@article {796, title = {A Second Order Multi-Stencil Fast Marching Method With a Non-Constant Local Cost Model}, journal = {IEEE Transactions on Image Processing}, volume = {28}, year = {2019}, month = {04/2019}, pages = {1967{\textendash}1979}, abstract = {

The fast marching method is widely employed in several fields of image processing. Some years ago a multi-stencil version (MSFM) was introduced to improve its accuracy by solving the equation for a set of stencils and choosing the best solution at each considered node. The following work proposes a modified numerical scheme for MSFM to take into account the variation of the local cost, which has proven to be second order. The influence of the stencil set choice on the algorithm outcome with respect to stencil orthogonality and axis swapping is also explored, where stencils are taken from neighborhoods of varying radius. The experimental results show that the proposed schemes improve the accuracy of their original counterparts, and that the use of permutation-invariant stencil sets provides robustness against shifted vector coordinates in the stencil set.

}, keywords = {Approximation algorithms, Differential equations, Eikonal equation, Frequency modulation, MSFM, Mathematical model, Silicon, Three-dimensional displays, Unmanned aerial vehicles, Vectors, axis swapping, difference equations, fast marching methods, finite difference methods, finite differences, image processing, iterative methods, least squares approximations, multi-stencil schemes, multistencil version, nonconstant local cost model, permutation-invariant stencil sets, second order multistencil fast marching method, stencil orthogonality, stencil set}, issn = {1057-7149}, doi = {10.1109/TIP.2018.2880507}, url = {https://ieeexplore.ieee.org/document/8531783/}, author = {S. Merino-Caviedes and Lucilio Cordero-Grande and M. T. P{\'e}rez and Pablo Casaseca-de-la-Higuera and M. Mart{\'\i}n-Fern{\'a}ndez and R. Deriche and C. Alberola-L{\'o}pez} } @article {628, title = {Efficient and Robust Image Restoration Using Multiple-Feature L2-Relaxed Sparse Analysis Priors}, journal = {IEEE Transactions on Image Processing}, volume = {24}, year = {2015}, month = {Dec}, pages = {5046-5059}, abstract = {

We propose a novel formulation for relaxed analysis-based sparsity in multiple dictionaries as a general type of prior for images, and apply it for Bayesian estimation in image restoration problems. Our formulation of a l2 -relaxed l0 pseudo-norm prior allows for an especially simple maximum a posteriori estimation iterative marginal optimization algorithm, whose convergence we prove. We achieve a significant speedup over the direct (static) solution by using dynamically evolving parameters through the estimation loop. As an added heuristic twist, we fix in advance the number of iterations, and then empirically optimize the involved parameters according to two performance benchmarks. The resulting constrained dynamic method is not just fast and effective, it is also highly robust and flexible. First, it is able to provide an outstanding tradeoff between computational load and performance, in visual and objective, mean square error and structural similarity terms, for a large variety of degradation tests, using the same set of parameter values for all tests. Second, the performance benchmark can be easily adapted to specific types of degradation, image classes, and even performance criteria. Third, it allows for using simultaneously several dictionaries with complementary features. This unique combination makes ours a highly practical deconvolution method.

}, keywords = {Bayes methods, Bayesian estimation, Convergence, Dictionaries, Estimation, Kernel, L2-relaxed L0 pseudo norm, L2-relaxed L0 pseudo-norm prior, L2-relaxed sparse analysis priors, Maximum likelihood estimation, Optimization, Redundancy, computational load, constrained dynamic method, deconvolution, deconvolution method, dynamically evolving parameters, estimation loop, fast constrained dynamic algorithm, image restoration, iterative marginal optimization, iterative methods, maximum a posteriori estimation, mean square error, mean square error methods, multiple representations, multiple-feature L2-relaxed sparse analysis priors, optimisation, robust tunable parameters, structural similarity terms}, issn = {1057-7149}, doi = {10.1109/TIP.2015.2478405}, author = {Javier Portilla and Antonio Trist{\'a}n-Vega and Ivan W. Selesnick} }