Uniform Color Spaces Based on CIECAM02 and IPT Color Difference Equations

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Theses

11-1-2008

Uniform color spaces based on CIECAM02 and IPT color difference equations

Yang Xue

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Uniform Color Spaces Based on CIECAM02 and IPT Color Difference Equations

Yang Xue

A thesis submitted for partial fulfillment of the requirements for the degree of Master in the Chester F. Carlson Center for Imaging Science of the College of Science Rochester Institute of Technology

November 2008

Signature of the Author______

Accepted By______

Coordinator, M.S. Degree Program Date

i CHESTER F. CARLSON CENTER FOR IMAGING SCIENCE COLLEGE OF SCIENCE ROCHESTER INSTITUTE OF TECHNOLOGY ROCHESTER, NEW YORK

CERTIFICATE OF APPROVAL

M.S. DEGREE THESIS

The M.S. Degree Thesis of Yang Xue has been examined and approved by the thesis committee as satisfactory for the thesis requirement for the Master of Science degree in Imaging Science.

______Prof. Roy S.Berns, Thesis Advisor

______Prof. Mark D. Fairchild, Committee Member

______Prof. Mitchell R. Rosen, Committee Member

THESIS RELEASE PERMISSION ROCHESTER INSTITUTE OF TECHNOLOGY COLLEGE OF SCIENCE CHESTER F. CARLSONCENTER FOR IMAGING SCIENCE

Title of Thesis: Uniform Color Spaces Based on IPT and CIECAM02 Color Difference Equations

I, Yang Xue, hereby grant permission to the Wallace Memorial Library of R.I.T. to reproduce my thesis in whole or in part. Any reproduction will not be for commercial use or profit.

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iii Uniform Color Spaces Based on CIECAM02 and IPT Color Difference Equations

Yang Xue

Submitted for partial fulfillment of the requirements for the degree of Master in the Chester F. Carlson Center for Imaging Science of the College of Science Rochester Institute of Technology

November, 2008

ABSTRACT

Color difference equations based on the CIECAM02 color appearance model and IPT color space have been developed to fit experimental data. There is no color space in which these color difference equations are Euclidean, e.g. describe distances along a straight line. In this thesis, Euclidean color spaces have been derived for the CIECAM02 and IPT color difference equations, respectively, so that the color difference can be calculated as a simple color distance. Firstly, the Euclidean line element was established, from which terms were derived for the new coordinates of lightness, chroma, and hue angle. Then the spaces were analyzed using performance factors and statistics to test how well they fit various data. The results show that the CIECAM02 Euclidean color space has performance factors similar to the optimized CIECAM02 color difference equation. To statistical significance, the CIECAM02 Euclidean color space had superior fit to the data when compared to the CIECAM02 color difference equation. Conversely, the IPT Euclidean color space performed poorer than the optimized IPT color difference equation. The main reason is that the line element for the lightness vector dimension could not be directly calculated so an approximation was used. To resolve this problem, a new IPT color difference equation should be designed such that line elements can be established directly.

iv ACKNOWLEDGEMENTS

I would like to express my gratitude to my advisor, Prof. Roy.S Berns, for his very helpful and

invaluable support and guidance to my research. Without his constructive advice and

encouragement, I would not finish my thesis.

I also want to thank my committee member: Prof. Mark Fairchild and Prof. Mitch Rosen.

Thanks for their helps in my study in RIT and taking time to review my thesis.

This research was supported by the E. I. du Pont de Nemours and Company Graduate

Fellowship in Color Science. Their support is greatly appreciated.

I am very honored to have opportunity to study in Munsell Color Science Laboratory. Being a

member in this big family is an unforgettable memory in my life. My special thanks to

everyone in MCSL: the faculty, staff, and my officemates.

Lastly, and most importantly, my deepest thanks are given to my parents and my husband.

Thanks for their understanding, support, and endless love. I love you all and you are the most

important part in my life.

v Table of Contents

1 Introduction...... 1 2. Background ...... 4 2.1 IPT Color Space ...... 7 2.2 CIECAM02 Color Space ...... 8 2.2.1 Forward Model...... 8 2.2.2 Inverse Model ...... 13 2.3 Color-Difference Equations with Improved Performance ...... 16 2.3.1 CIE94...... 16 2.3.2 CIEDE2000...... 18 2.3.3 DIN99 ...... 21 2.3.4 Modification of CIECAM02...... 22 2.4 Euclidean Spaces...... 23 2.4.1 A Euclidean Color Space by Thomsen ...... 23 2.4.2 Euclidean Color Space based on DIN99...... 24 3. RIT-Dupont Dataset...... 28 3.1 Background...... 28 3.2 Color-Difference Ellipsoids ...... 33 3.3 CIECAM02...... 36 3.4 IPT...... 39 4. Qiao and Berns Dataset ...... 44 4.1 Background...... 44 4.2 Plots of Qiao Experimental Results in Color Spaces ...... 46 5. Hung and Berns Dataset...... 50 5.1 Background...... 50 5.2 Hung Data Plots in CIELAB, CIECAM02, and IPT Color Spaces...... 51 5.3 Conclusion...... 53 6. Optimizing Color-Difference Formula ...... 54 6.1 Background...... 54 6.2 CIECAM02...... 55 6.3 IPT...... 61 7. Euclideanization...... 65 7.1 Theory ...... 65 7.2 Derivation of CIECAM02 Euclidean ...... 68 7.3 Derivation of IPT Euclidean ...... 72 8. Performance Evaluation...... 76

vi 8.1 Computation Experiment...... 76 8.2 Analysis ...... 81 9. Performance of Visual Dataset...... 85 9.1 Visualization of RIT-DuPont Dataset...... 85 9.2 Visualization of Qiao Dataset...... 100 9.3 Visualization of Hung and Berns Dataset ...... 106 10. Conclusions ...... 108 11. References ...... 110

vii 1 Introduction

A color appearance1 model is used to model how the human visual system perceives the color of an object under different viewing conditions and with different backgrounds. TC1-34 gave the following definition:2 “to be considered a color appearance model, a model must account for at least chromatic adaptation and have correlates of at least lightness, chroma, and hue.” As a simple color appearance model, the CIE 1976 (L* a* b*) color space (CIELAB)3 was designed to fit the Munsell color order system. It includes simple chromatic adaptation transforms and predictors of lightness, chroma, and hue. However, CIELAB cannot predict luminance dependent effects such as the Hunt effect and the Stevens effect. Moreover, CIELAB does not provide correlates for the absolute appearance attributes of brightness and colorfulness.

In 1997, the CIE recommended an interim color appearance model, CIECAM97s, which can predict corresponding colors for imaging application. In 2002, a revision of

CIECAM97s,4 CIECAM02,5,6 was developed to improve the performance accuracy and simplify the model.

Hue constancy is an important property of a color space. Compared with lightness and chroma, hue angle is more difficult to mathematically manipulate. In 1998, Ebner and

Fairchild derived the IPT color space,7 which models constant perceived hue. This model was designed to accurately predict hue without affecting other color appearance attributes.

1 In a uniform color space, the Euclidean distance between the coordinates of any two colors corresponds to a perceptual color difference. In other words, the calculated distance should predict visual color difference between the colors. For example, if there are two colors P1(x1,y1,z1) and P2(x2,y2,z2) in a uniform color space, the perceived difference magnitude, ΔE, between the two colors P1,P2 is the Euclidean distance:

2 2 2 ΔE = k (x1 − x2 ) + (y1 − y2 ) + (z1 − z2 ) ,

where k is a constant.

In CIELAB color space, the Euclidean distance between two colors is:

* * 2 * 2 * 2 * 2 * 2 * 2 ΔE ab = (ΔL ) + (Δa ) + (Δb ) = (ΔL ) + (ΔCab ) + (ΔHab ) ,

where ΔL*, Δa*, and Δb* is the difference in lightness, redness-greenness, and

* * yellowness-blueness, respectively. ΔCab and ΔHab is the difference in chroma and hue.

But, this Euclidean distance poorly relates to visual color difference8. Thus, several color difference formulas such as CMC,9 CIE94,9 and CIEDE200010 have been developed using the CIELAB color space. These formulas were optimized to fit visual experimental data sets. However, none of these associate with a uniform color space.

Color difference formulas based on the CIECAM02 color appearance model and IPT color space have been developed to fit experimental data from RIT.11 These color equations have the same forms as CIE94 and CIEDE2000. Thus, they are not

2 Euclidean. In this thesis, Euclidean color spaces based on the CIECAM02 and IPT color difference formulas were derived, so that color difference can be calculated as a simple color distance. In addition, two general approaches were given to evaluate the performance of their new spaces. One approach was to statistically evaluate how well the new Euclidean color spaces fit the experimental data. The other was to visualize experimental data in new Euclidean color spaces.

3 2. Background

Color appearance models are used to extend traditional colorimetry (such as CIE XYZ

and CIELAB9) and to predict observed appearance of objects under a wide variety of

viewing conditions.

The CIE XYZ color space is based on measurements of the human visual perception.

CIE used positive integers to formulate tristimulus values X,Y, and Z. These values do

not correspond to red, green, and blue. Y represents the human eye’s response to the

total power of a light source. The CIE XYZ color space is not a uniform color space.

After the introduction of CIE XYZ in 1931, much work went into searching for a

transformation to a uniform color space.9,12 CIE developed CIELAB color space in

1970s. The CIELAB color space takes the XYZ tristimulus values of a stimulus and the

reference white as input and outputs correlates to lightness, chroma, and hue. Although

the CIELAB color space is more uniform than CIE XYZ color space, it is not a uniform

color-difference space.8 Furthermore, it cannot predict luminance-dependent effects such

as the Hunt effect and the Stevens effect.1

In order to predict the observed appearance of objects, color appearance models must

take into account the tristimulus values of the object, its background, its surround, the

adapting stimulus and the luminance level.1 The output of color appearance models

should include perceptual attributes such as brightness, lightness, colorfulness, chroma,

saturation, and hue.

4 In 1996, Hunt presented twelve principles13 for consideration in establishing a single color appearance model at a CIE expert symposium ‘96, Colour Standards for Image

Technology. These principles are as follows:

1. The model should be as comprehensive as possible, so that it can be used in a variety of applications; but at this stage, only static states of adaptation should be included, because of the great complexity of dynamic effects.

2. The model should cover a wide range of stimulus intensities, from very dark object colors to very bright self-luminous color. This means that the dynamic response function must have a maximum, and cannot be a simple logarithmic or power function.

3. The model should cover a wide range of adapting intensities, from very low scotopic levels, such as occur in starlight, to very high photopic levels, such as occur in sunlight.

This means that rod vision should be included in the model; but because many applications will be such that rod vision is negligible, the model should be usable in a mode that does not include rod vision.

4. The model should cover a wide range of viewing conditions, including backgrounds of different luminance factors, and dark, dim, and average surrounds. It is necessary to cover the different surrounds because of their widespread use in projected and self- luminous displays.

5. For ease of use, the spectral sensitivities of the cones should be a linear transformation of the CIE x,y,z or x10, y10, z10 functions, and the V(λ) function should be used for the spectral sensitivity of the rods. Because scotopic photometric data is often unknown, methods of providing approximate scotopic values should be provided.

6. The model should be able to provide for any degree of adaptation between complete

5 and none, for cognitive factors, and for the Helson-Judd effect, as options.

7. The model should give predictions of hue (both as hue-angle, and as hue-quadrature), brightness, lightness, saturation, chroma, and colorfulness.

8. The model should be capable of being operated in a reverse mode.

9. The model should be no more complicated than is necessary to meet the above requirements.

10. Any simplified version of the model, intended for particular applications, should give the same predictions as the complete model for some specified set of conditions.

11. The model should give predictions of color appearance that are not appreciably worse than those given by the model that is best in each application.

12. A version of the model should be available for application to unrelated colors (those seen in dark surrounds in isolation from other colors).

After the symposium, Hunt and Luo provided two revised models. Fairchild provided a third alternative and Richter provided a fourth.1 These four alternatives were considered at the TC1-34 meeting in 1997. One of the Hunt and Luo alternatives was modified as the simple form of the CIE 1997 Interim Color Appearance Model, tentatively designated CIECAM97s. After that, a comprehensive version of the model,

CIECAM97c, was developed. In 2002, the CIE provided CIECAM02, which was revision to CIECAM97s. CIECAM02 can be used for comparison of device gamuts based on a set of perceptual attribute correlates. It is also used as the color space for gamut mapping. Besides, CIECAM02 can be useful for color management applications.

6 Prior to the development of CIECAM02, in 1997, Ebner and Fairchild derived the IPT color space to model constant perceived hue. The model accurately predicts hue without affecting other color appearance attributes.7

2.1 IPT Color Space

In the IPT color space,7 I, P, and T coordinates represent the lightness dimension, the red-green dimension and the yellow-blue dimension, respectively. IPT is also short for

Image Processing Transform since it is useful for transformations such as gamut mapping.

The model consists of a (3×3) matrix, followed by a nonlinearity and another (3×3) matrix. The model assumes input data is in CIEXYZ for the 1931 2-deg observer with an illuminant of D65. The parameters are shown in matrix form in equation 2-1.

⎡L ⎤ ⎡0.4002 0.7075 −0.0807⎤⎡ X D65⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ M = −0.228 1.1500 0.0612 Y ⎢ ⎥ ⎢ ⎥⎢ D65 ⎥

⎣⎢S ⎦⎥ ⎣⎢ 0.0 0.0 0.9184 ⎦⎥⎣⎢ ZD65 ⎦⎥ L'= L0.43; L ≥ 0 L'=−(−L)0.43; L < 0 M'= M 0.43; M ≥ 0 Eq.2-1 M'=−(−M)0.43; M < 0 S'= S 0.43; S ≥ 0 S'=−(−S)0.43; S < 0 ⎡I ⎤ ⎡0.4000 0.4000 0.2000 ⎤⎡ L' ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢P ⎥ = ⎢4.4550 −4.8510 0.3960 ⎥⎢ M'⎥ ⎣⎢T ⎦⎥ ⎣⎢0.8056 0.3572 −1.1628⎦⎥⎣⎢ S' ⎦⎥

7 The first (3×3) matrix, L,M, and S is very near the Hunt-Pointer-Estevez cone primaries normalized to D65.14 The compression factor (0.43) is nearly identical to that of the

RLAB color space for average surround conditions.14

This model is readily invertible. The range of lightness axis, I, is 0 to 1 and the range of the other two axes is -1 to 1 in the IPT color space.

2.2 CIECAM02 Color Space

In CIECAM02,5,6 due to different viewing conditions, the CIE considered the differences in color perception by transforming to and from a single reference white. The forward model transforms tristimulus values viewed under a wide range of viewing conditions to the corresponding perceptual attribute correlates as viewed under a reference white, in this case the equal-energy illuminant (illuminant E) with the perfect reflecting diffuser as the reference white. The inverse model transforms from this reference white to some other viewing condition.

2.2.1 Forward Model

Since the surround setting impacts the calculations significantly, two parameters, the constants c and F, are given to relate to the impact of the surround and the degree of adaptation.

8 Firstly, the recommended values for F, c, and Nc can be read from Table 2-I after the surround is selected. For intermediate surrounds these values can be linearly interpolated.

Table 2-I Viewing condition parameters for different surrounds.

Viewing Condition c Nc F Average Surround 0.69 1.0 1.0 Dim Surround 0.59 0.9 0.9 Dark Surround 0.525 0.8 0.8

Then the CAT0215 forward matrix converts the sample CIE 1931 tristimulus values to a long, medium and short wavelength sensitive space.

⎡R ⎤ ⎡ X⎤ ⎢ ⎥ ⎢ ⎥ G = M Y Eq.2-2 ⎢ ⎥ CAT 02⎢ ⎥ ⎣⎢B ⎦⎥ ⎣⎢Z ⎦⎥

⎡ 0.7328 0.4296 −0.1624⎤ ⎢ ⎥ M = −0.7036 1.6975 0.0061 Eq.2-3 CAT 02 ⎢ ⎥ ⎣⎢ 0.0030 0.0136 0.9834 ⎦⎥

The degree of adaptation to the white point, the D factor, is computed. The range of D value will be from 1 for complete adaptation to 0 for no adaptation. In practice, the minimum D value will not be less than 0.65 for a dark surround and will exponentially

15 converge to 1 for average surrounds with increasingly large values of LA.

⎛ ⎞ ⎡ −(LA +42) ⎤ ⎛ 1 ⎞ ⎜ ⎟ D = F⎢1 −⎜ ⎟e ⎝ 92 ⎠ ⎥ Eq.2-4 ⎣⎢ ⎝ 3.6⎠ ⎦⎥

After D factor is calculated, it is applied to weight chromatic adaptation:

9 ⎡⎛ D ⎞ ⎤ Rc = ⎢⎜ Y w ⎟ + (1 − D)⎥R ⎣⎝ Rw ⎠ ⎦ ⎡⎛ D ⎞ ⎤ Gc = ⎢⎜ Y w ⎟ + (1 − D)⎥G Eq.2-5 ⎣⎝ Gw ⎠ ⎦ ⎡⎛ D ⎞ ⎤ Bc = ⎢⎜ Y w ⎟ + (1 − D)⎥B ⎣⎝ Bw ⎠ ⎦ where Rw, Gw, and Bw are the RGB values computed for the white point using equations

2-2 and 2-3.

