In this paper, we define a distance for the HSL colour system. Next, the proposed distance is used for a fuzzy colour clustering algorithm construction. The presented algorithm is related to the well-known fuzzy c-means algorithm. Finally, the clustering algorithm is used as colour reduction method. The obtained experimental results are presented to demonstrate the effectiveness of our approach.
A colour image generally contains tens of thousands of colours. Therefore, most colour image processing applications first need to apply a colour reduction method before performing further sophisticated analysis operations such as segmentation. The using of colour clustering algorithm could be a good alternative for colour reduction method construction. In the framework of colour clustering procedure, we are faced with two colour comparison subject. We want to know how similar or how different two colours are. In order to do this comparison, we need to have a good coordinate system for colour representation and also, we need to define an efficient inter-colour distance measure in the considered system.
In this paper we will consider the particular case of colour
representation by the perceptual system HSL (hue,
saturation and luminosity)
In this paper, we propose a distance that minimizes the two disadvantages mentioned above. This distance is constructed as a quasi-linear combination of the three standard distances of H, S, L scalar components. Next, this distance is used for a fuzzy colour clustering construction.
In the following sections, the paper is thus organized: section 2 presents the particular form of the system HSL used in this paper; section 3 presents the new colour distance; section 4 presents the colour clustering algorithm based on the proposed distance and related to the fuzzy c-means algorithm; section 5 presents some experimental results while section 6 outlines the conclusions.
The most part of the colour images are represented by the
RGB colour system. However, this space presents the
following two important limitations: the difficulty to
determine colour features like the presence or the absence of
a given colour, and the inability of the Euclidean distance to
correctly capture colour differences in the RGB space.
Starting from the RGB system, there were defined other
systems for colour representation. One of them is the HSL
system where H is the hue, S is the saturation and L is
the luminosity. Colour space HSL is also commonly used in
image processing. As opposed to RGB system, HSL is
considered as natural representation colour system. The
HSL system belongs to the perceptual system category
because it is very close to the human colour perception. In
the HSL system, colour is decomposed according to
physiological criteria like hue, saturation and luminosity.
Hue refers to the pure spectrum colours and corresponds to
dominant colour as perceived by human. Saturation
corresponds to the relative purity or the quantity of white
light that is mixed with hue while luminosity refers to the
amount of light in a colour (Gonzales et al. 2007). A great
advantage of HSLsystem over the RGB lies in its capacity
to recognize the presence of colours in a given image.
There exist many formulae for H, S, L components
calculation. For the hue H , the most part of the definitions
are closed to the following formula that uses the atan2
function
B G 2R B G
H atan2 2 , 6 and H ( , ].
For the luminosity calculation, in this paper, the following
formula will be used
L
M 1 M m
M max(R, G, B) where and
m min(R,G, B) .
Regarding to the saturation calculation, many variants are
defined by the distance between max(R, G, B) and
min(R, G,.B) . In this paper, we will use for saturation
calculation, the following formula
2(M m) 1 | M 0.5 | | m 0.5 | We suppose
R,G, B [0,1] and it results that M, m, S, L [0,1] .
In the Cartesian coordinate system (x, y, z) R3 , for two vectors v1 (x1, y1, z1) , v2 (x2 , y2 , z2 ) one defines the Euclidean distance by: DE2 (v1, v2 ) (x1 x2 ) 2 ( y1 y2 ) 2 (z1 z2 ) 2 In order to obtain the variant of Euclidean distance for the cylindrical coordinate system, we use the following substitution: x cos( ) , y sin( ) , z l . (1) (2) (3) d 2(1,2) sin 21 2
2 The coordinate system HSL is a cylindrical one and for two colours Q1 (H1, S1, L1) and Q2 (H2 , S2 , L2 ) , the Euclidean distance becomes (Gonzales et al. 2007): DE2 (Q1, Q2) 4S1S2 d 2(H1, H2) d 2(S1, S2) d 2(L1, L2) The colour Euclidean distance DE has three terms: the distance between hues, the distance between saturations and the distance between luminosities. The distance between hues is multiplied by a factor that depends on the colour saturations. This factor has a multiplicative structure. Thus, when the saturation values increase, the hues distance influence increases in framework of distance DE . When the saturation values decrease, the hue distance influence decreases.
From here, the idea to multiply the luminosity distance with a similar factor comes up. This factor will have the following behaviour: when the saturation values increase, the luminosity distance influence decreases and when the saturation values decrease, the luminosity distance influence increases.
