基于 PDE 压缩有噪音图像的 PDE 图像的最佳内插数据 (Optimal Interpolation Data for PDE-based Compression of Images with Noise)

We introduce and discuss shape-based models 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 the $L^2$-norm between the images and their reconstructed counterparts using time-dependent PDE inpainting. We analyze the proposed models in the framework of the $\Gamma$-convergence from two different points of view. First, we consider a continuous stationary PDE model, obtained by focusing on the first iteration of the discretized time-dependent PDE, and get pointwise information on the "relevance" of each pixel by a topological asymptotic method. Second, we introduce a finite dimensional setting of the continuous model based on "fat pixels" (balls with positive radius), and we study by $\Gamma$-convergence the asymptotics when the radius vanishes. Numerical computations are presented that confirm the usefulness of our theoretical findings for non-stationary PDE-based image compression.

翻译：我们引入并讨论在以噪音压缩图像时寻找最佳内插数据的基于形状的模型。目的是通过使用基于时间的 PDE 油漆来尽量减少图像与其重建对等方之间在$L$2$- 诺姆中的数据匹配术语, 从而重建缺失的区域。我们从两个不同的角度分析在$\Gamma$- converggence框架内提议的模型。首先, 我们考虑一个连续的固定的 PDE 模型, 以离散的、取决于时间的 PDE 第一次迭代为焦点, 并获得关于每个像素的“ 相对比值” 的点性信息。其次, 我们引入一个基于“ 脂肪像素( 正半径的球) ” 的连续模型的有限维度设置, 我们用 $\Gamma$- convergence 来研究半径消失时的静态。数字计算证实了我们理论结果对于非静止的 PDE 图像压缩的实用性。