The Radon transform on the two-dimensional space, called X-ray transform, is reviewed. Some methods to reconstruct an X-ray image from its sinogram are investigated. Important details are lost when the shearlet transform is used to retrieve an image from its noisy Rdadon data. We present a new theorem to relate the shearlet and Radon transforms to come up with this challenge. An optimum threshold to use is obtained. The algorithm of fast finite shearlet transform is improved, and numerical results show the notable superiority of the proposed method over other existing algorithms

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