Abstract
This paper is devoted to the challenging problem of fine structure detection with applications to bituminous surfacing crack recovery. Drogoul (SIAM J Imag Sci 7(4):2700–2731, 2014) shows that such structures can be suitably modeled by a sequence of smooth functions whose Hessian matrices blow up in the perpendicular direction to the crack, while their gradient is null. This observation serves as the basis of the introduced model that also handles the natural dense and highly oscillatory texture exhibited by the images: We propose weighting \(\left| \frac{\partial ^2u}{\partial x_1^2}\right| ^2+\left| \frac{\partial ^2u}{\partial x_2^2}\right| ^2\), u denoting the reconstructed image, by a variable that annihilates great expansion of this quantity, making then a connection with the elliptic approximation of the Blake–Zisserman functional. Extending then the ideas developed in the case of first-order nonlocal regularization to higher-order derivatives, we derive and analyze a nonlocal version of the model, and provide several theoretical results among which there are a \(\varGamma \)-convergence result as well as a detailed algorithmic approach and an MPI implementation based on a natural domain decomposition approach.
Highlights
Segmentation is a cornerstone step in image processing that attempts to reproduce the ability of human beings to track down significant patterns and automatically gather them into relevant and identified structures
While a contour is classically viewed as the boundary of a non zero volume object and is defined as the set of intensity discontinuities with jump, fine structures exhibit discontinuities without jump
We emphasize that this paper is an extension of [18] not reduced to a simple supplement, containing no overlap with [18] and including significant theoretical and computational aspects. It completes [18] since it includes substantial results, such as a Γ -convergence result relating the local basis functional with the derived nonlocal one, and an exhaustive and substantive PDE analysis, in the viscosity solution theory framework, of the resulting evolution equation satisfied by the variable Q introduced hereafter
Summary
Segmentation is a cornerstone step in image processing that attempts to reproduce the ability of human beings to track down significant patterns and automatically gather them into relevant and identified structures. We emphasize that this paper is an extension of [18] not reduced to a simple supplement, containing no overlap with [18] and including significant theoretical and computational aspects It completes [18] since it includes substantial results, such as a Γ -convergence result (main result) relating the local basis functional with the derived nonlocal one, and an exhaustive and substantive PDE analysis, in the viscosity solution theory framework, of the resulting evolution equation satisfied by the variable Q introduced hereafter.
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