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Solving Continuous-Domain Problems Exactly with Multiresolution B-Splines

T. Debarre, J. Fageot, H. Gupta, M. Unser

Proceedings of the Forty-Fourth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'19), Brighton, United Kingdom, May 12-17, 2019, pp. 5122-5126.



We propose a discretization method for continuous-domain linear inverse problems with multiple-order total-variation (TV) regularization. It is based on a recent result that proves that such inverse problems have sparse polynomial-spline solutions. Our method consists in restricting the search space to splines with knots on a uniform grid, which results in a standard convex finite-dimensional problem. As basis functions for this search space, we use the B-splines matched to the regularization order, which are optimally localized. This leads to a well-conditioned, computationally feasible optimization task. Our proposed iterative multiresolution algorithm then refines the grid size until a desired level of accuracy is met and converges to sparse solutions of our inverse problem. Finally, we present experimental results that validate our approach.


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