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 norm: Working best overall • 퐿6 2/19 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 3/19 Problem of Max-RGB & Gray-World – Two extremes in the Minkoswki family norm • Mean( ) and Maximum( ) – Assuming the optimal illuminant estimate is between and 퐿1 퐿∞ 퐿1 퐿∞ 4/19 Background Modeling a color signal – Assuming illuminance is uniform over a scene – A Lambertian surface illuminated by a spectral distribution 퐸 휆 = (1) where : Spectral퐶 distribution휆 퐸 휆 푆 휆 : Lambertian surface 퐸 휆 : Color signal 푆 휆 퐶 휆 5/19 – Intensity on three sensors ( , , ) 푅 휆 퐸 휆 푅 퐺 퐵 퐺 휆 = 퐵 휆 푅 �휔 퐸 휆 푆 휆 푅 휆 푑휆 푆 휆 = Sensor response curve Or 퐺 �휔 퐸 휆 푆 휆 퐺 휆 푑휆 = Sensitivity function 퐵 �휔 퐸 휆 푆 휆 퐵 휆 푑휆 – An image represented by three N-dimensional vector :image • Given image 퐼 =퐼 [ , , , ] = [ , , , ]푇 푅 푅1 푅2 ⋯ 푅푁 = [ , , , ]푇 1 2 푁 퐺 퐺 퐺 ⋯ 퐺 푇 퐵 퐵1 퐵2 ⋯ 퐵푁 6/19 – One pixel intensity over the image = 푖 푖 푅 �휔 퐸 휆 푆 휆 푅 휆 푑휆 = (3) 푖 푖 퐺 �휔 퐸 휆 푆 휆 퐺 휆 푑휆 = 푖 푖 :image 퐵 �휔 퐸 휆 푆 휆 퐵 휆 푑휆 퐼 Position: 푖 7/19 Conventional algorithms – Max-RGB • Assuming that at least a white patch exist in an image max = = , , , 푖 푒 푖∈ 1 2 ⋯ 푁 푅 �휔 퐸 휆 푅 휆 푑휆 푅 max = = (7) , , , 푖 푒 푖∈ 1 2 ⋯ 푁 퐺 �휔 퐸 휆 퐺 휆 푑휆 퐺 max = = , , , 푖 푒 푖∈ 1 2 ⋯ 푁 퐵 �휔 퐸 휆 퐵 휆 푑휆 퐵 :image 퐼 1 8/19 – Gray-world • Assuming that a scene average is grey :image 퐼 = 푁 = 푆푖 휆 휇 푆 휆 � 푘 푖=1 푁 푆푖 휆 = 푁 = 푆푖 휆 휇 푅 � 퐸 휆 � 푅 휆 푑휆 푘푅푒 휔 푖=1 푁 = 푁 = (6) 푆푖 휆 휇 퐺 � 퐸 휆 � 퐺 휆 푑휆 푘퐺푒 휔 푖=1 푁 = 푁 = 푆푖 휆 휇 퐵 � 퐸 휆 � 퐵 휆 푑휆 푘퐵푒 휔 푖=1 푁 9/19 Minkowski family norm Minkowski norm – Definition of p-norm for = , , , 푇 푋 푋1 푋/2 ⋯ 푋푁 1 푝 = 푁 (8) 푝 푋 푝 � 푋푖 푖=1 – Example of 2 norm • Equal to Euclidean distance / 1 2 = 푁 = + + + 2 2 2 2 푋 2 � 푋푖 푋1 푋2 ⋯ 푋푁 푖=1 10/19 Mean of p-norm – Mean of p-norm for = , , , 푇 푋 푋1 푋2 ⋯ 푋푁 + + + = / = 푝 푝 푝 푝 (11) 푋 푝 푋1 푋2 ⋯ 푋푁 휇푝 푋 1 푝 푁 푁 – Property of Minkowski norm • Triangular inequality: Equation (8) in this paper • Monotonically increasing sequence / , 푋 푝 푋 푞 1 푝 ≤ 1 푖푖 푝 ≤ 푞 푁 푞 • Infinity norm 푁 = max 푖 11/19 푋 ∞ 0≤푖≤푁 푋 Proposed method Expression of Max-RGB & Gray-world with Minkowski norm – Max-RGB = ∞ 푅푒 휇 푅 퐺푒 휇∞ 퐺 푒 퐵 휇∞ 퐵 – Gray-World = 1 푅푒 휇 푅 퐺푒 휇1 퐺 푒 퐵 1 휇 퐵 12/19 – Order relationship between Max-RGB and Gray-World Max-RGB Gray-World 1 2 ∞ 휇 푅 ≤ 휇 푅 ≤ ⋯ ≤ 휇 푅 휇1 퐺 ≤ 휇2 퐺 ≤ ⋯ ≤ 휇∞ 퐺 1 2 ∞ 휇 퐵 ≤ 휇 퐵 ≤ ⋯ ≤ 휇 퐵 – Proposed method: (Shade of grey algorithm) • Assuming that the average of pixels raised to the power of p is gray (15) , = 푝 = 푝 푝 푝 푖 푖 푖 푅 �휔 퐸 휆 푆 휆 푅 휆 푑휆 �휔 퐸 휆 푆 휆 푅 휆 푑휆 = = 푝 13/19 푝 푖 푖 �휔 퐸 휆 휎 휆 푅 휆 푑휆 푅 – Extension of this formula to R,G,B , = 푝 = = 푝 푝 푖 푖 푝 푖 푖 푅 �휔 퐸 휆 푆 휆 푅 휆 푑휆 �휔 퐸 휆 휎 휆 푅 휆 푑휆 푅 , = 푝 = = (16) 푝 푝 푖 푖 푝 푖 푖 퐺 �휔 퐸 휆 푆 휆 퐺 휆 푑휆 �휔 퐸 휆 휎 휆 퐺 휆 푑휆 퐺 , = 푝 = = 푝 푝 푖 푖 푝 푖 푖 :image 퐵 �휔 퐸 휆 푆 휆 퐵 휆 푑휆 �휔 퐸 휆 휎 휆 퐵 휆 푑휆 퐵 – Shade of grey algorithm 퐼 • Assumption / 푆푖 휆 = 1 푝 = 푁 푝 푆푖 휆 휇푝 푆 휆 � 푘푝 푖=1 푁 14/19 / = 1 푝 = 푁 푝 푖 푝 푝 푝 푆 휆 푝 푒 휇 푅 � 퐸 휆 � 푅 휆 푑휆 / 푘 푅 휔 푖=1 푁 = 1 푝 = 푁 푝 푖 푝 푝 푝 푆 휆 푝 푒 휇 퐺 � 퐸 휆 � 퐺 휆 푑휆 / 푘 퐺 휔 푖=1 푁 = 1 푝 = 푁 푝 푆푖 휆 휇푝 퐵푝 � 퐸푝 휆 � 퐵 휆 푑휆 푘푝퐵푒 휔 푖=1 푁 where = , , , 푝 푝 푝 푇 , = , , , 푅푝 푅1 푅2 ⋯ 푅푁 푝 푝 푝 푇 , 퐺푝 = 퐺1 , 퐺2 , ⋯ , 퐺푁 푇 푝 푝 푝 퐵푝 퐵1 퐵2 ⋯ 퐵푁 15/19 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 16/19 – norm: Working best overall 퐿6 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 17/19 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. 18/19 Conclusion Shade of grey algorithm – Performance norm: Working best overall • Comparable to many advanced colour constancy algorithm for the • 퐿6 norm 6 algorithm • But, significant computational cost 19/19 .