Non-local functionals related to the total variation and connections with image processing. (English) Zbl 1398.49042
Motivated by applications in image processing, the authors present in this massive work new results concerning the approximation of the total variation of a given function via some nonlocal and nonconvex functionals of a specific form. The pointwise convergence brings some results, however too many pathologies can be observed, therefore, as the authors put it, “a reasonable theory of pointwise convergence […] seems out of reach”. Fortunately, employing De Giorgi’s \(\Gamma\)-convergence radically improves the situation, also leaving space for some open problems. Worth noticing are the numerous and sometimes delicate steps needed in the (quite technical) proofs of the main results. A section of the paper is dedicated to investigating some functionals related both to the theoretical statements presented before and to image processing. Three appendices dealing with proofs of some of the mentioned pathologies and to a special case of the pointwise convergence, respectively, close this 77-page long (but fascinating) paper.
Reviewer: Sorin-Mihai Grad (Vienna)
MSC:
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 26B30 | Absolutely continuous real functions of several variables, functions of bounded variation |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 28A75 | Length, area, volume, other geometric measure theory |
| 49M25 | Discrete approximations in optimal control |
Keywords:
total variation; bounded variation; non-local functional; non-convex functional; \(\Gamma \)-convergence; Sobolev spaces; image processing; pointwise convergenceReferences:
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