Color Morphological Image Segmentation on a New Opponent Color SpaceBased on the Notion of Critical Functions

Israel SantillanFacultad de Ingenierıa

Universidad Autonoma de QueretaroCerro de las Campanas, 76000, Qro., Mexico

santilis@gmail.com

Ivan R. Terol-VillalobosCIDETEQ

Parque Tecnologico Queretaro S/NSanfandila Pedro-Escobedo, 76700, Qro. Mexico

famter@ciateq.net.mx

Gilberto Herrera-RuizFacultad de Ingenierıa

Universidad Autonoma de QueretaroCerro de las Campanas, 76000, Qro. Mexico

gherrera@uaq.mx

Abstract

In the present paper color image segmentation, based onthe notion of critical functions, is investigated. The notionof critical functions is derived from the well-known water-fall transformation. In the one-dimensional case the criticalfunctions and the waterfall transformation provide similarresults. However, in the two-dimensional case, the criticalfunctions give better results for segmenting images than thewaterfall transformation, since they take into account thefusion of regional minima (or maxima) to transform the im-age, providing a good control of the regions merging pro-cess. Moreover, contrary to the waterfall transformationwhich is non-parametric, critical functions allow the intro-duction of image segmentation criteria to compute an opti-mal partition. To carry out our study, a color space basedon the opponent colors is proposed that we call Y O1O2

model. The main reason to propose a new color space isthat despite the numerous perceptually color spaces (HSV,HSL, HSI, among others) proposed in the literature, theyhave several drawbacks for color image processing. Theperformance in image segmentation of both proposals, thecolor model and critical functions, is illustrated with sev-eral examples. Also, some interesting properties of the colormodel Y O1O2 and the critical functions are shown.

1. Introduction

In the last years much research has been made on colorimage filtering and segmentation, as well as on color im-

age retrieval. This research has not only been focused onthe use of numerous color spaces that exist in the litera-ture, but also to propose new color spaces and new schemesfor color image processing. For color representation onehas the perceptual color spaces such as HSL (hue, satura-tion, luminance), HSV (hue, saturation, value) [21] and HSI(hue, saturation, intensity) [6] as well as perceptually uni-form color spaces such as L*u*v* and L*a*b* (luminanceL*, chrominance u*, v*, a*, and b*) and the CIECAM02[14]. The first class, the perceptual color spaces, are morefrequently used in image processing while the perceptu-ally uniform color spaces are mainly focused in measur-ing the difference between colors. However, even if theselast spaces were developed for measuring color differences,both spaces, L*u*v* and L*a*b have also been applied toimage processing. For example, in [10], a new frameworkfor chromatic filtering of color images was introduced. Thechromatic content of a color image is encoded in the CIEuv chromaticity coordinates, and colors are added accord-ing to the center of gravity law of additive color mixtures[8], whereas the achromatic content is encoded as CIE Ytristimulus value. In Lucchese et al [9] the same concept(gravity center law) is used for introducing a method forcolor image enhancement. These ideas [10, 9] were usedin [15] for the restoration of ancient chinese paintings andin [5] to propose a morphological multiscale contrast ap-proach for color images. On the other hand, in Hanburyand Serra [7], the use of the mathematical morphology inthe CIE L*a*b* space is discussed. In particular, the au-thors show that it is possible to impose a total order on thecolor vectors in this space by using a weighting function

