On the Range of the Attenuated Radon Transform in Strictly Convex Sets
On the Range of the Attenuated Radon Transform in Strictly Convex Sets
复制标题
严格凸集上衰减Radon变换的范围
DOI：
10.1090/s0002-9947-2014-06307-1
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发表时间：
2013
影响因子：
1.3
通讯作者：
A. Tamasan
中科院分区：
文献类型：
--
作者：
K. Sadiq;A. Tamasan
我们提出了新的充分必要条件，使 @×S 1 上的函数处于凸集 ⊂ R 2 中足够平滑函数支持的衰减 Radon 变换范围内。该方法基于与 Bukhgeim 意义上的 A 解析函数边界迹相关的显式希尔伯特变换。在本文中，我们关注平面上紧支撑函数的衰减Radon变换的范围表征。自（6）、（7）和（10）中的工作以来，欧几里德空间中非衰减（经典）Radon 变换的范围的必要和充分约束已经众所周知。这些约束称为 Cavalieri 或力矩条件，以角度变量表示。对于 Schwartz 类中的函数，由于 Paley-Wiener 型定理，它们本质上是唯一的。此外，Helgason 支持定理将条件扩展到紧致支持的平滑函数 (8)。然而，在紧支持函数的情况下，有可能获得本质上不同的范围条件，因为不止一个算子可以在Radon变换的范围内消除紧支持函数。这里的结果就是这样一个例子。平面内衰减Radon变换的反演方法首先出现在(1)和(14)中，并且在(13)、(3)、(5)、(2)中可以找到各种发展。对范围条件的兴趣源于它们在医学成像方法（例如单光子或正电子发射计算机断层扫描）中数据增强的应用（12）。对于欧几里德衰减氡变换，基于（14）中的反演过程的一些范围特征可以在（15）中找到。这些约束也是角度变量的约束。与上面现有的结果不同，我们的新表征是根据与 A 解析图相关的希尔伯特变换`a la
We present new necessary and sufficient conditions for a function on@×S 1 to be in the range of the attenuated Radon transform of a sufficiently smooth function support in the convex set ⊂ R 2 . The approach is based on an explicit Hilbert transform associated with traces of the boundary of A-analytic functions in the sense of Bukhgeim. In this paper we are concerned with the range characterization of the attenuated Radon transform of function of compact support in the plane. Necessary and sufficient constraints on range of the non-att enuated (clas- sical) Radon transform in the Euclidean space have been known since the works in (6), (7), and (10). These constraints, known as the Cavalieri or the moment conditions, are in terms of the angular variable. For function in the Schwartz class, they are essentially unique due to a Paley-Wiener type the- orem. Moreover, the Helgason support theorem extends the conditions to smooth functions of compact support (8). However, in the case of functions of compact support, it is possible to obtain essentially dif ferent range con- ditions since more than one operator can annihilate functions of compact support in the range of the Radon transform. The results here constitute one such example. Inversion methods of the attenuated Radon transform in the plane ap- peared first in (1), and (14), and various developments can be found in (13), (3), (5), (2). The interest in range conditions stems out fro m their appli- cations to data enhancement in medical imaging methods such as Single Photon, or Positron Emission Computed Tomography (12). For the Eu- clidean attenuated Radon transform, some range characterization based on the inversion procedure in (14) can be found in (15). These constraints are also in terms of the angular variable. Different from the existing results above, our new characterization is in terms of a Hilbert transform associated with the A-analytic maps ` a la