Efficient Path Openings and Closings

Path openings and closings are algebraic morphological operators using families of thin and oriented structuring elements that are not necessarily perfectly straight. These operators are naturally translation invariant and can be used in filtering applications instead of operators based on the more standard families of straight line structuring elements. They give similar results to area or attributebased operators but with more flexibility in the constraints. Trivial implementations of this idea using actual suprema or infima of morphological operators with paths as structuring elements would imply exponential complexity. Fortunately a linear complexity algorithm exists in the literature, which has similar running times as an efficient implementation of algebraic operators using straight lines as structuring elements. However even this implementation is sometimes not fast enough, leading practitioners to favour some attribute-based operators instead, which in some applications is not optimal. In this paper we propose an implementation of path-based morphological operators which has logarithmic complexity and comparable computing times with those of attribute-based operators.