Mathematical Sciences: Computed Tomography and Sampling

9404436法里达尼层析成像根据其大量的线积分重建广义密度函数f。除了最初在医学成像中的应用外，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；单点重建的标准公式需要在包含该点的某个平面内的所有直线上进行测量。局部层析成像产生与f相关的函数的线性组合的重建。这种重建是局部的；对于一个点的重建，只需要沿着非常接近该点的线进行衰减测量。这使得在不必扫描整个对象的情况下重建大对象内的感兴趣区域成为可能。术语计算机断层摄影术描述了一类用于成像不透明物体内部的方法。这些方法的特点是通过大量的间接测量来计算图像。在数学上，图像表示某个密度函数，而测量值是该函数的线积分。一个广为人知的例子是CAT扫描。这里的密度函数是X射线吸收系数，测量是通过记录穿过患者的细小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。这项研究涉及断层成像在医学成像(为冠状动脉树成像)、生物学研究(使用新兴的高分辨率Micro-CT扫描仪对组织的精细结构进行成像)和量子光学等领域的应用。这项研究项目的一部分涉及“局部层析成像”技术。普通的断层扫描需要扫描整个物体，而局部断层扫描只允许扫描感兴趣的区域。局部层析成像具有实用价值，因为它需要更便宜的扫描设备，减少X射线曝光，并使成像太大而不能完整扫描的物体成为可能。目前，局部层析成像允许人们识别不同特征的形状，但不能产生正确的密度差异。其目标是开发一种允许重建密度差异的方法。这将进一步增加局部层析成像在许多应用中的有用性。此外，将与梅奥诊所的研究人员合作，研究三维局部断层成像中的最佳采样、分辨率和重建算法问题，他们正在建造一台高分辨率的Micro-CT扫描仪。本研究的第二部分继续研究香农采样理论及其在计算机层析成像中的应用。香农采样理论在信号处理中具有重要意义。对非等距周期采样集上的非带限函数采样的混叠误差进行了研究，并研究了更一般类型的采样集。结果将应用于计算机断层扫描，从而产生误差估计和新的有效抽样方案。由此获得的对最佳采样和重建算法的深入了解将使我们能够充分利用当前层析扫描仪的能力，并对未来设备的设计产生影响。

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.