Multiplicative operator splittings in nonlinear diffusion from spatial splitting to multipl.pdfVIP

Journal of Mathematical Imaging and Vision 19: 33–48, 2003 c? 2003 Kluwer Academic Publishers. Manufactured in The Netherlands. Multiplicative Operator Splittings in Nonlinear Diffusion: From Spatial Splitting to Multiple Timesteps DANNY BARASH? AND TAMAR SCHLICK Department of Chemistry and the Courant Institute of Mathematical Sciences, New York University, New York 10012, USA barash@. MOSHE ISRAELI AND RON KIMMEL Computer Science Department, Technion, Israel Institute of Technology, Haifa 32000, Israel Abstract. Operator splitting is a powerful concept used in many diversed fields of applied mathematics for the design of effective numerical schemes. Following the success of the additive operator splitting (AOS) in performing an efficient nonlinear diffusion filtering on digital images, we analyze the possibility of using multiplicative operator splittings to process images from different perspectives. We start by examining the potential of using fractional step methods to design a multiplicative operator splitting as an alternative to AOS schemes. By means of a Strang splitting, we attempt to use numerical schemes that are known to be more accurate in linear diffusion processes and apply them on images. Initially we implement the Crank-Nicolson and DuFort-Frankel schemes to diffuse noisy signals in one dimension and devise a simple extrapolation that enables the Crank-Nicolson to be used with high accuracy on these signals. We then combine the Crank-Nicolson in 1D with various multiplicative operator splittings to process images. Based on these ideas we obtain some interesting results. However, from the practical standpoint, due to the computational expenses associated with these schemes and the questionable benefits in applying them to perform nonlinear diffusion filtering when using long timesteps, we conclude that AOS schemes are simple and efficient compared to these alternatives. We then examine the potential utility of using multiple timestep methods combine