Next, the viewing-condition-dependent constants are computed as:

1 k = Eq.2-6 5LA +1

4 4 2 1/3 FL = 0.2k (5LA ) + 0.1(1− k ) (5LA ) Eq.2-7

Y n = b Eq.2-8 Yw

The value n is a function of the luminance factor of the background. Its range is from 0 for a background luminance factor of zero to 1 for a background luminance factor equal to the luminance factor of the adapted white point. The n value can then be used to compute Nbb, Ncb and z, which are then used during the computation of several of the perceptual attribute correlates.

1 N = N = 0.725( )0.2 Eq.2-9 bb cb n

z =1.48 + n Eq.2-10

The Rc,Gc and Bc values are converted to the Hunt-Pointer-Estevez space before the post-adaptation non-linear response compression is applied.

10 ⎡R '⎤ ⎡R c ⎤ ⎢ ⎥ ⎢ ⎥ G' = M M −1 G Eq.2-11 ⎢ ⎥ HPE CAT 02⎢ c ⎥

⎣⎢B '⎦⎥ ⎣⎢B c ⎦⎥

⎡ 0.3897 0.6889 −0.0787⎤ ⎢ ⎥ M = −0.2298 1.1834 0.0464 Eq.2-12 HPT ⎢ ⎥ ⎣⎢ 0.0000 0.0000 1.0000 ⎦⎥

⎡ 1.0961 −0.2789 0.1827⎤ ⎢ ⎥ M −1 = 0.4544 0.4735 0.0721 Eq.2-13 CAT 02 ⎢ ⎥ ⎣⎢− 0.0096 −0.0057 1.0153⎦⎥

The Hunt-Pointer-Estevez provides better predictions of perceptual attribute correlates.

Blue constancy, a significant shortcoming for CIELAB, is improved by using a space closer to a cone fundamental space.

The post-adaptation non-linear response compression is then applied to the output.

0.42 ' 400(FL R'/100) Ra = 0.42 + 0.1 27.13+ (FL R'/100) 0.42 ' 400(FLG'/100) Ga = 0.42 + 0.1 Eq.2-14 27.13+ (FLG'/100) 0.42 ' 400(FL B'/100) Ba = 0.42 + 0.1 27.13+ (FL B'/100)

If any of the values of R’, G’ or B’ are negative, their absolute values must be used, and the quotient term in Equation 2-14 must be multiplied by a negative 1 before adding the value 0.1.

11 Temporary Cartesian representation and hue are calculated before computing eccentricity factor and perceptual attributes. These values are used to compute a preliminary magnitude t.

' ' ' a = Ra −12Ga /11+ Ba /11 ' ' ' a = (1 / 9)(Ra + Ga − 2Ba ) Eq.2-15 h = tan−1(b/a)

cos(hπ /180 + 2) + 3.8 e = Eq.2-16 t 4

Hue quadrature or H can be computed from linear interpolation of the data shown in

Table 2-II.

Table 2-II. Unique hue data for the calculation of Hue Quadrature

Red Yellow Green Blue Red i 1 2 3 4 5

hi 20.14 90.00 164.25 237.53 380.14

ei 0.8 0.7 1.0 1.2 0.8

Hi 0.0 100.0 200.0 300.0 400.0

If h

100(h'−hi )/ei H = Hi + Eq.2-17 (h'−hi )/ei + (hi+1 − h')/ei

Compute the achromatic response A:

' ' ' A = (2Ra + Ga + (1 / 20)Ba − 0.305)Nbb Eq.2-18

Lightness, J, is calculated from the achromatic signals of the stimulus, A, and white, Aw:

CZ J =100(A / Aw ) Eq.2-19

Compute Q or brightness:

4 J Q = (A + 4)F 0.25 Eq.2-20 w L c 100

12 Compute temporary magnitude quantity, T. This quantity is used for computing C.

2 2 (50000/13)NcNcbet a + b t = ' ' ' Eq.2-21 Ra + Ga + (21/20)Ba

Calculate chroma C:

J C = (1.64 − 0.29n )0.73 t 0.9 Eq.2-22 100

Calculate colorfulness M:

0.25 M = CFL Eq.2-23

Calculate saturation s:

M s =100 Eq.2-24 Q

Calculate corresponding Cartesian coordinates as necessary:

a = Ccos(h) c Eq.2-25 bc = Csin(h)

a = M cos(h) M Eq.2-26 bM = M sin(h)

a = scos(h) s Eq.2-27 bs = ssin(h)

2.2.2 Inverse Model

The viewing condition constants shown in Table 2-I and the viewing condition dependent parameters shown in equations 2-6 through 2-10 are used in the inverse model. The value of Aw is computed for the adapted white point using equations 2-2 through 2-23.

Then, if starting from Q, J can be calculated:

13 2 ⎛ cQ ⎞ J = 6.25⎜ 0.25 ⎟ Eq.2-28 ⎝ (Aw + 4)FL ⎠

If starting from M, C can be calculated:

M C = 0.25 Eq.2-29 FL

If starting from s, C can be calculated:

4 J Q = (A + 4)F 0.25 Eq.2-30 w L c 100

⎛ s ⎞ 2 Q C = ⎜ ⎟ 0.25 Eq.2-31 ⎝ 100⎠ FL

From Table 2-II,

(H − H )(e h − e h ) −100h e h'= i i+1 i i i+1 i i+1 Eq.2-32 (H − Hi )(ei+1 − ei ) −100ei+1

Then, t,e,p1, p2, and p3 are calculated:

1 ⎛ C ⎞ 0.9 t = ⎜ ⎟ Eq.2-33 ⎝ J /100(1.64 − 0.29n )0.73 ⎠

cos(hπ /180 + 2) + 3.8 e = Eq.2-34 t 4

1 ⎛ J ⎞ CZ A = A ⎜ ⎟ Eq.2-35 w⎝ 100⎠

(50000/13)N N e p = c cb t 1 t A p2 = + 0.305 Eq.2-36 Nbb

p3 = 21/20

Calculate a and b:

14 π h = h Eq.2-37 r 180 if |sin(hr)|≥ |cos(hr)|

p1 p4 = sin(hr ) p (2 + p )(460/1403) b = 2 3 p4 + (2 + p3 )(220/1403) × []cos(hr )/sin(hr ) − (27/1403) + p3 (6300/1403)

a = b[]cos(hr )/sin(hr )

if |sin(hr)|< |cos(hr)|

p1 p5 = cos(hr ) p (2 + p )(460/1403) a = 2 3 p5 + (2 + p3 )(220/1403) − [](27/1403) + p3(6300/1403) × []cos(hr )/sin(hr )

b = a[]sin(hr )/cos(hr )

’ ’ ’ Compute R a, G a, and B a:

460 451 288 R' = p + a + b a 1403 2 1403 1403 460 891 261 G' = p − a + b Eq.2-38 a 1403 2 1403 1403 460 220 6300 B' = p − a − b a 1403 2 1403 1403

Compute R’,G’ and B’:

1 ⎛ ' ⎞ 0.42 100 27.13Ra − 0.1 R' = ⎜ ⎟ F ⎜ ' ⎟ L ⎝ 400 − Ra − 0.1⎠ 1 ⎛ ' ⎞ 0.42 100 27.13Ga − 0.1 G' = ⎜ ⎟ Eq.2-39 F ⎜ ' ⎟ L ⎝ 400 − Ga − 0.1⎠ 1 ⎛ ' ⎞ 0.42 100 27.13Ba − 0.1 B' = ⎜ ⎟ F ⎜ ' ⎟ L ⎝ 400 − Ba − 0.1⎠

15 ’ ’ ’ If any of the values of (R a-0.1), (G a-0.1), and (B a-0.1) are negative, then the corresponding value R’, G’ or B’ must be made negative.

Compute Rc, Gc, and Bc:

⎡R c ⎤ ⎡R '⎤ ⎢ ⎥ ⎢ ⎥ G = M M −1 G' Eq.2-40 ⎢ c ⎥ CAT 02 HPE ⎢ ⎥

⎣⎢B c ⎦⎥ ⎣⎢B '⎦⎥

⎡1.9101 −1.1121 0.2019⎤ ⎢ ⎥ M −1 = 0.3709 0.6291 0.0000 Eq.2-41 HPT ⎢ ⎥ ⎣⎢0.0000 0.0000 1.0000⎦⎥

Compute R, G, and B and X,Y and Z:

R R = c ()YwD/Rw +1− D G G = c Eq.2-42 ()YwD/Gw +1− D B B = c ()YwD/Bw +1− D

⎡ X⎤ ⎡R ⎤ ⎢ ⎥ ⎢ ⎥ Y = M −1 G Eq.2-43 ⎢ ⎥ CAT 02⎢ ⎥ ⎣⎢Z ⎦⎥ ⎣⎢B ⎦⎥

2.3 Color-Difference Equations with Improved Performance

2.3.1 CIE94

When the CIE94 color-difference formula was recommended, CIE defined a general form for a weighted CIELAB color-difference equation:16

2 2 2 1/2 ⎡ * * * ⎤ 1 ⎛ ΔL ⎞ ⎛ ΔC ⎞ ⎛ ΔH ⎞ ΔE = ⎢⎜ ⎟ + ⎜ ab ⎟ + ⎜ ab ⎟ ⎥ Eq.2-44 kE ⎣⎢⎝ kL SL ⎠ ⎝ kC SC ⎠ ⎝ k H SH ⎠ ⎦⎥

16 where,

* * 2 * 2 Cab = (a ) + (b ) * * * ΔL = L1 − L2 * * * ΔC = C1 − C2 ⎛ Δh⎞ ΔH* = 2 C*C* sin⎜ ⎟ 1 2 ⎝ 2 ⎠

Δh = h1 − h2

In this equation, the lightness, chroma, and hue differences are computed from CIELAB.

This color-difference equation requires the definition of a set of reference conditions.

17 Coefficients KE, KL, KC , and KH represent parametric factors. If the color difference stimuli correspond to the set of reference conditions, parametric factors are unnecessary.

The positional functions SL, SC, and SH correct CIELAB’s lack of visual uniformity for the same set of reference conditions.

18,19 CIE used RIT-DuPont and Luo-Rigg dataset to develop SL, SC, and SH. RIT-DuPont and Luo-Rigg20 dataset were more consistent in their chroma position dependencies.

Chroma difference and hue difference were dependent consistently on chroma position.

It was suggested that the lightness function would be a constant of unity and the chroma and hue functions would both be linear functions dependent only on chroma position.

The RIT-DuPont and Luo-Rigg datasets were used to derive the other parameters for the

CIE94 color-difference equation.

17 1/2 ⎡⎛ * ⎞ 2 ⎛ * ⎞ 2 ⎛ * ⎞ 2⎤ * ΔL ΔCab ΔHab ΔE94 = ⎢⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ ⎥ ⎣⎢⎝ kL SL ⎠ ⎝ kC SC ⎠ ⎝ k H SH ⎠ ⎦⎥

SL =1 * Eq.2-45 SC =1+ 0.045Cab * SH =1+ 0.015Cab

kL = kC = k H =1 for reference conditions * * * * Cab = Cab,standard or Cab,1Cab,2

* When one of the color samples is the standard, SC and SH are calculated based on the C ab value of the standard. When neither sample is a standard, the geometric mean chroma is used to calculate SC and SH.

2.3.2 CIEDE2000

CIE published CIEDE200010 color-difference equation in Technical Report CIE 142-

2001. CIEDE2000 has the form:

1/2 ⎡ * 2 * 2 * 2 ⎤ ⎛ ΔL ⎞ ⎛ ΔC ⎞ ⎛ ΔH ⎞ ΔE = ⎢⎜ ⎟ + ⎜ ab ⎟ + ⎜ ab ⎟ +ΔR⎥ Eq.2-46 ⎣⎢⎝ kL SL ⎠ ⎝ kC SC ⎠ ⎝ k H SH ⎠ ⎦⎥ where,

* * ΔR = RT f (ΔC ΔH ),

ΔR is an interactive term between chroma and hue differences, and the RT function is intended to improve the performance of a color-difference equation for fitting chromatic differences in the blue region.

Given a pair of color values in CIELAB,

18 L'= L* a'= (1 + G)a* b'= b* C'= a'2 +b'2 Eq.2-47 h'= tan−1(b'/a'), where

⎛ 7 ⎞ C* G = 0.5⎜ 1− ab ⎟, ⎜ * 7 7 ⎟ ⎝ Cab + 25 ⎠

* * and where Cab is the arithmetic mean of the Cab values for a pair of samples.

ΔL* = L* − L* 2 1 * * * ΔC = C2 − C1

⎧ 0 C'C' = 0 ⎪ 1 2 ' ' ' ' ' ' ⎪ h2 − h1 C1C2 ≠ 0; h2 − h1 ≤180 ' ⎪ Δh = ⎨ ' ' ' ' ' ' ⎪ h2 − h1 − 360 C1C2 ≠ 0; ()h2 − h1 >180 ⎪ ' ' ' ' ' ' ⎩⎪ h2 − h1 + 360 C1C2 ≠ 0; ()h2 − h1 <−180

⎛ ' ⎞ * * * Δh ΔH = 2 C1 C2 sin⎜ ⎟ ⎝ 2 ⎠

Equation 2-43 can now be rewritten for CIEDE2000 as:

2 2 2 1/2 ⎡⎛ * ⎞ ⎛ * ⎞ ⎛ * ⎞ * * ⎤ ΔL ΔCab ΔHab ΔCabΔHab ΔE00 = ⎢⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + RT ⎥ Eq.2-48 ⎣⎢⎝ kL SL ⎠ ⎝ kC SC ⎠ ⎝ k H SH ⎠ kC SC k H SH ⎦⎥ where,

0.015(L' − 50)2 SL =1+ 20 + (L' − 50)2 and

SC =1+ 0.045C'

19 and

SH =1+ 0.015C'T , where

T =1− 0.17cos(h' − 30o ) + 0.24cos(2h') + 0.32cos(3h' + 6o ) − 0.2cos(4h' − 63o ) and

RT = −sin(2Δθ)RC , where

Δθ = 30exp{−[(h' − 275o )/25]2} and

C'7 R = 2 C C'7 +257

L', C', and h' are the arithmetic means of the L’, C’ and h’ values for a pair of samples.

L' = (L* + L*)/2 2 1 ' ' C' = (C2 − C1)/2

⎧ ' ' h1 + h2 ' ' ' ' ⎪ h1 − h2 ≤180; C1C2 ≠ 0 ⎪ 2 ⎪ h' + h' + 360 ⎪ 1 2 h' − h' >180; h' + h' < 360; C'C' ≠ 0 h' = ⎨ 2 1 2 ()1 2 1 2 ⎪ ' ' h1 + h2 − 360 ' ' ' ' ' ' ⎪ h1 − h2 >180; ()h1 + h2 ≥ 360; C1C2 ≠ 0 ⎪ 2 ⎪ ' ' ' ' ⎩ h1 + h2 C1C2 = 0

20 2.3.3 DIN99

CIEDE94 and CIEDE2000 color-difference equations do not include an associated color space. In CIEDE2000, there are a lightness correction, a chroma correction, a hue correction, an interactive correction between chroma and hue, and a rescaling of a*.

These corrections indicate that CIELAB is not a uniform color space. A potential uniform space candidate, DIN99 color space,21 was developed in 1999 and has been adopted as a German standard. Rohner and Rich22 presented the DCI-95 formula which

* * applies logarithmic transformations and rescaling of the CIELAB variables L and Cab

and calculates new coordinates using CIELAB hue angle hab . Datacolor International tested DCI-95 and found that the equation performed as well as CIE94 by making a modification to the internal parameters.23 Thus, based on the structure of DCI-95, the

chroma weighting function, SC , was added. Also, in order to fit the experimental data for neutral colors, the ratios of the red/green and yellow/blue coordinates were added.

The DIN99 color difference equation is:

1 2 2 2 1/2 ΔE99 = []ΔL99 +Δa99 +Δb99 kE where * L99 =105.51ln(1+ 0.0158L ) e = acos(16o ) + bsin(16o ) f = 0.7[bcos(16o ) − asin(16o )] Eq.2-49 G = e2 + f 2

C99 = ln(1+ 0.045G)/0.045 f h = arctan( ) 99 e

a99 = C99 cos(h99) b = C sin(h ) 99 99 99

21 This equation is not only a new color-difference equation but also has an associated color space.

2.3.4 Modification of CIECAM02

In 2006, Luo et al.24 derived three uniform color spaces based on CIECAM02 to fit large color differences, suprathreshold color differences and the combined large color difference. In these uniform color spaces, equal distances approximately represent equal color differences. The large color difference datasets include six data sets: Zhu et al.,25-27

OSA,28 Guan and Luo,29 BFDB,30 Pointer and Attridge,31 and Munsell.32,33

Suprathreshold sets were BFD, RIT-DuPont, Leeds, and Witt datasets.

Zhu et al. gave 144 large color difference pairs using CRT colors. The pairs were chosen along five scales in CIELAB color space: hue, lightness, chroma, a light series, and a

* dark series and had an average of 10 ΔE ab units. The OSA data were from the experiment in which the viewing condition was D65 and 10° observer conditions. The

* OSA data had 128 color pairs and the average was 14 ΔE ab units. The Guan data had

* 292 wool sample pairs and the average color difference is 11 ΔE ab units. The BFDB

* dataset had 238 nylon sample pairs and the average ΔE ab was 12 units. The Pointer &

* Attridge dataset had 1308 pairs and an average of 9 ΔE ab units. The datasets consisting of smaller color difference data included the BFDB, RIT-DuPont, Leeds, and Witt

* datasets and had 3657 sample pairs with an average of 2.6 ΔE ab units.

