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Image segmentation
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== Variational methods == The goal of variational methods is to find a segmentation which is optimal with respect to a specific energy functional. The functionals consist of a data fitting term and a regularizing terms. A classical representative is the [[Potts model]] defined for an image <math>f</math> by :<math> \operatorname{argmin}_u \gamma \| \nabla u \|_0 + \int (u - f)^2 \, dx. </math> A minimizer <math>u^*</math> is a piecewise constant image which has an optimal tradeoff between the squared L2 distance to the given image <math>f</math> and the total length of its jump set. The jump set of <math>u^*</math> defines a segmentation. The relative weight of the energies is tuned by the parameter <math>\gamma >0 </math>. The binary variant of the Potts model, i.e., if the range of <math>u</math> is restricted to two values, is often called Chan-[[Luminița Vese|Vese]] model.<ref>{{cite journal | last1 = Chan | first1 = T.F. | last2 = Vese | first2 = L. | author2-link= Luminița Vese | year = 2001 | title = Active contours without edges | journal = IEEE Transactions on Image Processing | volume = 10 | issue = 2| pages = 266–277 | doi=10.1109/83.902291| pmid = 18249617 | bibcode = 2001ITIP...10..266C | s2cid = 7602622 }}</ref> An important generalization is the [[Mumford–Shah functional|Mumford-Shah model]]<ref>[[David Mumford]] and Jayant Shah (1989): [https://dash.harvard.edu/bitstream/handle/1/3637121/Mumford_OptimalApproxPiece.pdf?sequence=1 Optimal approximations by piecewise smooth functions and associated variational problems], ''Communications on Pure and Applied Mathematics'', pp 577–685, Vol. 42, No. 5</ref> given by :<math> \operatorname{argmin}_{u, K} \gamma |K| + \mu \int_{K^C} |\nabla u|^2 \, dx + \int (u - f)^2 \, dx. </math> The functional value is the sum of the total length of the segmentation curve <math>K</math>, the smoothness of the approximation <math>u</math>, and its distance to the original image <math>f</math>. The weight of the smoothness penalty is adjusted by <math>\mu > 0</math>. The Potts model is often called piecewise constant Mumford-Shah model as it can be seen as the degenerate case <math>\mu \to \infty</math>. The optimization problems are known to be NP-hard in general but near-minimizing strategies work well in practice. Classical algorithms are [[Graduated optimization|graduated non-convexity]] and [[Mumford–Shah functional|Ambrosio-Tortorelli approximation]].
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