Grayscale Area Opening and Closing, their efficient implementation and applications

The filter that removes from a binary image its con-nected components with area smaller than a parameter is called area opening. From a morphological per-spective, this filter is an algebraic opening, and it can be extended to grayscale images. The properties of area openings and their dual area closings are recalled. In par-ticular, it was proved in [13] that the area opening of pa-rameter of an image I is the supremum of the grayscale images that are smaller than I and whose regional max-ima are of area greater than or equal to . This theorem is at the basis of an efficient algorithm for computing grayscale area openings and closings. Its implementa-tion involves scanning pixels in an order that depends both on their location and value. For this purpose, the use of pixel heaps is proposed. This data structure is shown to be both efficient and low in memory require-ments. In addition, it can be used in the computation of various other complex morphological transforms. The use of these area openings and closings is illustrated on image filtering and segmentation tasks 1 Notations, Definitions In this paper, the binary images or sets under study are subsets of a connected compact set M IR2 called the mask. Similarly, grayscale images are mappings from M to IR. For simplicity, only the 2-D case is considered here, but the notions and algorithms discussed generalize to arbitrary dimensions. By area is meant the Lebesgue measure in IR2. B denotes the morphological opening with respect to structuring element B. Let us first recall the notion of connected opening