The modified CIECAM02 models were:

22 ' ' 2 '2 '2 1/2 ΔE = [(ΔJ /KL ) +Δa +Δb ] where (1 +100c )J J' = 1 1+ c J 1 Eq.2-50 ln(1+ c M) M ' = 2 c2 a' = M ' cos(h) b' = M ' sin(h) where, J, M, h are the CIECAM02 lightness, colorfulness, and hue angle values respectively. The ΔJ’, Δa’, and Δb’ are the J’, a’, and b’ differences between the standard and sample in a pair. The KL, c1, and c2 coefficients had different values for CAM02-

LCD, CAM02-SCD, and CAM02-UCS. These values are listed in Table 2-III.

Table 2-III. The coefficients for modified CIECAM02 models.

CAM02-LCD CAM02-SCD CAM02-UCS KL 0.77 1.24 1.00 c1 0.007 0.007 0.007 c2 0.0053 0.0363 0.0228

2.4 Euclidean Spaces

2.4.1 A Euclidean Color Space by Thomsen

In 1999, Thomsen34 derived a Euclidean color space by integrating the chroma difference adjustment Sc based on CIE94. He assumed that the adjustment of the chroma difference in CIE94 was more important than the hue difference. The lightness L* in

Euclidean color space was not changed, only chroma was derived as:

* C dt ln(1+ 0.045C*) C*' = ∫ = Eq.2-52 0 1+ 0.045t 0.045

Thus, the Euclidean color space has the form:

23 L*' = L* ln(1+ 0.045C* ) a*' = a* 0.045C* Eq.2-53 ln(1+ 0.045C* ) b*' = b* 0.045C* C* = a*2 + b*2

The constants in Eq.2-53 were determined by minimizing the maximum disagreement with ΔE94:

L*' = L* ln(1+ 0.053C* ) a*' = a* 0.050C* Eq.2-54 ln(1+ 0.053C* ) b*' = b* 0.050C* C* = a*2 + b*2

In this color space, the small Euclidean distances agreed to within 10.5% with ΔE94.

2.4.2 Euclidean Color Space based on DIN99

Both Thomsen’s Euclidean and DIN99 color spaces have the same formula for chroma

C*, but the lightness L* in DIN99 color space has a logarithmic form. Cui and Luo, et al.23 presented three Euclidean color spaces by modifying DIN99 in 2001. They merged four datasets (VFD-P,35 RIT-DuPont, Leeds,36 and Witt37) to form a combined dataset for deriving new Euclidean color space. The first Euclidean color space, DIN99b, kept the same basic structure as the DIN99 formula. The coefficients in DIN99b were calculated by fitting the combined dataset. The DIN99b is:

24 1 2 2 2 1/2 ΔE99b = []ΔL99b +Δa99b +Δb99b kE where * L99b = 303.671ln(1+ 0.0039L ) e = acos(26o ) + bsin(26o ) f = 0.83[bcos(26o ) − asin(26o )] Eq.2-55 G = e2 + f 2

C99b = 23.0ln(1+ 0.075G) f h = arctan( ) + 26 99b e

a99b = C99b cos(h99b ) b = C sin(h ) 99b 99b 99b

In DIN99b color space, the color tolerance in blue region was not a circle, but ellipses.

Compared with ΔE00, DIN99b performed worse in the blue region. Thus, the second and the third Euclidean color space were derived to improve the performance in the blue region. In 1999, Kuehni pointed out that a single linear power for CIELAB was not correct. He gave a new formula that linearized a* and b* values and also included the blue region correction. According to his idea, Cui and Luo et al. replaced the tristimulus

' value X by X = c0 x − (c0 −1)Z , where c0 was an optimized coefficient. The second color space DIN99c and the third color space DIN99d were derived by using X’ equation with and without the rotation of a* and b* axies respectively. DIN99c is below:

25 1 2 2 2 1/2 ΔE99c = []ΔL99c +Δa99c +Δb99c kE where X'=1.1X − 0.1Z * L99c = 317.651ln(1+ 0.0037L ) e = a* f = 0.94b* Eq.2-56 G = e2 + f 2

C99c = 23.0ln(1+ 0.066G) f h = arctan( ) 99c e

a99c = C99c cos(h99c )

b99c = C99c sin(h99c )

In DIN99c, there is no rotation of a* and b* axes.

The DIN99d is:

1 2 2 2 1/2 ΔE99d = []ΔL99d +Δa99d +Δb99d kE where X'=1.12X − 0.12Z * Eq.2-57 L99d = 325.221ln(1+ 0.0036L ) e = a*cos(50o) + b*sin(50o ) f =1.14[−a*sin(50o ) + b*cos(50o )] G = e2 + f 2

C99d = 22.5ln(1+ 0.06G)

f h = arctan( ) + 50o 99d e

a99d = C99d cos(h99d )

b99d = C99d sin(h99d )

In DIN99d, there is a rotation of 500.

26 These three color spaces were tested by using the combined dataset. Overall, there was a slight improvement for the combined dataset, but for the blue region, DIN99c and

DIN99d gave a large improvement.

27 3. RIT-Dupont Dataset

3.1 Background

In 1989, Alman et al.18 at RIT designed an experiment (phase I)38 to develop a color tolerance dataset that could be used to test the performance of color-difference metrics. In this experiment, fifty color normal observers observed one color-difference anchor pair and 315 different color pairs in random order. Here, the color-difference of the anchor pair was defined as the visual color tolerance. The observer compared the total color difference of each pair with color tolerance of the anchor pair and decided if the former one was larger or smaller than the latter one.

The color-difference of the near-gray anchor pair was ∆E*ab 1.02, and the L*a*b* values of anchor pair were: L*1=49.53, a*1=-0.08, b*1=-5.65; L*2=48.89, a*2=0.17, b*2=-4.90.

The 317 color pairs had nine color centers: grey, four medium chroma hues (orange, yellow-green, blue-green, purple), and four high chroma hues (red, yellow, green, blue).

There were five vectors sampled at each color center. Five vector directions

(A,B,C,D,and E in blue color) are plotted in Fig.3-1.

L* b*

A D E C B a*

a* b*

Fig.3-1 Five vectors (A, B,C,D,and E) in CIELAB space

28 An acrylic-lacquer automotive coating was sprayed onto primed aluminum panels.

Samples were cut into 6.5 by 5.0 cm2 rectangles from the sprayed panels. The color difference pairs subtended a visual field of 10° by 5°. The viewing condition was filtered tungsten with a CCT near 6500K. Principal component analyses were used to determine each vector direction. Probit analysis was used to analyse the experimental response that was the population frequency of rejection decisions for each color-difference pair (visual difference greater than the anchor pair). The frequency-of-rejection data yielded sigmoidal response functions. The tolerance with rejection probability equal to 0.5

(denoted as T50) means the color difference visually equivalent to the anchor pair. The frequency distribution was compared to the assumed probability distribution by using a

χ 2test. The large number of high χ 2 values indicated the experimental data did not fit a cumulative-normal distribution model in all cases. Based on the T50 tolerance estimates, uncertainty was calculated by the 95% upper and lower fiducial limits. Although the median tolerance determinations were very precise, the design of five vectors in this experiment could not determine interaction between L* and a* (or b*). Thus, in 1999,

Berns et al.18 designed a second experimental phase. In phase II,39 the two vectors D and

E in phase I were replaced by four vectors which varied simultaneously in L*, a* and b*, shown in Fig.3-2.

29 L* L* G F A H I C B

a* a* b* b*

Fig.3-2 Seven vectors (A,B,C,D,E,F,G,H and I) in CIELAB space

Also, based on nine color centers in phase I, five color centers with higher L* values and five color centers with lower L* values were added. Totally, there were nineteen color centers that sampled the color gamut of the acrylic lacquer paint. The viewing conditions and anchor pair were the same as those in phase I. A total of 315 color-difference pairs were compared with color anchor pair by the fifty observers and the T50 was determined in Phase I. A second group of 50 observers judged 530 color-difference pairs in phase II.

Compared with phase I, the vector directions sampled lightness and chromaticness separately and together in phase II, which gave more difficulty to judge. Besides, since the color difference increased between the anchor pair color space location and the test pair location in phase II, the observer uncertainty increased. Thus, the phase II represented greater experimental uncertainty than the phase I. In order to pool the results in phase I and phase II, a median filtering algorithm was designed by M.R. Balonon-

Rosen.40 The filtering algorithm reduced the uncertainty in phase II, and did not affect phase I T50, because the visual results in phase I had high precision. As a result, 156 visual color tolerance with 19 color centers perceptually equivalent to 1 CIELAB color- difference unit were generated.

30 The 19 color centers should be considered as the nominal color centers because the vectors of these color centers have their own actual color centers, which are slightly different from the nominal color centers. Here, the average values ( x ) of actual color center for 19 color centers were calculated and used as a common origin for all the vectors at each center. Thus, each tolerance has: +T50, x , -T50. The RIT-DuPont color- tolerance vectors in CIELAB space are shown in Fig.3-3 – Fig.3-7.

RIT−Dupont Color−Difference Vectors in CIEL*a*b* Space

40 b*

−20

−80 −60 −40 −20 0 20 40 60 80 a* Fig.3-3 RIT-DuPont color-tolerance vectors in a*b* plane

RIT−Dupont Color−Difference Vectors in CIEL*a*b* Space

50 L*

−50 −40 −30 −20 −10 0 10 20 30 40 a*

31 Fig.3-4 RIT-DuPont color-tolerance vectors in a*L* plan

RIT−Dupont Color−Difference Vectors in CIEL*a*b* Space 100

50 L*

0 −20 0 20 40 60 80 b* Fig.3-6 RIT-DuPont color-tolerance vectors in b*L* plane

RIT−Dupont Color−Difference Vectors in CIEL*a*b* Space

50 L* 40

80 60 40 40 20 20 0 0 −20 −20 b* −40 a* Fig.3-7 RIT-DuPont color-tolerance vectors in CIELAB space

32 3.2 Color-Difference Ellipsoids

If a color space is uniform, the unit color differences at a color center will form a sphere.

In CIELAB space, the color tolerance is summarized as an ellipsoid, not a sphere.

Visualization of color difference ellipsoids is very useful to analyze the systematic variation in color difference behavior. Melgosa et al41 provided a method to obtain the ellipsoids that fit the RIT-Dupont data in 1997.

Mathematically, the ellipsoid equation in xyz coordinate can be described as:

2 2 2 b11x + 2b12 xy + b22 y + 2b13 xz + 2b23 yz + b33z =1 Eq.3-1

42 where bij is the coefficients of ellipsoid.

In CIELAB space, color discrimination35,43 at any color center can be represented as:

*2 * * *2 * * * * *2 *2 b11Δa + 2b12 Δa Δb + b22 Δb + 2b13Δa ΔL + 2b23Δb ΔL + b33ΔL = ΔEab Eq.3-2

* For constant ΔEab , this equation defines an ellipsoid. In order to fit RIT-Dupont dataset,

2 the six coefficients (bij) in Eq.3-2 will be calculated by minimizing the S function defined as:

2 2 *2 2 S = ∑(ΔV − ΔEab ) Eq. 3-3

Where, ΔV is the constant visual difference. The median tolerance T50 and first eigenvector provided by the RIT-Dupont dataset, were used to minimize the S2 function.

The average values of actual color center for 19 color centers were used so that each eigenvector has a common center.

From Eq.3-2, the equation of ellipses in a*b* plane can be written as:

33 2 2 *2 b11Δa + 2b12ΔaΔb + b22Δb = ΔEab Eq.3-4

The orientation of the major axis of the ellipses in Eq.3-4 with respect to the a* axis can be calculated as:

1 ⎛ 2b ⎞ θ = tan−1⎜ 12 ⎟ Eq.3-5 2 ⎝ b22 − b11 ⎠

The major and minor axes of the ellipses can be calculated as:

2sin(2θ) a = Eq.3-6 sin(2θ) × (b11 + b22) − 2b12

2sin(2θ) b = Eq.3-7 sin(2θ) × (b11 + b22) + 2b12

The RIT-DuPont ellipses in a*b*, a*L*, and b*L* planes are shown in Fig.3-8, Fig.3-10, and Fig.3-12, respectively. The RIT-DuPont ellipses with vectors a*b*, a*L*, and b*L* plane are shown in Fig.3-9, Fig.3-11, and Fig.3-13, respectively. Here, all the ellipses are enlarged 300%.

RIT−Dupont Visual data in CIEL*a*b* Space RIT−Dupont Color−Difference Ellipses and Vectors in CIEL*a*b* Space

80 80

60 60

40 40 b* b* 20 20

0 0

−20 −20

−40 −40

−80 −60 −40 −20 0 20 40 60 80 −80 −60 −40 −20 0 20 40 60 80 a* a* Fig.3-8 RIT-DuPont ellipses in a*b* plane. Fig.3-9 RIT-DuPont ellipses with vectors in a*b* plane.

34 RIT−Dupont Visual data in CIEL*a*b* Space RIT−Dupont Color−Difference Ellipses and Vectors in CIEL*a*b* Space

80 80

70 70

60 60

50 50 L* L*

40 40

30 30

20 20

−50 −40 −30 −20 −10 0 10 20 30 40 −50 −40 −30 −20 −10 0 10 20 30 40 a* a* Fig.3-10 RIT-DuPont ellipses in a*L* plane. Fig.3-11 RIT-DuPont ellipses with vectors in a*L* plane.

RIT−Dupont Visual data in CIEL*a*b* Space RIT−Dupont Color−Difference Ellipses and Vectors in CIEL*a*b* Space 100

90 90

80 80

70 70

60 60

50 50 L* L*

40 40

30 30

20 20

10 10

0 0 −20 0 20 40 60 80 −20 0 20 40 60 80 b* b* Fig.3-12 RIT-DuPont ellipses in b*L* plane. Fig.3-13 RIT-DuPont ellipses with vectors in b*L* plane.

In Fig.3-8, the ellipses in blue region tilt toward greenish blue at low chroma and toward reddish blue at high chroma. The neutral ellipses are smaller than those in the higher chroma regions. In addition, when the chroma increases, the ellipse size increases. The tilt relative to the L* direction is very slight in Fig.3-10 and Fig.3-12. In general, the RIT-

DuPont ellipsoids visualization indicates that CIELAB is not a color uniform space.

35 3.3 CIECAM02

The RIT-DuPont vectors 3D-CIECAM02 space and ab, aJ, and bJ plane of CIECAM02 space are shown in Fig.3-14-Fig.3-17, respectively. The RIT-DuPont ellipses ab, aJ, and bJ plane in CIECAM02 space are shown in Fig.3-18, Fig.3-20, and Fig.3-22, respectively.

The RIT-DuPont ellipses with vectors ab, aJ, and bJ plane in CIECAM02 space are shown in Fig.3-19, Fig.3-21, and Fig.3-23, respectively.

RIT−Dupont Color−Difference Vectors in CIECAM02 Space

50 J 40

10 −40 −20 60 0 40 20 20 0 −20 40 −40 b a Fig.3-14 RIT-DuPont vectors in CIECAM02 space.

RIT−Dupont Color−Difference Vectors in ab plane (CIECAM02 space)

20 b

−20

−40 −60 −40 −20 0 20 40 60 a Fig.3-15 RIT-DuPont vectors in ab plane (in CIECAM02 space).

36 RIT−Dupont Color−Difference Vectors in aJ plane (CIECAM02 Space) 80

J 20

−20

−40 −80 −60 −40 −20 0 20 40 60 a Fig.3-16 RIT-DuPont vectors in aJ plane (in CIECAM02 space)

RIT−Dupont Color−Difference Vectors in CIECAM02 Space

50 J

0 −40 −20 0 20 40 60 b Fig3-17 RIT-DuPont vectors in Jb plane (in CIECAM02 space).

37 RIT−Dupont Color−Difference Ellipses in CIECAM02 Space (from +T50 & −T50) RIT−Dupont Color−Difference Vectors and Ellipses in CIECAM02 Space

60 60

40 40

20 20 b b

0 0

−20 −20

−40 −40

−80 −60 −40 −20 0 20 40 60 −80 −60 −40 −20 0 20 40 60 a a Fig.3-18 RIT-DuPont ellipses in Fig.3-19 RIT-DuPont ellipses with ab plane (in CIECAM02 space). vectors in ab plane (in CIECAM02 space).

RIT−Dupont Color−Difference Ellipses in CIECAM02 Space (from +T50 & −T50) RIT−Dupont Color−Difference Vectors and Ellipses in CIECAM02 Space

80 80

70 70

60 60

50 50 J J

40 40

30 30

20 20

10 10 −40 −30 −20 −10 0 10 20 30 40 −40 −30 −20 −10 0 10 20 30 40 a a Fig.3-20 RIT-DuPont ellipses in Fig.3-21 RIT-DuPont ellipses with aJ plane (in CIECAM02 space). vectors in aJ plane (in CIECAM02 space).

RIT−Dupont Color−Difference Ellipses in CIECAM02 Space (from +T50 & −T50) RIT−Dupont Color−Difference Vectors and Ellipses in CIECAM02 Space

90 90

80 80

70 70

60 60

50 50 J J

40 40

30 30

20 20

10 10

0 0

−40 −20 0 20 40 60 −40 −20 0 20 40 60 b b Fig.3-22 RIT-DuPont ellipses in Fig.3-23 RIT-DuPont ellipses with bJ plane (in CIECAM02 space). vectors in bJ plane (in CIECAM02 space).

