Variational Methods in Image Segmentation pp 151-164 | Cite as

# Properties of the Approximating Image in the Mumford-Shah Model

## Abstract

^{1}-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 =u

_{K}, where u is assumed to be the minimum point of the two-dimensional part of the Mumford-Shah energy,

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 K_{n} “tends to” K (in a sense which will be discussed), what can be said about the convergence of UK_{n}, associated with K_{n}, 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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