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and a lexicographical order. However, representations ofcolor in terms of hue, saturation and brightness coordinatesare used in image processing since these models are moreintuitive (i.e. for color mixing), simpler to calculate, and donot require any calibration information. Nevertheless, theserepresentations suffer from some defects, such as the pres-ence of unstable singularities and non-uniform distributionsof their components. Recently, Serra [20] proposed variousnew perceptual color spaces in order to eliminate some ofthe drawbacks of these spaces; in particular, luminance andsaturation do not verify any norm (L1, L2 or the pseudo-norm Max-Min). A consequence of the use of saturationand luminance definitions that verify the concept of normis that new colors are not created. These new color mod-els were used by Angulo and Serra [1] for segmenting im-ages. Finally, in Sarifuddin and Missaoui [17], a new per-ceptually uniform color space, called HCL (hue, chroma,luminance), was proposed and used for content-based im-age and video retrieval by taking into account that humanvision reacts non linearly (logarithmic). This color modelwas successfully applied for vehicle license plate recogni-tion in [12] and for fingerprint recognition in [4]. Here,a new color space, called Y O1O2, is introduced based onthe opponent colors concept. It is well-accepted that hu-man vision processes images based on the opponent colors,R-G, (2B-(R+G)); therefore, this color codification is usedto specify the saturation and hue, whereas the luminance iscodified according to bit-mixing taking into account the or-der green, red and blue. One perceives better the green colorthan the red color and the red color better than the blue. Thiscolor codification provides an appropriate color specifica-tion for color image processing. This is particularly illus-trated in this paper in image segmentation using an hierar-chical approach based on the critical functions notion. Crit-ical functions work by a flattening process of regional min-ima (or maxima). The flattening process is stopped to forma critical function at a given level of the hierarchy, when theminima (or maxima) disappear or become another feature.The neighborhood analysis of the minima (or maxima) onthe critical function is easily carried out in order to take adecision, according to some homogeneity criteria, to deter-mine whether a set of minima (or maxima) can be mergedto form a feature in another level of hierarchy or they re-main intact. This paper is organized as follows. In Section2, the concept of morphological filter and filters by recon-struction are presented. In Section 3, the new color spaceY O1O2 is introduced illustrating some interesting proper-ties of the color model. Next, in Section 4, the notion ofcritical function is proposed and analyzed. In the same Sec-tion, some homogeneity color criteria are introduced in thecritical functions to control the merging process.

2 Some Basic Concepts of Morphological Fil-tering

The basic morphological filters ([19], [22]) are the mor-phological opening γµB and the morphological closing ϕµB

with a given structuring element; where, in this work, B isan elementary structuring element (3x3 pixels) that containsits origin. B is the transposed set (B = {−x : x ∈ B}) andµ is an homothetic parameter. The morphological openingand closing are given, respectively, by

γµB(f)(x) = δµB(εµB(f))(x)

ϕµB(f)(x) = εµB(δµB(f))(x) (1)

where the morphological erosion εµB and dilation δµB

are expressed as: εµB(f)(x) = ∧{f(y) : y ∈ µBx} andδµB(f)(x) = ∨{f(y) : y ∈ µBx}. ∧ is the inf operator and∨ is the sup operator. Another class of filters is composedby the opening and closing by reconstruction. These filtersare built using the geodesic dilation and erosion. Where thegeodesic dilation and the geodesic erosion of size one aregiven by δ1

f (g) = f∧δ(g) with g ≤ f and ε1f (g) = f∨ε(g)

with g ≥ f , respectively. When filters by reconstructionare built, the basic geodesic transformations, the geodesicdilation, and the geodesic erosion of size 1 are iterated un-til idempotence is reached: R(f, g) = limn→∞ δn

f (g) andR∗(f, g) = limn→∞ εn

f (δµ(f)). When the function g isequal to the erosion or the dilation of the original function,we obtain the opening and the closing by reconstruction:

γµ(f) = R(f, εµ(f)) ϕµ(f) = R∗(f, δµ(f)) (2)

3 Color Model Based on Opponent Colors

3.1 Luminance definition

Different approaches exist to extend the morphologicaloperators from gray level to color; marginal, reduced andvectorial processing. From these approaches, it is preferableto use a purely vectorial approach to process the image all atonce using the information of three channels. But one mustdefine a well suited vectorial lattice structure such that avectorial ordering relation is introduced. An interesting vec-torial ordering relation is given by the lexicographic orderthat enables us to have a total ordering. Then, one codifiesthe three bytes of the (R,G,B) vector in a 24-bit representa-tion by establishing the importance of the color. One givesto each byte, corresponding to a color channel, the most,the middle or the least significant byte in the 24 bit repre-sentation. This approach induces an important dissymmetry

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(a) (b) (c)

(d) (e) (f)

(g) (h)

Figure 1. a) Original image; b) and c) Morpho-logical opening and closing size 5; d) and e)Opening and closing by reconstruction size5; f), g), and h) Luminance, saturation andhue, respectively.

