Nonlinear smoothing of N-dimensional data using successive over-relaxation method

Local smoothing of N-dimensional data can be performed in many ways. This paper is oriented to local penalization and its minimization which generates a system of nonlinear equations. This approach enables to realize trade-off between denoising, edge, and structure preserving. This is mainly useful in the case of discontinuous signals and images. Various penalization strategies can be used for this task, but only constrained penalizations (Tukey, Welsch, Andrews) are successful. Novel nonlinear method is inspired by successive over-relaxation scheme for linear systems of equations, but it is applied to nonlinear root-finding problem. The method is designed to be stable for several smoother types. Root bracketing inside inner loop is included in the procedure and extends the stability range in many applications. Numerical experiments are performed on 1D signal and 2D image. Optimum relaxation factors are found experimentally for maximum rate of convergence. The main results of experimental part are: preference of Tukey method in the case of discontinuous signal, similarity of proposed methods in the case of continuous signal, and efficiency of Tukey method followed by watershed transform in the case of image segmentation. Selected smoothers are recommended mainly for signals and images with discontinuities and can be useful in signal and image enhancement, analysis, segmentation, and classification.