New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction

This paper describes newly invented multiscale transforms known under the name of the ridgelet (Appl. Comput. Harmonic Anal. 6 (1999) 197) and the curvelet transforms (In: Cohen et al. (Eds.), Curves and Surfaces, Vanderbilt University Press, Nashville, TN, 2000, pp. 105–120; Candès and Donoho available at http://www-stat.stanford.edu/donoho/Reports/1998/curvelets.zip, 1999). These systems combine ideas of multiscale analysis and geometry. Inspired by some recent work on digital Radon transforms (Averbuch et al. available at http://www-stat.stanford.edu/~donoho/Reports/index.html), we then present very effective and accurate numerical implementations with computational complexities of at most N log N . In the second part of the paper, we propose to combine these new expansions with the Total Variation minimization principle for the reconstruction of an object whose curvelet coefficients are known only approximately: quantized, thresholded, noisy coefficients, etc. We set up a convex optimization problem and seek a reconstruction that has minimum Total Variation under the constraint that its coefficients do not exhibit a large discrepancy from the data available on the coefficients of the unknown object. We will present a series of numerical experiments which clearly demonstrate the remarkable potential of this new methodology for image compression, image reconstruction and image ‘de-noising’.