Nonperiodic Sampling Theory, Deconvolution, and Wavelet Applications to Tomography

这个项目将研究抽样理论中的新结果在某些反卷积方程中的应用。例如，多年来一直知道，一个两个变量的函数可以完全从它在三个边长成对无理相关的正方形上的平均值中恢复出来。由于平方特征函数卷积是光电探测器阵列(例如电子相机)对图像进行遥感的标准模型，这一事实表明可以从放置在距图像三个不同距离的光电探测器阵列感测到的低分辨率图像中恢复高分辨率图像。主要研究人员和合作者的最新结果表明，对大小无理相关的规则网格的并集进行抽样的理论之间存在着很强的联系。这种非均匀抽样的几个方面有待调查。具体地说，(1)应用于高分辨率层析成像的高维采样和应用于旋转正方形上的平均值的去卷积，(2)用于频带受限于圆域的函数的采样，其应用于由具有圆形孔的光电传感器模糊的图像的去卷积，以及(3)当模糊函数是光滑样条线时，应用于去模糊的高阶采样。研究的第二个重点是小波在层析成像中的应用。主要研究人员和合作者已经证明，小波变换是分析CT的主要数学模型Radon变换的局部方面的有效理论和实用工具。主要研究人员建议通过研究从CT图像的基于小波的边缘分析中直接恢复图像中的边缘信息来继续这一分析。这将具有构建原始图像的准确边缘图像的优点，而不会损失任何重建或反投影方案所固有的SNR。小波的自然降噪特性也应该有助于降低测量CT数据中的噪声。第一个涉及到数字图像的超分辨率问题。任何数字黑白图像都由图像强度为固定灰度级的各个像素组成。在这种情况下，小于一个像素的比例的图像的特征对于成像设备是不可见的。例如，在纽约市的卫星图像中，像素大小对应于一个城市街区，就不可能区分给定街区中的不同建筑。所有的建筑(特征)将在该区块上模糊为单一的灰色阴影。这样的图像是假想的“完美图像”的模糊版本。我们的目标总是尽可能地接近这幅完美的图像。已经观察到，某些经典的数学结果可以解释为，理论上可以获取几个不同像素大小的模糊图像，并以这样一种方式组合它们，以准确地恢复完美的图像。完美的去模糊在任何现实系统中都是不可能的，但通过以特定的方式组合几个模糊图像来显著提高图像的分辨率是可能的。首席研究人员观察到，这里描述的去模糊问题实际上与研究得很好的信号和图像处理问题--从样本进行内插--密切相关，从而为这个项目做出了贡献。首席研究人员的工作表明，有可能提供非常简单的去模糊公式。目前的研究项目集中在寻找在计算机上实现这些公式的实用方案。本项目的第二个重点是小波分析技术在层析成像问题中的应用。断层成像“指的是像医用CAT扫描仪这样的设备，可以对物体内部形成非常锐利(比X射线锐利得多)的非侵入性图像。这种成像技术也被用于工业应用，如检测金属结构中的裂纹。主要研究人员建议直接从基于小波的图像边缘分析中恢复图像中的边缘数据。这将具有构建原始图像的准确边缘图像的优势，而不会像标准重建方案所固有的那样失真。

9971697WalnutThis project will study applications of new results in sampling theory to certain deconvolution equations. For example, it has been known for many years that a function of two variables can be recovered completely from its averages on three squares whose sidelengths are pairwise irrationally related. Since convolution by characteristic functions of squares is a standard model for remote sensing of images by arrays of photodetectors (e.g., electronic cameras), this fact suggests that a high-resolution image can be recovered from low-resolution images sensed by arrays of photodetectors placed at three distinct distances from the image. Recent results of the principal investigator and collaborators indicate a strong link between the theory of sampling on unions of regular grids of irrationally related sizes. Several aspects of this type of nonuniform sampling are to be investigated. Specifically, (1) higher dimensional sampling with applications to high resolution tomography and to deconvolution from averages over rotated squares, (2) sampling for functions bandlimited to circular domains with applications to deconvolution of images blurred by photosensors with circular apertures, and (3) higher order sampling with applications to deblurring when the blurring function is a smooth spline. A second focus of the research is in wavelet applications to tomography. The principal investigator and collaborators have shown that the wavelet transform is a valid theoretical and practical tool for analyzing local aspects of the Radon transform, the primary mathematical model in CT. The principal investigator proposes to continue this analysis by studying the recovery of edge information in images directly from a wavelet-based edge analysis of their CT images. This would have the advantage of constructing an accurate edge picture of the original image without the loss of SNR inherent in any reconstruction or backprojection scheme. The natural noise-reduction properties of wavelets should also help in reducing noise in the measured CT data.This research has two foci. The first is related to the problem of super-resolution of digital images. Any digital black-and-white image consists of individual pixels on which the image intensity is a fixed shade of gray. In this case, features of the image at scales smaller than one pixel in size are invisible to the imaging apparatus. For example, in a satellite image of New York City in which the pixel size corresponds to one city block, it will be impossible to distinguish different buildings in a given block. All of the buildings (features) will be blurred into a single shade of gray on that block. Such an image is a blurred version of a hypothetical "perfect image." The goal is always to get as close as possible to this perfect image. It has been observed that certain classical mathematical results can be interpreted as saying that it is theoretically possible to take several blurred images of different pixel sizes and combine them in such a way as to recover the perfect image exactly. Perfect deblurring is not possible in any realistic system but it is possible to significantly improve the resolution in an image by combining several blurred images in a specified way. The principal investigator has contributed to this program by observing that the deblurring problem here described is actually very closely related to the well-studied signal and image processing problem of interpolation from samples. The principal investigator's work has demonstrated that it is possible to provide very simple deblurring formulas. The current research project focuses on finding practical schemes to implement these formulas on computers. The second focus of this project is on applications of the technique of wavelet analysis to problems in tomographic imaging. The term tomographic imaging" refers to devices such as medical CAT scanners that form very sharp (much sharper than x-rays) non-invasive images of the inside of an object. Such imaging techniques are also used in industrial applications such as detection of cracks in metal structures. The principal investigator proposes to study the recovery of edge data in images directly from a wavelet-based edge analysis of their tomographic data. This would have the advantage of constructing an accurate edge picture of the original image without the distortion inherent in standard reconstruction schemes.