European Mathematical Society - g. david
https://euro-math-soc.eu/author/g-david
enSingular Sets of Minimizers for the Mumford-Shah Functional
https://euro-math-soc.eu/review/singular-sets-minimizers-mumford-shah-functional
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Image processing is ranked among the most topical sources of inspiration for recent mathematical analysis. The Mumford Shah functional has been proposed as a model for image segmentation. Given a bounded measurable function g on an n-dimensional domain G (the most important case is n=2) that represents the original image, we look for another function u that represents a simplified image. The balance between simplicity and fidelity is expressed by the functional J, which is composed from three summands. The first part of J is the fidelity, which is represented by the distance of u from g measured by the square of the L2-norm. The second part is the (n-1)-dimensional Hausdorff measure of the singular set K of u. In particular, we believe that the singular set found here describes the boundaries between objects that are displayed on the picture. The function u is allowed to jump across K whereas it is assumed to be smooth outside K. The last summand is the Dirichlet integral of u outside of K, which measures the smoothness of u. The major problem in dimension 2 is the celebrated Mumford-Shah conjecture, which claims that if u minimizes J then the singular set K is a finite union of C1 arcs. This would help very much in understanding the planar case.<br />
Even less complete is the knowledge in the higher dimensional case. The open problems are of a fine nature, which is perhaps not so exciting for technically oriented readers but which stimulates the development of mathematics. The importance of the functional does not consist only of its interpretation but more in the fact that it represents a whole class of free boundary problems whose theory will profit from methods developed for this particular model case. This is also the main motivation of the author. Hence, he does not write a handbook of image segmentation for practitioners, but instead a monograph on the Mumford-Shah Theory.<br />
For the definition of a global minimiser, the fidelity part is omitted (hence this is independent of data). There are four types of global minimisers of the functional: a constant function, a function attaining two values separated by a line, a function attaining three values separated by a propeller (a union of three half-lines emanating from the same point forming angles of 120 degrees), and finally the so called cracktip, a special nonconstant solution with a singular half-line. The Mumford-Shah conjecture is equivalent to the conclusion that each minimiser at each point looks locally like one of these fundamental global minimisers.<br />
This discussion is one of the most important achievements of the book. Although the book is focused on the regularity theory in dimension 2, the existence issue and general dimension problems are also treated. The text is comprehensible for graduate students. The author endows the book with his enthusiasm and the presentation sometimes seems like a fascinating adventure. Readers who know the field will surely profit from the development and find inspiration for new inventions in the theory of free boundary problems. The book has been awarded the Ferran Sunyer i Balaguer 2004 prize.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jama</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/g-david" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">g. david</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-basel-progress-mathematics-vol-233" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">birkhäuser, basel: progress in mathematics, vol. 233</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2005</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">3-7643-7182-X</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">EUR 108</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/49-calculus-variations-and-optimal-control" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">49 Calculus of variations and optimal control</a></li></ul></span>Fri, 30 Sep 2011 11:35:23 +0000Anonymous39756 at https://euro-math-soc.euCracktip is a Global Mumford-Shah Minimizer
https://euro-math-soc.eu/review/cracktip-global-mumford-shah-minimizer
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>One of the most topical models studied in the modern calculus of variations is the Mumford-Shah functional. It is motivated by the question of best approximation by piecewise smooth functions. Similar problems are studied in mathematical theory of image processing. A competitor for the Mumford-Shah functional is a function u of two variables, which is smooth except for a singular set, where it can jump. The Mumford-Shah functional is a sum of two terms: The first term is the ordinary Dirichlet integral over the regular part of the domain of u. The second term is the length (precisely, one-dimensional Hausdorff measure) of the singular part of the domain. Any minimizer of the Mumford-Shah functional must be a harmonic function in the regular part. The cracktip is a canonical example of a harmonic function in the complement of the half-line y=0, x<0. In polar coordinates, u(r cos t, r sin t) = const r 1/2 sin (t/2). In his paper from 1991, E. De Giorgi raised the conjecture that the cracktip could be a global minimizer of the Mumford-Shah functional. Although the conjecture was supported by experiments, the question of a rigorous proof became a famous problem. Now, the solution of the problem is presented by Alexis Bonnet and Guy David. The fact that they need the extent of a monograph to describe the proof certifies that the problem was really considerably deep. The method of the proof exploits a careful analysis of the harmonic conjugate to the competitor and its level set. Blow up techniques and monotonicity of the energy functional are also used. In the existence part, weak compactness properties in SBV are bypassed and all is done in the framework of strong minimizers and competitors with closed singular sets. The presentation is well organized so that the reader can recognize where to learn the strategy of the proof and where to look for particular technical details. The aim of the monograph is to give an evidence that the problem is solved. Certainly, the book is a valuable source of inspiration for researchers, which try to attack problems of a similar nature.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jama</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/bonnet" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">a. bonnet</a></li><li class="vocabulary-links field-item odd"><a href="/author/g-david" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">g. david</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/soci%C3%A9t%C3%A9-math%C3%A9matique-de-france" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">société mathématique de france</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2001</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">ISBN 2-85629-108-2</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">FRF 300</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/49-calculus-variations-and-optimal-control" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">49 Calculus of variations and optimal control</a></li></ul></span>Sun, 04 Sep 2011 14:13:30 +0000Anonymous39620 at https://euro-math-soc.eu