The von Kries Hypothesis and a Basis for Color Constancy

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one can algorithmically grow the compatible illuminant set. to include additional non black illuminants as well 16 7. This analysis however is still incomplete because given. color data the analysis does not reveal a method for com. Figure 1 The 3xIxJ measurement tensor The tensor can be. puting the color space to begin with, sliced in three ways to produce the matrices j i and k. While these limitations have been well documented a. more complete characterization of the conditions for von. Kries compatibility has yet to be established As a result 1 E1 E2 E D E1 E2 s t R R. the development of more powerful systems for choosing op p R E2 D E1 E2 p R E1. timized color bases has been slow This paper addresses. these issues by answering the following questions 2 R1 R2 R D R1 R2 s t E E. p R2 E D R1 R2 p R1 E, 1 What are the necessary and sufficient conditions that. sensors illuminants and materials must satisfy to be In the case that D and D are linear and hence identified. exactly von Kries compatible and what is the structure with matrices is just matrix vector multiplication If D is. of the solution space linear we say that the world supports linear adaptive color. constancy If D is linear we say the world supports linear. 2 Given measured spectra or labeled color observations relational color constancy D being linear does not imply. how do we determine the color space that best sup D is linear and vice versa If both D and D are linear we. ports diagonal color constancy say the world supports doubly linear color constancy. We observe that the joint conditions are impositions only In particular we shall be interested in the case when D. on the sensor measurements not the physical spectra This and D are both furthermore diagonal under some choice of. allows the von Kries compatibility conditions to be suc color basis It is proven in 4 that for a fixed color space D. cinctly formulated as rank constraints on an order 3 mea is diagonal if and only if D is diagonal So the two notions. surement tensor Our analysis leads directly to an algorithm of color constancy are equivalent if either D or D is diago. that given labeled color data computes a locally optimal nal and we say the world supports diagonal color constancy. choice of color basis in which to carry out diagonal color the doubly modifier is unnecessary The equivalence is. constancy computations The proposed framework also uni nice because we as biological organisms can likely learn. fies most existing analyses of von Kries compatibility to achieve definition 1 but seek to achieve definition 2 for. Given a set of illuminants Ei i 1 I reflectances,Rj j 1 J and sensor color matching functions. We define two notions of color constancy The first def k k 1 2 3 we define a measurement data tensor see Fig. inition captures the idea that a single adjustment to the sen ure 1. sors will map all material colors seen under an illuminant Z. E1 to reference colors under a possibly chosen standard Mkij k Ei Rj d 1. illuminant E2 The second definition also known as rela. tional color constancy captures the idea that surface colors For fixed values of j we get 3xI matrices j. have a fixed relationship between each other no matter what Mkij that map illuminants expressed in the Ei i 1 I. overall illumination lights the scene As stated these two basis to color vectors expressed in the sensor basis Like. definitions are not interchangeable One being true does not wise for fixed values of i we get 3xJ matrices i. imply the other Mkij that map surface reflectance spectra expressed in the. To define the issues formally we need a bit of notation Rj j 1 J basis to color vectors We can also slice the. Let R be the smallest linear subspace of L2 functions en tensor by constant k to get IxJ matrices k Mkij. closing the spectral space of materials of interest Let E be Since color perception can depend only on the eye s. the smallest linear subspace of L2 functions enclosing the trichromatic color measurements worlds i e sets of illu. spectral space of illuminants of interest Let p R E be the minant and material spectra giving rise to the same mea. color in the sensor basis of material reflectance R R surement tensor are perceptually equivalent To understand. under illumination E E In the following D and D diagonal color constancy therefore it is sufficient to ana. are operators that take color vectors and map them to color lyze the space of measurement tensors and the constraints. vectors D is required to be independent of the material R that these tensors must satisfy This analysis of von Kries. likewise D is required to be independent of the illuminant compatible measurement tensors is covered in section 2 1. E The denotes the action of these operators on color vec Given a von Kries compatible measurement tensor e g. tors an output from