Collaborative Research: Mathematical Aspects of Interior Problem of Tomography

计算机断层扫描（CT）中一个长期存在的范例是CT检测器应该足够大以覆盖患者的整个横截面。这一要求的原因是，根据经典理论，从截断的CT数据重建图像（即所谓的“内部问题”）是非唯一的。因此，截断数据反演的许多方面尚未在文献中得到研究。在这个项目中，pi将从理论上分析用最小先验知识从截断数据重建图像的问题。将讨论恒定衰减的透射CT和发射断层扫描（SPECT）。pi将研究二维和三维重建的稳定性问题，表征内部问题中出现的积分变换的零空间，并分析某些奇异Sturm-Liouville问题的特征函数。pi还将为内部问题的数值解开发有效的算法。这些算法将使用模拟和真实数据来实现和测试。目前，内部断层扫描正处于直接应用于一般医学影像诊断的风口浪尖。理论、算法、计算和技术进步的融合给了pi希望，在不久的将来，内部断层扫描可以从“绘图板”走向实用的医学成像。虽然仍有一些尚未解决的研究问题，但最近获得的结果表明，这些问题可以有相当高的信心得到解决。如果成功，该研究将提供理论上合理且实际适用的二维和三维重建算法，用于从截断的数据中对物体（例如患者）内感兴趣的区域进行成像。拟议研究的更广泛影响在于，它有望改善临床医学和生物医学的实践，以及工业无损检测。最直接的好处是减少了病人的x射线剂量；其他好处包括更便宜的扫描仪、改进的时间分辨率、增加的扫描仪吞吐量、成像更大对象的能力、降低的系统成本等。

One of the long-standing paradigms in computed tomography (CT) is that CT detectors should be large enough to cover the entire cross-section of the patient. The reason for this requirement is that, according to the classical theory, image reconstruction from truncated CT data (the so-called "interior problem") is non-unique. As a consequence, many aspects of truncated data inversion have not been studied in literature. In this project the PIs will theoretically analyze the problem of image reconstruction from truncated data with minimal prior knowledge. Both transmission CT and emission tomography (SPECT) with constant attenuation will be addressed. The PIs will study the questions of stability of the reconstructions in 2D and 3D, characterize the null-space of an integral transform arising in the interior problem, and analyze eigenfunctions of certain singular Sturm-Liouville problems. The PIs will also develop efficient algorithms for numerical solution of the interior problem. These algorithms will be implemented and tested using both simulated and real data. At present, interior tomography is at the cusp of being directly applicable in general medical imaging diagnostics. The confluence of theoretical, algorithmic, computational, and technological advances gives the PIs hope that interior tomography can move from the "drawing boards" to practical medical imaging in the not so distant future. While there are still some open research problems, the results obtained recently show that the problems can be solved with a fairly high degree of confidence. If successful, this study will provide theoretically justified and practically applicable two and three-dimensional reconstruction algorithms for imaging a region of interest inside an object (e.g., the patient) from truncated data. The broader impact of the proposed research lies in the promise it holds for improvements in the practice of clinical medicine and biomedical science generally, as well as for industrial non-destructive testing. The most direct benefit is the reduction in the x-ray dose to the patient; other benefits include less expensive scanners, improved temporal resolution, increased scanner throughput, capability of imaging larger objects, reduced system cost, etc.