Nonclassical PDO and Some Practical Problems of Local Tomography

伪微分算子理论（PDO）是一个强大的工具，已成功地应用于数学的各个领域，特别是断层摄影。在经典理论中，pdo被假定为具有无穷可微符号。尽管这种情况经常发生，但是关于平滑性的假设是非常严格的，并且不允许人们解决重要的实际问题。这一假设特别受限制的两种情况如下：有限角度局部断层扫描和非光滑衰减的单光子发射计算机断层扫描（SPECT）。在这里，人们感兴趣的是对一个分布的波前的恢复，只知道对该分布应用带有非光滑符号的PDO的结果。已知的结果对pdo，其符号有有限的平滑，不适用，因为发生符号的奇异性的特定性质。提出的研究的主要目标如下：(1)建立在断层扫描中出现的具有非光滑符号的某些类型的pdo的理论；(2)研究了衰减系数不光滑情况下有限角度层析数据和SPECT数据的分析算法。在有限角度数据的情况下，我们的目标是能够以最小的失真恢复尽可能多的可见奇点。这对于位于能见度范围边界附近的奇点尤其重要。在SPECT的情况下，我们的目标是证明局部断层扫描可以在非光滑衰减的情况下使用；(3)通过数值实验对算法进行验证。提出的研究的意义是双重的。首先，预期的结果将促进我们对具有非光滑符号的pdo如何变换分布波前的理解。第二，预期结果将有助于解决重要的实际问题。SPECT是一种常用的无创医学成像工具。它用于大脑、心脏和其他类型的医学成像。SPECT是基于重建病人体内放射性核素的分布，知道从病人在一个平面上沿不同方向发射的光子的数量。根据所使用的放射性药物的种类，分布提供了有关患者的不同类型的医疗信息。分析SPECT数据的主要困难来自这样一个事实：通常，在病人体内发射的光子是以一种复杂的、不均匀的方式衰减的。在这种情况下，没有已知的反演公式。因此，即使光子在病人体内的衰减是不均匀的，实际上所有当前的算法都必须假设衰减系数为零或常数。这导致了衰减伪影，这仍然是SPECT图像分析的一个重大障碍。即使是基于局部层析成像的思想并考虑可变衰减的最新方法，也不能考虑非光滑衰减的情况。我们的结果将提供更准确的定量SPECT图像，即使衰减系数变化从点到点，并有不连续。看来我们的研究将是第一个从理论上探讨不连续衰减对SPECT图像的影响。有限角度局部层析成像的相关研究在涉及大型设备无损评估的应用中具有重要意义，其中扫描设备不能围绕物体旋转360度。例如，扫描位于龙门架、喷气发动机、涡轮机等上的火箭的出口锥。

Alexander Katsevich ABSTRACT Theory of pseudo-differential operators (PDO) is a powerful tool which has been successfully used in diverse areas of mathematics and, in particular, in tomography. In the classical theory, PDOs are assumed to have infinitely differentiable symbols. Although this is frequently the case, the assumption about smoothness is very restrictive and does not allow one to address important practical problems. Two cases where this assumption is especially restrictive are as follows: limited-angle local tomography and Single Photon Emission Computed Tomography (SPECT) with nonsmooth attenuation. Here one is interested in recovery of the wave front of a distribution knowing only the result of applying a PDO with nonsmooth symbol to that distribution. Known results on PDOs, symbols of which have limited smoothness, do not apply because of specific nature of the singularities of the occurring symbols. Main objectives of the proposed research are as follows (1) To develop a theory of certain classes of PDOs with nonsmooth symbols which occur in tomography; (2) To develop algorithms for analysis of the limited-angle tomographic data and SPECT data when the attenuation coefficient is not smooth. In the case of the limited-angle data, our goal is to be able to recover as much of the visible singularities with minimal distortions as possible. This is especially important for the singularities which are located close to the boundary of the visibility range. In the case of SPECT, our goal is to demonstrate that local tomography can be used in the case of nonsmooth attenuation; and (3) To test the algorithms on numerical experiments. Significance of the proposed research is twofold. First, the expected results will advance our understanding of how PDOs with nonsmooth symbols transform wave fronts of distributions. Second, the expected results will help solve important practical problems. SPECT is a popular tool for noninvasive medical imaging. It is used in cerebral, cardiac, and other kinds of medical imaging. SPECT is based on reconstructing distribution of radionuclides inside a patient knowing counts of photons emitted from the patient along different directions on a plane. Depending on what kind of the radiopharmaceutical is used, the distribution provides different types of medical information about the patient. The main difficulty in analyzing the SPECT data comes from the fact that, in general, the photons emitted inside a patient are attenuated in a complicated, nonuniform manner. In this case, no inversion formula is known. Therefore, even though the photons are attenuated nonuniformly inside the patient, practically all current algorithms must assume that the attenuation coefficient is either zero or a constant. This results in attenuation artifacts, which remain a significant obstacle in the analysis of SPECT images. Even most recent approaches, that are based on the idea of local tomography and take into account variable attenuation, fail to consider the case of nonsmooth attenuation. Our results will provide for more accurate quantitation of SPECT images even when the attenuation coefficient varies from point to point and has discontinuities. It appears that our study will be the first to investigate theoretically effects of discontinuous attenuation on SPECT images. Research related to limited-angle local tomography is of importance in applications involving nondestructive evaluation of big pieces of equipment, where a scanning device cannot be rotated 360 degrees around the object. Examples are scanning of exit cones of rockets located on a gantry, jet engines, turbines, etc.