Multiscale Signal and Image Processing using Singularity Grammars

许多真实世界的信号和图像都是自然地以多尺度方式处理的，就像我们离开一个物体来查看它的大致外观并放大来查看它的细节一样。小波变换的多尺度数据表示自然地匹配重要的信号和图像奇异性结构，例如边缘、脊线、突发、纹理和其他瞬变。这个项目的重点是用小波来刻画这些关键结构。由此产生的新的信号和图像处理算法将影响AdiVerse的一系列重要问题，从下一代互联网中计算机网络流量的管理到对复杂数据(如同时包含文本和图像的文档)的无缝处理。由于小波是基础，它们可以复制任意函数，从高度结构化的现实世界信号和图像到完全非结构化的噪音。用语言学的术语来说，小波词汇可能太有表现力了。为了对真实世界的信号和图像进行有用的建模，词汇表的范围必须缩小，方法是施加一组约束--一种语法--来捕捉奇点的显著结构。虽然许多研究都集中在开发新的小波词汇上，但当前基于小波的处理面临的最大挑战在于语法建模。这个项目研究了一种新的多尺度建模的统计方法，旨在匹配奇异性丰富的数据。正在考虑的语法包括有效和可训练的概率图(隐马尔可夫树模型)和乘法金字塔级联。新的语法设计是通过使用贝索夫空间和多重分形分析对奇点的精细结构进行详细研究来指导的。在这些新模型的基础上，出现了用于信号和图像估计、检测/分类、分割、融合和合成的新算法。

Many real-world signals and images are naturally processed in a multiscale fashion, just as we move away from an object to view its general appearance and zoom in to see its fine details. The multiscale data representation of the wavelet transform naturally matches important signal and image singularity structures such as edges, ridges, bursts, textures, and other transients. This project focuses on characterizing these key structures using wavelets. The resulting new signal and image processing algorithms will impact adiverse range of important problems, from the management of computer network traffic in the next-generation Internet to seamless handling of complicated data such as documents containing both text and images.Since wavelets form a basis, they can reproduce arbitrary functions, from highly structured real-world signals and images to completely unstructured noise. In linguistic terms, the wavelet vocabulary canbe too expressive. To perform useful modeling of real-world signals and images, the vocabulary's scope must be narrowed by imposing a set of constraints -- a grammar -- that capture the salient structures ofsingularities. While much research has concentrated on developing new wavelet vocabularies, the most important current challenges to wavelet-based processing lie in grammatical modeling. This projectstudies a new statistical approach to multiscale modeling designed to match singularity-rich data. Grammars under consideration include efficient and trainable probabilistic graphs (hidden Markov tree models) and multiplicative pyramidal cascades. New grammar design is being guided by a detailed study of the fine structure of singularities using Besov spaces and multifractal analysis. From these new models follow new algorithms for signal and image estimation, detection/classification, segmentation, fusion, and synthesis.