Sparse Object Representations by Digital Distance Functions

Abstract

In this paper, some methods for representing objects using path-based distances are considered. The representations can be used as anchor points when extracting medial representations of the objects. The distance transform (DT) is obtained by labeling each object element with the distance to the background. By local operations on the DT, different sets of anchor points can be obtained. We present two different methods based on local operations and prove that the representations are reversible, when this is the case. The methods are defined for weighted distances based on neighborhood sequences, which includes for example the well known cityblock and chessboard distances.

A Streaming Distance Transform Algorithm for Neighborhood-Sequence Distances

“…These two distances are known as city-block, or d 4 , and chessboard, or d 8 . d 4 and d 8 are highly rotation dependent and have been generalized to combine both (or more) neighborhoods, either simultaneously using different weights (weighted, or chamfer, distances), either by changing the neighborhood depending on the travelled distance (neighborhood-sequence, or NS, distances). For NS-distances, used in the following, the neighborhood used at a given step of the path is driven by B, a sequence of 1 and 2: − −− → p i−1 p i ∈ N B(i) .…”

“…Due to the limited number of allowed steps and weights in the paths, a 3 × 3 neighborhood around each point is enough to extract reversible representations of objects, that can be used for, e.g., skeletonization, object decomposition, and resolution pyramids. For details, see [4] and the references therein.…”