CIF: Small:Sparse and Geometric Representations of Images and Multidimensional Signals

摘要：自然图像和多维信号最基本的特征之一是由于平滑区域之间的平滑边界而存在几何规律性。这类信号的当前表示，如曲波、轮廓波、剪切波和表面波，都是在频域中构造的，这导致了具有大空间支持和吉布斯振荡的基函数。本文研究了具有几何规律的多维信号的稀疏表示。这些表示允许从粗到细的连续逼近，并且是数字友好的。研究人员将重点放在基于真多维和几何提升方案的空间域构建上，这将导致新一代几何小波。此外，还探讨了低相干性的几何平铺字典。该研究的目的是在连续域和离散域之间建立精确的联系，连续域具有几何规律性，离散域在采样网格上定义输入信号和变换。最后，该研究开发了新的处理算法，利用获得的几何规则信号的知识并开发了稀疏表示。

Project: Sparse and Geometric Representations of Images and Multidimensional SignalsAbstract:One of the most fundamental characteristics of natural images and multidimensional signals is the presence of geometric regularities due to smooth boundaries between smooth regions. Current representations for this class of signals such as curvelets, contourlets, shearlets, and surfacelets, are constructed in the frequency domain, which lead to basis functions with large spatial support and Gibbs oscillations. This research develops new sparse representations for multidimensional signals with geometric regularities. These representations allow successive approximation from coarse to fine and will be digital friendly.The investigators focus on spatial domain constructions based on true multidimensional and geometric lifting schemes that would lead to a new generation of geometric wavelets. In addition, geometric tiling dictionaries with low coherence are explored. The research aims for a precise connection between the continuous domain, where geometric regularity is characterized, and the discrete domain, where input signals and transforms are defined on sampling grids. Finally, the research develops new processing algorithms that exploit the gained knowledge of geometrically regular signals and developed sparse representations.