Partitioning 2D Images into Prototypes of Slope Region

Image = Structure + Few Colors

“…The use of combinatorial maps supports parallel processing [23] with time complexity of O(log(d)), where d is the diameter of the longest object in the image. The approach is reinforced by a concrete theory of cellular decomposition into cells called slope regions 1 [2,15]. The TIIP algorithm is explained in two distinct algorithms: (a) the bottomup construction of the pyramid (REDUCE) and (b) the top-down expansion of the higher level of the pyramid (EXPAND), terminologies as introduced in [6].…”

Characterizing slope regions

“…Observe that there can be many holes (as mentioned above) inside the slope region and that the outer boundary of the region is surrounded by exactly two monotonic paths connecting the 1-max m 1 and the 1-min m 2 . Prototype 2 ("horizontal slope region") mentioned in Batavia et al (2019a) can also be obtained from the general prototype presented in Fig. 5, by replacing the monotonic paths in the outer boundary by paths representing level curves.…”

“…Besides, we introduced a technique to orient monotonic paths with level curves in the primal graph and provided necessary and sufficient conditions for both primal and dual graph to merge two slope regions. In Batavia et al (2019a), we introduced a second prototype with the aim of characterizing slope regions with the constraint that the outer boundary is a level curve. We provided necessary conditions for their existence and we showed that a well-composed 2D digital image can be partitioned into slope regions categorized into one of the two previously described prototypes.…”