116 research outputs found
A Relaxation result for energies defined on pairs set-function and applications
We consider, in an open subset Ω
of RN, energies depending on the perimeter of a subset
E С Ω
(or some equivalent surface integral) and on a function u which is defined only on
E. We compute the lower semicontinuous envelope of such energies. This relaxation has
to take into account the fact that in the limit, the “holes”
Ω \ E may collapse into a
discontinuity of u, whose surface will be counted twice in the relaxed energy. We discuss
some situations where such energies appear, and give, as an application, a new proof of
convergence for an extension of Ambrosio-Tortorelli’s approximation to the Mumford-Shah
functional
Random finite-difference discretizations of the Ambrosio-Tortorelli functional with optimal mesh size
We propose and analyze a finite-difference discretization of the
Ambrosio-Tortorelli functional. It is known that if the discretization is made
with respect to an underlying periodic lattice of spacing , the
discretized functionals -converge to the Mumford-Shah functional only
if , being the elliptic approximation
parameter of the Ambrosio-Tortorelli functional. Discretizing with respect to
stationary, ergodic and isotropic random lattices we prove this
-convergence result also for , a regime at which
the discretization with respect to a periodic lattice converges instead to an
anisotropic version of the Mumford-Shah functional.Comment: 36 pages, 6 figures. Added some numerical example
Finite Difference Approximation of Free Discontinuity Problems
We approximate functionals depending on the gradient of and on the
behaviour of near the discontinuity points, by families of non-local
functionals where the gradient is replaced by finite differences. We prove
pointwise convergence, -convergence, and a compactness result which
implies, in particular, the convergence of minima and minimizers.Comment: 39 pages. to appear on Proc. Royal Soc. Edinb. Ser.
A new proximal method for joint image restoration and edge detection with the Mumford-Shah model
International audienceIn this paper, we propose an adaptation of the PAM algorithm to the minimization of a nonconvex functional designed for joint image denoising and contour detection. This new functional is based on the Ambrosio–Tortorelli approximation of the well-known Mumford–Shah functional. We motivate the proposed approximation, offering flexibility in the choice of the possibly non-smooth penalization, and we derive closed form expression for the proximal steps involved in the algorithm. We focus our attention on two types of penalization: 1-norm and a proposed quadratic-1 function. Numerical experiments show that the proposed method is able to detect sharp contours and to reconstruct piecewise smooth approximations with low computational cost and convergence guarantees. We also compare the results with state-of-the-art re-laxations of the Mumford–Shah functional and a recent discrete formulation of the Ambrosio–Tortorelli functional
Discrete stochastic approximations of the Mumford-Shah functional
We propose a -convergent discrete approximation of the Mumford-Shah
functional. The discrete functionals act on functions defined on stationary
stochastic lattices and take into account general finite differences through a
non-convex potential. In this setting the geometry of the lattice strongly
influences the anisotropy of the limit functional. Thus we can use
statistically isotropic lattices and stochastic homogenization techniques to
approximate the vectorial Mumford-Shah functional in any dimension.Comment: 47 pages, reorganized versio
Unilateral gradient flow of the Ambrosio-Tortorelli functional by minimizing movements
Motivated by models of fracture mechanics, this paper is devoted to the
analysis of unilateral gradient flows of the Ambrosio-Tortorelli functional,
where unilaterality comes from an irreversibility constraint on the fracture
density. In the spirit of gradient flows in metric spaces, such evolutions are
defined in terms of curves of maximal unilateral slope, and are constructed by
means of implicit Euler schemes. An asymptotic analysis in the Mumford-Shah
regime is also carried out. It shows the convergence towards a generalized heat
equation outside a time increasing crack set.Comment: accepted in Ann. Inst. H. Poincar\'e, Anal. Nonli
Joint Image Reconstruction and Segmentation Using the Potts Model
We propose a new algorithmic approach to the non-smooth and non-convex Potts
problem (also called piecewise-constant Mumford-Shah problem) for inverse
imaging problems. We derive a suitable splitting into specific subproblems that
can all be solved efficiently. Our method does not require a priori knowledge
on the gray levels nor on the number of segments of the reconstruction.
Further, it avoids anisotropic artifacts such as geometric staircasing. We
demonstrate the suitability of our method for joint image reconstruction and
segmentation. We focus on Radon data, where we in particular consider limited
data situations. For instance, our method is able to recover all segments of
the Shepp-Logan phantom from angular views only. We illustrate the
practical applicability on a real PET dataset. As further applications, we
consider spherical Radon data as well as blurred data
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