Recently, the modern imaging techniques of positron emission tomography and of single photon emission computed tomography are two of the most important tools for studying the functional characteristics of the brain, and they now play a vital role in several areas of clinical medicine, including nuclear medicine emission imaging. The basic mathematical problems associated with these techniques are the reconstruction of the Radon transform and the so-called generalized Radon transform, respectively. The singular value decomposition of the generalized Radon transform is developed by means of operator theory of function spaces, which are square integrable on R" with known weight functions. Furthermore, the inversion formula and characteristics of its range are also obtained.%近年来,正电子和单光子发射断层成像的现代图像处理技术不仅是医学上研究大脑功能特征的两个重要工具,而且它们在临床医学、核医学等领域发挥着重要的作用.而与这些图像处理技术紧密相关的需研究的数学问题就是Radon变换和广义Radon变换的重建和反演.该文利用带权的平方可积函数空间上算子理论得到一类广义Radon变换的奇异值分解,从而导出了广义Radon变换的反演公式以及值域的特征.
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