A Direct Algorithm for Optimization Problems with the Huber Penalty

We present a direct (noniterative) algorithm for 1-D quadratic data fitting with neighboring intensity differences penalized by the Huber function. Applications of such an algorithm include 1-D processing of medical signals, such as smoothing of tissue time concentration curves in kinetic data analysis or sinogram preprocessing, and using it as a subproblem solver for 2-D or 3-D image restoration and reconstruction. dynamic programming was used to develop the direct algorithm. The problem was reformulated as a sequence of univariate optimization problems, for k = 1, ⋯ ,N, where N is the number of data points. The solution to the univariate problem at index k is parameterized by the solution at k + 1, except at k = N. Solving the univariate optimization problem at k = N yields the solution to each problem in the sequence using back-tracking. Computational issues and memory cost are discussed in detail. Two numerical studies, tissue concentration curve smoothing and sinogram preprocessing for image reconstruction, are used to validate the direct algorithm and illustrate its practical applications. In the example of 1-D curve smoothing, the efficiency of the direct algorithm is compared with four iterative methods: the iterative coordinate descent, Nesterov's accelerated gradient descent algorithm, FISTA, and an off-the-shelf second order method. The first two methods were applied to the primal problem, the others to the dual problem. The comparisons show that the direct algorithm outperforms all other methods by a significant factor, which rapidly grows with the curvature of the Huber function. The second example, sinogram preprocessing, showed that robustness and speed of the direct algorithm are maintained over a wide range of signal variations, and that noise and streaking artifacts could be reduced with almost no increase in computation time. We also outline how the proposed 1-D solver can be used for imaging applications.