Optimal approximations by piecewise smooth functions and associated variational problems
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Cites background from "Optimal approximations by piecewise..."
...While most of these ideas go back to the 1970s (and earlier), the 1980s brought in the use of Markov Random Fields [10] and variational formulations [ 17 ], [2], [14]....
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10,404 citations
Cites background or methods or result from "Optimal approximations by piecewise..."
...We expect, of course, to have existence of minimizers of the energy , due to several general results: our model is a particular case of the minimal partition problem, for which the existence has been proved in [ 18 ] (assuming that is continuous on ), and also in [16] and [17], for more general data . Also, the existence for the general Mumford‐Shah segmentation problem has been proved in [5]....
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...The Mumford‐Shah functional for segmentation is [ 18 ]...
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...The stopping term is based on Mumford‐Shah segmentation techniques [ 18 ]....
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...A reduced form of this problem is simply the restriction of to piecewise constant functions , i.e., on each connected component of . Therefore, as it was also pointed out by D. Mumford and J. Shah [ 18 ], on each connected component . The reduced case is called the minimal partition problem....
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5,318 citations
Cites background from "Optimal approximations by piecewise..."
...Their energy is a modi.ed form of the common Mumford-Shah energy [Mumford and Shah 1989], which evolves the contour until a region with homogeneous .ow vectors is achieved....
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References
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"Optimal approximations by piecewise..." refers background in this paper
...A closely related functional defined for functions g and f on a lattice instead of on a plane domain R was first introduced by D. and S. Geman [6], and has been studied by Blake and Zisserman [4], J. Marroquin [lo] and others....
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8,965 citations
"Optimal approximations by piecewise..." refers methods in this paper
...If there were no boundaries at all, then Green's function for this operator and the whole plane is obtained from the so-called " modified Bessel's function of the second kind " KO (cf. Whittaker and Watson [14], Section 17.71, pp. 373-374): where K , ( r ) (defined for r > 0) is the solution of...
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5,248 citations
"Optimal approximations by piecewise..." refers background or methods in this paper
...6 in Grisvard [7] (applied to gp and f r separately),...
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...By the trace theorem (see [7]), the restriction of u E Wp?(W) to dW defines a continuous linear map w,"(w) -+ w;(aw) c w;(aw)....
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...By Sobolev imbedding theorem, u E Hi(Q2) where q > 2, and by the trace theorems (see [7]),...
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...by the trace formula in Sobolev spaces (see [7], Section 1....
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...However, the behavior of the solutions of elliptic boundary value problems in domains with " corners " has been studied by several people, starting with a classic paper of Kondratiev [9] and recently surveyed in a book of Grisvard [7]....
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1,498 citations