Abstract
The component trees provide a high-level, hierarchical, and contrast invariant representations of images, suitable for many image processing tasks. Yet their definition is ill-formed on multivariate data, e.g., color images, multi-modality images, multi-band images, and so on. Common workarounds such as marginal processing, or imposing a total order on data are not satisfactory and yield many problems, such as artifacts, loss of invariances, etc. In this paper, inspired by the way the Multivariate Tree of Shapes (MToS) has been defined, we propose a definition for a Multivariate min-tree or max-tree. We do not impose an arbitrary total ordering on values; we use only the inclusion relationship between components. As a straightforward consequence, we thus have a new class of multivariate connected openings and closings.