Nonlinear diffusion filtering is usually performed with explicit schemes. They are only stable for very small time steps, which leads to poor efficiency and limits their practical use. Based on a recent discrete nonlinear diffusion scale-space framework we present semi-implicit schemes which are stable for all time steps. These novel schemes use an additive operator splitting (AOS) which guarantees equal treatment of all coordinate axes. They can be implemented easily in arbitrary dimensions, have good rotational invariance and reveal a computational complexity and memory requirement which is linear in the number of pixels. Examples demonstrate that, under typical accuracy requirements, AOS schemes are at least ten times more efficient than the widely-used explicit schemes.

The completion of interrupted lines or the enhancement of flow-like structures is a challenging task in computer vision, human vision, and image processing. We address this problem by presenting a multiscale method in which a nonlinear diffusion filter is steered by the so-called interest operator (second-moment matrix, structure tensor). An m-dimensional formulation of this method is analysed with respect to its well-posedness and scale-space properties. An efficient scheme is presented which uses a stabilization by a semi-implicit additive operator splitting (AOS), and the scale-space behaviour of this method is illustrated by applying it to both 2-D and 3-D images.

In most cases nonlinear diffusion filtering is implemented by means of explicit finite difference schemes. These algorithms are not very efficient, since they are only stable for small time steps. We address this problem by presenting unconditionally stable semi-implicit schemes which are based on an additive operator splitting (AOS). They are very efficient since they can be implemented by recursive filtering, and their separability allows a straightforward implementation in any dimension. We analyse their behaviour on a parallel computer and demonstrate that parallel AOS schemes on a modern shared-memory multiprocessor system with 8 processors allow a speed-up of two orders of magnitude in comparison to the widely-used explicit scheme on a single processor. c fl1997 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to server...

. Poor efficiency is a typical problem of nonlinear diffusion filtering, when the simple and popular explicit (Euler-forward) scheme is used: for stability reasons very small time step sizes are necessary. In order to overcome this shortcoming, a novel type of semi-implicit schemes is studied, so-called additive operator splitting (AOS) methods. They share the advantages of explicit and (semi-)implicit schemes by combining simplicity with absolute stability. They are reliable, since they satisfy recently established criteria for discrete nonlinear diffusion scale-spaces. Their efficiency is due to the fact that they can be separated into one-dimensional processes, for which a fast recursive algorithm with linear complexity is available. AOS schemes reveal good rotational invariance and they are symmetric with respect to all axes. Examples demonstrate that, under typical accuracy requirements, they are at least ten times more efficient than explicit schemes. 1 Introduction Although non...

The goal of this paper is to present segmentation algorithms which combine regularization by nonlinear partial differential equations (PDEs) with a watershed transformation with region merging. We develop efficient algorithms for two wellfounded PDE methods. They use an additive operator splitting (AOS) leading to recursive and separable filters. Further speed-up can be obtained by embedding AOS schemes into a pyramid framework. Examples are presented which demonstrate that the preprocessing by these PDE techniques eases and stabilizes the segmentation. The typical CPU time for segmenting a 256 2 image on a workstation is less than 2 seconds. Key Words: Nonlinear diffusion, additive operator splitting, Gaussian pyramid, watershed segmentation, region merging CR Subject Classification: I.4.6, I.4.3, I.4.4. 1 Introduction Segmentation is one of the bottlenecks of many image analysis and computer vision tasks ranging from medical image processing to robot navigation. Ideally it sho...