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  "body": "\n## Example\n\nLet $\\Omega \\subseteq \\R^2$ be a bounded open set, the image domain.\nLet $g \\in L^\\infty(\\Omega)$ be a grey-level image, and let $\\alpha, \\beta \u003e 0$ be parameters.\nFor a relatively [[closed set|0ZCK8DQYN1]] $K \\subseteq \\Omega$ and a function $u \\in C^1(\\Omega \\setminus K)$, the *Mumford–Shah functional* is\n$$E(u, K) = \\int_{\\Omega \\setminus K} \\abs{\\nabla u}^2 \\, dx + \\alpha \\int_{\\Omega \\setminus K} (u - g)^2 \\, dx + \\beta \\, \\mathcal{H}^1(K).$$\nHere $\\mathcal{H}^1$ is the one-dimensional [[Hausdorff measure|Q1TGZK8RN6]], the length of $K$.\nA minimizing pair $(u, K)$ gives a smoothed image $u$ and a set of edges $K$.\n\nCheck on a simple image: let $D$ be a closed disc of radius $r$ in $\\Omega$, and let $g = \\mathbf{1}_D$.\nThe pair $u = g$, $K = \\partial D$ has no gradient term and no fidelity term, so $E(u, K) = 2 \\pi \\beta r$.\nThe pair $u = 0$, $K = \\emptyset$ has energy $\\alpha \\pi r^2$.\nSo for $\\alpha \u003e 2\\beta/r$ the first pair is cheaper than the second, and for $\\alpha \u003c 2\\beta/r$ the second is cheaper.\nLarge weights on fidelity favour keeping the edge, and small weights favour erasing it.\n\n## Remarks\n\nThe functional was proposed by David Mumford and Jayant Shah for image segmentation.\nIt is a free discontinuity problem: the unknown set $K$ is where $u$ may jump.\nEnnio De Giorgi, Michele Carriero and Antonio Leaci proved the existence of minimizers.\nTheir method first solves a weak form, where $u$ is a special function of bounded variation and $K$ is its jump set.\nLuigi Ambrosio and Vincenzo Maria Tortorelli approximated it, in the sense of [[Γ-convergence|NW9MM370A5]], by elliptic functionals of two smooth functions, which is the basis of many numerical methods.\n\n## Explanations\n\n### L1 human\n\nAn image has smooth regions separated by sharp edges.\nThe Mumford–Shah functional looks for a clean image $u$ that stays close to the given image $g$.\nThe clean image must be smooth away from a set of edges, and the total length of the edges must be small.\nEach wish has a weight, and the best compromise gives both a cleaned image and a drawing of its contours.\nFor a white disc on a black background, the answer depends on the weights: either the circle is kept as an edge, or the disc is wiped out.\n",
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