Directional morphological filtering

We show that a translation invariant implementation of min/max filters along a line segment of slope in the form of an irreducible fraction dy/dx can be achieved at the cost of 2+k min/max comparisons per image pixel, where k=max(|dx|,|dy|). Therefore, for a given slope, the computation time is constant and independent of the length of the line segment. We then present the notion of periodic moving histogram algorithm. This allows for a similar performance to be achieved in the more general case of rank filters and rank-based morphological filters. Applications to the filtering of thin nets and computation of both granulometries and orientation fields are detailed. Finally, two extensions are developed. The first deals with the decomposition of discrete disks and arbitrarily oriented discrete rectangles, while the second concerns min/max filters along gray tone periodic line segments.