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A Fast Thresholded Landweber Algorithm for Wavelet-Regularized Multidimensional Deconvolution

C. Vonesch, M. Unser

IEEE Transactions on Image Processing, vol. 17, no. 4, pp. 539-549, April 2008.



We present a fast variational deconvolution algorithm that minimizes a quadratic data term subject to a regularization on the ℓ1-norm of the wavelet coefficients of the solution. Previously available methods have essentially consisted in alternating between a Landweber iteration and a wavelet-domain soft-thresholding operation. While having the advantage of simplicity, they are known to converge slowly. By expressing the cost functional in a Shannon wavelet basis, we are able to decompose the problem into a series of subband-dependent minimizations. In particular, this allows for larger (subband-dependent) step sizes and threshold levels than the previous method. This improves the convergence properties of the algorithm significantly. We demonstrate a speed-up of one order of magnitude in practical situations. This makes wavelet-regularized deconvolution more widely accessible, even for applications with a strong limitation on computational complexity. We present promising results in 3-D deconvolution microscopy, where the size of typical data sets does not permit more than a few tens of iterations.


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