between the different components of the image. It is prefer-able to mix the bits of the three bytes corresponding to thecomponents of the image [3]. Thus, one first takes the firstbit of one component, followed by the first bit of the secondcomponent and finally the first bit of the last component.The process is repeated until the 8 bits of the three chan-nels are codified. Nevertheless, different combinations canbe taken created. Here, one proposes the choice given bythe human visual perception. Consider the following orderin the bit mixing where the green channel has the greatestimportance and the blue component the smallest. In otherwords, a 24-bit image is built from the 3 color componentsG7R7B7 · · · G0R0B0 where Gj , Rj , Bj are the bit codifi-cation of the (R,G,B) color with Rj , Gj , Bj ∈ {0, 1}. Un-der this ordering the color components are weighted closelyto the luminance Y of the European standard for televisiongiven by L = 0.299R + 0.587G + 0.144B. In fact, the bitmixing is weighted by Y = 0.2857R+0.5714G+0.1429B(4/7 for the green, 2/7 for the red and 1/7 for the blue),which is closely linked to the cone perception in the eye’sretina. Moreover, this bijective mapping ensures that theone-dimensional representation contains all the informationof the three color components. For a point p of the im-age, the bit mixing codification is given in this case byI(p) =

∑7j=0 23j+2Gj +

∑7j=0 23j+1Rj +

∑7j=0 23jBj .

In Fig. 1 some examples of morphological filters usingthe bit mixing approach are illustrated. Figures 1 (b) and (c)show the morphological opening and closing, respectively,of the original image in Fig. 1 (a), whereas in Figs. 1(d) and

(e) the respective opening and closing by reconstruction, areillustrated.

3.2 Saturation and hue definitions interms of opponent colors

Many color models in the literature have different lumi-nance and saturation definitions. In Serra [20], a very inter-esting study has shown the main drawbacks of luminanceand saturation definitions of the perceptual spaces such asthe HSI, HSV and HSL. In particular, the main conclusionof this last study shows that the luminance and saturationmust be defined by means of a norm L1, L2, or the semi-norm Max − Min. In Serra [18], the author shows thatthe norm L1 presents several advantages from a practicalpoint of view; however, we show here that the semi-normMax −Min has other interesting characteristics. First, inthe chromatic plane the Max−Min becomes a norm. Sec-ond, this definition is equivalent to a linear combination ofthe form:

|R−G|+ |G−B|+ |B −R|)/3 =

(2/3)(max(R, G,B)−min(R, G, B))

Now, let us introduce our saturation definition in termsof opponent colors, (2B − (R + G)) and (R − G), thatis closely a representation of Max − Min saturation. Infact, take the expressions O1 = (R − G) and O2 =2B − (R + G) and compute the norm L1: c = |O1|+|O2|

2 =|R−G|+|2B−(R+G)|

2 , for R > G > B to get |R − G| +|2B − (R + G)|= (R − G) + (R − B) + (G − B), andthen 2R − 2B = Max(R, G,B) − Min(R,G, B). Sim-ilarly, for the other combinations where the blue compo-nent is not the middle element, for example: B > R > Gone has R − G + 2B − (R + G), thus 2B − 2G =Max(R,G, B) − Min(R, G, B). In fact, when B is themedian element in either R > B > G or G > B > Rconfigurations, two possible expressions can be computed.For example, for R > B > G one obtains: |R−G|+ |2B−(R+G)| = |R−G|+ |(R−B)− (B−R)|. If (R−B) >(B − R), then one has 2R − 2B = 2(Max(R,G, B) −Med(R, G, B) otherwise, the saturation is given by 2B −2G = 2(Med(R,G, B) − Min(R,G, B)). Nevertheless,it is clear that for blue values close to the Max value or tothe Min value, the saturation |O1|+|O2|

2 is equal to the semi-norm Max(RGB)−Min(R, G, B). Finally, the hue defi-nition is given in terms of degrees in the perceptual spaces.Unlike all color spaces where the origin (zero degrees) is al-ways assigned to the red color, here the blue color is taken asthe origin. This decision was made by recalling that the bluecolor is less sensitive in human vision perception. However,this will enable us to be coherent with the selection of the

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(a) (b)

(c)

