Approximation of noisy data using multivariate splines and finite element methods

We compare a recently proposed multivariate spline based on mixed partial derivatives with two other standard splines for the scattered data smoothing problem. The splines are defined as the minimiser of a penalised least squares functional. The penalties are based on partial differentiation operators, and are integrated using the finite element method. We compare three methods to two problems: to remove the mixture of Gaussian and impulsive noise from an image, and to recover a continuous function from a set of noisy observations.

[1]  Ron Kikinis,et al.  Registration of 3-d intraoperative MR images of the brain using a finite-element biomechanical model , 2000, IEEE Transactions on Medical Imaging.

[2]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[3]  Bishnu P. Lamichhane,et al.  Finite Element Techniques for Removing the Mixture of Gaussian and Impulsive Noise , 2009, IEEE Transactions on Signal Processing.

[4]  B. Lamichhane.  Removing a mixture of Gaussian and impulsive noise using the total variation functional and split Bregman iterative method , 2015 .

[5]  Fugen Zhou,et al.  A Discontinuous Finite Element Method for Image Denoising , 2006, ICIAR.

[6]  Markus Hegland,et al.  Fitting multidimensional data using gradient penalties and the sparse grid combination technique , 2009, Computing.

[7]  G. Wahba.  Spline models for observational data , 1990 .

[8]  Bishnu P. Lamichhane,et al.  A new multivariate spline based on mixed partial derivatives and its finite element approximation , 2014, Appl. Math. Lett..

[9]  Martin Rumpf,et al.  An Adaptive Finite Element Method for Large Scale Image Processing , 2000, J. Vis. Commun. Image Represent..

[10]  Armin Iske,et al.  Adaptive image approximation by linear splines over locally optimal delaunay triangulations , 2006, IEEE Signal Processing Letters.

[11]  M. Hutchinson.  A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines , 1989 .

[12]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[13]  H. Triebel,et al.  Topics in Fourier Analysis and Function Spaces , 1987 .

[14]  Erkan Besdok,et al.  Impulsive Noise Suppression from Images by Using Anfis Interpolant and Lillietest , 2004, EURASIP J. Adv. Signal Process..

[15]  G. Burton.  Sobolev Spaces , 2013 .

[16]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[17]  Dennis D Cox,et al.  Convergence Rates for Multivariate Smoothing Spline Functions. , 1982 .