Quaternion colour representations and derived total orderings for morphological operators

The definition of morphological operators for colour images needs a total ordering between the colour points. A colour can be represented according to different algebraic structures, in particular in this paper we focus on real quaternions. The paper presents two main contributions. On the one hand, we have studied different alternatives to introduce the scalar part to obtain full colour quaternions. On the other hand, several total lexicographic orderings for quaternions according to their different decompositions have been defined. These quaternionic orderings have been characterised in order to identify the most useful to define morphological operators for colour images. The theoretical results are illustrated with examples of processed images. Introduction Let ci = (ri,gi,bi) be the triplet of the red, green and blue intensities for the pixel i of a digital colour image. The definition of morphological operators for colour images needs a total ordering ≤Ω between the colour points, i.e., for any pair of unequal points ci and c j it should be possible to verify if ci ≤Ω c j or if ci ≥Ω c j, in order to be able to compute the erosion (as the infimum) and the dilation (as the supremum) of any subset of RGB colour set T ; or in other words, (T ,≤Ω) must be a complete lattice. In the literature, many techniques have been proposed on the extension of mathematical morphology to colour images. An exhaustive state-of-the-art has been reported in recent papers [3] [4]. The point ci can be represented according to different algebraic structures, in particular in this paper we focus on real quaternions [10]. We explore here the way to build colour quaternions from a RGB triplet and the different alternatives to define total orderings based on the specific properties of two quaternion representations (polar form and parallel/perpendicular decomposition), characterising and identifying the most useful to define nonlinear morphological operators. The theoretical results are illustrated with examples of processed images. Quaternion-based colour operations, such as colour Fourier transform, colour convolution and linear filters, have been studied mainly by [7, 12, 8] and by [6, 5]. From RGB colours to colour quaternions: The choice of the scalar part Quaternions and product of quaternions A quaternion q ∈ H may be represented in hypercomplex form as q = a+bi+c j+dk, (1) where a, b, c and d are real. A quaternion has a real part or scalar part, S(q) = a, and an imaginary part or vector part, V (q) = bi+ c j+dk, such that the whole quaternion may be represented by the sum of its scalar and vector parts as q = S(q)+V (q). A quaternion with a zero real/scalar part is called a pure quaternion. (a) (b) Figure 1. RGB colour spaces: (a) Unitary cube C, (b) Centered cube, representing the nine dominant colours. Bianca Carmen Sweets Figure 2. Original colour images, f(x), used in the paper. The addition of two quaternions, q,q′ ∈ H, is defined as follows q+q′ = (a+a′)+(b+b′)i+(c+c′) j+(d +d′) j. The addition is commutative and associative. The quaternion result of the product of two quaternions is defined as q′′ = qq′ = (aa′−bb′−cc′−dd′)+(ab′ +ba′ +cd′−dc′)i+ (ac′ +ca′ +db′−bd′) j+(ad′ +a′d+bc′−cb′)k (2) which can be also written in terms of dot and cross product of vectors:q′′ = qq′ = S(q′′) +V (q′′), with S(q′′) = S(q)S(q′)− V (q) · V (q′), and V (q′′) = S(q)V (q′) + S(q′)V (q) + V (q) × V (q′), where · and × represent the dot product vector and the cross product vector respectively. The multiplication of quaternions is not commutative, i.e., qq′ 6= q′q; but it is associative. Colour quaternions According to the previous works on the representation of colour by quaternions, we consider the gray-centered RGB colour-space [7]. In this space, the unit RGB cube is translated so that the coordinate origin Ô(0,0,0) represents mid-gray (middle point of the gray axis or half-way between black and white), see Fig 1(b). Then, a colour can be represented by a pure quaternion: ci = (r,g,b) ⇒ qi = ir̂ + jĝ + kb̂, where ĉ = (r̂, ĝ, b̂) = (r−1/2,g−1/2,b−1/2). It should be remarked that in this centered RGB color space the black colour quaternion and the white colour quaternion play r, b=0.5 g , b = 0 .5 Saturation as scalar part of colour quaternions 50 100 150 200 250 50