Nonperiodic Sampling Theory, Deconvolution, and Wavelet Applications to Tomography

9971697核桃本项目将研究应用抽样理论的新成果，以某些反卷积方程。 例如，多年来人们已经知道，一个二元函数可以完全从它在三个平方上的平均值中恢复出来，这些平方的边长是两两无理相关的。由于平方的特征函数的卷积是用于通过光电探测器阵列（例如，电子照相机），这一事实表明可以从由放置在距图像三个不同距离处的光电探测器阵列感测的低分辨率图像中恢复高分辨率图像。 最近的主要研究者和合作者的结果表明，强有力的联系理论之间的抽样工会的规则网格的不合理的相关大小。 这种类型的非均匀采样的几个方面进行了研究。 具体而言，（1）应用于高分辨率层析成像和从旋转平方上的平均值进行反卷积的高维采样，（2）应用于由具有圆形孔径的光电传感器模糊的图像的反卷积的带限到圆形域的函数的采样，以及（3）当模糊函数是平滑样条时应用于去模糊的高阶采样。研究的第二个重点是在小波层析成像的应用。主要研究者和合作者已经表明，小波变换是一个有效的理论和实践工具，用于分析局部方面的Radon变换，在CT中的主要数学模型。 主要研究者建议继续这种分析，通过研究直接从基于小波的CT图像边缘分析中恢复图像中的边缘信息。 这将具有构造原始图像的精确边缘图片而不损失任何重建或反投影方案中固有的SNR的优点。 小波的自然降噪特性也有助于降低CT测量数据中的噪声。 第一个问题涉及数字图像的超分辨率问题。 任何数字黑白白色图像都由单个像素组成，其图像强度是固定的灰色阴影。在这种情况下，在尺寸上小于一个像素的尺度上的图像的特征对于成像装置是不可见的。 例如，在像素大小对应于一个城市街区的纽约市的卫星图像中，将不可能区分给定街区中的不同建筑物。 所有的建筑物（特征）将被模糊成该块上的单一灰色阴影。这样的图像是一个模糊的版本的一个假设的“完美的图像。“我们的目标总是尽可能接近这个完美的形象。 已经观察到，某些经典的数学结果可以解释为，理论上可以拍摄不同像素大小的几个模糊图像，并以精确地恢复完美图像的方式将它们联合收割机组合。 完美的去模糊在任何现实系统中都是不可能的，但是通过以指定的方式组合多个模糊图像来显著提高图像的分辨率是可能的。 首席研究员通过观察到这里描述的去模糊问题实际上与经过充分研究的样本插值信号和图像处理问题密切相关，从而为该计划做出了贡献。 首席研究员的工作已经证明，有可能提供非常简单的去模糊公式。 目前的研究项目的重点是寻找实用的方案，在计算机上实现这些公式。本计画之第二个焦点是小波分析技术在层析成像问题上的应用。 术语“断层成像”是指诸如医用CAT扫描仪之类的设备，其形成物体内部的非常清晰（比X射线清晰得多）的非侵入性图像。 这种成像技术也用于工业应用，例如检测金属结构中的裂纹。 主要研究者建议研究直接从基于小波的断层数据边缘分析中恢复图像中的边缘数据。 这将具有构造原始图像的精确边缘图像而没有标准重建方案中固有的失真的优点。

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.