论文标题
图像分割的Ambrosio-Tortorelli近似方案的$γ$ - 融合
$Γ$-Convergence of an Ambrosio-Tortorelli approximation scheme for image segmentation
论文作者
论文摘要
给定图像$ u_0 $，最小化mumford-shah功能的目的是将图像域的分解为子域，而分段平滑近似$ u $ $ u $ u_0 $，以使$ u $ $ u $在每个子域中都平稳变化。由于Mumford-Shah功能是高度平滑的，因此可以考虑诸如Ambrosio tortorelli近似等正规化，这是Mumford-Shah函数用于图像分割的最有效效率的近似之一。虽然在最小化功能时，在大量应用中已经取得了非常令人印象深刻的数值结果，但目前在分段平滑设置中最小化功能的分析结果，这是这项工作的目标。我们的主要结果是Mumford-Shah功能的Ambrosio-Tortorelli近似值的$γ$ - 可用于分段光滑近似。这需要引入适当的功能空间。由于我们的$γ$ convergence结果，我们可以推断相应功能的最小化器的收敛性。
Given an image $u_0$, the aim of minimising the Mumford-Shah functional is to find a decomposition of the image domain into sub-domains and a piecewise smooth approximation $u$ of $u_0$ such that $u$ varies smoothly within each sub-domain. Since the Mumford-Shah functional is highly non-smooth, regularizations such as the Ambrosio-Tortorelli approximation can be considered which is one of the most computationally efficient approximations of the Mumford-Shah functional for image segmentation. While very impressive numerical results have been achieved in a large range of applications when minimising the functional, no analytical results are currently available for minimizers of the functional in the piecewise smooth setting, and this is the goal of this work. Our main result is the $Γ$-convergence of the Ambrosio-Tortorelli approximation of the Mumford-Shah functional for piecewise smooth approximations. This requires the introduction of an appropriate function space. As a consequence of our $Γ$-convergence result, we can infer the convergence of minimizers of the respective functionals.
