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

小行星9500909 该项目包括两个部分。 首先，研究者提出申请 新的成果，在采样理论做高分辨率信号处理， 组合给定信号的几个低分辨率版本。这是 Berenstein和其他人观察到， 问题（称为多传感器反卷积）是唯一可解的，但不适定，并且难以数值求解。调查员发现 正则格并上的Shannon型抽样公式 不相称的密度，它提供了简单的解决方案， 特殊情况下的多通道反卷积问题。 以下 1）通过CT进行的锐化测量 扫描仪通过结合测量从几个低分辨率 探测器 这可能最终导致廉价的测量设备， 分辨率可媲美昂贵的高分辨率机器。2)一个最近 一种提高工业层析成像分辨率的方案 这个问题涉及到对不相称的规则网格的联合进行采样 尺寸 调查员的采样结果可能会导致一个完整的解决方案 解决这个问题3)需要新技术来实现“超分辨率”， 电子成像，因为像素分辨率的理论极限很快被 被达到。 几个低分辨率的高分辨率反褶积 图像是解决这个问题的自然方法。 第二，调查人员和 其他人已经成功地表明，小波是一种自然的工具， 从局部Radon变换数据恢复图像的边缘特征。 的 研究者建议开发这些局部算法， 设想的目标：1）找到恢复局部密度和边缘的方法 图像的特征，并开发与现有算法竞争的算法。 局部层析成像算法和2）识别图像的波前集 使用小波，并最终将技术应用于 在稀薄氡 变换 该项目的第一部分涉及提高分辨率， 例如电子照相机的遥感装置。 例如，假设， CCD相机的分辨率为1毫米，也就是说，它可以 在一个侧面上至少有1 mm的场景特征。 特征 比它小的物体会被模糊到周围的环境中贝伦斯坦和其他人 观察到一些经典数学导致了一种可能的方法， 无需设计昂贵的高分辨率设备。 如果一个 用和第一次一样的相机拍了几张现场的照片， 它的分辨率比第一个稍差（这也可能是 通过重新定位相同的相机来实现），那么理论上可以 将原始场景的特征恢复到任意分辨率。 在实际操作中， 任意的解决方案是不可能的，但似乎真实的增加， 可以实现分辨率。 调查人员制定了一种方法， 这个问题从抽样理论的角度来看。 使用此 方法，研究人员已经能够产生数值稳定 一维模型问题的分辨率提高了20倍。 还需要做更多的工作，但设想了以下应用：1） 通过组合CT扫描仪产生高分辨率测量 从几个低分辨率扫描仪测量和2）增加 直接在特定工业层析成像问题中解决 研究者新的采样结果的应用。 的第二部分 该项目涉及“局部”CT扫描。 在现有的CT扫描仪中， 患者身体的整个切片必须暴露于辐射，即使 医生只关注一个小区域。研究者 建议使用一种新的强大的信号处理技术， 小波找到有效的算法获得一个小面积的图像 同时仅将感兴趣的区域暴露于辐射。是 希望这些技术能与现有的所谓的当地技术竞争， 层析成像算法

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.