Shades of Gray and Colour Constancy
IS&T/SID Twelfth Color Imaging Conference pp. 37-41, 2004
Graham D. Finlayson and Elisabetta Trezzi
Presented by Jung-Min Sung
School of Electrical Engineering and Computer Science Kyungpook National Univ.
Abstract Proposed method
– Max-RGB & Gray-World • Instantiations of Minkowski norm
– Optimal illuminant estimate • 𝐿6 norm: Working best overall
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Introduction Categories of color constancy
– Representing an image by illuminant invariant descriptors – Color constancy methods
• Physical-based algorithm • Statistic-based algorithm
− Max-RGB, Gray-World, Gray-Edge • Gamut constrained algorithm • Probability-based algorithm
− Markov Random Field, Conditional Random Field • Learning-based algorithm
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Problem of Max-RGB & Gray-World – Two extremes in the Minkoswki family norm
• Mean(𝐿1) and Maximum(𝐿∞) – Assuming the optimal illuminant estimate is between 𝐿1 and 𝐿∞
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Modeling a color signal – Assuming illuminance 𝐸 𝜆 is uniform over a scene – A Lambertian surface illuminated by a spectral distribution
𝐶 𝜆 = 𝐸 𝜆 𝑆 𝜆
where 𝐸 𝜆 : Spectral distribution 𝑆 𝜆 : Lambertian surface 𝐶 𝜆 : Color signal
Background
(1)
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– Intensity on three sensors (𝑅,𝐺,𝐵)
𝑅 = � 𝐸 𝜆 𝑆 𝜆 𝑅 𝜆 𝑑𝜆𝜔
𝐺 = � 𝐸 𝜆 𝑆 𝜆 𝐺 𝜆 𝑑𝜆𝜔
𝐵 = � 𝐸 𝜆 𝑆 𝜆 𝐵 𝜆 𝑑𝜆𝜔
– An image represented by three N-dimensional vector
• Given image 𝐼
𝑅 = [𝑅1,𝑅2,⋯ ,𝑅𝑁]𝑇 𝐺 = [𝐺1,𝐺2,⋯ ,𝐺𝑁]𝑇 𝐵 = [𝐵1,𝐵2,⋯ ,𝐵𝑁]𝑇
Sensor response curve Or
Sensitivity function
𝐼:image
𝑆 𝜆
𝐸 𝜆 𝑅 𝜆 𝐺 𝜆 𝐵 𝜆
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– One pixel intensity over the image
𝑅𝑖 = � 𝐸 𝜆 𝑆𝑖 𝜆 𝑅 𝜆 𝑑𝜆𝜔
𝐺𝑖 = � 𝐸 𝜆 𝑆𝑖 𝜆 𝐺 𝜆 𝑑𝜆𝜔
𝐵𝑖 = � 𝐸 𝜆 𝑆𝑖 𝜆 𝐵 𝜆 𝑑𝜆𝜔
𝐼:image
Position: 𝑖
(3)
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Conventional algorithms – Max-RGB
• Assuming that at least a white patch exist in an image
max𝑖∈ 1,2,⋯,𝑁
𝑅𝑖 = � 𝐸 𝜆 𝑅 𝜆 𝑑𝜆𝜔
= 𝑅𝑒
max𝑖∈ 1,2,⋯,𝑁
𝐺𝑖 = � 𝐸 𝜆 𝐺 𝜆 𝑑𝜆𝜔
= 𝐺𝑒
max𝑖∈ 1,2,⋯,𝑁
𝐵𝑖 = � 𝐸 𝜆 𝐵 𝜆 𝑑𝜆𝜔
= 𝐵𝑒
𝐼:image
1
(7)
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– Gray-world • Assuming that a scene average is grey
𝜇 𝑆 𝜆 = �𝑆𝑖 𝜆𝑁
𝑁
𝑖=1
= 𝑘
𝜇 𝑅 = � 𝐸 𝜆 �𝑆𝑖 𝜆𝑁
𝑁
𝑖=1
𝑅 𝜆 𝑑𝜆𝜔
= 𝑘𝑅𝑒
𝜇 𝐺 = � 𝐸 𝜆 �𝑆𝑖 𝜆𝑁
𝑁
𝑖=1
𝐺 𝜆 𝑑𝜆𝜔
= 𝑘𝐺𝑒
𝜇 𝐵 = � 𝐸 𝜆 �𝑆𝑖 𝜆𝑁
𝑁
𝑖=1
𝐵 𝜆 𝑑𝜆𝜔
= 𝑘𝐵𝑒
(6)
𝐼:image
𝑆𝑖 𝜆
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Minkowski family norm Minkowski norm
– Definition of p-norm for 𝑋 = 𝑋1,𝑋2,⋯ ,𝑋𝑁 𝑇
𝑋 𝑝 = � 𝑋𝑖 𝑝𝑁
𝑖=1
1/𝑝
– Example of 2 norm
• Equal to Euclidean distance
𝑋 2 = � 𝑋𝑖 2𝑁
𝑖=1
1/2
= 𝑋12 + 𝑋22 + ⋯+ 𝑋𝑁2
(8)
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Mean of p-norm – Mean of p-norm for 𝑋 = 𝑋1,𝑋2,⋯ ,𝑋𝑁 𝑇
