realizations on parallel and distributed architectures appear. Geodesic active contours lead to equations that resemble mean curvature motion. For this application, a novel AOS scheme is presented that uses harmonic averaging and does not require reinitializations of the distance function in each iteration step. Its accuracy is evaluated in case of mean curvature motion.

AMS subject classification. Primary 35-K35, 65-M06, 68-W10, 94-A08
Keywords. Nonlinear diffusion, variational methods, image processing, computer vision, finite difference methods, parallel algorithms, mean curvature motion

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Acta Mathematica Universitatis Comenianae
Institute of Applied Mathematics
Faculty of Mathematics, Physics and Informatics
Comenius University
842 48 Bratislava, Slovak Republic
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