Median-Tree: An Efficient Counterpart of Tree-of-Shapes

Representing an image through a tree structure as provided with a morphological hierarchy enables efficient image analysis and processing methods operating directly on the tree structure. Max-tree and min-tree can be built with efficient algorithms but they only focus on brighter and darker components of the image respectively. Conversely, the Tree-of-Shapes is a self-complementary image representation that provides access to all regional extrema of the image (both brighter and darker components), but its computation is more time-consuming. In this paper, we introduce a new, simple and efficient tree structure called median-tree. It relies on a median image that is straightforwardly constructed by subtracting the median pixel value from an image to decompose it into positive and negative parts. The median tree can then be obtained by applying the efficient max-tree algorithms available in the literature on this median image. We show through theoretical and experimental studies that the median-tree offers similar characteristics to the Tree-of-Shapes, but comes with a considerably lower construction complexity.