优化空间和色调数据以实现基于PDE的修复
Optimising Spatial and Tonal Data for PDE-based Inpainting
摘要：
最近的一些有损信号和图像压缩方法只存储少数选定的像素，并通过用偏微分方程（PDE）修复来填充丢失的结构。合适的算子包括拉普拉斯算子、双调和算子和边缘增强各向异性扩散（EED）。这种方法的质量在很大程度上取决于所保存的数据的选择。优化这些数据的域和codomain引起了具有挑战性的数学问题，应在我们的工作中解决。在1D的情况下，我们证明的结果，提供洞察这个问题的难度，我们给出的证据表明，分裂成空间和色调（即函数值）优化几乎不会恶化的结果。在2D设置中，我们提出了通用算法，即使指定的数据是非常稀疏的，实现高重建质量。为了优化空间数据，我们使用概率稀疏化，然后进行非局部像素交换，以避免陷入糟糕的局部最优值。在此空间优化之后，我们执行修改函数值的色调优化，以减少全局重建误差。对于均匀扩散修复，这归结为一个最小二乘问题，我们证明它有一个唯一的解决方案。我们证明，它可以有效地找到一个梯度下降的方法，加速快速显式扩散（FED）周期。我们的框架允许指定所需的修补掩模的先验密度。此外，对于稀疏修复问题，它比其他数据优化方法更通用，因为它也可以扩展到EED等非线性修复运算符。这被用来实现具有最先进质量的重建。我们还对基于偏微分方程的图像压缩方法进行了广泛的文献调查。
英文摘要：
Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. We also give an extensive literature survey on PDE-based image compression methods.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
