Canonical Encoding of the Combinatorial Pyramid

This paper presents a novel framework to en-code a combinatorial pyramid. A combinatorial pyramidis a hierarchy of successively reduced combinatorial maps.Important properties of the combinatorial pyramids such astopology preservation, the process global and local featureswithin the same data structure, etc. made them useful forimage processing and pattern recognition tasks. Their ad-vantages have been widely proved in the literature. Never-theless, the main disadvantage of this approach is the highrate of memory requirement. A combinatorial map of an im-age maybe stored in an array of size approximately equal tofour times the number of pixels of the image. Furthermore,every level of the combinatorial pyramid stores a differentcombinatorial map. In respond to this problem a canoni-cal encoding of the combinatorial pyramid is provided. Itconsists of a single array where its elements are orderedwith respect to the construction history of the pyramid. Inthis manner the memory consumptions are equal to the sizeof the initial combinatorial map and do not depend on thenumber of pyramid´s levels. In addition, this canonical en-coding allows the whole reconstruction of the pyramid inboth directions: from the base to the top level and from thetop to the base level, without additional information.