Multiscale Methods for Signal Processing

我们提出了一个多尺度信号处理方法的研究计划。我的计划由以下两个互补的组成部分组成： 分析：自适应多尺度正交（AMO）信号和图像分析的基础。 合成：用于信号和图像合成的基于模式的仿真方法。 小波在信号处理中的成功是基于小波有效地表示分段规则信号的能力。对于这样的信号，离散小波变换表示是稀疏的，在这个意义上，许多变换系数小到足以被忽略。我们一直在开发新的AMO基地，提供非常稀疏的信号表示，稀疏比小波分段规则信号。我们将发展和完善AMO基地，使他们可以有效地用于各种各样的信号和图像，并重新审视小波与AMO基地的主要应用。这项工作应该对许多应用产生重大影响，例如医学成像（疾病诊断，筛查协议），医学图像（MRI和超声）的压缩和降噪。 自然资源（如石油、水、矿石）开采的运营决策是由资源空间分布的随机建模指导的。在这方面，确定自然资源复杂空间分布的特点和建立模型的能力对于自然资源的评估和开发至关重要。多点统计量的正确描述对于正确模拟油藏中的通道和裂缝等连通结构尤为重要。我们一直在开发一种模拟方法，产生的图像，在许多视觉和定量方面，即形态和连接，是上级竞争多点模拟方法。我们的方法考虑到条件数据，即已知的地质在几个位置，和多点统计，即复杂的图案几何形状。我们建议进一步发展我们的方法，考虑到地球物理测量所定义的约束。

We propose a research program on multiscale methods for signal processing. My program is composed of the following two complementary components: Analysis: Adaptive Multiscale Orthogonal (AMO) bases for signal and image analysis. Synthesis: Patterns-based Simulation methods for signal and image synthesis. The success of wavelets in signal processing is based on the ability of wavelets to efficiently represent piecewise regular signals. For such signals, the discrete wavelet-transform representation is sparse in the sense that many transform coefficients are small enough to be neglected. We have been developing new AMO bases that provide exceptionally sparse representations for a signal, much sparser than wavelets for piecewise regular signals. We will develop and refine AMO bases so that they can be used effectively for a wide variety of signals and images, and revisit the main applications of wavelets with AMO bases. This work should have a major impact on a multitude of applications, e.g. medical imaging (diagnosis of disease, screening protocols), compression and denoising of medical images (MRI and ultrasound). Operational decisions made for the exploitation of natural resources (e.g, oil, water, ore) are guided by a stochastic modeling of the resource spatial distribution. In this context, the ability to characterize and model the complex spatial distribution of natural resources is essential for their assessment and exploitation. A correct description of multipoint statistics is particularly important to correctly simulate connected structures such as channels and fractures in oil reservoirs. We have been developing a simulation method that produces images that in many visual and quantitative aspects, i.e. morphology and connectivity, are superior to competing multipoint simulation approaches. Our method takes into account conditioning data, i.e. known geology at several locations, and multipoint statistics, i.e. complex patterns geometry. We propose to develop further our method to take into account constraints defined by geophysical measurements.