Figure 2. a) Original function; b) waterfallTransformation containing 3 minima; c) Firstand second critical functions Ψ1(f) and Ψ2(f)with 10 and 3 minima, respectively

opponent colors to define the saturation. In our case, thehue is specified as follows:

h = arctan(√

3O1

O2) = arctan(

√3

(R−G)(2B − (R + G)

) (3)

The images in Figs 1 (f)-(h) show the luminance Y, thesaturation c, and the hue h, respectively. Linked to this colorspace the following color distance [16] between two colorsk1 = (Y1, c1, h1) and k2 = (Y2, c2, h2) will be used:

d(k1, k2) =√

dy2 + c21 + c2

1 − (2c1c2 ∗ cos(dh)) (4)

where, dy = Y1 − Y2 and dh = h1 − h2.

4 Color Image Segmentation

Image segmentation consists in partitioning the imageinto different meaningful regions with homogeneous char-acteristics. In most cases, the segmentation of color im-ages demonstrates to be more useful than the segmenta-tion of monochrome images, because color image revealsmore features than monochrome image. A natural questionthat arises is whether one must focus on the traditional toolfor segmenting images called the watershed transform [13].Here, one illustrates some drawbacks of this transformationand propounds a segmentation method based on a mergingprocess using the notion of critical functions.

4.1 Watershed, waterfall and critical func-tions

Image segmentation based on the use of the watershedtransformation has proved to be an efficient method [13].Its main drawback consists in the over-segmentation pro-duced by the watershed transformation if applied directlyon the images to be segmented. A solution to prevent thisover-segmentation consists in a prior selection of a set ofmarkers locating the regions to be extracted. Unfortunately,because of the complexity of the regions in a real scene,this marking is often difficult and sometimes impossible.In Beucher’s work [2] another approach for eliminating theover-segmentation problem was proposed. This approachis based on a hierarchical segmentation of the image aim-ing at merging the catchment basins of the watershed imagebelonging to almost homogeneous regions. Both, the water-shed transformation and the image reconstruction R∗(f, g)are used. Consider a function f in a one-dimensional ex-ample as illustrated in Fig. 2 (a) containing 10 minima.Among the various minima of the function f, only three areinteresting because they mark regions of higher importancein the image. The other minima can be produced by noise.Consider that the function f is bound and let W(f) be its wa-tershed transformation. W(f) is the set of the watershed linesof f. Let us to build a new function g with W(f)

g(x) =

f(x) iff x ∈ W (f)

1 iff x ∈ W c(f)

This function g is obviously greater than f. Let us nowreconstruct f from g. It is easy to see that the minima ofthe resulting image correspond to the significant markers ofthe original image f. In fact, the reconstruction R∗(f, g)fills in (partially) each catchment basin with a plateau at aheight equal to the minimum height of the watershed linesurrounding this catchment basin. Therefore, if there ex-ists an adjacent catchment basin where the correspondingheight is lower than the previous one, the waterfall will notbe symmetrical and the plateau generated in the basin willnot be a minimum. Moreover, the watershed of R∗(f, g)produces the catchment basins associated with these signif-icant markers. Figure 2(b) shows the output function with3 minima computed by the waterfall transformation. Thisprocedure, the watershed followed by a morphological im-age reconstruction, is iterated until a set of homogeneouszones is achieved according to the particular image prob-lem. Recently, a fast implementation of the waterfall wasproposed in [11]. The method does not require the settingof parameters. To illustrate the waterfall algorithm appliedto color images, the gray-level image in Fig. 3(b) was com-puted by the color distance (defined in Section(3)) from theoriginal image in Fig. 3(a). Before computing the color

