论文标题
对Perona-Malik功能的二阶正规化的楼梯现象的定量变分分析
A quantitative variational analysis of the staircasing phenomenon for a second order regularization of the Perona-Malik functional
论文作者
论文摘要
我们考虑了perona-malik在维度上的功能，即一个积分函数，其lagrangian是相对于衍生物的凸 - concave，其凸式化相同为零。我们通过添加取决于二阶导数乘以小系数的术语来近似和正规化功能。我们研究了最小值和最小化器的渐近行为，因为这个小参数消失了。特别是，我们表明，最小化器表现出所谓的阶梯现象，即它们开发出一种微观结构，看起来像是在适当的尺度上的分段常数函数。我们的分析依赖于伽马连接的结果，用于重新验证的功能，爆破技术以及局部最小化器在极限问题上的表征。这种方法可以扩展到更一般的模型。
We consider the Perona-Malik functional in dimension one, namely an integral functional whose Lagrangian is convex-concave with respect to the derivative, with a convexification that is identically zero. We approximate and regularize the functional by adding a term that depends on second order derivatives multiplied by a small coefficient. We investigate the asymptotic behavior of minima and minimizers as this small parameter vanishes. In particular, we show that minimizers exhibit the so-called staircasing phenomenon, namely they develop a sort of microstructure that looks like a piecewise constant function at a suitable scale. Our analysis relies on Gamma-convergence results for a rescaled functional, blow-up techniques, and a characterization of local minimizers for the limit problem. This approach can be extended to more general models.
