Radon transform, computer tomography, and ridgelet transform

We are interested in efficient and high quality reconstructions of digital $N \times N$ medical images from their Radon transform. The standard reconstruction algorithm, the filtered backprojection, ensures a good quality of the images at the expense of ${\cal O}( N^3 )$ arithmetic operations. Fourier reconstruction methods reduce the number of arithmetic operations to ${\cal O}( N^2 \log N )$ . Unfortunately, the straightforward Fourier reconstruction algorithm suffers from unacceptable artifacts so that it is useless in practice. A better quality of the reconstructed images can be achieved by our algorithm based on NFFTs. For details see [47,46,48] and the directory applications/radon.

Another application of the discrete Radon transform is the discrete Ridgelet transform, see e.g. [10]. A simple test program for denoising an image by hard thresholding the ridgelet coefficients can be found in applications/radon. It uses the NFFT-based discrete Radon transform and the translation-invariant discrete Wavelet transform based on MATLAB toolbox WaveLab850 [11]. See [39] for details.