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distance, the color image was filtering by an alternating se-quential filter using opening an closing by reconstruction bytaking into account the total order given by the bit-mixingcodification. This filtering process removes the noise of theimage. Instead of mapping the color image according toa color distance, one can take into account the bit-mixingimage associated with the Y O1O2 color space. Neverthe-less, in the present work, the gray-level image computed bythe color distance is used. Then, after computing the mor-phological gradient (Grad(f) = δµB(f) − εµB(f)) of theimage in Fig. 3(b), the waterfall transformation was appliedto the Grad(f). The images in Figs. 3(c) and (d) show thewatersheds computed on the gradient of the original imageand on the waterfall after 3 iterations. Observe the reduc-tion in the number of regions after 3 iterations of the water-fall. To obtain the color image in Fig. 3(e), the mean colorwas computed from the original image in Fig. 3(a) for eachelement of the partition (Fig. 3(d)) and it was used to af-fect each partition element. The number of color connectedcomponents (connected regions with the same color) in Fig.3(e), permits to have a parameter representing the reductionin complexity of the output image. For example, the orig-inal image has 148, 276 connected components, while theoutput image has 412. The image in Fig. 3 (f) was obtainedafter 5 iterations of the waterfall transformation and con-tains 131 connected components. The main drawbacks ofthe waterfall algorithm is the fact of not requiring param-eters, then the segmentation can not be controlled. There-fore, another algorithm using the notion of critical functionsis proposed for obtaining an hierarchical segmentation. Thealgorithm proposed here is simpler than that proposed byBeucher [2]. Let us introduce the notion of a critical func-tions of the minima. The name of critical functions is in thesense of regional minima (or maxima) modifications. First,let us define an operator that increases the gray-level of theminima of a function f (Min(f )) given by the following op-erator:

h(f)(x) =

f(x) + 1 iff x ∈ Min(f)

f(x) otherwise

Now, we define an operator that increases the gray-levelof the minima if and only if the minima will not disappear.

Ψ1(f)(x) =

f(x) + 1 iff x ∈ Min(f) ∩Min(h)

f(x) otherwise

In the second step one has Ψ2(f)(x) = Ψ1{Ψ1(f)}(x).At the nth step when the stability is reached, the firstcritical function of the minima is obtained; Ψ1(f)(x) =Ψn(f)(x) = Ψ1{Ψn(f)}(x). To obtain the second criticalfunction one must increase the minima by one and then ap-ply again the critical function Ψ2(f) = ΨnhΨ1(f) . Figure

(a) (b) (c)

(d) (e) (f)

(g) (h)

Figure 3. a) Original image; b) Distance im-age; c) and d) Watersheds of the originalfunction and of the waterfall after 3 iterations;e) and f) Output images computed by the wa-terfall transformation with 3 and 4 iterations,respectively; g) and h) Output images com-puted by the critical functions for levels 3 and4 of the hierarchy, respectively.

2 (c) shows the first and the second critical function Ψ1(f).Although the sizes of the minima of the second critical func-tion and the output image computed by the waterfall trans-formation are different, both algorithms detect in this casethe same minima. Let us know introduce the hierarchy gen-erated by the critical functions. First, we remember thata pyramid of operators is a family {Γλ} depending on apositive parameter λ, with µ ≥ λ ≥ 0, where there existsν ≥ 0, such that Γµ(f) = ΓνΓλ(f). Then, a compositionof critical functions defines a pyramid which implies thateach level resumes the past.

Property 1 Let {Ψi(f)} be a family of critical functions.The family of critical functions {Ψi(f)}, given by Ψi(f) =ΨnhΨi−1(f) with Ψ1(f) = Ψ(f), defines a pyramid ofadditive law. This means, for each µ ≥ λ ≥ 0, there existsν ≥ 0, such that Ψµ(f) = ΨνhΨλ(f), with µ = ν + λ,where: Ψi(f) ≤ · · · ≤ Ψ2(f) ≤ Ψ1(f)

By duality, the critical function of the maxima is defined byΨ∗i (f) = [Ψi(f c)]c. Consider the minima of a critical func-tion at level n of the hierarchy, given by Ψn(f). The minimafor each element of this family are the zones in which afterthe arithmetical addition of one gray level, one of the fol-lowing situations may occur: a) the minima disappear andb) the minima are merged with adjacent minima and theyremain minima at an upper gray level. Figures 3 (g) and(h) illustrate two levels of the hierarchy (Ψ3 and Ψ4). Thecritical functions were computed on the gradient of the im-

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(a) (b) (c)

(d) (e) (f)

Figure 4. a) Original image; b), c) and d) Threehierarchy levels (Ψ2, Ψ3, Ψ4, respectively) ofthe critical functions; e) and f) Two levels ofidempotent critical functions (Ψm1 and Ψm2 ,)

age in Fig. 3 (b). Once the critical functions are obtained,the watershed is applied to the critical function to illustratethe segmented image. The output images have 179 and 39color connected components, respectively. The critical out-put images are easily obtained by using hierarchical queues(this data structure has been shown to be extremely efficientin mathematical morphology). Unlike the waterfall trans-formation, one can introduce neighborhood relation criteriato decide the minima that will be attenuated by the operator.This will be shown in the next section.

