Theory of Nonlinear Subdivision and Multiscale Transforms

在过去的25年里，快速多尺度算法，如金字塔变换和细分方法导致了巨大的成功，在数据和几何处理，并在一般的科学计算。虽然线性多尺度分析是在一个成熟的状态，没有那么多的是已知的非线性情况下。非线性是自然产生的，例如在数据自适应算法中，在图像和几何处理中，鲁棒去噪，或者由于分析对象本身的非线性约束，这些约束需要在所有尺度上保留。该项目旨在为这种方案的收敛性，极限平滑性和稳定性的核心问题开发一致的理论。重点是稳定性问题，这是相关的非线性数据和几何压缩算法。单变量和更复杂的多变量理论都将受到攻击。工作将包括实际相关方案的案例研究，如用于高效几何处理的正常多分辨率。案例研究也被认为是重要的，以帮助形成非线性多尺度方法的最终理论。

Over the past 25 years, fast multiscale algorithms such as pyramid transforms and subdivision methods lead to tremendous successes in data and geometry processing, and in scientific computing in general. While linear multiscale analysis is in a mature state, not so much is known in the nonlinear case. Nonlinearity arises naturally, e.g. in data-adaptive algorithms, in image and geometry processing, robust denoising, or due to nonlinear constraints on the analyzed objects themselves that need to be preserved on all scales.The project aims at developing a consistent theory for the core questions of convergence, limit smoothness, and stability of such a scheme. Focus is on the stability problem which is relevant for nonlinear data and geometry compression algorithms. Both the univariate and the more complicated multivariate theory will be attacked. Work will include case studies for practically relevant schemes, such as normal multiresolution used for efficient geometry processing. Case studies are also considered important to help shape the final theory on nonlinear multiscale methods.