Introducing the Dahu Pseudo-Distance

Abstract

The minimum barrier (MB) distance is defined as the minimal interval of gray-level values in an image along a path between two points, where the image is considered as a vertex-valued graph. Yet this definition does not fit with the interpretation of an image as an elevation map, i.e. a somehow continuous landscape. In this paper, based on the discrete set-valued continuity setting, we present a new discrete definition for this distance, which is compatible with this interpretation, while being free from digital topology issues. Amazingly, we show that the proposed distance is related to the morphological tree of shapes, which in addition allows for a fast and exact computation of this distance. That contrasts with the classical definition of the MB distance, where its fast computation is only an approximation.

Equivalence between Digital Well-Composedness and Well-Composedness in the Sense of Alexandrov on n-D Cubical Grids

“…A third approach which need well-composedness is the Dahu pseudo-distance [18] where we compute the distance between two points in an image based on the distance in the tree of shapes between the two shapes containing these two points (see Figure 50). This pseudodistance has been shown as being a good approximation of the well-known minimum-barrier distance (MBD).…”

“…Fig. 50: Computation of the Dahu pseudo-distance [18]: the distance between the components B and F depends on the number of level lines that are crossed in the image when we go from the interior of B to the interior of F in the domain of the image (see the left side), this distance is easily computable by computing the length of the path joining the corresponding nodes B and F in the tree (see the right side). A direct application of this Dahu pseudo-distance can be found in [29] (see Figure 51) and corresponds to a saliency-based detection of identity documents captured by smartphones.…”

A Tutorial on Well-Composedness

“…Using these separated Jordan curves, we can naturally induce a hierarchy [28] Fig. 82: In the raster scan order: the initial image, its level lines, the Dahu pseudo-distance, and its saliency map [64]. in the image: the components given by the interior of these curves (whatever the chosen connectivity) are either nested or disjoint.…”

“…This distance has been shown very efficient to proceed to salient object segmentation [177,195]. A continuous version of this distance, the Dahu pseudo-distance, has then been proposed in [64]; note that this is not a distance but a pseudo-distance in the sense that it does not satisfy the identity of indiscernibles. Besides, this distance leads to topological issues in its naive version at saddle points, and for this reason, a slightly modified version of this distance has been proposed to overcome these issues: starting from a vertex-valued graph u, Géraud et al compute first the self-dual interpolation u DWC to get rid of possible saddle 0D points (note that saddle faces still exist), and then they compute its selfdual span-based immersion into the Khalimsky grids to obtain a self-dual continuous representation of the initial signal.…”