The approximate discrete Radon transform (ADRT) is a hierarchical multiscale approximation of the Radon transform. In this paper, we factor the ADRT into a product of linear transforms that resemble convolutions, and derive an explicit spectral decomposition of each factor. We further show that this implies - for data lying in the range of the ADRT - that the transform of an $N \times N$ image can be formally inverted with complexity $\mathcal{O}(N^2 \log^2 N)$. We numerically test the accuracy of the inverse on images of moderate sizes and find that it is competitive with existing iterative algorithms in this special regime.

翻译：近似离散Radon变换（ADRT）是Radon变换的一种分层多尺度近似。本文通过将ADRT分解为一系列类卷积线性变换的乘积，推导出每个因子的显式谱分解。我们进一步证明，对于位于ADRT值域内的数据，该分解意味着$N \times N$图像的变换形式可通过$\mathcal{O}(N^2 \log^2 N)$复杂度实现形式上的逆变换。通过对中等尺寸图像的数值测试，我们发现该逆变换在此特殊条件下的精度与现有迭代算法具有可比性。