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Mathematical Discontinuities in CIEDE2000 Color Difference

Computations

Gaurav Sharma*, Wencheng Wu+, Edul N. Dalal+, Mehmet U. Celik*

*University of Rochester+Xerox Corporation

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Outline

• Color Difference Equations• CIEDE2000 Computation• Sources of Discontinuity• Discontinuity Visualization• Discontinuity Magnitude Characterization

Maximum (reasonable) magnitude

• Conclusions + workarounds

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Color Difference Equations

• Quantitative evaluation of color differences• Main uses:

Quantitative color error evaluationAlgorithm/parameter optimization

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Color Difference Equations: Desirable Attributes

• Perceptual uniformityEqual numerical differences correspond to equal perceived differences

• Mathematical properties:Continuity and differentiability

- Taylor series/small-error approximation- Gradient based optimization

Symmetry - reference/sample distinction un-necessary

Correspondence to a distance metric- Underlying “uniform” color space

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CIE 1976 CIELAB Color Space• “Uniform” color space

Based on ANLAB, in turn on Munsell• Transformation of 1931 CIEXYZ tristimulus

coordinates• Nonlinearity: Cube-root with linear end

segment

• Transformation carefully designedContinuous first derivatives [Pauli1976]

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CIELAB Based Color Difference Fomulae

• 1976: ∆E*ab Color difference

Euclidean distance betw. points in CIELAB space222222 ******* HCLbaLE ab ∆+∆+∆=∆+∆+∆=∆

L*

b*

a*

∆E ∆L*

∆C* ∆H*

• CMC and CIE ‘94 color difference Eqns.Chroma/Hue dependent weights for ∆L*, ∆C*, ∆H*Greater uniformity w.r.t. experimental dataRetain continuity of first derivatives

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CIEDE2000

• a* Axis Scalinga* -> a’

• Decomposition• Hue, Chroma Dependent Weighting• Cross Term (blue hue nonlinearity)

L*

b*

a’

∆E ∆L’

∆C’ ∆H’

• CIEDE2000 Color Difference is discontinuous

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CIEDE2000 Hue & Hue Weighting Functions

• sample chroma values• hue angle difference• mean hue angle• mean chroma value (arithmetic)

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Mean Hue/Hue Difference Computation

• Mean: Bi-sector of smaller angle betw h1, h2

• Difference: Smaller angle + direction gives sign

1h’h’∆

*b1

2

2

h’

h’

12

a’

Discontinuous Operations

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Mean Hue Discontinuityb *

a’

2h’

h’12

3

13

1

ε/2

ε/2

• 180o discontinuity in mean hue

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Hue-difference Discontinuity

• 180o (Sign) discontinuity in hue difference

= −π+ε/2

*

a’

h’13

∆

12h’∆

2

3

1

ε/2

ε/2

= π−ε/2

b

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Discontinuity Characterization

• Where does it occur ?

• How big is it (magnitude) ?

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Discontinuity Locations

• Discontinuity for points 180o apart in hue

• 6-D Space of input values

• 5-D manifold in 6-D space

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Discontinuity Locations

• Discontinuity loci in plane

h2

h = h + 1802 1

h1

2 1h = h − 180

0 360

0

360

180

180

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Visualization

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Discontinuity Magnitude

• Main contribution mean hue discontin. in

• Minor contribution from hue diff. discontin.Sign change ofContributes through rotation term

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Discontinuity Magnitude Bounds

• CIEDE2000 intended for small color differences

• Colors under 5 units apartDiscontinuity magnitude under 0.2374

- Non-negligible, not too large

Occurs for 143o hue sample

• Increasing distance: sharp rise

o

*

a *

2R = 2.5

R1 = 2.51

143

b

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Conclusions

• CIEDE2000 color difference is a discontinuous function

• Discontinuity for colors 180o apart in hue• Discontinuity magnitude small in small error

practical applicationsUnder 0.238 for color under 5 units apart

• Serious limitation forTaylor series/small error approximationsGradient based optimization

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Potential workarounds/fixes

• Use formula asymetricallyMajor discontinuity due to mean hue eliminated

• Symmetrize if nesc by averaging color differences

• Discontin in Rotation term remainsHarder to fix

- Probably requires different functional format and re-optimization of parameters

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Additional Information

• Upcoming paper in Color Research and Application (Feb 2005)

includes detailed algorithmic statement of CIEDE2000 computationAdditional test data

- Several available implementations+ Agreement over CIE draft test data, disagreement over

other data!!

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Acknowledgements

• Thanks for suggestions/comments to:Mike BrillAnonymous reviewers

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Questions