Multidimensional Signal Processing in the Presence of Homeomorphic Deformations

从信号的变形形式估计信号的需求出现在广泛的工程问题中。这种变形可能是由于固有的物理效应或由于测量过程造成的。例如，考虑从不同的视点和距离拍摄的单个对象的两幅图像。这些图像是彼此的变形版本。任何比较这些图像的尝试都需要转换其中一个或两个，以便图像中的相应元素显示在相同的坐标上。在一维情况下，考虑在存在(可能是时变的)延迟和多普勒的情况下测量的信号的情况，其导致时间轴的失真和信号形状的相应变形。这些问题的难点在于它们的非线性和高维性质，本研究为一类信号变形问题提供了基本的数学解决方案。该解决方案涉及可能的变换族的特定参数模型(或者更确切地说，它们的逆)。利用该模型，将原问题转化为低维空间中线性方程组形式的等价问题。换言之，该方法将一个计算量较大的估计问题转化为变形模型未知参数中的一组线性方程。在仿射变换的情况下，得到了变形参数的精确闭合解。在更一般的情况下，同胚变形的解是近似的，但可以达到高精度。研究的重点是评估该方法在存在噪声和其他不确定性的情况下的性能，以及该算法所使用的某些基函数的优化设计。这项研究应用于广泛的图像和视频处理系统，特别是医学成像系统和基于视频/图像的安全系统。%由于预算和执行期大大减少，将不再详细介绍该技术的具体应用。

The need to estimate signals from their deformed versions arises in a wide range of engineering problems. Such deformations may be due to inherent physical effects or due to the measurement process. As an example consider two images of a single object taken from different viewpoints and distances. These images are deformed versions of each other. Any attempt to compare these images requires transforming one or both so that corresponding elements in the images appear at the same coordinates. In the one-dimensional case consider the case of a signal measured in the presence of (possibly time-varying) delay and Doppler which causes a distortion of the time axis and a corresponding deformation of the signal shape. The difficulty of these problems is their non-linear and high-dimensional nature.This research provides a fundamental mathematical solution to a class of signal deformation problems. The solution involves a certain parametric model of the family of possible transformations (or rather, their inverses). Using this model the original problem is transformed into an equivalent problem expressed in the form of a linear system of equations in the low dimensional space. In other words, the method converts a computationally prohibitive estimation problem into a set of linear equations in the unknown parameters of the deformation model. In the case of affine transforms an exact closed form solution of the deformation parameters is obtained. In the more general case of homeomorphic deformations the solution is approximate, but high accuracy is achievable. The focus of the research is on evaluating the performance of this approach in the presence of noise and other uncertainties, and the optimal design of certain basis functions used by the algorithm. This research has applications to a wide range of image and video processing systems, in particular to medical imaging systems, and video/image based security systems.%%%%%%%%%%%%%%%%%%%%%%%%%% 3. A level of effort statement.At the recommended level of support, the PI will make every effort to address all of the issues related to the basic operation of the proposed technique as described in the proposal, with emphasis on its performance in the presence of noise and various uncertainties, and the optimal design of the basis functions. Because of the greatly reduced budget and period of performance the specific applications of the technique will not be addressed in any detail.