Topology preserving alternating sequential filter for smoothing two-dimensional and three-dimensional objects

Abstract

We introduce the homotopic alternating sequential filter as a new method for smoothing two-dimensional (2D) and three-dimensional (3D) objects in binary images. Unlike existing methods, our method offers a strict guarantee of topology preservation. This property is ensured by the exclusive use of homotopic transformations defined in the framework of digital topology. Smoothness is obtained by the use of morphological openings and closings by metric disks or balls of increasing radius, in the manner of an alternating sequential filter. The homotopic alternating sequential filter operates both on the object and on the background, in an equilibrated way. It takes an original image X and a control image C as input, and smooths X "as much as possible" while respecting the topology of X and geometrical constraints implicitly represented by C. Based on this filter, we introduce a general smoothing procedure with a single parameter which allows to control the degree of smoothing. Furthermore, the result of this procedure presents small variations in response to small variations of the parameter value. We also propose a method with no parameter for smoothing zoomed binary images in 2D or 3D while preserving topology.