4.2 Idempotent critical functions based oncolor distance

Similarly to the waterfall transformation, the criticalfunctions do not have a manner to control the fusion pro-cess described above. Thus, if these transformations areiterated until the stability is achieved, the output image willbe composed of only one region (the whole image). It isclear that Ψi(f) 6= Ψi−1(f) at all level i of the hierarchy,and Ψi(f) = Ψi−1(f) when there are no more regions to beremoved. Then, to compute a good partition of the image,one requires to control the merging process in such a waythat at a given level of the hierarchy Ψi(f) = Ψi−1(f), withΨi(f) containing the principal regions of the image. Theseregions correspond to different levels of the hierarchy. Away of computing an idempotent critical function consistsin introducing a distance criterion that permits to take thedecision of merging the regions (minima). Figure 4 illus-trates the output images computed with and without dis-tance criterion. The images in Figs. 4(b)-(d) were computed

with 3, 4, and 5 critical functions, respectively, without us-ing a distance criterion. The number of regions (color con-nected components) in the original image is 408,456, whilethat in the output images in Figs. 4(b)-(d) are 3220, 456 and65, respectively. Observe, that several important regionsof the children’s faces (mouths, noses, eyes),particularly inFig. 4(d), have been removed. By introducing a gray-leveldistance between the minima to be merged, the images inFigs. 4(e) and (f) were obtained by applying idempotentcritical functions Ψmn with distance criteria mn of 20 and30, respectively (until stability is reached). The output im-ages contain 505 and 203 regions. Even if some regionsof the children’s faces were preserved, they would be de-graded. Thus, the color distance introduced in Section(3) isapplied. In this case the criterion based on the color measureis used to take the decision of whether a minimum can bemerged. This means that if a minimum is merged with otherminima to form a single one a color distance must be com-puted between the color of the original image of this min-imum and color of the neighboring regions. If the largestcolor distance is above a given value, then the minima willbe preserved, otherwise they will be merged. Images inFigs. 5(b) and (c), show the output images computed byboth criteria, gray and color distances, respectively. Theimage in Fig. 5(b), computed by a gray-level distance con-tains 116 regions, while the image in Fig. 5(c) has 106 re-gions. Observe in Fig. 5(b), that zones with color and with-out color are merged since a gray-level distance was used.Conversely, when a color distance is applied (Fig. 5(c)), thismerging process does not occur. Finally, a last example us-ing the color distance criterion is shown in Fig. 5(d) and (e).The image in Fig. 5(d) was computed using a criterion valueof 30, whereas the image in Fig. 5(e) was obtained using acriterion value of 35. The number of regions of these outputimages is 357 and 209, respectively. In particular, comparethe image in Fig. 5(e) containing 209 regions, with that inFig. 4(e) with 203 regions. Even though the number of re-gions is practically similar, the output image using a colordistance preserves better the features than when a gray-leveldistance was used.

5 Conclusion

In the present paper the notion of critical functions forsegmenting color images was investigated. The color imagesegmentation study was carried out on a new color spacebased on opponent colors Y O1O2. The performance of thecritical functions for segmenting images was compared withthe waterfall transformation. In fact, critical functions pro-vide better results for segmenting images than the waterfalltransformation, since they take into account the fusion ofregional minima (or maxima) to transform the image, en-abling a good control of the merging process of the regions.

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(a) (b) (c)

(d) (e)

Figure 5. a) Original image; b), c) Idempotentcritical functions using gray-level and colordistances, respectively; d) and e) Two hierar-chy levels of idempotent critical functions

Moreover, contrary to the waterfall transformation that isnon-parametric, the use of critical functions permits to eas-ily introduce image segmentation criteria in order to com-pute an image partition. The performance of both proposals,the color model and critical functions, for image segmen-tation was illustrated with several examples. Also, someinteresting properties of the color model Y O1O2 and thecritical functions were shown.

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