38 As we known, for a perfect agreement between the RIT-Dupont dataset and a uniform color space, all ellipses should be circles with constant radius. RIT-DuPont ellipses in

CIELAB space are smaller in the neutral region and increase in size when chroma increases. In Fig.3-18, the sizes of ellipses in CIECAM02 color space are closer. For the

No.11 color center (light bluish green), the ellipses in ab plane behave like a circle. Also, the ellipses in blue region do not tilt as much as the ellipses in CIELAB color space. They tilt slightly toward greenish blue at low chroma and toward to reddish blue at high chroma. But, all ellipses are orientated more or less toward the origin, which are similar to CIELAB space. There is a ellipsis near neutral in Fig.3-18 which is larger than those in the other region. This indicates that the pattern of ellipses in CIECAM02 is similar to that of CIELAB except that CIECAM02 is more uniform than CIELAB in some color positions.

3.4 IPT

The RIT-DuPont vectors in IPT space and PT, IT, and IT plane of IPT space are shown in

Fig.3-24-Fig.3-27, respectively. The RIT-DuPont ellipses IP, IT, and PT plane in IPT space are shown in Fig.3-28, Fig.3-30, and Fig.3-32, respectively. The RIT-DuPont ellipses with vectors IP, IT, and PT plane in IPT space are shown in Fig.3-29, Fig.3-31, and Fig.3-33, respectively.

39 RIT−Dupont Color−Difference Vectors in IPT Space

I 50

−20 80 60 0 40 20 20 0 40 −20

T P Fig.3-24 RIT-DuPont vectors in IPT space.

RIT−Dupont Color−Difference Vectors in PT plane (IPT space)

40 T

−20

−60 −40 −20 0 20 40 60 80 P Fig.3-25 RIT-DuPont vectors in PT plane (in IPT space).

40 RIT−Dupont Color−Difference Vectors in IP plane (IPT Space)

I 50

−40 −30 −20 −10 0 10 20 30 40 P Fig.3-26 RIT-DuPont vectors in IP plane (in IPT space).

RIT−Dupont Color−Difference Vectors in IT plane (IPT Space) 100

I 50

−20 0 20 40 60 80 T Fig.3-27 RIT-DuPont vectors in IT plane (in IPT space).

41 RIT−Dupont Color−Difference Ellipses in PT plane (IPT Space) RIT−Dupont Color−Difference Ellipses&Vectors in PT plane (IPT Space)

80 80

60 60

40 40 T T 20 20

0 0

−20 −20

−40 −40 −60 −40 −20 0 20 40 60 80 −60 −40 −20 0 20 40 60 80 P P Fig.3-28 RIT-DuPont Fig.3-29 RIT-DuPont ellipses in PT plane (in IPT space). ellipses and vectors in PT plane (in IPT space).

RIT−Dupont Color−Difference Ellipses in IP plane (IPT Space) RIT−Dupont Color−Difference Ellipses&Vectors in IP plane (IPT Space)

80 80

70 70

60 60

I 50 I 50

40 40

30 30

20 20

−40 −30 −20 −10 0 10 20 30 40 −40 −30 −20 −10 0 10 20 30 40 P P Fig.3-30 RIT-DuPont Fig.3-31 RIT-DuPont ellipses and ellipses in IP plane (in IPT space). vectors in in IP plane (in IPT space).

RIT−Dupont Color−Difference Ellipses in IT plane (IPT Space) RIT−Dupont Color−Difference Ellipses&Vectors in IT plane (IPT Space)

100 100

90 90

80 80

70 70

60 60

I 50 I 50

40 40

30 30

20 20

10 10

0 0 −40 −20 0 20 40 60 80 −40 −20 0 20 40 60 80 T T Fig.3-30 RIT-DuPont Fig.3-33 RIT-DuPont ellipses and ellipses in IT plane (in IPT space). vectors in in IT plane (in IPT space).

42 Similar to the behavior of the ellipses of No.11 color center in CIECAM02 color space, the ellipses in IPT color space also looks like a circle. The ellipses in the blue region of

IPT space tilt toward to origin more than those CIELAB and CIECAM02 color spaces.

Thus, in IPT space, all ellipses are orientated toward the origin. The ellipses in neutral region are not as large as those in CIECAM02 color space. From this point, IPT space performs better than CIECAM02 and CIELAB space.

In general, the visualization of RIT-Dupont data in CIELAB, CIECAM02 and IPT color space indicates that none of these color spaces is uniform. CIECAM02 and IPT color space perform as well as CIELAB. In some case, they are more uniform than CIELAB color space.

43 4. Qiao and Berns Dataset

4.1 Background

In 1998, Qiao, et al44 designed an experiment that sampled hue circles at two lightness

(L*=40 and 60) and two chroma levels (C*ab=20 and 40) and also included three of the five CIE recommended colors (red, green, and blue). A Fujix Pictrography 3000 digital printer and glossy photographic papers were used to produce sample pairs. The gamut of the printer was tested and it was found that the greatest C*ab that could be produced at constant lightness and for all hues was limited to 40 due to the irregular shape of the color gamut. Since for a given color center, color difference pairs needed to be formed with

* predefined differences in ΔHab with minimal differences in lightness and chroma, the candidate color difference pairs were constrained by Eq.4-1:

ΔL* < 0.6

* ΔCab < 0.6 Eq.4-1

* ΔHab * < 0.9 ΔEab

The sample pairs were measured with a Milton Roy ColorScan 45/0 spectrophotometer, the identical instrument used for the RIT-DuPont dataset.

Based on the color gamut data, the highest C*ab of a hue circle meeting the experimental design was 40 with an L*=40. The chroma was reduced to 35 in order to minimize quantization limitations near the edge of the color gamut. Thus the first CIELAB hue

44 circle was defined at L*=40 and C*ab=35. A second hue circle was defined at the same lightness of 40, but a reduced chroma of 20. The third hue circle was defined at a higher lightness of 60 with the reduced chroma of 20. Also, the four chromatic CIE- recommended color centers were included to the experiment. Only the red, green, and blue centers were evaluated because the yellow center was outside the color gamut of the

Fujix printer. Thus, 39 color centers were generated.

* For each color center, 9 to 10 color-difference pairs with increasing ΔHab were selected from the sorted sample data. Each color sample was cut into a 2.5” x 2.0” rectangle, where the cutter was oriented 450 to the plane of the surface to minimize viewing of the paper support. Samples were affixed adjacently onto the L*=50 gray cardboard forming color-difference pairs.

Forty-five observers participated in the experiment. The CIELAB values of the anchor pair were: L*1=49.03, a*1=0.93, b*1=2.27; L*2=49.35, a*2=0.10, b*2=2.79. The color

* difference in CIELAB was ΔEab =1.03. Totally there were 393 color difference pairs.

Probit analysis was used to analyze the frequency of rejection data. Through the analysis, forty-four visual tolerances varying in hue at two lightness and chroma levels were chosen. The Qiao data are plotted in CIELAB space in Fig.4-1 and 4-2.

45 Qiao data in CIEL*a*b* space (L*=40,C*=20,C*=35) Qiao data in CIEL*a*b* space (L*=60, C*=20) 40 40

30 30

20 20

10 10

b* 0 b* 0

−10 −10

−20 −20

−30 −30

−40 −40 −40 −30 −20 −10 0 10 20 30 40 −40 −30 −20 −10 0 10 20 30 40 a* a* Fig.4-1 Qiao data in a*b* plane Fig.4-2 Qiao data (Lightness=40, Chroma=20 and 35) (Lightness=60, Chroma=20) in a*b* plane.

4.2 Plots of Qiao Experimental Results in Color Spaces

The color difference of Qiao data in CIELAB, CIECAM02 and IPT color space are plotted in Fig.4-3 – 4-5, respectively.

Qiao Data in CIEL*a*b* Color Space

3.5 L40C20 L40C35 L60C20

2.5

ΔH 2

1.5

0.5 0 50 100 150 200 250 300 350 Hue Angle Fig.4-3 Color difference of Qiao data in CIELAB

46 Qiao Data in CIECAM02 Color Space 3.5 L40C20 L40C35 L60C20

2.5

ΔH 2

1.5

0.5 0 50 100 150 200 250 300 350 Hue Angle

Fig.4-4 Color difference of Qiao data in CIECAM02

Qiao Data in IPT Color Space 3.5 L40C20 L40C35 L60C20

2.5

ΔH 2

1.5

0.5 0 50 100 150 200 250 300 350 Hue Angle Fig.4-5 Color difference of Qiao data in IPT

0 In CIELAB color space, for L=40,C*ab=20, the colors with hue angle around 100 and

3000 have almost the unit color difference. The color difference from 00 to 400 agrees with

0 0 0 that from 130 to 160 . For L=40, C*ab=35, only color difference at around 30 has the unit color difference. The maximum color difference centered around 2300 and 2500.

47 Generally, there is close agreement between 1300 and 2200 for the three datasets. The

0 0 tolerances for L=40,C*ab=20 between 0 and 30 changes in the direction opposite to that of the other two datasets. Apparently, CIELAB color tolerances have a hue angle dependency.

The mean value ( ΔE ) and standard deviation ( δ(ΔE) ) of Qiao color differences in

CIELAB, CIECAM02 and IPT color spaces are listed in Table 4-I.

Table 4-I. ΔE and δ(ΔE) in Three Color Spaces L40C20 L40C35 L60C20

ΔEab 1.343 2.075 1.351

ΔECIECAM 02 1.509 2.144 1.525

ΔE IPT 1.168 1.764 1.301

δ(ΔEab ) 0.353 0.748 0.363

δ(ΔECIECAM 02) 0.303 0.630 0.512

δ(ΔE IPT ) 0.259 0.537 0.396

The comparisons among the three datasets in CIELAB, CIECAM02, and IPT color space are plotted in Fig.4-6 – 4-8.

L40C20 in CIEL*a*b*, CIECAM02 and IPT Color Space 2.5 CIEL*a*b* IPT CIECAM02

ΔH 1.5

0.5 0 50 100 150 200 250 300 350 Hue Angle

48 Fig.4-6 Color difference of L40C20 in three color spaces

L40C35 in CIEL*a*b*, CIECAM02 and IPT Color Space

3.5 CIEL*a*b* IPT CIECAM02

2.5

ΔH 2

1.5

0.5 0 50 100 150 200 250 300 350 Hue Angle Fig.4-7 Color difference of L40C35 in three color spaces

L60C20 in CIEL*a*b*, CIECAM02 and IPT Color Space 2.5 CIEL*a*b* IPT CIECAM02

ΔH 1.5

0.5 0 50 100 150 200 250 300 350 Hue Angle Fig.4-8 Color difference of L60C20 in three color spaces

Clearly, the Qiao color difference in IPT color space is closest to the unit color difference.

However, there is evidence that CIELAB, CIECAM02, and IPT color tolerances have a hue-angle dependency. Thus, it also means that the color difference equation in these color spaces should contain a hue-angle dependent function.

49 5. Hung and Berns Dataset

5.1 Background

In 1995, Hung and Berns45 designed an experiment to determine twenty-four loci of constant perceived hue using a CRT. A BARCO CalibratorTM 19” high-resolution color monitor was set up with a maximum white near D65 and a luminance of 50 cd/m2.

Twelve reference stimuli (red, red-yellow, yellow, yellow-green, green, green-cyan, cyan, cyan-blue, blue, blue-magenta, magenta, magenta-red, white) were selected as the reference. These twelve stimuli had the highest chroma for the display. The test stimuli were used to sample the CRT color gamut in two ways: at constant lightness (CL) and at varying lightness (VL). Each CL test stimuli increased C*uv by 25% compared with the reference stimulus. The VL test stimuli sampled the CRT gamut in 10 L* increments between 20 and 90. There were three CL test stimuli and 8 VL test stimuli for each of the

12 reference hues. Each test stimulus was along a locus with constant C*uv and L* and variable hue.

Twenty-four color patches were displayed on the CRT having three rows by eight columns. There were eight 20-mm square stimuli in each row. The gray patches with equal lightness to the test stimulus position were put on the first row. The variable hue patches were on the second row and the reference patches were on the bottom row respectively. For a given test stimulus position, the corresponding reference stimulus was displayed on the third row, and a neutral patch with equal lightness as the test stimuli was displayed on the first row. For the test patches on the second row, the lightness (L*) and chroma (C*uv) were equal to the test stimuli position. If the C*uv was out of display

50 gamut, the C*uv was replaced by the new C*uv that was on the gamut boundary. The right four patches were with increased huv and the left four patches were with decreased huv.

The color-difference of the center pair was 2.0ΔE*uv relative to the test stimulus position.

The color-difference of each successive patch increased by 4.0ΔE*uv from its neighbor.

The color that was picked up on the middle row by the observer and its opposite-sided color was displayed below the bottom row.

There were nine observers in a darkened room that participated in the experiment. The viewing condition was under Illuminant C and the background corresponded to a luminance factor of 0.2. The observer’s task was to choose one color stimulus closest in hue to the reference stimulus. Then, the selected color became the new center point of the eight positions. Thus, the color difference between each patch changed to 5/8ΔE*uv of the previous display. Totally, 132 color positions were represented in the experiment.

5.2 Hung Data Plots in CIELAB, CIECAM02,

and IPT Color Spaces

The Hung dataset in CIELAB, CIECAM02, and IPT color spaces are plotted in Fig.5-1 –

5-6, respectively.

51 Constant Lightness in CIEL*a*b* Varying Lightness in CIEL*a*b*

80 80

60 60

40 40

20 20

b* 0 b* 0

−20 −20

−40 −40

−60 −60

−80 −80

−100 −50 0 50 100 −100 −50 0 50 100 a* a* Fig.5-1 Constant hue loci from the constant Fig.5-2 Constant hue loci from the variable lightness experiment in CIELAB space. lightness experiment in CIELAB space.

Constant Lightness in CIECAM02 Varying Lightness in CIECAM02

60 60

40 40

20 20

0 0 b bc

−20 −20

−40 −40

−60 −60

−80 −80 −80 −60 −40 −20 0 20 40 60 80 −80 −60 −40 −20 0 20 40 60 80 ac a Fig.5-3 Constant hue loci from the constant Fig.5- 4 Constant hue loci from the variable lightness experiment in CIECAM02 space. lightness experiment in CIECAM02 space.

Constant Lightness in IPT Varying Lightness in IPT 100 100

80 80

60 60

40 40

20 20

0 0 T T

−20 −20

−40 −40

−60 −60

−80 −80

−100 −100

−100 −50 0 50 100 −100 −50 0 50 100 P P

Fig.5-5 Constant hue loci from the Fig.5-6 Constant hue loci from the constant lightness experiment in IPT space. variable lightness experiment in IPT space.

52 In Fig.5-1, the lines of constant hue in the blue region tend to shift to purple as chroma is decreased. This means the lack of blue constancy in CIELAB color space. The curve shift in the blue region is shown in Fig.5-3. However, there is an improvement in hue uniformity and hue constancy in CIECAM02, even in the blue region. In Fig.5-5 and

Fig.5-6, the IPT color space has the minimum degree of spread for all the hue angles.

Comparing Fig.5-4 and Fig.5-6 with Fig.5-2, the curve in cyan-blue region in

CIECAM02 and IPT space is straighter than the CIELAB color space.

In CIELAB space, L*, a*, and b* are calculated from tristimulus values and reference white point. The reference white point values are used in a von Kries transformation. But, this chromatic adaptation is performed in XYZ tristimulus space, not a physiological cone space. This causes hue shift problem1, especially effects hue constancy in blue area46. The perceptual attributes of CIECAM02 and IPT color spaces are calculated from the Hunt-

Pointer-Estevez primaries, not from tristimulus values directly. Using transform to Hunt-

Pointer-Estevez cone fundamentals improves the hue constancy in CIECAM02 and IPT.

5.3 Conclusion

The Hung data can be used to evaluate the hue linearity of color spaces. The visualization of Hung data in CIELAB, CIECAM02 and IPT color spaces shows the poor blue linearity in CIELAB color space. Also, CIECAM02 and IPT color spaces result in hue linearity improvement.

53 6. Optimizing Color-Difference Formula

6.1 Background

The CIE used the RIT-DuPont and Luo-Rigg datasets to derive the color-difference equation, CIE94. In 2007, Berns used the RIT-DuPont color tolerance dataset (19 color centers and156 color pairs) and the Qiao, et al. hue dataset (44 color pairs) reference to optimize color-difference equations in CIECAM02 and IPT color space. Eq.6-1 shows the optimization equation for IPT:

1/2 ⎡⎛ ⎞ 2 ⎤ ⎢⎜ ⎟ ⎥ ⎢ ⎥ ⎜ ⎟ 2 2 ⎢⎜ ΔI ⎟ ⎛ ΔC ⎞ ⎛ ΔH ⎞ ⎥ ΔEoptimized = βe ⎢⎜ β ⎟ + ⎜ ⎟ + ⎜ ⎟ ⎥ Eq.6-1 ⎛ − ⎞ L ,2 ⎜ 1+ β C ⎟ ⎜ 1+ β C ⎟ ⎢⎜ ⎟ ⎝ C,1 pt ⎠ ⎝ H,1 pt ⎠ ⎥ ⎢ ⎜ I ⎟ ⎥ ⎜ βL,0 + βL,1 ⎟ ⎢⎜ ⎜ 100⎟ ⎟ ⎥ ⎣⎝ ⎝ ⎠ ⎠ ⎦

The same form of the equation was used for CIECAM02.

