A novel set of image morphological operators using a modified vector distance measure with color pixel classification☆
Introduction
Mathematical morphology (MM) introduced by Matheron [1] and Serra [2], is a theory for image analysis based on set theory. Originally developed for binary images, it was later generalized for grayscale images as well [3]. MM facilitates the quantitative analysis or description of the geometric structure of images that depends on small synthetic images called structuring elements. On the other hand, MM can also be described on a lattice structure using algebraic tools [4]
The direct extension of MM to color images is not simple since there is no natural ordering on a set of multivariate data. The ordering procedure is highly related to the definition of an appropriate color ordering. According to Barnett [5], there are four types of vector ordering methods: marginal (M), conditional (C), partial (P) and reduced (R) ordering approaches. In M-ordering, the components of the color vectors are ordered independently (pointwise ordering). The C-ordering, also known as lexicographic ordering, produces an ordered set of the color vectors according to the ordering of one component or more. Here, some marginal components were selected sequentially according to different conditions. The P-ordering is based on the partition of the vectors into groups, such that the groups can be distinguished with respect to rank or extremeness. The R-ordering performs the ordering according to some scalars, computed from the components of each vector with respect to different measure criteria (distances or projections onto a reduced space).
A wide variety of works tackled the ordering issue [6]. The work of Aptoula and Lefèvre [7], represents a relatively recent and comprehensive work in this field. Approaches that are based on C- and combined C- and R-orderings were implemented to preserve the input color vectors. Examples can be found in [8], [9], [10], [11]. Sartor and Weeks [9] have implemented a combination of R- and C-ordering where the concept of a reference color is introduced to provide the primary ordering criterion. However, the morphological operators in respect to a reference color and the implemented operators were restricted to red erosion, green opening, etc. Al-Otum [10], proposed a corrected componentwise morphological algorithm with the end of preventing the appearance of new vectors based on implementing a Mahalanobis distance-based error function. Another approach is implemented in the polar version of the C–Y color space [11], where a saturation based combination of the hue and intensity planes is proposed. Here, the vectorial pixels contained within the SE are reduced into scalars by means of a combined weighted vector-Euclidean distance. Velasco-Forero and Angulo [12], constructed morphological operators using a P-ordering of the multivariate images based on the statistical depth functions. The main drawback in the proposed approach is the difficulty to select the number of random projections which was empirically set to 1000. Lezoray [13], proposed R-ordering morphological operators relying on a complete lattice, the latter is dynamically obtained by manifold learning with Laplacian Eigenmaps.
Other approaches implemented the so-called pseudo-morphology that does not require an underlying order among the image data and focuses on directly computing the minimum and maximum of a given set [14]. This can be accomplished by estimating two pseudo-extrema based on Chebyshev inequality. The framework embeds a parameter which allows controlling the linear versus non-linear behavior of the probabilistic pseudo-morphological operators. Lei et al. [15] presented vector morphological operators based on fuzzy extremum estimation. Three novel marginal components were developed, in the lexicographical ordering, by means of the quaternion description.
Angulo [16], proposed a generalization of distance- and lexicographical-based approaches, allowing the extension of morphological operators to color images. Aptoula and Lefèvre [17], focused on the application of the morphological framework to the problem of color description. In particular, they produced an ordering approach that is based on the so called global descriptors. The proposed approach includes the color specific granulometries and two multiresolution histograms based on morphological levelings and watersheds.
In this work, a novel set of color image morphological operators is proposed and is based on using a modified Vector Distance measure with an efficient preprocessing color Pixel Classification procedure (denoted as VDPC). VDPC starts by classifying the input color pixels, into different categories, based on the characteristics of the human color perception properties. Using V- and S-components in the HSV color model, color pixels are classified into three categories: Chromatic, Bright Chromatic and Pseudo-Gray pixels. Next, the color distance is calculated for each input color pixel based on its category, and, a combined R- and C-ordering technique is used to obtain basic morphological operators. The designed morphological operators were extended to be implemented for developing vector-based tools for noise suppression, texture analysis, shape analysis, edge detection, skeletonization, and color image compression.
The arrangement of this paper is as follows. Section 2 provides preliminaries on image morphological operators as well as on selected color spaces. In Section 3, the proposed VDPC algorithm is introduced and explained for the basic morphological operations: dilation, erosion, opening and closing. An extension, to other morphological operators is also developed in this section, including: boundary extraction, noise filtration, image enhancement, etc. The validity of the VDPC algorithm is checked in Section 4. Finally, in Section 5, a summary and concluding remarks are provided.
Section snippets
Color space representation
The most direct way to manipulate color images is to work on the RGB color space. A color image f is a vector function f(x) = (fR(x); fG(x); fB(x)) ∊ Z3, x ∊ Z2, where fR(x), fG(x) and fB(x) are the red, green, and blue channels at the pixel x respectively. However, the RGB color representation has some drawbacks: (1) components are strongly correlated, (2) the lack of human interpretation, and, (3) nonuniformity. A polar representation with the variables Luminance, Saturation and Hue allows to solve
VDPC pixel classification step
In fact, classic color MM methods did not take into account the properties of the Human Vision System and treated color pixels similarly. In order to assess the color pixel behavior, it would be suitable to classify color pixels according to their human color perception properties. In this work, a recursive HSV-space is going to be used for the pixel classification step. We choose this color space, due to its proven segmentation performance and for the fact that it allows for fast and efficient
Conclusions and future recommendations
In this work, a novel vector-based approach to color image morphology using a Vector-based Distance measure with Pixel Classification is considered (denoted as VDPC) and is based on extending mathematical morphology to color images by treating multichannel data as vectors. VDPC implements the HSV color space for color pixel classification into three categories based on their grayness content and vector behavior. A modified color distance is calculated, based on the color pixel category, and a
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This paper has been recommended for acceptance by M.T. Sun.