We consider a shape optimization based method for finding the best interpolation data in the compression of images with noise. The aim is to reconstruct missing regions by means of minimizing a data fitting term in an $L^p$-norm between original images and their reconstructed counterparts using linear diffusion PDE-based inpainting. Reformulating the problem as a constrained optimization over sets (shapes), we derive the topological asymptotic expansion of the considered shape functionals with respect to the insertion of small ball (a single pixel) using the adjoint method. Based on the achieved distributed topological shape derivatives, we propose a numerical approach to determine the optimal set and present numerical experiments showing, the efficiency of our method. Numerical computations are presented that confirm the usefulness of our theoretical findings for PDE-based image compression.

翻译：我们考虑一种基于形状优化的方法，用于在含噪图像压缩中寻找最佳插值数据。其目标是通过最小化原始图像与基于线性扩散PDE修复的重建图像之间的$L^p$范数数据拟合项，来重建缺失区域。将问题重新表述为集合（形状）上的约束优化后，我们利用伴随方法推导了所考虑的形状泛函相对于插入小球（单个像素）的拓扑渐近展开。基于获得的分布式拓扑形状导数，我们提出了一种数值方法来确定最优集合，并通过数值实验展示了该方法的有效性。数值计算结果证实了我们的理论发现对于基于PDE的图像压缩的实用性。