Fast Computation of Greyscale Path Openings

Path openings are morphological operators that are used to preserve long, thin, and curved structures in images. They have the ability to adapt to local image structures, which allows them to detect lines that are not perfectly straight. They are applicable in extracting cracks, roads, and more. Although path openings are very efficient to implement for binary images, the greyscale case is more problematic. This study provides an analysis of the main existing greyscale algorithm, and shows that although its time complexity can be quadratic in the number of pixels, this is optimal in terms of the output (if the full opening transform is created). Also, it is shown that under many circumstances the worst-case running time is much less than quadratic. Finally, a new algorithm is provided, which has the same time complexity, but is simpler, faster in practice and more amenable to parallelization.