We find a new and simple inversion formula for the 2D Radon transform (RT) with a straight use of the shearlet system and of well-known properties of the RT. Since the continuum theory of shearlets has a natural translation to the discrete theory, we also obtain a computable algorithm that recovers a digital image from noisy samples of the 2D Radon transform which also preserves edges. A very well-known RT inversion in the applied harmonic analysis community is the biorthogonal curvelet decomposition (BCD). The BCD uses an intertwining relation of differential (unbounded) operators between functions in Euclidean and Radon domains. Hence the BCD is ill-posed since the inverse is unstable in the presence of noise. In contrast, our inversion method makes use of an intertwining relation of integral transformations with very smooth kernels and compact support away from the origin in the Fourier domain, i.e. bounded operators. Therefore, we obtain, at least, the same asymptotic behavior of mean-square error as the BCD (and its shearlet version) for the class of cartoon-like functions. Numerical simulations show that our inverse surpasses, by far, the inverse based on the BCD. Our algorithm uses only fast transformations.

我们找到了一个新的，简单的二维Radon变换（RT）的反演公式，直接使用了来回波系统和RT的众所周知的属性。由于小波的连续理论对离散理论具有自然的转化，因此我们还获得了一种可计算算法，该算法可从2维Radon变换的噪声样本中恢复数字图像，并保留边缘。在应用谐波分析领域中，一个非常著名的RT反演是双正交曲波分解（BCD）。BCD使用欧几里德域和Radon域中函数之间的微分（无界）算符交织在一起的关系。因此，由于在存在噪声的情况下逆函数不稳定，因此BCD处于不适状态。相反，我们的反演方法利用积分变换的交织关系，该变换具有非常平滑的内核和紧凑的支持，远离傅里叶域的原点，即有界算子。因此，对于类卡通函数，我们至少获得了与BCD（和其来回波形式）相同的均方误差渐近行为。数值模拟表明，到目前为止，我们的逆值已经超过了基于BCD的逆值。我们的算法仅使用快速变换。