Properties of the Approximating Image in the Mumford-Shah Model
In Section 13.1, we explain some elementary properties satisfied by u, namely the elliptic equation -Δu + u = g and we answer the following question: If Kn “tends to” K (in a sense which will be discussed), what can be said about the convergence of UKn, associated with Kn, towards u = uk ?
In Section 13.2, we look for the effect on the energy of u of a certain kind of modification of K: when some part of K has been erased. Then the new approximating function v satisfies I(v) ≥ I(u), and we give precise estimates on I(v) — I(u) which relate this “jump of energy” to the geometry of K. Section 13.3 is devoted to an accurate estimate of the gradient of u, Vu, as a function of the distance to K. When coupled with the “jump of energy” estimates of Section 13.2, this estimate will prove a basic tool to understand the geometry of minimal edge sets K.
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