Mathematical Sciences: Computed Tomography and Sampling

小行星9404436 层析成像从大量的线积分中重建广义密度函数f。 除了最初用于医学成像外，它还发现了越来越多的科学和工业应用。 普通的层析成像不是局部的;在单个点重建的标准公式需要在包含该点的某个平面内的所有线上进行测量。 局部层析成像产生与f相关的函数的线性组合的重建。 该重建是局部的;对于在一点处的重建，仅需要沿非常接近该点的线沿着进行衰减测量。 这使得有可能在大对象内重建感兴趣区域，而不必扫描整个对象。 术语计算机断层扫描描述了一类用于对不透明物体的内部进行成像的方法。 这些方法的特点是，图像是从大量的间接测量计算。 在数学上，图像表示某个密度函数，测量值是该函数的线积分。 一个著名的例子是CAT扫描。 这里的密度函数是X射线吸收系数，测量值是通过记录穿过患者的细X射线束的衰减来获得的。 除了最初在医学成像中的应用外，计算机断层扫描还发现了越来越多的科学和工业应用。 该研究涉及断层扫描在医学成像（冠状动脉树成像），生物研究（使用高分辨率Micro-CT扫描仪的新兴技术对组织的精细结构进行成像）和量子光学等领域的应用。 该研究项目的一部分涉及“局部断层摄影”技术。 虽然普通断层扫描需要扫描整个对象，但局部断层扫描允许仅扫描感兴趣的区域。 局部层析成像具有实际意义，因为它需要更便宜的扫描设备，减少X射线曝光，并且可以对太大而不能整体扫描的对象进行成像。 目前，局部层析成像允许识别不同特征的形状，但不能产生正确的密度差异。 我们的目标是开发一种方法，允许重建的密度差异。 这将进一步增加局部层析成像对于许多应用的有用性。此外，三维局部断层扫描的最佳采样、分辨率和重建算法问题将与正在建造高分辨率Micro-CT扫描仪的马约诊所的研究人员合作进行研究。 本研究的第二部分继续工作在香农采样理论及其应用计算机断层扫描。 Shannon采样理论在信号处理中具有重要意义。 本文将研究在非等距但周期性的采样集上对非带限函数进行采样的混叠误差，并研究更一般的采样集类。 结果将被应用到计算机断层扫描，导致误差估计和新的有效的采样方案。 因此获得的最佳采样和重建算法的更深入的理解将允许充分利用当前的断层扫描仪的能力，并对未来设备的设计有影响。

9404436 Faridani Tomography produces the reconstruction of a generalized density function f from a large number of its line integrals. Beyond its initial use in medical imaging, it has found a growing number of scientific and industrial applications. Ordinary tomography is not local; the standard formulas for reconstruction at a single point require measurements at all lines within some plane containing the point. Local tomography produces the reconstruction of a linear combination of functions related to f. This reconstruction is local; for reconstruction at a point, attenuation measurements are needed only along lines very close to that point. This makes it possible to reconstruct a region of interest within a large object without having to scan the whole object. The term computed tomography describes a class of methods for imaging the interior of opaque objects. Characteristic of these methods is that the image is computed from a large number of indirect measurements. Mathematically, the image represents some density function, and the measurements are line integrals of this function. A well-known example is CAT scans. Here the density function is the x-ray absorption coefficient, and the measurements are obtained by recording the attenuation of thin x-ray beams traversing the patient. Beyond its original application in medical imaging, computed tomography has found a growing number of scientific and industrial applications. The research involves applications of tomography in areas such as medical imaging (imaging the coronary arterial tree), biological research (imaging the fine structure of tissue using the emerging technology of high-resolution Micro-CT scanners), and quantum optics. Part of this research project concerns the technique of `local tomography'. While ordinary tomography requires scanning the whole object, local tomography allows for scanning only a region of interest. Local tomography is of practical interest sin ce it requires less expensive scanning equipment, reduces x-ray exposure, and makes it possible to image objects too large to be scanned in their entirety. At present, local tomography allows one to identify the shapes of different features, but does not produce the correct density differences. The goal is to develop a method which allows reconstruction of density differences. This will further increase the usefulness of local tomography for many applications. Furthermore, questions of optimal sampling, resolution, and reconstruction algorithms in three-dimensional local tomography will be investigated in collaboration with researchers at the Mayo Clinic, who are building a high-resolution Micro-CT scanner. The second part of this research continues work in Shannon sampling theory and its application to computed tomography. Shannon sampling theory is of fundamental importance in signal processing. The aliasing error for sampling non-bandlimited functions on non-equidistant but periodic sampling sets will be investigated, and more general classes of sampling sets will be studied. The results will be applied to computed tomography, resulting in error estimates and new efficient sampling schemes. The deeper understanding of optimal sampling and reconstruction algorithms thus obtained will allow to fully exploit the capabilities of current tomographic scanners and have implications for the design of future equipment.