A color tree of shapes with illustrations on filtering, simplification, and segmentation

Abstract

The Tree of Shapes (ToS) is a morphological tree that provides a high-level, hierarchical, self-dual, and contrast invariant representation of images, suitable for many image processing tasks. When dealing with color images, one cannot use the ToS because its definition is ill-formed on multivariate data. Common workarounds such as marginal processing, or imposing a total order on data are not satisfactory and yield many problems (color artifacts, loss of invariances, etc.) In this paper, we highlight the need for a self-dual and contrast invariant representation of color images and we provide a method that builds a single ToS by merging the shapes computed marginally, while guarantying the most important properties of the ToS. This method does not try to impose an arbitrary total ordering on values but uses only the inclusion relationship between shapes. Eventually, we show the relevance of our method and our structure through some illustrations on filtering, image simplification, and interactive segmentation.