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1
Mathematical Discontinuities in CIEDE2000 Color Difference
Computations
Gaurav Sharma*, Wencheng Wu+, Edul N. Dalal+, Mehmet U. Celik*
*University of Rochester+Xerox Corporation
2
Outline
• Color Difference Equations• CIEDE2000 Computation• Sources of Discontinuity• Discontinuity Visualization• Discontinuity Magnitude Characterization
Maximum (reasonable) magnitude
• Conclusions + workarounds
3
Color Difference Equations
• Quantitative evaluation of color differences• Main uses:
Quantitative color error evaluationAlgorithm/parameter optimization
4
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