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Variational versus PDE-based approaches in mathematical image processing. (English) Zbl 1144.35401
Alama, Stanley (ed.) et al., Singularities in PDE and the calculus of variations. Selected papers of the CRM workshop, Montreal, Canada, July 17– 21, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4350-5/pbk). CRM Proceedings and Lecture Notes 44, 113-126 (2008).
Summary: In mathematical image processing we are often presented with amazing examples of image enhancement algorithms. Yet, when applied to different noisy images, they can produce unwanted effects. The analysis of such algorithms lags behind their intuitive development. Two essentially different models have found wide recognition: a variational approach according to Mumford and Shah and an approach via nonlinear diffusion equations. One of these equations is nonparabolic and was suggested by Perona and Malik. In this short survey I point out a connection between these two seemingly unrelated approaches and explain some connections with total variation flow.
For the entire collection see [Zbl 1135.35003].

MSC:
35K55 Nonlinear parabolic equations
68U10 Computing methodologies for image processing
35J20 Variational methods for second-order elliptic equations
49J45 Methods involving semicontinuity and convergence; relaxation
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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