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.

中文翻译：

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基于非局部拉普拉斯算式的沥青路面裂缝恢复模型及其MPI实现

本文致力于解决具有挑战性的精细结构检测问题，并将其应用到沥青表面裂缝恢复中。Drogoul（SIAM J Imag Sci 7（4）：2700–2731，2014）表明，可以通过一系列光滑函数对此类结构进行适当建模，这些光滑函数的Hessian矩阵沿与裂纹垂直的方向爆炸，而其梯度为零。该观察结果作为引入的模型的基础，该模型也处理图像显示的自然密集和高度振荡的纹理：我们建议对\（\ left | \ frac {\ partial ^ 2u} {\ partial x_1 ^ 2} \ right加权| ^ 2 + \ left | \ frac {\ partial ^ 2u} {\ partial x_2 ^ 2} \ right | ^ 2 \），u用一个消除该数量极大扩展的变量表示重建图像，然后与Blake-Zisserman函数的椭圆逼近联系起来。然后将在一阶非局部正则化的情况下发展的思想扩展到高阶导数，我们导出并分析模型的非局部版本，并提供一些理论结果，其中有一个\（\ varGamma \）-收敛结果以及基于自然域分解方法的详细算法方法和MPI实现。