Efficient incremental computation of attributes based on locally countable patterns in component trees

In this paper, we propose a collection of pattern sets to compute bit-quad matchings in component tree structures of grayscale images. The proposed patterns are designed to be used in an efficient algorithm which runs in linear time by counting incrementally each pattern matching per pixel. It is worth emphasizing that the proposed algorithm computes any attribute derived from 2×2 bit-quads such as area, perimeter, Euler number, among others. Finally, this algorithm is applied for circular object segmentation using shape ultimate opening with attribute of circularity computed efficiently by counting matchings from this collection of pattern sets.