Properties of the Approximating Image in the Mumford-Shah Model
Jean‐Michel Morel, Sergio Solimini · Birkhäuser Boston eBooks · 1995
In this chapter, we assume that a closed 1-set K with finite ℋ1 -measure has been associated with an image g as a possible ”edge set“. We shall not assume that K is minimal with respect to the Mumford-Shah functional because we wish to focus on the following question: Given K, what is to be said of u =uK, where u is assumed to be the minimum point of the two-dimensional part of the Mumford-Shah energy, 1 $$I(u) = \int_{\Omega \backslash K} {(| abla u{|^2} + {{(u - g)}^2}?} $$ 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.