Intersecting Color Manifolds

Logvinenko’s color atlas theory provides a structure in which a complete set of color-equivalent material and illumination pairs can be generated to match any given input RGB color. In chromaticity space, the set of such pairs forms a 2-dimensional manifold embedded in a 4-dimensional space. For singleilluminant scenes, the illumination for different input RGB values must be contained in all the corresponding manifolds. The proposed illumination-estimation method estimates the scene illumination based on calculating the intersection of the illuminant components of the respective manifolds through a Hough-like voting process. Overall, the performance on the two datasets for which camera sensitivity functions are available is comparable to existing methods. The advantage of the formulating the illumination-estimation in terms of manifold intersection is that it expresses the constraints provided by each available RGB measurement within a sound theoretical foundation. Introduction Logvinenko’s color atlas [1] provides a way to enumerate a complete and unique set of color-equivalent stimuli (material reflectance and illuminant spectral pairs that are indistinguishable) for all possible RGB tristimulus values [2]. Specifically, it provides a mechanism by which we can generate a unique set of illuminant and material spectra that completely covers the entire color space without any redundancy. In his theory, both the material reflectance and illuminant spectra are each specified by 3 parameters, so a color-equivalent stimulus is specified by 6 parameters. For a given tristimulus value, the set of colorequivalent stimuli defines a 3-dimensional manifold, which he terms the material-lighting-invariance manifold. Using this theoretical structure, we propose an illuminationestimation method. For an image of a scene under a single illuminant, two different RGBs from two different pixels define two different material-lighting-invariance manifolds. By assumption they have at least the single scene illuminant in common. To find the common illuminants, the material-lightinginvariance manifold is projected onto the 2D illuminant space. The common illuminants must lie within the intersection of the projected manifolds. However, the intersection is not, in general, a single value, but rather a set of values. Intersecting the sets of illuminants defined by the RGBs from other pixels further constrains the range of possible scene illuminants. Tests with real images show that the method’s performance is comparable to that of other well-known methods. An advantage of the proposed method is that it is founded on the theoretical principles of the color atlas and exploits precisely the theoretical constraints the atlas provides. Many strategies have been proposed for estimating the chromaticity of the scene illumination. The present approach has similarities to Forsyth’s gamut mapping method [4] and the voting methods (Color by Correlation [5] and Sapiro’s Illuminant Voting [6]). It relates to gamut mapping in that it represents a set of constraints and derives information from their intersection. It relates to Color by Correlation and Illuminant Voting in that the intersection is implemented via voting for candidate illuminants. Background The camera or eye’s response to light with the spectral power distribution P(λ) light reflected from a Lambertian non-specular surface material with reflectance S(λ) is modeled in the standard way as the triplet φi (i=1, 2, 3): φi = P(λ)S(λ)Ri (λ)dλ λmin λmax ∫ i =1, 2,3 (1) where Ri(λ) is the spectral sensitivity function of a sensor class. Logvinenko [2] defines a light-color atlas Ap as a subset of light spectral functions (strictly positive ones) such that given any color stimulus (object material reflectance x illuminated by a given light) there is a unique element p in the light-color atlas such that, illuminating x by p, this element results in a metameric match to the given color stimulus. There is a similar definition for the object-color atlas Ax. An object-color atlas is defined as a set of object material functions Ax such that for each color stimulus there is a unique element in Ax such that, if illuminated with the same light, this element results in a metameric match to the given color stimulus. Logvinenko further defines a general color atlas in terms of an object-color atlas Ax and a light-color atlas Ap. The color atlas A is such that for any object illuminated by any light, there is a unique element p from Ap and a unique element x from Ax that is colorequivalent to the input pair. Color equivalence means being indistinguishable in multiple-illuminant scenes; however, in the special case of single-illuminant scenes it corresponds to metamerism [2]. For an arbitrary color stimulus (x’, p’) therefore there is a unique pair (x, p) in the color atlas that is color equivalent to it. By virtue of the definition of a color atlas, there will be a one-to-one map between any two atlases. The set of all color-equivalent object/light pairs is called the object-color set. Logvinenko refers to the object-color set along with the set of coordinates systems defined by the family of color atlases as the object-color manifold. For any given color stimulus (x, p) its coordinates in the general color atlas can be determined by 2-step color matching [2]. Step 1 is to find the unique element am from the object-color atlas such that 166 ©2011 Society for Imaging Science and Technology φi(x, p) = φi(am, p) i=1,2,3 (2) Step 2 is then to find the unique element al from the lightcolor atlas such that φi(am, p)= φi(am, al) i=1,2,3 (3) Since the object-color atlas and the light-color atlas are each 3-dimensional, the resulting coordinates of the general color atlas are 6-dimensional. The object reflectances and light spectra of the color atlases can be represented in terms of rectangular functions [2] that are a mixture of uniform gray and a rectangular component that takes only values 0 and 1, with at most 2 transitions between 0 and 1. An algorithm for computing these functions from CIE XYZ is described by Godau et al. [3]. These rectangular functions are very unlike typical reflectance and illuminant spectral functions and hence are not very suitable for our purposes. However, Logvinenko also proposes other parameterizations of the color atlas, one of which is based on a Gaussian representation of spectra. The Gaussian representation is given in terms of a 3parameter set of spectral reflectance functions gm(λ;km,σm,μm) and a similar 3-parameter set of spectral power distribution functions gl(λ;kl,σl,μl), both of which are Gaussian-like (see equations 1926 of [2]) functions, where k, σ, and μ indicate the height, standard deviation and center (peak) of the Gaussian. The functions are not strictly Gaussians in that they wraparound from one end of the visual spectrum to the other. They also differ from the inverse Gaussians used by Macleod et al. [7]. Any given sensor response triplet φi (i=1,2,3) can be decomposed into a sextuplet (km,σm,μm,kl,σl,μl) representing a Gaussian-like material reflectance lit by a Gaussian-like illuminant such that gl (λ;k,σ l ,ml )gm (λ;k,σ m ,mm )Ri(λ)dλ = φ i i =1,2,3 λm in λm ax