Fast Multi-Level Reconstruction of Biomedical Images Using Wavelet Sparsity Constraints

M. Unser, C. Vonesch, M. Guerquin-Kern, D. Van De Ville

Approximation and Optimization in Image Restoration and Reconstruction (AOIRR'09), Île de Porquerolles, French Republic, June 8-12, 2009.

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Wavelet-domain ℓ₁-regularization is a powerful approach for solving inverse problems. In their 2004 landmark paper, Daubechies et al. proved that one could solve such linear inverse problems by means of a "thresholded Landweber" (TL) algorithm [1]. While this iterative procedure is simple to implement, it is known to converge slowly. Here, we present a multilevel version of the algorithm that is inspired from the multigrid techniques used for solving PDEs, but with one important difference: instead of cycling through coarser versions of the problem (REDUCE part of multigrid), the multilevel algorithm cycles through the successive wavelet subspaces. The method works with arbitrary wavelet representations; it typically yields a 10-fold speed increase over the standard TL algorithm, while providing the same restoration quality. We illustrate the applicability of the method to three biomedical image reconstruction problems: the deconvolution of 3D fluorescence micrographs [2], the global reconstruction of dynamic PET from time measurements [3], and the reconstruction of magnetic resonance images from arbitrary (non-uniform) k-space trajectories. We present experimental results with real data sets in all three cases.

References

1. I. Daubechies, M. Defrise, C. De Mol, "An Iterative Thresholding Algorithm for Linear Inverse Problems with a Sparsity Constraint," Communications on Pure and Applied Mathematics, vol. 57, no. 11, pp. 1413-1457, November 2004.

2. C. Vonesch, M. Unser, "A Fast Multilevel Algorithm for Wavelet-Regularized Image Restoration," IEEE Transactions on Image Processing, vol. 18, no. 3, pp. 509-523, March 2009.

3. J. Verhaeghe, D. Van De Ville, I. Khalidov, Y. D'Asseler, I. Lemahieu, M. Unser, "Dynamic PET Reconstruction Using Wavelet Regularization with Adapted Basis Functions," IEEE Transactions on Medical Imaging, vol. 27, no. 7, pp. 943-959, July 2008.

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