One-Dimensional Openings, Granulometries and Component Trees in ${cal O}(1)$ Per Pixel

We introduce a new, efficient and adaptable algorithm to compute openings, granulometries and the component tree for one-dimensional (1-D) signals. The algorithm requires only one scan of the signal, runs in place in O(1) per pixel, and supports any scalar data precision (integer or floating-point data). The algorithm is applied to two-dimensional images along straight lines, in arbitrary orientations. Oriented size distributions can thus be efficiently computed, and textures characterized. Extensive benchmarks are reported. They show that the proposed algorithm allows computing 1-D openings faster than existing algorithms for data precisions higher than 8 bits, and remains competitive with respect to the algorithm proposed by Van Droogenbroeck when dealing with 8-bit images. When computing granulometries, the new algorithm runs faster than any other method of the state of the art. Moreover, it allows efficient computation of 1-D component trees.