The traditional approaches to computerized tomography (CT) depend on the samples of Radon transform at multiple angles. In optics, the real time imaging requires the reconstruction of an object by the samples of Radon transform at a \emph{single angle} (SA). Driven by this and motivated by the connection between Bin Han's construction of wavelet frames (e.g \cite{Han1}) and Radon transform, in refinableshift-invariant spaces (SISs) we investigate the SA-Radon sample based reconstruction problem. We have two main theorems. The fist main theorem states that, any compactly supported function in a SIS generated by a general refinable function can be determined by its Radon samples at the designed angle. Motivated by the extensive application of positive definite (PD) functions to interpolation of scattered data, we also investigate the SA reconstruction problem in a class of (refinable) box-spline generated SISs. Thanks to the PD property of the Radon transform of such spline, our second main theorem states that, the reconstruction of compactly supported functions in these spline generated SISs can be achieved by the Radon samples at a \emph{generic angle}. Numerical simulation is conducted to check the result.

中文翻译：

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通过单角度Radon样本精确地变换不变空间中函数的精确恢复

计算机断层扫描（CT）的传统方法取决于多个角度的Radon变换样本。在光学领域，实时成像需要通过Radon变换的样本以\ emph {单角度}（SA）重建对象。在此基础上，并以Bin Han的小波帧构造（例如\ cite {Han1}）与Radon变换之间的联系为动力，在可精化的不变移空间（SIS）中，我们研究了基于SA-Radon样本的重构问题。我们有两个主要定理。第一个主定理指出，由一般可细化函数生成的SIS中任何紧密支持的函数都可以通过其Radon样本在设计角度来确定。由于正定（PD）函数广泛应用于分散数据的插值，我们还研究了一类（细化）盒样条生成的SIS中的SA重建问题。由于这种样条的Radon变换的PD特性，我们的第二个主要定理指出，这些样条生成的SIS中紧支持功能的重构可以通过\ emph {generic angle}的Radon样本来实现。进行数值模拟以检查结果。