In the CIELAB color space, +T50 and –T50 of the RIT-DuPont dataset is symmetric.

However, this symmetric relation did not maintain in the CIECAM02 and IPT color spaces. Thus, only the x and +T50 coordinates were used in developing CIECAM02 color-difference and IPT color-difference equations.

s * A coefficient of variation (CV), ΔE pt , was calculated as a performance metric. The x * ΔE pt objective function minimized CV for the 200-sample dataset with the constraint that the average color difference was unity.

54 6.2 CIECAM02

CIECAM02 JCh (lightness, chroma, hue angle) coordinates were calculated for the D65 illumination at 1500 lx, background reflectance Yb=100, and an average surround.

The optimized CIECAM02 color-difference equation is shown in Eq.6-2.11

⎡ 2 2 2⎤1/2 ⎛ ΔJ ⎞ ⎛ ΔC ⎞ ⎛ ΔH ⎞ ΔECIECAM 02,JCH = ⎢⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ ⎥ ⎣⎢⎝ kJ SJ ⎠ ⎝ kC SC ⎠ ⎝ kH SH ⎠ ⎦⎥ ⎛ ⎞ 2 J Eq.6-2 SJ = 0.5 +1.25⎜ ⎟ ⎝ 100⎠

SC =1+ 0.02C jch

SH =1+ 0.01C jch

Since there is an assumption that lightness and chromaticness is independent in CIELAB and CIECAM02, the three axes in CIECAM02 color space could be rotated relative to each other so that the CIECAM02 color-difference equation can be optimized to match neutral color tolerances. Berns derived a rotation matrix (Eq.6-3) to improve the medium gray ellipsoid. In another words, a rotation matrix was optimized where the objective function was minimizing the medium gray CV. The final optimized rotation matrix is given in Eq.6-4.

⎛ J '⎞ ⎛ β1,1 β1,2 β1, 3 ⎞ ⎛ J⎞ ⎜ a'⎟ = ⎜ β β β ⎟ ⎜ a⎟ Eq.6-3 ⎜ ⎟ ⎜ 2,1 2,2 2,3⎟ ⎜ ⎟ ⎜ ⎟ ⎝ b'⎠ ⎝ β3,1 β2,2 β3,3 ⎠ ⎝ b⎠

⎛ J'⎞ ⎛ 1.000 0.015 −0.032⎞⎛ J⎞ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ a'⎟ = ⎜ −0.166 0.934 −0.086⎟⎜ a⎟ Eq.6-4 ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ b'⎠ ⎝ −0.060 −0.088 0.892 ⎠⎝ b⎠

55 The medium gray was used to evaluate the performance of color-difference equations.

The magnitudes of each vector for the medium gray tolerance are listed in Table 6-I.

Table 6-I. Medium gray tolerance vector

VECTOR CIELAB CIEDE2000 CIECAM02OPT CIECAM02OPT_ROT A 0.92 0.75 0.99 1.04 B 0.85 1.14 0.95 1.00 C 1.30 1.13 1.12 1.01 D 0.91 1.07 1.09 1.01 E 1.25 1.27 0.96 1.01 F 1.01 0.97 1.13 0.98 G 0.99 0.94 0.92 0.98 H 0.85 0.85 0.90 0.98 I 0.91 0.88 0.93 0.98 CV 0.16 0.17 0.09 0.02

It indicates that the neutral tolerance in CIELAB and CIEDE2000 is not spherical and the optimized CIECAM02 color-difference equation had an improvement. Also, rotated

CIECAM02 gave the best performance.

The optimized CIECAM02 color-difference equation ellipses in ab, aJ, and bJ plane are shown in Fig.6-1 - 6-3.

CIECAM02 Difference Dquation Ellipses CIECAM02 Difference Dquation Ellipses

60 70

40 60

50 20 J b

40 0

−20 20

−40 10 −60 −40 −20 0 20 40 60 −40 −30 −20 −10 0 10 20 30 40 a a Fig.6-1 CIECAM02 Color-Difference Fig.6-2 CIECAM02 Color-Difference Equation Ellipses in ab Plane Equation Ellipses in aJ Plane

56 CIECAM02 Difference Dquation Ellipses 90

50 J

0 −40 −20 0 20 40 60 b Fig.6-3 CIECAM02 Color-Difference Equation Ellipses in bJ Plane

The rotated CIECAM02 color-difference equation ellipses in ab, aJ, and bJ plane are plotted in Fig.6-4 – 6-6.

Rotated CIECAM02 Difference Equation Ellipses in Rotated CIECAM02 Space Rotated CIECAM02 Difference Equation Ellipses in Rotated CIECAM02 Space 80

60 70

40 60

50 20 J b

−20 20

−40 10 −60 −40 −20 0 20 40 60 −40 −30 −20 −10 0 10 20 30 40 a a Fig.6-4 Rotated CIECAM02 Color-Difference Fig.6-5 Rotated CIECAM02 Color-Difference Equation Ellipses in ab Plane Equation Ellipses in aJ Plane

Rotated CIECAM02 Difference Equation Ellipses in Rotated CIECAM02 Space 90

50 J

0 −40 −20 0 20 40 60 b Fig.6-6 Rotated CIECAM02 Color-Difference Equation Ellipses in bJ Plane

57 CIECAM02 color-difference equation ellipses and CIECAM02 RIT-Dupont ellipses at middle gray color center are shown in Fig.6-7 - 6-9. The ellipses are also enlarged triple times.

CIECAM02 Difference Equation(blue) VS. RIT−DuPont Visual Data(red) for Middle Gray CIECAM02 Difference Equation(blue) VS. RIT−DuPont Visual Data(red) for Middle Gray

6 52

4 51

2 50 J b 49 0

−2

−4 46 −8 −6 −4 −2 0 2 4 −6 −5 −4 −3 −2 −1 0 1 2 3 a a Fig.6-7 ΔECIECAM 02 Ellipses VS. RIT-Dupont Fig.6-8 ΔECIECAM 02 Ellipses VS. RIT-Dupont Ellipses at Middle gray in ab Plane Ellipses at Middle gray in aJ Plane

CIECAM02 Difference Equation(blue) VS. RIT−DuPont Visual Data(red) for Middle Gray 54

50 J 49

−4 −2 0 2 4 6 b Fig.6-9 ΔECIECAM 02 Ellipses VS. RIT-Dupont Ellipses at Middle gray in bJ Plane

Fig.6-10 – Fig.6-12 is the rotated CIECAM02 color-difference equation ellipses and rotated CIECAM02 RIT-Dupont ellipses at middle gray color center.

58 Rotated CIECAM02 Difference Equation(blue) VS. RIT−DuPont Visual Data(red) for Middle Gray Rotated CIECAM02 Difference Equation(blue) VS. RIT−DuPont Visual Data(red) for Middle Gray 53

4 51

2 50 J b

0 49

48 −2

−4

46 −8 −6 −4 −2 0 2 4 −6 −5 −4 −3 −2 −1 0 1 2 a a Fig.6-10 ΔERot _ CIECAM 02 VS. RIT-Dupont Fig.6-11 ΔERot _ CIECAM 02 VS. RIT-Dupont Ellipses at Middle gray in ab Plane Ellipses at Middle gray in aJ Plane

Rotated CIECAM02 Difference Equation(blue) VS. RIT−DuPont Visual Vata(red) for Middle Gray

50 J 49

−4 −3 −2 −1 0 1 2 3 4 5 6 b Fig.6-12 ΔERot _ CIECAM 02 VS. RIT-Dupont Ellipses at Middle gray in bJ Plane

In Fig.6-7 – Fig.6-12, it indicates that the rotated CIECAM02 color difference equation ellipses fit the visual color tolerance better than CIECAM02 color difference equation ellipses. Due to rotated CIECAM02 color difference equation was designed to optimize the middle gray ellipsoid, its tolerance at middle gray is more like a sphere than

CIECAM02.

The CIECAM02 color-difference equation ellipses and CIECAM02 RIT-Dupont ellipses in ab, aJ, and bJ planes are plotted in Fig.6-13 – Fig.6-15.

59 CIECAM02 Difference Equation(blue) VS. RIT−DuPont Visual Data(red) CIECAM02 Difference Equation(blue) VS. RIT−DuPont Visual Data(red)

60 70

40 60

20 50 b J

40 0

30 −20

−40

10 −80 −60 −40 −20 0 20 40 60 80 −40 −30 −20 −10 0 10 20 30 40 a a Fig.6-13 ΔECIECAM 02 Ellipses VS. Fig.6-14 ΔECIECAM 02 Ellipses VS. RIT-Dupont Ellipses in ab Plane RIT-Dupont Ellipses in aJ Plane

CIECAM02 Difference Equation(blue) VS. RIT−DuPont Visual Data(red)

50 J

−40 −20 0 20 40 60 b Fig.6-15 ΔECIECAM 02 Ellipses VS. RIT-Dupont Ellipses in bJ Plane

The rotated CIECAM02 color-difference equation ellipses and CIECAM02 RIT-Dupont ellipses in ab, aJ, and bJ planes are shown in Fig.6-16 – Fig.6-18.

Rotated CIECAM02 Difference Equation(blue) VS. RIT−DuPont Visual Data(red) Rotated CIECAM02 Difference Equation(blue) VS. RIT−DuPont Visual Data(red)

60 70

40 60

20 50 J b

40 0

30 −20

−40

10 −80 −60 −40 −20 0 20 40 60 −40 −30 −20 −10 0 10 20 30 40 a a Fig.6-16 ΔERotated _ CIECAM 02 Ellipses Fig.6-17 ΔERotated _ CIECAM 02 Ellipses VS. RIT-Dupont Ellipses in ab Plane VS. RIT-Dupont Ellipses in aJ Plane

Rotated CIECAM02 Difference Equation(blue) VS. RIT−DuPont Visual Data(red)

50 J

−40 −20 0 20 40 60 b Fig.6-18 ΔERotated _ CIECAM 02 Ellipses VS. RIT-Dupont Ellipses in bJ Plane

Generally, the agreement between CIECAM02 color equation and visual results is close to that between rotated CIECAM02 color equation and visual results. For the yellow, moderate greenish blue, and grayish yellow green center, both equations are very similar to the visual experimental ellipses. For thee blue and strong orange yellow center, the agreement between equations and visual results was very poor.

6.3 IPT

Eq.6-5 is the optimized IPT color-difference equation.11

1/2 ⎡⎛ ⎞ 2 ⎛ ⎞ 2 ⎛ ⎞ 2⎤ ΔI ΔCpt ΔH pt ΔEIPT = ⎢⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ ⎥ ⎣⎢⎝ kI SI ⎠ ⎝ kC SC ⎠ ⎝ k H SH ⎠ ⎦⎥ ⎛ ⎞ 3.5 I Eq.6-5 SJ = 0.7 +1.6⎜ ⎟ ⎝ 100⎠

SC =1+ 0.03C pt

SH =1+ 0.01C pt

61 2 2 Cpt = t + p where, Eq.6-6 −1⎛ t ⎞ h pt = tan ⎜ ⎟ ⎝ p⎠

ΔI = I2 − I1

ΔCpt = Cpt,2 − Cpt,1 Eq.6-7

⎛ hpt,2 − hpt,1 ⎞ ΔH pt = 2 Cpt,2Cpt,1 sin⎜ ⎟ ⎝ 2 ⎠

The optimized IPT color-difference equation ellipses in PT, IP, and IT planes are shown in Fig.6-19 – Fig.6-21.

IPT Color Difference Equation Ellipses IPT Color Difference Equation Ellipses

80 80

70 60

60 40 I T 50

−20 20

−60 −40 −20 0 20 40 60 80 −30 −20 −10 0 10 20 30 40 P P Fig.6-19 IPT Color-Difference Fig.6-20 IPT Color-Difference Equation Ellipses in PT Plane Equation Ellipses in IP Plane

IPT Color Difference Equation Ellipses

I 50

−20 0 20 40 60 80 T Fig.6-21 IPT Color-Difference Equation Ellipses in IT Plane

62 The IPT color difference equation ellipses and RIT-Dupont ellipses at the middle gray color center are plotted in Fig.6-22 – Fig.6-24.

IPT Color Difference Equation(blue) VS. RIT−DuPont Visual Data(red) for Middle Gray IPT Color Difference Equation(blue) VS. RIT−DuPont Visual Data(red) for Middle Gray 7

59.5 6

59 5 58.5 4 58

3 57.5 I T 2 57

1 56.5

0 56

−1 55.5

55 −2

54.5 −3 −6 −4 −2 0 2 4 −4 −3 −2 −1 0 1 2 3 P P Fig.6-22 ΔEIPT VS. RIT-Dupont Fig.6-23 ΔEIPT VS. RIT-Dupont Ellipses at Middle gray in PT Plane Ellipses at Middle gray in IP Plane

IPT Color Difference Equation(blue) VS. RIT−DuPont Visual Data(red) for Middle Gray

I 57

−3 −2 −1 0 1 2 3 4 5 6 T Fig.6-24 ΔE IPT VS. RIT-Dupont Ellipses at Middle gray in IT Plane

IPT color difference equation ellipses fit the experimental color tolerance at the middle gray color center better than CIECAM02 color difference equation ellipses did, especially the ellipses in the IP plane.

Fig.6-25 – Fig.6-27 is the comparison between IPT color difference equation ellipses and

RIT-Dupont ellipses.

63 IPT Color Difference Equation(blue) VS. RIT−DuPont Visual Data(red) IPT Color Difference Equation(blue) VS. RIT−DuPont Visual Data(red)

80 80

70 60

60 40 I T 50 20

40 0

30 −20

−40 −60 −40 −20 0 20 40 60 80 −30 −20 −10 0 10 20 30 40 P P Fig.6-25 ΔEIPT VS. Fig.6-26 ΔEIPT VS. RIT-Dupont Ellipses in PT Plane RIT-Dupont Ellipses in IP Plane

IPT Color Difference Equation(blue) VS. RIT−DuPont Visual Data(red)

100

I 50

0 −40 −20 0 20 40 60 80 T Fig.6-27 ΔEIPT VS. RIT-Dupont Ellipses in IT Plane

For some color centers, such as light brown, moderate reddish brown, moderate yellow, the IPT color difference equation ellipses fit the RIT-DuPont ellipses very well. For the blue center, there was an improvement to fit the visual experimental result.

64 7. Transformations to Euclidean Color-Difference Spaces

7.1 Theory

In Euclidean space, the shortest distance between two points is a straight line. Is the shortest distance between two points in curved surfaces still a straight line in a non-

Euclidean space? In Fig.7-1, there are points A, B, and C on a mountain. We can see that points A and B are locally in a plane, and points A and C are on a curved surface. Now, the question is what are the shortest distances on the mountain from points A to B, and from points A to C, respectively? Since points A and B are in a plane, the shortest distance between points A and B on the mountain is a Euclidean distance. However, shortest distance on the mountain between points A and C is a curve, not a straight line, which is shown in Fig.7-2.

Fig.7-1 Point A, B, and C on the mountain

Fig.7-2 The distance between Point A and C on the mountain

A geodesic line is the shortest path between two points in a curved space. The line element47,48,49 is used to characterize the properties of a Riemann space. A Riemann space is a subset of a Euclidean space in which tensors are used to describe distance, angle, and curvature. It is shown as curve C with two end points A and B in Fig.7-3.

B Fig.7-3 Curve C between point A and B

In order to calculate the length of curve C, it is divided into many n segments, shown in

Fig.7-4. Their lengths are summed.

C A

Fig.7-4 Curve C and its segments

47 Assume curve C is represented by X(t) , a≤t≤b. X(tm) is a chord with vertices m, m =

0,1,2,…,n; t0 = a

66 l(Z) = l1 + l2 + ...+ ln where lm = X m − Xm−1 Eq.7-1

th lm is the length of the m chord.

• • Here we define X = dx /dt . Because X is continuous, the mean value theorem of differential calculus can be applied:

lm = (tm − tm−1)Vm Eq.7-2 where, Vm is a vector with components

• • • X1(tm1),X 2 (tm2 ),X3(tm 3)(tm−1 < tmj < tm ) Eq.7-3

• • Thus, lm = (tm − tm−1)( X m + ηm ) where ηm = Vm − X m Eq.7-4

Now, sum up the chords from 1 to n,

n • n l(Zn ) = ∑(tm − tm−1)( X m ) + ∑(tm − tm−1)ηm Eq.7-5 m=1 m=1

• ηm is the difference of the distances of the points X m and Vm from the origin and is, at

• most, equal to the distance Dm = Vm − X m between these points because of the triangle

• inequality. X is continuous, so, with an ε > 0, a δ(ε) > 0 can be found such that

Dm < ε when tmj − tm < δ ( j =1, 2, 3) Eq.7-6

Thus if the maximum chord length is less than δ, then

n n n

∑(tm − tm−1)ηm ≤ ∑(tm − tm−1)Dm < ε∑(tm − tm−1) = ε(b − a) Eq.7-7 m=1 m=1 m=1

67 This indicates that as the maximum chord length approaches zero the last sum in Eq.7-5 approaches zero while the first sum in Eq.7-5 approaches the integral:

b • • b • l = ∫ X X dt = ∫ X dt Eq.7-8 a a

Now, we replace the fixed value a and b in Eq.7-8 by t0 and a variable t respectively, then l becomes a function of t. Thus, the arc length of curve C s(t) is

t • • ~ s(t) = ∫ x• xdt. Eq.7-9 t0

It has a simple geometric interpretation. If t>t0, then s(t) is the length of the portion of C with initial point x(t0) and terminal point x(t). If t

In 3-D space, we may write symbolically dx=(dx1,dx2,dx3) and

2 2 2 2 ds = dx • dx = dx1 + dx2 + dx3 . Eq.7-10

Here, ds in Eq.7-10 is called the line element of C.

