Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 596348, 16 pages

http://dx.doi.org/10.1155/2015/596348

## Multivariate Self-Dual Morphological Operators Based on Extremum Constraint

^{1}School of Electronic and Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China^{2}School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710072, China

Received 2 January 2015; Revised 15 June 2015; Accepted 21 June 2015

Academic Editor: Babak Shotorban

Copyright © 2015 Tao Lei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Self-dual morphological operators (SDMO) do not rely on whether one starts the sequence with erosion or dilation; they treat the image foreground and background identically. However, it is difficult to extend SDMO to multichannel images. Based on the self-duality property of traditional morphological operators and the theory of extremum constraint, this paper gives a complete characterization for the construction of multivariate SDMO. We introduce a pair of symmetric vector orderings (SVO) to construct multivariate dual morphological operators. Furthermore, utilizing extremum constraint to optimize multivariate morphological operators, we construct multivariate SDMO. Finally, we illustrate the importance and effectiveness of the multivariate SDMO by applications of noise removal and segmentation performance. The experimental results show that the proposed multivariate SDMO achieves better results, and they suppress noises more efficiently without losing image details compared with other filtering methods. Moreover, the proposed multivariate SDMO is also shown to have the best segmentation performance after the filtered images via watershed transformation.

#### 1. Introduction

One of the main issues for mathematical morphology is its extension to the multivariable case [1]. So far, a lot of effort has been made to design multivariate morphological operators which can be applied in multichannel images [2–5]. The key point of constructing morphological operators for multichannel images consists in the definition of the ordering scheme for multivariate data. Thus, researchers have proposed many vector orderings, which can be divided into four categories: marginal ordering, lexicographical ordering, partial ordering, and reduced ordering in which the lexicographical ordering is one of the most popular algorithms [6]. Louverdis presented multivariate morphological operators based on lexicographical ordering in HSV color space [7]. As lexicographical ordering is most suitable for the situations in which an order of “importance” exists on the available channels, the vector ordering based on the combinations of the different vector distances (e.g. Mahalanobis distance, Euclidean distance) and lexicographical ordering was proposed by Angulo [8]. Recently, Aptoula et al. proposed *α*-trimmed lexicographical extremum estimation algorithms, based on the new lexicographical ordering, and constructed the new multivariate morphological operators [9, 10]. In order to make further improvement on the performance of multivariate morphological operators for color image filtering and segmentation, Angulo studied quaternion properties and applied quaternion decomposition to color image representation and then proposed a new lexicographical ordering based on quaternion decomposition, which can provide better results in image processing [11]. According to [10, 11], Lei et al. proposed vector ordering based on fuzzy extremum estimation algorithm, and then constructed multivariate morphological operators. The proposed operators have a high performance on color image filtering [12].

In recent years, various complex mathematical tools, such as machine learning [13], rand projection depth function [14], principal component analysis [15], hypercomplex [16], probabilistic extrema estimation [17], and group-invariant frames [18], are employed to develop multivariate morphological theory and improve the performance on color image filtering and segmentation. Lezoray proposed nonlocal vector mathematical morphology. First, unsupervised vector ordering was proposed according to dictionary learning and machine learning. And then the improved multivariate morphological operators were proposed [19]. Velasco-Forero studied multivariate mathematical morphology based on random projection depth function and proposed vector ordering algorithm based on P-ordering. The proposed algorithm solved the problem of extending morphological theory to multichannel images [14].

A lot of multivariate morphological operators have been proposed, but few researches on the properties have been reported. Most of the morphological operators occur in pairs of dual operators, such as erosion/dilation and opening/closing. In the binary case, erosion on the background of an image is equivalent to the dilation on its foreground [20]. Also in the grayscale case, the complement of erosion on the original image is equivalent to dilation on its complement. The traditional morphological operators will lead to a heavy deviation of the gray values of filtered images [21]. Therefore, the final results depend on whether erosion or the dual dilation is used as the first operator in the sequence. For example, images operated by those morphological operators of which the first operator is erosion will tend to be darker; in contrast, images operated by operators of which the first operator is the dual dilation will have the opposite results. In order to treat the image foreground and background identically, self-dual morphological operators can be applied in image processing [22]. Bouaynaya et al. studied SDMO in the case of the general theory of lattice morphology and spatially variant morphology and illustrated the importance of the self-duality property by an application to speckle noise removal in radar images [23]. Self-dual morphological operators have been widely used in binary and grayscale images. However, it is difficult to extend the applications of SDMO to multichannel images.

To address this issue, we study the existing multivariate morphological operators and find that vector ordering is the key factor for the duality. By defining a pair of SVO in RGB color space, we construct multivariate dual morphological operators and multivariate SDMO. However, it is well known that VMF (vector median filters) is able to provide better results than multivariate morphological operators for color image filtering. Therefore, multivariate SDMO based on extremum constraint are finally presented. The experimental results demonstrate that the proposed multivariate SDMO performs better than the popular VMFs. As image filtering is the foundation of image segmentation and understanding, we applied watershed transformation to the images filtered by various filters. The segmentation results show that the proposed multivariate SDMO also has the best segmentation performance.

The structure of this paper is organized as follows. Section 2 introduces the traditional self-dual morphological operators. Section 3 analyzes the reasons why the existing multivariate morphological operators are not dual and then gives the solution of the duality of multivariate morphological operators. Multivariate self-dual morphological operators based on extremum constrain are presented in Section 4. Section 5 presents the experimental results and analysis, and Section 6 gives conclusions of this paper.

#### 2. Self-Dual Morphological Operators

Before entering a general discussion on the construction of self-dual operators, the related concepts of self-dual morphological transformation are proposed.

*Definition 1. *The complement of an image , denoted by , is defined for each pixel as the maximum value of the data type used for storing the image minus the value of the image at position : where , .

*Definition 2. *Two transformation and are dual with respect to complementation if applying to an image is equivalent to applying to the complement of the image and taking the complement of the result:

*Definition 3. *A transformation is self-dual with respect to complementation , if its dual transformation with respect to the complementation is itself:

The median operation is a self-dual neighborhood image transformation since it replaces each pixel value by the median value of the original image pixels located in a window centered at the considered pixel. The detailed proof is proposed as below. For color images, Zanoguera and Meyer studied vector leveling which are self-dual vector morphological filers and proposed a convenient algorithm for calculating a nonseparable rotation-invariant vector leveling because the algorithm which derives from the definition is rather costly. The proposed algorithm is applied to integer images without compromising convergence issues and provides better results than those obtained when independently applying a scalar leveling to the three color components [24].

The traditional morphological filtering operators always have the problem that the filtered image is brighter or darker than the original image. Fortunately, self-dual morphological transformation can solve the problem; it can maintain the gray value of the filtered image. Based on dual and self-dual transformation, Heijmans proposed morphological median operators, which are also called self-dual morphological operators [22], to address the problem that the classical morphological filters are unfair for the foreground and background of an image.

Proposition 4. *Let be a pair of dual transformation, and let and denote morphological erosion and dilation, respectively. Let be an increasing transformation and let be the dual transformation of . Thus, morphological median denoted by can be represented as follows:**Or**where denotes identity operator. We can demonstrate that is a self-dual transformation.*

*Self-dual morphological filtering operators are applied in image filtering, and the results are shown in Figure 1, where is the self-dual filtering operators based on opening-closing and closing-opening and the structuring element is a disk of size 5 × 5, , .*