Variational image reconstruction from arbitrarily spaced samples: a fast multiresolution spline solution.
We propose a novel method for image reconstruction from nonuniform samples with no constraints on their locations. We adopt a variational approach where the reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms: (1) the sum of squared errors at the specified points and (2) a quadratic functional that penalizes the lack of smoothness. We search for a solution that is a uniform spline and show how it can be determined by solving a large, sparse system of linear equations. We interpret the solution of our approach as an approximation of the analytical solution that involves radial basis functions and demonstrate the computational advantages of our approach. Using the two-scale relation for B-splines, we derive an algebraic relation that links together the linear systems of equations specifying reconstructions at different levels of resolution. We use this relation to develop a fast multigrid algorithm. We demonstrate the effectiveness of our approach on some image reconstruction examples.
Arigovindan
Muthuvel
M
Ecole Polytechnique Federale de Lausanne, Biomedical Imaging Group, Swiss Federal Institute of Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland. muthuvel.arigovindan@epfl.ch
Sühling
Michael
M
Hunziker
Patrick
P
Unser
Michael
M
D000465:Algorithms; D001185:Artificial Intelligence; D003198:Computer Simulation; D007089:Image Enhancement / Q000379:methods*; D007090:Image Interpretation, Computer-Assisted / Q000379:methods*; D016247:Information Storage and Retrieval / Q000379:methods*; D015233:Models, Statistical; D009716:Numerical Analysis, Computer-Assisted; D010363:Pattern Recognition, Automated / Q000379:methods*; D015203:Reproducibility of Results; D018401:Sample Size; D012680:Sensitivity and Specificity; D012815:Signal Processing, Computer-Assisted; D013382:Subtraction Technique