We can generalize (4) using two real and positive parameters and , by the following: DP2 (Q1, Q2 ) d 2 (H1, H 2 ) d 2 (L1, L2 ) d 2 (S1, S2 ) The parameter will be related to the colour chromaticity while the parameter will be related to the colour achromaticity. Before the construction of parameters and , we define the index of chromaticity c and the index of achromaticity a , having the following properties: (4) (5) index of chromaticity c :[0,1] [0,1], (i) c(0) 0 (ii) c(1) 1 We will use for the chromaticity index the following particular function:
DE2 (v1, v2 ) 41 2 d 2 (1, 2 ) d 2 (1, 2 ) d 2 (l1, l2 ) if S1 S2 then c(S1) c(S2 ) .
c(S ) S index of achromaticity a :[0,1] [0,1] , (i) a(0) 1 , (ii) a(1) 0 . Using (6), (7), (8) and (9) for the two multipliers and , it results the following particular forms:
DP2 (Q1, Q2 ) S1 S2 sin 2 H1 H 2 2 (1 S1 ) (1 S 2 ) (L1 L2 ) 2 (S1 S2 ) 2
Let there be n colours Q1, Q2 ,...,Qn that must be clustered
into k sets. Each cluster j is characterized by the
membership coefficients w j1, w j2 ,...,w jn for the considered
n colours and the cluster center defined by the colour
q j h j , s j , l j . For colour clustering, we will construct an
algorithm that is similar to the fuzzy c-means algorithm
The membership function wij is calculated using formula (13), where the distance DP is calculated with formula (12). i [1, n] j [1, k]
, wij 1
2 k DP Qi , q j 1 1 m1 DP Qi , qm
m j The functions wij verify the condition of the partition of wi1 wi2 ... wik 1 The cluster center components h j , s j , l j are calculated with formulae (14), (15) and (16).
n n wij Ci sin H i wij Ci cosH i
, i1 n wij Ci i1 n wij Ci i1 unity, namely: i [1, n] , j [1, k] , h j atan2 i1 with Ci Si . j [1, k] , with Ai 1 Si . j [1, k] ,
s j i1 l j i1 n wij Ai Li n wij Ai i1 n wij Si n wij i1 (13) (14) (15) (16) where is a fuzzification-defuzzification parameter and also, (1,1.5).
We have applied this method to images: “red-green” (figure 1a), “flower” (figure 2a), “house” (figure 3a), “parrots” (figure 4a) and “bird” (figure 5a). The clustered images obtained using the new distance can be seen in figures 1b, 2b, 3b, 4b, 5b and those obtained using the Euclidean distance can be seen in figures 1c, 2c, 3c, 4c, 5c.
Figure 1a shows a synthetic image. Using the proposed distance, it was obtain the 3-colour representation shown in figure 1b while figure 1c shows the image obtained using the Euclidean distance. In the first case, the black region has a small area, while in the second case, it has a large one. This fact proves the influence of achromatic parameter that was used in the proposed distance.
In the case of image “flower”, using the Euclidean distance it was obtained two clusters for background. These two clusters have the same hue but the luminosities are different (figure 2c). Using the proposed distance, one obtained only one cluster for the entire background (figure 2b).
For image “house”, using the Euclidean distance, the white and bright blue colours were not separated (figure 3c). In the case of image “parrots”, the yellow and red colours were not separated (figure 4c) and the clustering is different from that obtained using the proposed distance (figure 4b). For the image “bird”, using the Euclidean distance, the orange and grey colours were not separated (figure 5c).
The experimental results illustrate that our distance performs well compared to Euclidean distance. In the same time, the obtained results show the helpfulness of using this novel inter-colour distance measure in colour clustering algorithms.
In this paper one presents an enhancement of fuzzy c-means algorithm for the particular case of colour clustering. It was used the perceptual colour system HSL for colour representation. The main step is represented by definition of a new distance in the HSL colour space. In this construction, there were used two multipliers that make the balance between the hue weight and luminosity weight in the framework of this three-term colour distance. We can conclude that the new inter-colour distance “ DP ” defined by (12), the two multipliers “ ” defined by (8) and “ ” defined by (9), the particular form of index of chromaticity “ c ” defined by (6) and achromaticity “ a ” defined by (7) construct a new and useful framework for colour clustering procedures. The obtained experimental results were compared with those obtained by using the Euclidean distance. This comparison shows the efficiency of the proposed clustering algorithm using for the colour reduction method.
Figure 1: The image „red-green” (a) and its 3-colour representation based on proposed HSL distance (b) and based on HSL Euclidean distance (c).
Figure 2: The image „flower” (a) and its 3-colour representation based on proposed HSL distance (b) and based on HSL Euclidean distance (c).