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 single angle (SA). Driven by this and motivated by the connection between Bin Han's construction of wavelet frames (e.g [13]) and Radon transform, in refinable shift-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 an appropriate 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 samples of Radon transform at almost every angle. Numerical simulation is conducted to check the result.
翻译:在光学中,实时成像要求用一个单一角度(SA)来重建雷达变形样品产生的一个物体。受此驱动,并受Bin Han建造波子框架(如[13])和拉登变形之间在可翻新的变换空间(如SIS)中的连接的驱动,我们调查了SA-Radon基于样本的重建问题。我们有两个主要理论。拳头主论点指出,通过一般可再造功能生成的光子变形样品在SIS中产生的任何紧固支持的功能都可以由它的雷达样品在适当角度加以确定。由于广泛应用正确定功能对分散数据进行内插,我们还在可翻新的变换式空间(如SIS)中调查了SA-Radon变形(如SIS)中的变形问题。由于Radon变形的PD属性,我们的第二个主要理论指出,这些变形功能的缩压后功能的重塑,可以通过这些变形模型在每一个变形角进行。