𝜇𝑝 𝑋 =𝑋 𝑝
𝑁1/𝑝 =𝑋1𝑝 + 𝑋2
𝑝 + ⋯+ 𝑋𝑁𝑝
𝑁
𝑝
– Property of Minkowski norm
• Triangular inequality: Equation (8) in this paper • Monotonically increasing sequence
𝑋 𝑝
𝑁1/𝑝 ≤𝑋 𝑞
𝑁1𝑞
, 𝑖𝑖 𝑝 ≤ 𝑞
• Infinity norm
𝑋 ∞ = max
0≤𝑖≤𝑁𝑋𝑖
(11)
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Proposed method Expression of Max-RGB & Gray-world with Minkowski norm
– Max-RGB
𝑅𝑒𝐺𝑒𝐵𝑒
=𝜇∞ 𝑅𝜇∞ 𝐺𝜇∞ 𝐵
– Gray-World
𝑅𝑒𝐺𝑒𝐵𝑒
=𝜇1 𝑅𝜇1 𝐺𝜇1 𝐵
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– Order relationship between Max-RGB and Gray-World
𝜇1 𝑅 ≤ 𝜇2 𝑅 ≤ ⋯ ≤ 𝜇∞ 𝑅
𝜇1 𝐺 ≤ 𝜇2 𝐺 ≤ ⋯ ≤ 𝜇∞ 𝐺
𝜇1 𝐵 ≤ 𝜇2 𝐵 ≤ ⋯ ≤ 𝜇∞ 𝐵
– Proposed method: (Shade of grey algorithm) • Assuming that the average of pixels raised to the power of p is gray
𝑅𝑝,𝑖 = � 𝐸 𝜆 𝑆𝑖 𝜆 𝑅 𝜆 𝑑𝜆𝜔
𝑝
= � 𝐸 𝜆 𝑝 𝑆𝑖 𝜆 𝑝𝑅 𝜆 𝑑𝜆𝜔
= � 𝐸𝑝 𝜆 𝜎𝑖 𝜆 𝑅 𝜆 𝑑𝜆𝜔
= 𝑅𝑖𝑝
Max-RGB Gray-World
(15)
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– Extension of this formula to R,G,B
𝑅𝑝,𝑖 = � 𝐸 𝜆 𝑆𝑖 𝜆 𝑅 𝜆 𝑑𝜆𝜔
𝑝
= � 𝐸𝑝 𝜆 𝜎𝑖 𝜆 𝑅 𝜆 𝑑𝜆𝜔
= 𝑅𝑖𝑝
𝐺𝑝,𝑖 = � 𝐸 𝜆 𝑆𝑖 𝜆 𝐺 𝜆 𝑑𝜆𝜔
𝑝
= � 𝐸𝑝 𝜆 𝜎𝑖 𝜆 𝐺 𝜆 𝑑𝜆𝜔
= 𝐺𝑖𝑝
𝐵𝑝,𝑖 = � 𝐸 𝜆 𝑆𝑖 𝜆 𝐵 𝜆 𝑑𝜆𝜔
𝑝
= � 𝐸𝑝 𝜆 𝜎𝑖 𝜆 𝐵 𝜆 𝑑𝜆𝜔
= 𝐵𝑖𝑝
– Shade of grey algorithm • Assumption
𝜇𝑝 𝑆 𝜆 = �𝑆𝑖 𝜆 𝑝
𝑁
𝑁
𝑖=1
1/𝑝
= 𝑘𝑝
(16)
𝐼:image
𝑆𝑖 𝜆
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𝜇𝑝 𝑅𝑝 = � 𝐸𝑝 𝜆 �𝑆𝑖 𝜆 𝑝
𝑁
𝑁
𝑖=1
𝑅 𝜆 𝑑𝜆𝜔
1/𝑝
= 𝑘𝑝𝑅𝑒
𝜇𝑝 𝐺𝑝 = � 𝐸𝑝 𝜆 �𝑆𝑖 𝜆 𝑝
𝑁
𝑁
𝑖=1
𝐺 𝜆 𝑑𝜆𝜔
1/𝑝
= 𝑘𝑝𝐺𝑒
𝜇𝑝 𝐵𝑝 = � 𝐸𝑝 𝜆 �𝑆𝑖 𝜆 𝑝
𝑁
𝑁
𝑖=1
𝐵 𝜆 𝑑𝜆𝜔
1/𝑝
= 𝑘𝑝𝐵𝑒
where 𝑅𝑝 = 𝑅1𝑝,𝑅2
𝑝,⋯ ,𝑅𝑁𝑝 𝑇
, 𝐺𝑝 = 𝐺1𝑝,𝐺2
𝑝,⋯ ,𝐺𝑁𝑝 𝑇
, 𝐵𝑝 = 𝐵1𝑝,𝐵2
𝑝,⋯ ,𝐵𝑁𝑝 𝑇
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Experimental evaluation Evaluation by using angular error
– Using two databases • Data set suggested Barnard et al. • One consisting of 321 images of a variety of 32 scenes • Another of 220 images of a variety of 22 scenes • Both groups taken under 11 coloured illuminant\ • Comparison measure
− Angular error: Equation (18) in this paper − Distance error in the chromaticity space: Equation (19) in this paper
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– 𝐿6 norm: Working best overall
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Fig. 2. The figure shows the angular error of the group A images for 30 values of p
Fig. 3. The figure shows the angular error of the group B images for 30 values of p
Table 1. Results for the p shade of grey algorithm on two databases considered: the firsts two columns are the mean of angular errors and the lasts two report the distance error in the chromaticities space.
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Conclusion Shade of grey algorithm
– Performance • 𝐿6 norm: Working best overall • Comparable to many advanced colour constancy algorithm for the
norm 6 algorithm • But, significant computational cost
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