7.2 Derivation of CIECAM02 Euclidean Color Difference

Space

As was mentioned before, in Cartesian coordinates, a line element is

ds = dx 2 + dy 2 + dz2 Eq.7-11

In cylindrical coordinates, it is shown in Fig 7-5,

Fig 7-5 Cylinder coordinate (x,y,z)

x = rcosϕ y = rsinϕ Eq.7-12 z = z

Thus, the line element in cylindrical coordinates is

ds = dx 2 + dy 2 + dz2 = (cosϕ)2 dr2 + (sinϕ)2 dr2 + r2 (cosϕ)2 dϕ 2 + r2 (sinϕ)2 dϕ 2 + dz2 Eq.7-13 = dr2 + r2dϕ 2 + dz2

CIECAM02 color space can be considered as cylindrical coordinates:

r = C,ϕ = hab,z = J and the square of the line element is

ds2 = dC2 + C 2dh 2 + dJ 2

⎛ ⎞ 2 ⎛ ⎞ 2 ⎛ ⎞ 2 2 ΔCab ΔH ΔJ ΔECIECAM 02,JCH = ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ ⎝ kC SC ⎠ ⎝ kH SH ⎠ ⎝ kJ SJ ⎠ 2 2 ⎛ ∧ ⎞ 2 ⎛ dC ⎞ C dhab ⎛ dJ ⎞ =⎜ ab ⎟ + ⎜ ab ⎟ + ⎜ ⎟ Eq.7-14 k S ⎜ k S ⎟ k S ⎝ C C ⎠ ⎝ H H ⎠ ⎝ J J ⎠ ∧ ∧ habπ where, dH =C dhab, dhab = ab 180

In the new Euclidean space:

69 ∧ 2 ∧ 2 2 2 2 hnewπ ds = dC + C dhnew + dJ , where h new = Eq.7-15 new new new 180

From Eq 7-14, we have

1 dCnew = dCab kC SC ∧ ∧ Cab Cnewdhnew = dhab Eq.7-16 k H SH 1 dJnew = dJ kJ SJ

Therefore,

c ab 1 Cnew = ∫ dCab 0 kcSc c ab 1 = ∫ dt Eq.7-17 0 1+ 0.02t = 50ln(1+ 0.02C )

J 1 Jnew = ∫ dJ 0 kJ SJ J 1 = dt ∫ t 0 0.5 +1.25( )2 100 Eq.7-18 200 2 J 1 5 = ∫ d( t) 5 5t 100 2 0 1+ ( )2 100 2 200 2 5 = arctan J 5 100 2

∧ ⎛ ∧ ⎞ ∂hnew ∂hnew Cnewdhnew = Cnew⎜ dCab + dhab ⎟ Eq.7-19 ⎝ ∂Cab ∂hab ⎠

From Eq 7-16, we have

70 ∧ ⎛ ⎞ ∧ ∂hnew ∂hnew Cab Cnewdhnew = Cnew⎜ * dCab + dhab ⎟ = dhab Eq.7-20 ⎝ ∂Cab ∂hab ⎠ SH

Thus

∂h P = new = 0, ∂C ab Eq.7-21 ∂h C C Q = new = ab = ab ∂hab CnewSH Cnew (1 + 0.01Cab )

The color distance becomes independent of the actual distance, thus the integrability

condition ∂P /∂hab = ∂Q/∂Cab must be satisfied. Because ∂P /∂hab = 0, ∂Q/∂Cab must also equal zero. Consequently, Q must be constant. Also, since P=∂hnew/∂Cab=0, hnew is independent of Cab. So, we can rewrite Eq. 7-21 as

Cab hnew = hab = khab Eq.7-22 Cnew (1 + 0.01Cab )

Assuming constant k is not equal to 1, then new hue angle hnew is rotated by k times based on hue angle hab. When hue angle hab is large, then the rotation is large, and vice versa. To avoid this asymmetrical rotation, it is reasonable to set k equal to 1.

200 2 2 * Here, let J = k arctan J , C = k 50ln(1+ 0.02C ), k1 and k2 are 1 5 100 5 new 2 ab calculated by performing optimization, in which the coefficient of variation for the 200 sample dataset (RIT-Dupont dataset and Qiao dataset) is minimized.

Thus, the new Euclidean difference is:

71 200 2.5 J new = k J 0.99 × arctan J 2.5 100

Cnew = kC 0.94 × 50ln(1 + 0.02Cab )

hnew = hab , where Eq.7-23

anew = Cnew cosh new

bnew = Cnew sinh new

7.3 Derivation of IPT Euclidean

The optimized IPT color difference equation is below:

1/2 ⎡ 2 2 2⎤ ⎛ ΔI ⎞ ⎛ ΔC ⎞ ⎛ ΔH ⎞ ΔE = ⎢⎜ ⎟ + ⎜ pt ⎟ + ⎜ pt ⎟ ⎥ pt ⎢ k S k S ⎜ k S ⎟ ⎥ ⎣⎝ I I ⎠ ⎝ C C ⎠ ⎝ H H ⎠ ⎦ ⎛ ⎞ 3.5 I Eq.7-24 SI = 0.7 +1.6⎜ ⎟ ⎝ 100⎠

SC =1+ 0.03C pt

SH =1+ 0.01C pt

ΔI ΔI ΔI = = I1 + I2 3.5 1.6 I2 − I1 3.5 0.7 +1.6( ) 0.7 + 3.5 (I1 + ) 200 100 2 Eq.7-25 ΔI ΔI = ≈ 3.5 1.6 ΔI 3.5 1.6 0.7 + (I − ) 0.7 + * I1 1003.5 1 2 1003.5

So,

I 1 I = dt Eq.7-26 ∫ t 0 0.7 +1.6( )3.5 100

72 Because the result of the above integration is very complicated, we simply the above equation as

ΔI ΔI ≈ Eq.7-27 1.6 3.5 0.7 + * I 1003.5 1

ΔI ΔI ≈ + Eq.7-28 3.2 3 3.2 4 1.4 + * I 1.4 + * I 1003 1 1004 1

The above relationship is shown in Fig.7-6.

1.8 y=1/(0.7+1.6*(x/100)3.5) y=1/(1.4+3.2*(x/100)3)+1/(1.4+3.2*(x/100)4) 1.6

1.4

1.2 y

0.8

0.6

0.4 0 10 20 30 40 50 60 70 80 90 100 x Fig.7-6 The Plot of Eq.7-27(blue) VS. Eq.7-28(red) Thus,

I 1 I 1 I = dt + dt Eq.7-29 ∫ 3.2 ∫ 3.2 0 1.4 + * t 3 0 1.4 + * t 4 1003 1004

Since the integration of Eq.7-29 is very complicated, here, we use a logarithmic function to approximate the integration result. This integration result is shown below:

73 120 y=integral of 1/(1.4+3.2*(x/100)3)+1/(1.4+3.2*(x/100)4) y=326.7936*log(1+0.003*x);

100

60 y

−20 0 10 20 30 40 50 60 70 80 90 100 x Fig.7-7 The Integration Result (red) VS. Approximate Function (blue)

From Fig.7-7, we use the approximate function to replace the original function. Thus,

I = 326.7936ln(1+ 0.03I ) Eq.7-30

From Eq.7-24,

ΔC ΔC ΔC = ≈ Eq.7-31 1+ 0.03ΔC 1+ 0.03C

So,

c 1 1 C = ∫ dt = ln(1+ 0.03C) Eq.7-32 0 1+ 0.03t 0.03

Using the same method in CIECAM02, we have

74 Inew = kI 0.22 × 326.7936ln(1+ 0.03I) 1 C = k 0.92 × ln(1+ 0.03C) new c 0.03 h = h new pt Eq.7-33 pnew = Cnew coshnew tnew = Cnew sinhnew

2 2 2 1/2 ΔE pt =Δ[]Inew +Δpnew +Δtnew

75 8. Quantitative Performance Evaluation

8.1 Computational Experiment

Three factors were calculated to evaluate the Euclidean color spaces and other color spaces. The first one was PF/3 derived by Guan and Luo.50 There are four statistical measures used to evaluate the comparisons between visual color difference and ΔE from different color difference equations. Since these four different statistical measures gave different results, Guan and Luo combined the four measures into one value (PF) . PF is shown in Eq.8-1.

PF =100(γ + VAB + CV /100 − r) Eq.8-1

51 52 where, VAB was derived by Schultz , γ and CV were proposed by Coates et al., and r represents the correlation coefficient.

1 2 ∑(Xi − fYi ) CV = N ×100 Eq.8-2 X

∑ XiYi and f = 2 ∑Yi

2 1 ⎛ ⎛ X ⎞ ⎛ X ⎞⎞ log (γ) = ⎜ log i − log i ⎟ Eq.8-3 10 ∑⎜ 10⎜ ⎟ 10⎜ ⎟⎟ N ⎝ ⎝ Yi ⎠ ⎝ Yi ⎠⎠

2 1 []Xi − ()FYi VAB = ∑ Eq.8-4 N XiFYi

1/2 ⎡ n ΔE n ΔE ⎤ and F = ⎢∑ A /∑ B ⎥ Eq.8-5 ⎣ i=1 ΔEB i=1 ΔEA ⎦

76 N∑(XiYi ) − ∑ X i ∑Yi r = Eq.8-6 2 2 ⎡ 2 ⎤⎡ 2 ⎤ NX− X NY− Y ⎣⎢ ∑ i ()∑ i ⎦⎥⎣⎢ ∑ i ()∑ i ⎥⎦

In some cases the correlation coefficient (r) was inconsistent with the other measures. So, only three measures were used. The PF/3 is:

PF /3 =100[(γ −1) + VAB + CV /100]/3 Eq.8-7

For the perfect agreement between two datasets, CV=0, VAB=0, γ=1, and PF/3=0.

The second metric was the maximum disagreement,53 which evaluated the percentage of disagreement between two color difference equations. For perfect agreement, MD=0.

⎡ ⎤ max(ΔE1,ΔE 2) MD(ΔE1,ΔE2) = ⎢ −1⎥ ×100% Eq.8-8 ⎣ min(ΔE1,ΔE2) ⎦

Although the PF/3 is used widely, it cannot indicate the significance of difference between the two formulae or spaces tested. Thus, the third metric was used based on statistical F test.54

The testing hypothesis is described below, for which VM is given:

N 2 ∑(ΔVi − aM ΔE Mi) V = i=1 , M ∈ {A,B} Eq.8-9 M N −1

(1) Define the null and alternate hypotheses

H0: VA=VB (which means two formulae without significant difference)

(HA: VA≠ٛ VB (which means two formulae with significant difference

77 Calculate the F value as F ٛ =VA/VB (2)

Reject the hypothesis (H0) when F < FC or if F ٛ >1/FC (3)

where Fc = F(dfA ,df B ,0.975) is the lower critical value of the two-tailed F distribution

with 95% confidence level and dfA and dfB are the degrees of freedom. N is the number

of samples in the dataset, and dfA = df B = N −1 in this study. VA and VB represent the residual error variances after scaling correction for Models A and B, respectively. The formula of aM is:

∑(ΔEM iΔVi ) a i Eq.8-10 M = 2 ∑(ΔE M i ) i

The results can be divided into five categories as shown below:

;Model A is significantly better than model B when F <ٛ FC ●

;Model A is significantly poorer than model B when F ٛ >1/ FC ●

;Model A is insignificantly better than model B when FC ≤ ٛ F ٛ <1 ●

;Model A is insignificantly poorer than model B when 1 < ٛ F ٛ ≤1/ FC ●

.Model A is equal to model B when Fٛ =1 ●

In order to evaluate the performance of the Euclidean color spaces described in Chapter 7, the Euclidean CIECAM02 space (Eq.7-27) and Euclidean IPT space (Eq.7-36) was compared with the optimized CIECAM02 color-difference equation (Eq.6-2) and the optimized IPT color difference equation (Eq.6-5), respectively. 100,000 positions in

CIECAM02 were randomly selected as color standards. For each position, a random direction and random distance scaled between 0 and 5 CIECAM02 units were used to

78 define 100,000 sample colors. Color differences were calculated using Eq.6-2, Eq.7-27 and Eq.6-5, and Eq.7-36. A scatter plot is shown in Fig.8-1 and Fig.8-2.

PF/3 and the average disagreement for the Euclidean CIECAM02 color space was 2.10 and 1.75% respectively. In the Euclidean IPT color space, PF/3 and the average disagreement is 16.93 and 19.50% respectively.

ΔE EuclideanCIECAM02 VS. ΔE CIECAM02 5

4.5

3.5

2.5

1.5 Euclidean CIECAM02 Color Difference

0.5

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Optimized CIECAM02 Color Difference Fig.8-1 Scatter plot comparing ΔEEuclidean-CIECAM02 and ΔECIECAM02 for Random100,000 color-difference pairs.

ΔE EuclideanIPT VS. ΔE IPT 8

3 Euclidean IPT Color Difference

0 0 1 2 3 4 5 6 7 8 Optimized IPT Color Difference Fig.8-2 Scatter plot comparing ΔEEuclidean-IPT and ΔEIPT for Random100,000 color-difference pairs.

79 ΔE00, ΔELab, ΔECIECAM02,ΔERotate-CIECAM02,ΔEEuclidean-CIECAM02, ΔERotate-Euclidean-CIECAM02,

ΔEIPT, ΔEEuclidean-IPT were calculated by using RIT-Dupont and Qiao,et al. datasets, 200 color pairs and ΔV=1.02. PF/3 values and MD (maximum-disagreement) values between

ΔE and ΔV are shown in Table 8-I. In Table 8-I, the maximum MD and MD standard deviation are also listed.

Table 8-I. PF/3 and MD values using RIT-Dupont and Qiao, et al. datasets.

PF/3 MDmean MDmax MDstd

ΔEab 35.71 47.00% 324.07% 0.50

ΔE94 25.72 21.85% 120.73% 0.21

ΔE00 22.88 19.54% 89.09% 0.17

ΔECIECAM02 22.65 30.51% 109.63% 0.23

ΔERotate-CIECAM02 22.70 31.48% 111.65% 0.24

ΔEEuclidean-CIECAM02 22.67 19.83% 112.64% 0.18

ΔERotateEuclideanCIECAM02 21.79 18.96% 85.62% 0.17

ΔEIPT 22.54 20.65% 108.98% 0.19

ΔEEuclidean-IPT 24.12 30.13% 134.62% 0.24

The results of the F tests are given in Table 8-II. The Fc and 1/ Fc values were 0.757 and

1.321, respectively for 199 degree of freedom. In each table, the corresponding values above the diagonal are the inverse of the values under the diagonal.

80 Table 8-II. The F test results using RIT-Dupont and Qiao, et al. datasets.

ΔEab ΔE94 ΔE00 ΔECAM ΔERCAM ΔEECAM ΔERECAM ΔEIPT ΔEEIPT

ΔEab 1.878 2.324 2.545 2.68 2.702 2.805 2.405 2.314

ΔE94 0.532 1.238 1.355 1.428 1.439 1.494 1.281 1.232

ΔE00 0.430 0.808 1.095 1.154 1.163 1.207 1.035 0.995

ΔECAM 0.393 0.738 0.913 1.054 1.062 1.102 0.945 0.909

ΔERCAM 0.373 0.700 0.867 0.949 1.008 1.046 0.897 0.862

ΔEECAM 0.370 0.695 0.860 0.942 0.992 1.038 0.890 0.856

ΔERECAM 0.357 0.670 0.829 0.907 0.956 0.963 0.857 0.825

ΔEIPT 0.416 0.781 0.966 1.058 1.115 1.123 1.166 0.962

ΔEEIPT 0.432 0.812 1.005 1.100 1.159 1.168 1.212 1.040

NOTE. The Fc and 1/ Fc values are 0.757 and 1.321 respectively.

8.2 Analysis

The PF/3 between ΔECIECAM02 and ΔEEuclidean-CIECAM02 for 100,000 random color pairs was

2.1. The average disagreement ( MD ) was only 1.75%, the maximum disagreement

( MDmax ) was 13.8%, and standard deviation ( MDstd ) was 0.02. These indicate that the

Euclidean CIECAM02 color space well approximated the optimized CIECAM02 color- difference equation. In Figure 8-1, Euclidean distances in Euclidean CIECAM02 color space are plotted against ΔECIECAM02. Most of points are lying close to the line that has slope equal to 1. However, in Euclidean IPT color space, the PF/3 between ΔEIPT and

ΔEEuclidean-IPT for 100,000 random color pairs was 16.93, the average disagreement ( MD)

was 19.5%, the maximum disagreement ( MDmax ) was 48.46%, and standard deviation

( MDstd ) was 0.10. Also, in Fig 8-2, it indicates that Euclidean IPT does not approximate the IPT color-difference equation very well. More color pairs in Euclidean IPT color space gave larger color differences than the optimized IPT color-difference equation. In

Eq.7-24, the exponent in SI was 3.5, which complicated the integral of SI. So, there are

81 two steps to simply the integration (Eq.7-32 to Eq.7-34), which may affect the precision of the Euclidean IPT color space.

