Approximation of functional depending on jumps by elliptic functional via t‐convergence

Abstract

We show how it is possible to approximate the Mumford-Shah (see [29]) image segmentation functional uE W 1 . * ( Q \ K ) , K C Q closedin Q by elliptic functionals defined on Sobolev spaces. The heuristic idea is to consider functionals Sh( u, z ) with z ranging between 0 and I and related to the set K. The minimizing zh are near to 1 in a neighborhood of the set K, and far from the neighborhood they are very small. The neighborhood shrinks as h + +co .For a similar approach to the problem compare Kulkarni; see [25]. The approximation of Sh to 9 takes place in a variational sense, the De Giorgi I'-convergence. L. AMBROSIO AND V. M. TORTORELLIwhere a, / 3 > 0 are fixed parameters, and g E L*(fl). In the case n = 2 this functional has been suggested by Mumford-Shah (see [29]) for a variational a p proach to image segmentation. The function g in (2) represents the image of a group of objects given by a camera, with discontinuities along the edges of the objects. By minimizing the functional (2) one tries to distinguisl? the discontinuities due to the edges and shadows from the discontinuities due to noise and small irregularities. The functional penalizes large sets K, and outside the set K the function u is required to be close to g and W1.2. In the papers [29], [ 301 Mumford-Shah conjecture that 9 has minimizers, and they study their behavior under some a priori regularity assumptions. Recently, the Mumford-Shah conjecture concerning existence of minimizers has been proved for general integers n by De Giorgi-Camiero-Leaci; see [ 17 1. In the case n = 2 a different proof has been discovered by Dal Maso-Morel-Solimini; see [ 131. The idea in common to both proofs is to use a weak formulation of the problem by setting where S, is the discontinuity set of u in an approximate sense, and u varies in a special class of functions of bounded variation, denoted by S B V ( Q ) . This class consists of all functions of bounded variation such that the distributional derivative is absolutely continuous with respect to Lebesgue measure plus an ( n -1 )-dimensional measure. Since (see [2], [3], [17]) u E SBV(O), %-'(S,,\K) = 0, the domain SBV( a) contains the original domain of 8. By means of general compactness and lower semicontinuity theorems in SBV( a ) (see [I], [2], [ 31) it can be shown that G has minimizers in SBV( Q). The regularity theorems in [ 13 1, [ 171 show that u E C'(Q\K), , xn-I (s, -n WJ = 0, for any minimizer u so that, by setting K = f7 9, we recover a minimum of 9. The problem of finding effective algorithms for computing the minimizers of G is still widely discussed; see [7], [20], [22], [23], [24], [35]. In this paper we suggest an approach to this problem by approximating in a variational sense, defined by l'-convergence (see, for instance, [ 5 1, [ 121, [ 141, [ 15 I), the functional G by elliptic functionals. The outcome of recent numerical simulations, made by R. March in Pisa, has been very encouraging. Anyway, independently of the possible computational applications of this result, it is interestin...

Edge Strength Functions as Shape Priors in Image Segmentation

“…Unknown edge set Γ makes the minimization mathematically difficult. A convenient approximation is suggested by Ambrosio and Tortorelli in [1] where they introduce a smooth edge indicator function v(x, y) which is more convenient than the original edge indicator. On the edges, v(x, y) → 1 and on the smooth regions v(x, y) → 0.…”

“…be the principal components computed by Karhunen-Loeve Transformation, then a possible shape from this ensemble has 1 The alignment algorithm proposed in [11] is used in the experiments. …”

“…[6], we present a new method for shape prior incorporation into segmentation process. Backbone of our method is Ambrosio-Tortorelli [1] approximation (AT) of Mumford-Shah functional. We employ a by-product of this functional, a smooth edge indicator function which is known as the "edge strength function" as a distance-like surface which embeds the shape boundary.…”

Signal segmentation and denoising algorithm based on energy optimisation

“…This method was further developed mathematically by Mumford et al [5][6][7] for image segmentation and smoothing and was subsequently approximated and implemented by using different approaches e.g. [8][9][10][11][12][13][14][15] of three terms: (1) fidelity term indicating that the smoothed image should be as close as possible to the original image, (2) smoothing term requiring that the smoothed image should be as smooth as possible and (3) contour length removing the unnecessary contours and smoothing the contour representing discontinuities as much as required. Mumford and Shah conjectured that there exists a minimiser for their functional, although the proof for this conjecture is still an open problem.…”

A Variational Framework for Joint Image Registration, Denoising and Edge Detection

“…The E u 0 AT [u, v] denotes the Ambrosio-Tortorelli (AT) approximation functional proposed in [7,8]. This functional is originally designed to approximate the Mumford-Shah model [9] for image segmentation.…”