Collaborative Research: Mathematical Aspects of Interior Problem of Tomography

合作研究：层析成像内部问题的数学方面

• 批准号: 1211164
• 负责人: Alexander Katsevich
• 金额: $ 19.09万
• 依托单位: The University of Central Florida Board of Trustees
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-15 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211164&HistoricalAwards=false
• 关键词: Collaborative; Research; Mathematical; Aspects; Interior

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.

计算机断层扫描 (CT) 的长期范例之一是 CT 探测器应足够大以覆盖患者的整个横截面。提出这一要求的原因是，根据经典理论，截断的 CT 数据的图像重建（所谓的“内部问题”）是非唯一的。因此，文献中尚未研究截断数据反演的许多方面。在这个项目中，PI 将从理论上分析用最少的先验知识从截断数据重建图像的问题。将讨论具有恒定衰减的透射 CT 和发射断层扫描 (SPECT)。 PI 将研究 2D 和 3D 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 问题的本征函数。 PI 还将开发有效的算法来数值求解内部问题。这些算法将使用模拟和真实数据来实施和测试。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让 PI 希望在不远的将来，内部断层扫描能够从“绘图板”转向实际的医学成像。虽然仍然存在一些开放的研究问题，但最近获得的结果表明这些问题可以以相当高的置信度得到解决。如果成功，这项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，用于根据截断的数据对对象（例如患者）内的感兴趣区域进行成像。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实践以及工业无损检测。最直接的好处是减少了患者的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、对更大物体成像的能力、降低的系统成本等。