In Table 8-I, as expected, the optimized CIECAM02 and IPT color difference equations performed slightly better than ∆E00 color difference equations, because the first two models were optimized by using RIT-DuPont and Qiao dataset. These data were a part of data used to derive ∆E00. It shows in Table 8-I that the rotated Euclidean CIECAM02 color space gives the best performance. The Euclidean CIECAM02 color space and the rotated CIECAM02 color difference equation have as a good performance as the rotated

Euclidean CIECAM02 color space. Thus, the rotated CIECAM02 color difference equation, the Euclidean CIECAM02 color space, and the rotated Euclidean CIECAM02 color spaces improved the performance of the optimized CIECAM02 color difference equation. However, as was discussed in the previous paragraph, the Euclidean IPT color space did not give such improvement. As expected, ΔE*ab gave the worst performance.

Although ∆E00 has good performance, it was derived based on CIELAB, and is not associated with a uniform color space. The interactive term between chroma and hue differences in ∆E00 improved the performance for blue color, but it makes it difficult to derive a Euclidean space based on ∆E00.

In Table 8-II, all the values in the first column are smaller than 0.757, which means that these eight models significantly perform better than ΔE*ab. As the F-test results of the rotated CIECAM02 color difference equation, the Euclidean CIECAM02 color space and the rotated Euclidean CIECAM02 color space in Table 8-II were not significantly

82 different, their performances are nearly same. The optimized IPT color difference equation and the Euclidean IPT color space was optimized by using RIT-DuPont and

Qiao data, thus, their performances are very close, which is shown in Table 8-II. In order to compare these two models in the whole IPT color space, the 100,000 random color pairs were used to sample the color space. As we mentioned previously, this type of global test indicated that the Euclidean IPT color space did not improve the optimized

IPT color difference equation.

In this thesis, the lightness in new Euclidean color space was derived by integrating the color difference equation along the lightness axis, which is reasonable, because the lightness difference is independent of chroma and hue. In the new Euclidean color space, the hue angle did not change in order to avoid the asymmetrical hue angle rotation (see

Eq.7-22). However, the hue difference is dependent of chroma. Volz HG transformed

CIEDE2000 to Euclidean color space47. In his method new hue angle is dependent on original chroma and hue angle. Since this Euclidization was only done in the first quadrant, there is no rotation issue. Phillipp Urban also presented a way to transform color difference equation to Euclidean color space45. His main idea is to combine a 1-D

LUT for the lightness and a 2-D LUT for chroma and hue coordinates. The mean disagreement between the distances calculated by color difference equations and

Euclidean color difference is below 3%. Cui and Luo, et al52 derived three Euclidean color spaces based on the DIN99 color difference equation. Unlike CIECAM02 and IPT color difference equation, DIN99 equation is not only a color difference equation, but

83 also is a color space. One of the DIN99 Euclidean color spaces has 260 hue angle offset which corresponded to the CIELAB hue angle.

84 9. Qualitative Performance of Visual Dataset

9.1 Visualization of RIT-DuPont Dataset

In the previous chapter, the different metrics were calculated to evaluate different formulae. In this chapter, the visual performance of CIECAM02, rotated CIECAM02,

Euclidean CIECAM02, IPT, Euclidean IPT will be shown and discussed. All the ellipses are enlarged triple times.

RIT-DuPont visual ellipses in Euclidean CIECAM02 color space in ab, Ja, and Jb planes are plotted in Fig.9-1 – Fig.9-3.

RIT−Dupont Color−Difference Ellipses in Euclidean CIECAM02 Space RIT−Dupont Color−Difference Ellipses in Euclidean CIECAM02 Space

110 40

100 30

20 80

10 70 b J

0 60

50 −10

40 −20 30

−30 20 −50 −40 −30 −20 −10 0 10 20 30 40 50 −60 −40 −20 0 20 40 60 a a Fig.9-1 RIT-DuPont Ellipses in Fig.9-2 RIT-DuPont Ellipses in Euclidean CIECAM02 Space (in ab Plane) Euclidean CIECAM02 Space (in aJ Plane)

RIT−Dupont Color−Difference Ellipses in Euclidean CIECAM02 Space

110

100

70 J

20 −40 −20 0 20 40 60 b Fig.9-3 RIT-DuPont Ellipses in Euclidean CIECAM02 Space (in bJ Plane)

85 RIT-DuPont visual ellipses in rotated Euclidean CIECAM02 color-difference in ab, Ja, and Jb plane are plotted in Fig.9-4 – Fig.9-6.

RIT−DuPont Visual Ellipsis in Rotated Euclidean CIECAM02 Color Space RIT−DuPont Visual Ellipsis in Rotated Euclidean CIECAM02 Color Space

40 110

100 30

90 20 80

10 70 J b

0 60

50 −10

40 −20 30

−30 20 −50 −40 −30 −20 −10 0 10 20 30 40 −60 −40 −20 0 20 40 60 a a Fig.9-4 RIT-DuPont Ellipses in Rotated Fig.9-5 RIT-DuPont Ellipses in Rotated Euclidean CIECAM02 Space (in ab Plane) Euclidean CIECAM02 Space (in aJ Plane)

RIT−DuPont Visual Ellipsis in Rotated Euclidean CIECAM02 Color Space

110

100

70 J

20 −40 −20 0 20 40 60 b Fig.9-6 RIT-DuPont Ellipses in Rotated Euclidean CIECAM02 Space (in bJ Plane)

As discussed in Chapter 3, RIT-DuPont visual ellipses in CIELAB are long and thin and oriented toward the origin. In Euclidean CIECAM02 color space and rotated Euclidean

CIECAM02 color space, RIT-DuPont visual ellipses are almost the same, especially in aJ and bJ planes, and much closer to circles than those of CIELAB, CIECAM02, and IPT.

Also, the sizes of ellipses in Euclidean CIECAM02 color space and rotated Euclidean

CIECAM02 color space are much closer to each other than in CIELAB, CIECAM02, and

86 IPT. This corresponds to stretching CIECAM02 space by the Riemannian transformation.

In rotated Euclidean CIECAM02 color space, the visual tolerance in medium gray, light brown, dark reddish orange, and moderate greenish blue color center were more uniform than those in the Euclidean CIECAM02 color space.

RIT-Dupont ellipses with vectors in Euclidean CIECAM02 and rotated Euclidean

CIECAM02 color space are plotted in Fig.9-7 – Fig.9-12, respectively.

RIT−Dupont Visual Ellipses with Vectors in ab plane (Euclidean CIECAM02 Space) RIT−Dupont Visual Ellipses&Vectors in aJ plane (Euclidean CIECAM02 Space)

110 40

100 30

20 80

10 70 J b

0 60

50 −10

40 −20 30

−30 20 −50 −40 −30 −20 −10 0 10 20 30 40 50 −60 −40 −20 0 20 40 60 a a Fig.9-7 RIT-DuPont Ellipses with Vectors in Fig.9-8 RIT-DuPont Ellipses with Vectors in Euclidean CIECAM02 Space (in ab Plane) Euclidean CIECAM02 Space (in aJ Plane)

RIT−Dupont Visual Ellipses with Vectors in bJ plane (Euclidean CIECAM02 Space)

110

100

70 L

20 −40 −20 0 20 40 60 b Fig.9-9 RIT-DuPont Ellipses with Vectors in Euclidean CIECAM02 Space (in bJ Plane)

87 RIT−Dupont Visual Ellipsis with Vectors in ab plane (Rotataed Euclidean CIECAM02 Space) RIT−Dupont Visual Ellipsis with Vectors in aJ plane (Rotataed Euclidean CIECAM02 Space)

40 110

100 30

90 20 80

10 70 b J

0 60

50 −10

40 −20 30

−30 20 −50 −40 −30 −20 −10 0 10 20 30 40 −60 −40 −20 0 20 40 60 a a Fig.9-10 RIT-DuPont Ellipses with Vectors in Fig.9-8 RIT-DuPont Ellipses with Vectors in Rotated Euclidean CIECAM02 Space(ab Plane) Rotated Euclidean CIECAM02 Space(aJ Plane)

RIT−Dupont Visual Ellipsis with Vectors in bJ plane (Rotataed Euclidean CIECAM02 Space)

110

100

70 J

20 −40 −20 0 20 40 60 b Fig.9-9 RIT-DuPont Ellipses with Vectors in Rotated Euclidean CIECAM02 Space (in bJ Plane)

From Fig.9-4, Fig.9-6, Fig.9-9, and Fig.9-12, it indicates that the ellipsoids did not fit the vectors very well for the light gray color center. Whether the optimized ellipsoids fit the vectors well or not depends on the position of each vectors, which is a limitation.

Euclidean CIECAM02 color-difference ellipses and Euclidean CIECAM02 RIT-Dupont ellipses at middle gray color center are shown in Fig.9-13 – Fig.9-15. The semi-axes of each ellipsis are three times larger than those of ellipsis, whose color difference is equal to 1.

88 Euclidean CIECAM02 Difference Equation(B) VS. RIT−DuPont Visual Data(R) for Middle Gray Euclidean CIECAM02 Difference Equation(B) VS. RIT−DuPont Visual Data(R) for Middle Gray

6 86

85 4

2 J b 83

−2 81

−4 80

−8 −6 −4 −2 0 2 4 −6 −5 −4 −3 −2 −1 0 1 2 3 a a Fig.9-13 ΔE EucCIECAM 02 Ellipses VS. Fig.9-14 ΔE EucCIECAM 02 Ellipses VS. RIT-Dupont Ellipses at Middle Gray in ab Plane RIT-Dupont Ellipses at Middle gray in aJ Plane

Euclidean CIECAM02 Difference Equation(B) VS. RIT−DuPont Visual Data(R) for Middle Gray

J 83

−4 −2 0 2 4 6 b Fig.9-15 ΔE EucCIECAM 02 Ellipses VS. RIT-Dupont Ellipses at Middle gray in bJ Plane

Fig.9-16 – Fig.9-18 is the rotated Euclidean CIECAM02 color-difference equation ellipses and rotated Euclidean CIECAM02 RIT-Dupont ellipses for the middle gray color center.

89 Rotated Euclidean CIECAM02 Difference Equation(B) VS. RIT−DuPont Visual Data(R) for Middle Gray Rotated Euclidean CIECAM02 Difference Equation(B) VS. RIT−DuPont Visual Data(R) for Middle Gray

6 86

85 4

84 2 J b 83

−2 81

−4 80

−8 −6 −4 −2 0 2 4 −6 −5 −4 −3 −2 −1 0 1 2 3 a a Fig.9-16 ΔE RotEucCIECAM 02 Ellipses VS. Fig.9-17 ΔE RotEucCIECAM 02 Ellipses VS. RIT-Dupont Ellipses at Middle Gray in ab Plane RIT-Dupont Ellipses at Middle Gray in aJ Plane

Rotated Euclidean CIECAM02 Difference Equation(B) VS. RIT−DuPont Visual Data(R) for Middle Gray

J 83

−4 −2 0 2 4 6 b Fig.9-18 ΔERotEucCIECAM 02 Ellipses VS. RIT-Dupont Ellipses at Middle gray in bJ Plane

The ratio of major axis to minor axis of RIT-Dupont middle gray ellipses in ab, Ja, and Jb plane in Euclidean CIECAM02 space is 1.41, 1.00, 1.19, respectively. In rotated

Euclidean CIECAM02 space, the ratio is 1.18, 1.00, 1.24, respectively. The middle gray ellipsoid in rotated Euclidean CIECAM02 space is much closer to sphere and fit RIT-

DuPont visual data better than Euclidean CIECAM02 space.

The Euclidean CIECAM02 color-difference ellipses and Euclidean CIECAM02 RIT-

Dupont ellipses in ab, aJ, and bJ plane are plotted in Fig.9-19 – Fig.9-21.

90 Euclidean CIECAM02 Difference Equation(B) VS. RIT−DuPont Visual Data(R) Euclidean CIECAM02 Difference Equation(B) VS. RIT−DuPont Visual Data(R)

110 40

100 30

20 80

10 70 J b

0 60

50 −10

40 −20 30

−30 20 −50 −40 −30 −20 −10 0 10 20 30 40 50 −60 −40 −20 0 20 40 60 a a Fig.9-19 ΔEEucCIECAM 02 Ellipses VS. Fig.9-20 ΔEEucCIECAM 02 Ellipses VS. RIT-Dupont Ellipses in ab Plane RIT-Dupont Ellipses in aJ Plane

Euclidean CIECAM02 Difference Equation(B) VS. RIT−DuPont Visual Data(R)

110

100

70 J

20 −40 −20 0 20 40 60 b Fig.9-21 ΔE EucCIECAM 02 Ellipses VS. RIT-Dupont Ellipses in bJ Plane

The rotated Euclidean CIECAM02 color-difference ellipses and rotated Euclidean

CIECAM02 RIT-Dupont ellipses in ab, aJ, and bJ plane are plotted in Fig.9-22 – Fig.9-

23.

Rotated Euclidean CIECAM02 Difference Equation(B) VS. RIT−DuPont Visual Data(R) Rotated Euclidean CIECAM02 Difference Equation(B) VS. RIT−DuPont Visual Data(R)

40 110

100 30

90 20 80

10 70 J b

0 60

50 −10

40 −20 30

−30 20 −50 −40 −30 −20 −10 0 10 20 30 40 −60 −40 −20 0 20 40 60 a a Fig.9-22 ΔE RotEucCIECAM 02 Ellipses VS. Fig.9-23 ΔE RotEucCIECAM 02 Ellipses VS.

91 RIT-Dupont Ellipses in ab Plane RIT-Dupont Ellipses in aJ Plane

Rotated Euclidean CIECAM02 Difference Equation(B) VS. RIT−DuPont Visual Data(R)

110

100

70 J

20 −40 −20 0 20 40 60 b Fig.9-24 ΔERotEucCIECAM 02 Ellipses VS. RIT-Dupont Ellipses in bJ Plane

It is shown in Fig.9-19 – Fig.9-24 that color-difference ellipses in Euclidean CIECAM02, rotated Euclidean CIECAM02 and Euclidean IPT color space are circles as expected. The visual color tolerance gives good fit to color-difference equation ellipses in both

Euclidean CIECAM02 color space and rotated Euclidean CIECAM02 color space. These visual results are consistent with the statistic and evaluations in the previous chapter.

RIT-DuPont visual ellipses in Euclidean IPT color space in PT, IP, and IT plane are shown in Fig.9-25 – Fig.9-27.

RIT−Dupont Color−Difference Ellipses in PT plane (Euclidean IPT Space) RIT−Dupont Color−Difference Ellipses in IP plane (Euclidean IPT Space)

20 70

I 60 T

0 50

−10 40

30 −20

−30 −20 −10 0 10 20 30 −40 −30 −20 −10 0 10 20 30 40 P P Fig.9-19 RIT-DuPont Visual Fig.9-20 RIT-DuPont Visual

92 Ellipses in Euclidean IPT Space (in PT Plane) Ellipses in Euclidean IPT Space (in IP Plane)

RIT−Dupont Color−Difference Ellipses in IT plane (Euclidean IPT Space)

I 60

−30 −20 −10 0 10 20 30 40 T Fig.9-21 RIT-DuPont Visual Ellipses In Euclidean IPT Space (in IT Plane)

The ellipses in Euclidean IPT color space are not tilted so much. The neutral ellipses in

Euclidean CIECAM02 space has large ratio of major axis to minor axis, but, in Euclidean

IPT space, this ratio decreased dramatically. However, generally, the ellipses in IPT space are not close to circles and the sizes of the ellipses are very different from each other. The approximate integration in deriving the Euclidean IPT color space affected the performance of Euclidean IPT color space adversely. Clearly, in Fig.9-19 – Fig.9-21, visually the RIT-DuPont ellipses in Euclidean IPT color space are not as good as those in

Euclidean CIECAM02 color space and rotated Euclidean CIECAM02 color space.

Fig.9-22 – Fig.9-24 is RIT-Dupont ellipses with vectors in Euclidean IPT color space.

RIT−Dupont Color−Difference Ellipses&Vectors in PT plane (Euclidean IPT Space) RIT−Dupont Color−Difference Ellipses&Vectors in IP plane (Euclidean IPT Space)

20 70

I 60 T

0 50

−10 40

30 −20

−30 −20 −10 0 10 20 30 −40 −30 −20 −10 0 10 20 30 40 P P

93 Fig.9-22 RIT-DuPont Visual with Vectors Fig.9-23 RIT-DuPont Visual with Vectors Ellipses In Euclidean IPT Space (in PT Plane) Ellipses In Euclidean IPT Space (in IP Plane) RIT−Dupont Color−Difference Ellipses&Vectors in IT plane (Euclidean IPT Space)

I 60

−30 −20 −10 0 10 20 30 40 T Fig.9-24 RIT-DuPont Visual Ellipses with Vectors In Euclidean IPT Space (in IT Plane)

The ratio of major axis to minor axis of RIT-Dupont middle gray ellipses in PT, IP, and

IT plane in Euclidean IPT space is 1.87, 1.14, 1.62, respectively.

Fig.9-25 – Fig.9-27 is the Euclidean IPT color difference ellipses and RIT-Dupont ellipses at middle gray color center.

