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 $\Gamma$-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 $\Gamma$-convergence result, we can infer the convergence of minimizers of the respective functionals.

翻译：给定图像 $u_0$，极小化 Mumford-Shah 泛函的目标是找到图像域的子域分解以及 $u_0$ 的一个分片光滑近似 $u$，使得 $u$ 在每个子域内光滑变化。由于 Mumford-Shah 泛函是高度非光滑的，可以考虑诸如 Ambrosio-Tortorelli 逼近等正则化方法，该方法是图像分割中计算效率最高的 Mumford-Shah 泛函逼近之一。尽管在极小化该泛函时，大量应用中已取得令人印象深刻的数值结果，但目前对于分片光滑情形下该泛函极小元尚无解析结果，而本文旨在填补这一空白。我们的主要结果是分片光滑近似下 Mumford-Shah 泛函的 Ambrosio-Tortorelli 逼近的 Γ-收敛性。这需要引入合适的函数空间。作为我们 Γ-收敛性结果的推论，我们可以推断出相应泛函极小元的收敛性。