On lexicographical ordering in multivariate mathematical morphology

Abstract

Since mathematical morphology is based on complete lattice theory, a vector ordering method becomes indispensable for its extension to multivariate images. Among the several approaches developed with this purpose, lexicographical orderings are by far the most frequent, as they possess certain desirable theoretical properties. However, their main drawback consists of the excessive priority attributed to the first vector dimension. In this paper, the existing solutions to solving this problem are recalled and two new approaches are presented. First, a generalisation of α-modulus lexicographical ordering is introduced, making it possible to accommodate any quantisation function. Additionally, an input specific method is suggested, based on the use of a marker image. Comparative application results on colour noise reduction and texture classification are also provided.

Morphological sharpening of color images

“…Lexicographical ordering [18] is especially suitable for arranging pixels (vectors) in the context of color Mathematical Morphology, in combination with image data where a natural or artificial priority order exists among the different bands. The lexicographical ordering preserves the input vectors.…”

“…In the case of multidimensional vectors we can use this approach in a very similar manner by applying it to each dimension in an iterative mode [18]. Having a set X of k vectors and wishing to find the maximum of it, we can start from the first dimension, sorting according to this dimension in an increasing order and then the [α × k] greatest vectors are kept (α ∈ (0, 1]) and considered as the new set X.…”

Patch-Based Mathematical Morphology for Image Processing, Segmentation and Classification

“…Comparing color vectors to define color complete lattice being already difficult [3], one easily see that defining a complete lattice for patches' vectors is much more challenging (a classical lexicographic ordering [2] being obviously of no interest). One way to define an ordering relation between vectors of a set T is to use the framework of h-orderings [10].…”

Morphological Description of Color Images for Content-Based Image Retrieval

“…In order to realize our choice of color ordering among the available rich variety, we focused on two criteria, first its theoretical stability and second its suitability to the color space under consideration. For these reasons, we have chosen to use primarily the lexicographical ordering (i.e., a total ordering leading to theoretically valid morphological operators) and more specifically its quantization based variant [18], which renders it not only more flexible but takes into account the particular relations among the dimensions of LSH as well (e.g., the redundancy of hue when a pixel is not saturated "enough").…”

“…Thus, we conform here to previous works [13] and use in the lexicographical cascade, where the first two components are subquantized nonuniformly according to functions modeling the interchannel relations of the color space (3) In particular, we have used the exact configuration presented in [18] concerning luminance and saturation to produce quantized luminance and saturation (note that any eventual equivalences among distinct color vectors may be avoided by using the lexicographical cascade once more with the original pixel values). As to the hue, however, since it is a -periodical angular value, it has been decided to use a reference based hue ordering [19] (4) where hues are ordered with respect to their distance to a reference value (to be defined depending on the application), with the closer ones being considered greater.…”