Euclidean IPT Color Difference Equation(B) VS. RIT−DuPont Visual Data(R) for Middle Gray Euclidean IPT Color Difference Equation(B) VS. RIT−DuPont Visual Data(R) for Middle Gray

4 73.5

73 3

72.5 2 I

T 72 1

71.5

−1 70.5

−2 −4 −3 −2 −1 0 1 2 3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 P P Fig.9-25 Euclidean IPT Fig.9-26 Euclidean IPT Color-Difference Ellipses in PT Plane Color-Difference Ellipses in IP Plane

94 Euclidean IPT Color Difference Equation(B) VS. RIT−DuPont Visual Data(R) for Middle Gray 74.5

73.5

72.5

I 72

71.5

70.5

69.5 −1 0 1 2 3 4 T Fig.9-27 Euclidean IPT Color-Difference Ellipses in IT Plane

Fig.9-28 – Fig.9-30 is the comparison between Euclidean IPT color difference ellipses and RIT-Dupont ellipses.

Euclidean IPT Color Difference Equation(blue) VS. RIT−DuPont Visual Data(red) Euclidean IPT Color Difference Equation(blue) VS. RIT−DuPont Visual Data(red)

10 I T 60

0 50

−10 40

−20 30 −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 40 P P Fig.9-28 ΔE EucIPT VS. Fig.9-29 ΔE EucIPT VS. RIT-Dupont Ellipses in PT Plane RIT-Dupont Ellipses in IP Plane

Euclidean IPT Color Difference Equation(blue) VS. RIT−DuPont Visual Data(red)

I 60

−30 −20 −10 0 10 20 30 40 T Fig.9-30 ΔE EucIPT VS. RIT-Dupont Ellipses in IT Plane

95 Fig.9-31 – Fig.9-45 is unity color tolerance ellipses in the different color spaces. Fig.9-31

* – Fig.9-33 is ΔE94 =1 ellipses projected in a*b*, a*L*, and b*L* plane.

CIE94 Color Difference Equation CIE94 Color Difference Equation 100 80

90 60

80 40 70

20 60 b* 0 L* 50

40 −20

30 −40

−60 10

−80 0 −100 −80 −60 −40 −20 0 20 40 60 80 100 −60 −40 −20 0 20 40 60 a* a* * * Fig.9-31 ΔE94 =1 Fig.9-32 ΔE94 =1 ellipses projected in a*b* plane ellipses projected in a*L* plane

CIE94 Color Difference Equation 100

L* 50

0 −60 −40 −20 0 20 40 60 b* * Fig.9-33 ΔE94 =1 ellipses projected in b*L* plane

In Fig.9-31, all the ellipses predicted by CIE94 point toward the origin and become longer and thinner as the chroma increases. RIT-DuPont experimental ellipses in

CIELAB space indicate that the ellipses are orientated toward to the origin except in blue area. The ellipses in blue region have an anticlockwise rotation. Obviously, CIE94 equation gives errors in predicting chroma difference for blue color. Also, in RIT-DuPont ellipses in a*b* plane (see Fig.3-8), for the origin area colors, the ellipses are not constant circles, but orientated toward around 900. But, as shown in Fig.9-33, CIE94 equation

96 wrongly predicts the color ellipses in that area. It can be seen in Fig.9-32 and Fig.9-33 that the major axis of ellipses in L*a*, L*b* plane increases as chroma increases.

Fig.9-34 – Fig.9-36 is ΔE 00 =1 ellipses projected in a*b*, a*L*, and b*L* planes.

CIE2000 Color Difference Equation CIE2000 Color Difference Equation 100 100

90 80

60 80

40 70

20 60

0 50 L* b*

−20 40

−40 30

−60 20

−80 10

−100 0 −100 −50 0 50 100 −60 −40 −20 0 20 40 60 a* a* Fig.9-34 ΔE 00 =1 Fig.9-35 ΔE 00 =1 ellipses projected in a*b * plane ellipses projected in a*L * plane

CIE2000 Color Difference Equation 100

50 L*

0 −60 −40 −20 0 20 40 60 b* Fig.9-36 ΔE 00 =1 ellipses projected in b*L * plane

Since ΔE 00 was derived to improve the performance for blue colors centers, there is a hue

0 angle 275 rotation in Fig.9-34. Besides, a scaling factor for a* scale in ΔE 00 improved the performance for gray colors, which is shown in Fig.9-34 that the ellipses for gray

colors are not constant circles. ΔE 00 was optimized to fit the combined visual datasets, including Qiao dataset, Luo-Rigg dataset, and RIT-DuPont-Witt dataset, it gives the best

97 prediction in CIELAB color space. But, the shortcoming is the CIELAB space is not a

uniform space and the interactive term between chroma and hue differences in ΔE 00 makes it difficult derive a Euclidean space mathematically.

Fig.9-37 – Fig.9-39 is ΔECIECAM 02 =1 ellipses projected in ab, aJ, and bJ plane.

CIECAM02 Color Difference Equation CIECAM02 Color Difference Equation 80 100

90 60

80 40 70

20 60 J b 0 50

40 −20

30 −40 20

−60 10

−80 0 −100 −80 −60 −40 −20 0 20 40 60 80 100 −60 −40 −20 0 20 40 60 a a Fig.9-37 ΔECIECAM 02 =1 Fig.9-38 ΔECIECAM 02 =1 ellipses projected in ab plane ellipses projected in aJ plane

CIECAM02 Color Difference Equation 100

J 50

0 −60 −40 −20 0 20 40 60 b Fig.9-39 ΔECIECAM 02 =1 ellipses projected in bJ plane

Overall, the chromaticity ellipses are dependent on the chroma location. The ellipses are smaller in the neutral region and elongate as the chroma increases. Comparing with

* ΔE94 =1 and ΔE 00 =1 chromaticity ellipses, the ellipses in Fig.9-37 are much closer to circle, which also can be found in Fig.3-18.

98 Fig.9-40 – Fig.9-42 is ΔERotCIECAM 02 =1 ellipses projected in ab,aJ, bJ planes.

Rotated CIECAM02 Color Difference Equation Rotated CIECAM02 Color Difference Equation 80 100

90 60

80 40 70

20 60 b 0 J 50

40 −20

30 −40 20

−60 10

−80 0 −100 −80 −60 −40 −20 0 20 40 60 80 100 −60 −40 −20 0 20 40 60 a a Fig.9-40 ΔERotCIECAM 02 =1 Fig.9-41 ΔERotCIECAM 02 =1 ellipses projected in ab plane ellipses projected in aJ plane

Rotated CIECAM02 Color Difference Equation 100

J 50

0 −60 −40 −20 0 20 40 60 b Fig.9-42 ΔE RotCIECAM 02 =1 ellipses projected in bJ plane

Because a rotation matrix was used to minimize the mid-gray coefficient of variation, the

ΔERotCIECAM 02 =1 ellipses in ab plane are rotated in comparison with Fig.9-37.

ΔEIPT =1 ellipses projected in PT, IP, and IT plane are plotted in Fig.9-43 – Fig.9-45.

99 IPT Color Difference Equation IPT Color Difference Equation 100 80

90 60

40 70

20 60 I T 0 50

40 −20

30 −40

−60 10

−80 0 −100 −80 −60 −40 −20 0 20 40 60 80 100 −60 −40 −20 0 20 40 60 P P Fig.9-43 ΔEIPT =1 Fig.9-44 ΔEIPT =1 ellipses projected in PT plane ellipses projected in IP plane

IPT Color Difference Equation 100

I 50

0 −60 −40 −20 0 20 40 60 T Fig.9-45 ΔE IPT =1 ellipses projected in IT plane

* ΔEIPT =1 ellipses are similar with ΔE94 =1 ellipses. Due to the RIT-DuPont visual ellipses in blue region in IPT space tilt toward to origin, IPT color difference equation predicts

* visual data better than ΔE94 does.

9.2 Visualization of Qiao Dataset

The color difference of Qiao data in Euclidean CIECAM02, rotated Euclidean

CIECAM02, and Euclidean IPT color space are plotted in Fig.9-46 – Fig.9-49, respectively.

100 Qiao Data in Rotated CIECAM02 Color Space 3.5

2.5

2 ΔH

1.5

0.5 L40C20 L40C35 L60C20 0 0 50 100 150 200 250 300 350 Hue Angle Fig.9-46 ΔH of Qiao data in rotated CIECAM02

Qiao Data in Euclidean CIECAM02 Color Space 2.5

1.5

ΔH

0.5

L40C20 L40C35 L60C20 0 0 50 100 150 200 250 300 350 Hue Angle Fig.9-47 ΔH of Qiao data in Euclidean CIECAM02

101 Qiao Data in Rotated Euclidean CIECAM02 Color Space 2.5

1.5

ΔH

0.5

L40C20 L40C35 L60C20 0 0 50 100 150 200 250 300 350 Hue Angle Fig.9-48 ΔH of Qiao data in rotated Euclidean CIECAM02

Qiao Data in Euclidean IPT Color Space 2

1.8

1.6

1.4

1.2

ΔH 1

0.8

0.6

0.4

0.2 L40C20 L40C35 L60C20 0 0 50 100 150 200 250 300 350 Hue Angle Fig.9-49 ΔH of Qiao data in Euclidean IPT

Fig.9-50 – Fig.9-52 is the comparisons among those models.

102 Qiao Data Comparison (L40C20) 2.5

1.5

1 ΔH

0.5

IPT Euclidean IPT −0.5 CIECAM02 Rotated CIECAM02 Euclidean CIECAM02 Rotated Euclidean CIECAM02 −1 0 50 100 150 200 250 300 350 Hue Angle Fig.9-50 Color difference of L40C20

Qiao Data Comparison (L40C35) 3.5

2.5

1.5 ΔH

0.5

0 IPT Euclidean IPT CIECAM02 −0.5 Rotated CIECAM02 Euclidean CIECAM02 Rotated Euclidean CIECAM02 −1 0 50 100 150 200 250 300 350 Hue Angle Fig.9-51 Color difference of L40C35

103 Qiao Data Comparison (L60C20) 3

2.5

1.5

ΔH 1

0.5

0 IPT Euclidean IPT CIECAM02 −0.5 Rotated CIECAM02 Euclidean CIECAM02 Rotated Euclidean CIECAM02 −1 0 50 100 150 200 250 300 350 Hue Angle Fig.9-52 Color difference of L60C20

Fig.9-50 is the ΔH of these six models (IPT, Euclidean IPT, CIECAM02, Rotated

CIECAM02, Euclidean CIECAM02, Rotated Euclidean CIECAM02) for L40C20 change in the similar pattern as hue angle changes. However, they have unity ΔH at different hue angles. For L40C35, their behaviors are almost same, except that of ΔH the rotated

Euclidean CIECAM02 decreases from 1.5 to around 1 as the hue angle is larger than 300

degree. Table 9-I is the mean value and coefficient of variation (CV), δΔE ΔE , of Qiao color differences in six models. The average ΔH in Euclidean IPT color space is closest to the unit color difference. But, in general, the coefficients of variation of these six models are very close. The rotated CIECAM02 and rotated Euclidean CIECAM02 color spaces slightly improved the uniformity of color space. Table 9-II is the F-test result using Qiao data. Since the Fc and 1/ Fc values are 0.510 and 1.961 respectively, none of these six models has the significant better result and they have almost the same performance. The rotated CIECAM02 and rotated Euclidean CIECAM02 color spaces performed slightly better than the others, which agree with the CV results.

104 Table 9-I. ΔE and CV(ΔE) in Color Spaces L40C20 L40C35 L60C20 Average

ΔECIECAM 02 1.509 2.144 1.525 1.726

ΔERotCIECAM02 1.578 2.232 1.586 1.797

ΔEIPT 1.168 1.764 1.301 1.411

ΔEEucCIECAM 02 1.227 1.582 1.245 1.351

ΔERotEucCIECAM 02 1.276 1.608 1.306 1.397

ΔEEucIPT 0.803 1.075 0.878 0.919

CV(ΔECIECAM 02) 0.201 0.294 0.336 0.277

CV(ΔERotCIECAM02) 0.189 0.273 0.322 0.261

CV(ΔE IPT ) 0.222 0.304 0.304 0.277

CV(ΔEEucCIECAM 02 ) 0.249 0.294 0.326 0.290

CV(ΔERotEucCIECAM 02) 0.237 0.272 0.305 0.271

CV(ΔEEucIPT ) 0.255 0.323 0.318 0.299

Table 9-II. The F test results using Qiao, et al. datasets.

ΔECAM ΔERCAM ΔEECAM ΔERECAM ΔEIPT ΔEEIPT

ΔECAM 1.158 0.994 1.149 0.879 0.889

ΔERCAM 0.863 0.858 0.993 0.759 0.767

ΔEECAM 1.005 1.165 1.156 0.884 0.893

ΔERECAM 0.869 1.007 0.864 0.765 0.773

ΔEIPT 1.137 1.317 1.131 1.307 1.011

ΔEEIPT 1.125 1.303 1.118 1.293 0.989

NOTE. The Fc and 1/ Fc values are 0.510 and 1.961 respectively.

In Fig.9-53 – Fig.9-60, the Euclidean IPT color space has the best hue linearity among rotated CIECAM02, Euclidean CIECAM02, rotated Euclidean CIECAM02, and

Euclidean IPT color spaces. The linearity of rotated CIECAM02 and rotated Euclidean

CIECAM02 color space is very similar, which agrees with the result of the F tests.

105 9.3 Visualization of Hung and Berns Dataset

Hung and Berns dataset in rotated CIECAM02, Euclidean CIECAM02, rotated Euclidean

CIECAM02, and Euclidean IPT color space are plotted in Fig.9-53 – Fig.9-60, respectively.

Constant Lightness in Rotated CIECAM02 Color Space Varying Lightness in Rotated CIECAM02 Color Space

60 60

40 40

20 20

0 0 b b

−20 −20

−40 −40

−60 −60

−80 −80

−100 −80 −60 −40 −20 0 20 40 60 80 −100 −80 −60 −40 −20 0 20 40 60 80 a a Fig.9-53 Constant hue loci Fig.9-54 Constant hue loci (from CL) in Rotated CIECAM02 (from VL) in Rotated CIECAM02

Constant Lightness in Euclidean CIECAM02 Color Space Varying Lightness in Euclidean CIECAM02 Color Space 40 40

30 30

20 20

10 10

0 0 b b

−10 −10

−20 −20

−30 −30

−40 −40

−50 −40 −30 −20 −10 0 10 20 30 40 50 −40 −20 0 20 40 60 a a Fig.9-55 Constant hue loci Fig.9-56 Constant hue loci (from CL) in Euclidean CIECAM02 (from VL) in Euclidean CIECAM02

106 Constant Lightness in Rotated Euclidean CIECAM02 Color Space Varying Lightness in Rotated Euclidean CIECAM02 Color Space

40 40

30 30

20 20

10 10

0 0 b b

−10 −10

−20 −20

−30 −30

−40 −40

−50 −40 −30 −20 −10 0 10 20 30 40 50 −40 −20 0 20 40 60 a a Fig.9-57 Constant hue loci (from CL) Fig.9-58 Constant hue loci (from VL) in Rotated Euclidean CIECAM02 in Rotated Euclidean CIECAM02

Constant Lightness in Euclidean IPT Color Space Varying Lightness in Euclidean IPT Color Space

30 30

20 20

10 10

T 0

−10 −10

−20 −20

−30 −30

−40 −30 −20 −10 0 10 20 30 40 50 −40 −30 −20 −10 0 10 20 30 40 50 P P Fig.9-59 Constant hue loci Fig.9-60 Constant hue loci (from CL) in Euclidean IPT (from VL) in Euclidean IPT

In Euclidean CIECAM02 (Eq.7-23) and Euclidean IPT (Eq.7-33), hue angle is unchanged in CIECAM02 and IPT color space, and only chroma is changed. So, Fig.9-55 and Fig.9-

56 are similar with Fig.5-3 and Fig.5-4, respectively.

107 10. Conclusions

Optimized CIECAM02 and IPT color difference equations were developed to fit experimental data from RIT. These color equations are in similar fashion to CIE94 and

CIEDE2000, but not Euclidean. The fundamental focus of this thesis was to derive

Euclidean color spaces based on the CIECAM02 and IPT color difference equations, respectively, in which color differences can be calculated as a simple color distance. In order to derive a Euclidean color space, the Euclidean line element was established, from which terms were derived for the new coordinates of lightness, chroma, and hue angle.

The visual performance, mathematical performance (CV and PF/3) and statistical F test were used to evaluate how well the new Euclidean color spaces fit the experimental data.

The results show that CIECAM02 Euclidean color space has nearly the same performance factors as CIECAM02 color difference equation. Both Euclidean CIECAM02 color space and the rotated Euclidean CIECAM02 color space improve the performance of the optimized CIECAM02 color difference equation. Conversely, the Euclidean IPT color space does not give such improvement. The main reason is that the line element for the lightness dimension, I, could not be directly calculated and an approximation was used.

To improve the Euclidean IPT performance, a new IPT color difference equation should be designed so that line elements can be established directly. Also, in the new Euclidean color space (Eq.7-27 and Eq.7-36), the hue angle does not change, and this may affect the hue linearity for the Hung and Berns data. In the future, a function of hue angle could be

108 considered and used to create new metric for better performance of a new Euclidean color space.

In Euclidean CIECAM02 and Euclidean IPT color space, geodesics are straight line, thus, large color difference data set should be used to test in both Euclidean color spaces.

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