Mathematical Sciences: New Results in Sampling and Wavelet Applications in Tomography

9500909核桃本项目由两部分组成。首先，研究者建议将采样理论中的新结果应用于高分辨率信号处理，通过结合给定信号的几个低分辨率版本。Berenstein等人观察到，这个问题的数学模型（称为多传感器反褶积）是唯一可解的，但不适定式，难以在数值上求解。研究者发现了密度不相等的规则格并的香农型抽样公式，为特殊情况下的多通道反褶积问题提供了简单的解。设想以下应用：1)结合几个低分辨率探测器的测量结果，通过CT扫描仪进行锐化测量。这可能最终导致廉价的测量设备的分辨率可与昂贵的高分辨率机器相媲美。最近提出的一种提高工业层析成像问题分辨率的方案涉及对大小不相称的规则网格的联合进行采样。调查人员的抽样结果可以彻底解决这个问题。3)电子成像的“超分辨率”需要新技术，因为像素分辨率的理论极限正在迅速达到。多幅低分辨率图像的高分辨率反卷积是解决该问题的一种自然方法。其次，研究者和其他人已经成功地证明了小波是从局部Radon变换数据中恢复图像边缘特征的天然工具。研究者提出开发这些局部算法的目标如下：1)找到恢复局部密度和图像边缘特征的方法，并开发一种与现有局部层析成像算法竞争的算法；2)使用小波识别图像的波前集，并最终将这些技术应用于弱Radon变换。该项目的第一部分涉及提高电子相机等遥感设备的分辨率。例如，假设CCD相机的分辨率为1毫米，也就是说，它可以区分场景的一侧至少为1毫米的特征。比这更小的特征被模糊到周围的环境中。贝伦斯坦和其他人观察到，一些经典的数学方法可以在不设计昂贵的高分辨率设备的情况下提高分辨率。如果有人用与第一张相同的相机拍摄了几张场景图像，但分辨率略低于第一张（这也可以通过重新定位相同的相机来实现），那么理论上他可以恢复原始场景的特征到任意分辨率。在实践中，任意分辨率是不可能的，但似乎可以实现分辨率的实际增加。研究者从抽样理论的角度阐述了解决这个问题的方法。使用这种方法，研究者已经能够在一维模型问题中产生数值稳定的二十倍分辨率增加。虽然还需要做更多的工作，但可以设想以下应用：1)通过结合几个低分辨率扫描仪的测量结果，从CT扫描仪产生高分辨率测量结果；2)通过直接应用研究者的新采样结果，提高特定工业断层扫描问题的分辨率。该项目的第二部分涉及“局部”CT扫描。在现有的CT扫描仪中，即使医生只对一小部分感兴趣，病人身体的整个切片也必须暴露在辐射下。研究人员建议使用一种新的强大的信号处理技术，称为小波，以找到有效的算法，在只将感兴趣的区域暴露于辐射的情况下获得患者身体一小部分的图像。希望这些技术将与现有的所谓的局部断层扫描算法竞争。

9500909 Walnut The project consists of two parts. First, the investigator proposes to apply new results in sampling theory to do high-resolution signal processing by combining several low-resolution versions of a given signal. It was observed by Berenstein and others that a mathematical model of this problem (called multisensor deconvolution) is uniquely solvable but ill- posed, and difficult to solve numerically. The investigator has found Shannon-type sampling formulas on unions of regular lattices with incommensurate densities, which provide simple solutions to the multichannel deconvolution problem in special cases. The following applications are envisioned: 1) Sharpening measurements taken by CT scanners by combining measurements from several low-resolution detectors. This could ultimately lead to inexpensive measuring devices with resolution comparable to expensive high-resolution machines. 2) A recently proposed scheme to increase resolution in an industrial tomography problem involves sampling on unions of regular grids of incommensurate size. The investigator's sampling results could lead to a complete solution to this problem. 3) New techniques are required to do "superresolution" in electronic imaging since the theoretical limit of pixel resolution is rapidly being reached. Deconvolution at high-resolution of several low-resolution images is a natural approach to this problem. Second, the investigator and others have successfully shown that wavelets are a natural tool for recovering edge features of an image from local Radon transform data. The investigator proposes to develop these local algorithms with the following goals envisioned: 1) Find ways to recover locally density as well as edge features of an image, and develop an algorithm competitive with existing local tomography algorithms and 2) Identify the wavefront set of an image using wavelets, and ultimately apply the techniques to the at tenuated Radon transform. The first part of the project is concerned with increasing the resolution of remote sensing devices such as electronic cameras. Suppose, for example, that a CCD camera has a resolution of 1 millimeter, that is, it can distinguish features of a scene that are at least 1 mm on a side. Features smaller than that are blurred into their surroundings. Berenstein and others observed that some classical mathematics led to a possible way to increase resolution without designing an expensive high-resolution device. If one took several images of the scene with cameras identical to the first but which had slightly poorer resolutions than the first (this could also be achieved by repositioning the same camera), then in theory one could recover features of the original scene to arbitrary resolution. In practice, arbitrary resolution is not possible, but it seems that real increases in resolution can be achieved. The investigator has formulated an approach to this problem from the point of view of sampling theory. Using this approach, the investigator has been able to produce numerically stable twenty-fold increases in resolution in a one-dimensional model problem. More work is required, but the following applications are envisioned: 1) Producing high-resolution measurements from CT scanners by combining the measurements from several low-resolution scanners and 2) Increasing the resolution in a specific industrial tomography problem by a direct application of the investigator's new sampling results. The second part of the project is concerned with "local" CT scans. In existing CT scanners, an entire slice of a patient's body must be exposed to radiation even if the doctor is interested in looking at only a small region. The investigator proposes to use a new and powerful signal processing technique called wavelets to find efficient algorithms for obtaining an image of a small area of a patient's body while only exposing the area of interest to radiation. It is hoped that these techniques will be competitive with existing so-called local tomography algorithms.