从径向不完整球面Radon数据中进行图像重建
Image reconstruction from radially incomplete spherical Radon data
AI总结：
本文研究了球面Radon变换在球面上的反演问题，提出在径向不完整数据下可唯一恢复球壳中的图像函数，并展示了在不同支持区域的重建方法和算法。
摘要：
我们研究了具有球面中心的球面Radon变换的反演问题，此类反演在医学、雷达和声呐成像中的图像重建问题中至关重要。在径向不完整数据的情况下，我们证明球面Radon变换可以唯一地反演恢复球壳中的图像函数。我们的结果适用于图像函数的支持区域在数据采集球内、外以及球的两侧的情况。此外，除了唯一性结果外，我们的证明方法还为所有这些情况提供了重建公式。我们基于我们的反演公式提出了一个稳健的计算算法，并在多个数值示例中展示了其准确性和效率。
英文摘要：
We study inversion of the spherical Radon transform with centers on a sphere (the data acquisition set). Such inversions are essential in various image reconstruction problems arising in medical, radar and sonar imaging. In the case of radially incomplete data, we show that the spherical Radon transform can be uniquely inverted recovering the image function in spherical shells. Our result is valid when the support of the image function is inside the data acquisition sphere, outside that sphere, as well as on both sides of the sphere. Furthermore, in addition to the uniqueness result our method of proof provides reconstruction formulas for all those cases. We present a robust computational algorithm based on our inversion formula and demonstrate its accuracy and efficiency on several numerical examples.
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