the algorithm in section 3 one may also. Without loss of generality let an be vectors of length. I corresponding to the illuminant axis of the measurement. tensor let bn be vectors of length J corresponding to the. material axis of the tensor and let cn be vectors of length. Figure 2 Core tensor form a 3x3x3 core tensor is padded with 3 corresponding to the color sensor axis of the tensor Let. zeros The core tensor is not unique the vectors an make up the columns of the matrix A vec. tors bn make up the columns of the matrix B and vectors. cn make up the columns of the matrix C Then the de. be interested in the constraints such a tensor places on the composition above may be restated as a decomposition into. possible spectral worlds This analysis is covered in section the matrices A B C each with N columns. Proof Theorem 1 First suppose the measurement ten, 2 1 Measurement Constraints sor supports generalized diagonal color constancy Then by. The discussion in this section will always assume generic Lemma 1 there exists a color basis under which each k. configurations e g color measurements span three dimen is rank 1 as a matrix This means each k can be writ. sions color bases are invertible Proofs not essential to the ten as an outer product k ak bk In this color basis. main exposition are relegated to Appendix A then the measurement tensor is a rank 3 tensor in which the. matrix C following notation above is just the identity We. Proposition 1 A measurement tensor supports doubly lin. also point out that an invertible change of basis on any of. ear color constancy iff a change of basis for illuminants. A B C does not affect the rank of a tensor so the origi. and materials that reduces it to the core tensor form of Fig. nal tensor before color basis change was also rank 3 For. the converse case we now suppose the measurement ten. More specifically as is apparent from the proof of Propo sor is rank 3 Since C is in the generic setting invertible. sition 1 in Appendix A 1 if a single change of illuminant multi linearity gives us. basis makes all the j slices null past the third column 3. the measurement tensor supports linear relational color con X. C 1 cn an bn 3,C cn an bn, stancy Likewise a change of material basis making all the n 1 n 1.
i slices null past the third column implies the measure. ment tensor supports linear adaptive color constancy Sup The right hand side of Equation 3 is a rank 3 tensor with. port for one form of linear constancy does not imply support each k slice a rank 1 matrix By Lemma 1 the tensor. for the other must then support diagonal color constancy. The following lemma provides a stepping stone to our. In the proof above note that the columns of C exactly rep. main theoretical result and is related to some existing von. resent the desired color basis under which we get perfect. Kries compatibility results see section 4, diagonal color constancy This theorem is of algorithmic. Lemma 1 A measurement tensor supports generalized di importance because it ties the von Kries compatibility crite. agonal color constancy iff there exists a change of color ba ria to quantities best rank 3 tensor approximations that are. sis such that for all k k is a rank 1 matrix computable via existing multilinear methods. This leads to our main theorem characterizing the space 3 Color Basis for Color Constancy. of measurement tensors supporting generalized diagonal. color constancy Given a measurement tensor M generated from real. world data we would like to find the optimal basis in which. Theorem 1 A measurement tensor supports generalized to perform diagonal color constancy computations To do. diagonal color constancy iff it is a rank 3 tensor 2 this we first find the closest von Kries compatible measure. An order 3 tensor 3D data block T is rank N if ment tensor with respect to the Frobenius norm We then. N is the smallest integer such that there exist vectors return the color basis that yields perfect color constancy un. an bn cn n 1 N allowing decomposition as the sum of der this approximate tensor. outer products denoted by By Theorem 1 finding the closest von Kries compatible. measurement tensor is equivalent to finding the best rank. X 3 approximation Any rank 3 tensor may be written in the. T cn an bn 2 form of equation 2 with N 3 We solve for M s best. n 1 rank 3 approximation decomposition into A B C via Tri. 2 There exist measurement tensors supporting generalized diagonal linear Alternating Least Squares TALS 8 For a rank 3. color constancy with rank less than 3 but such examples are not generic tensor TALS forces A B and C to each have 3 columns It. then iteratively fixes two of the matrices and solves for the. third in a least squares sense, Repeating these computations in lockstep guarantees. convergence to a local minimum A B C can be used to. reconstruct the closest von Kries compatible tensor and the. columns of C exactly represent the desired color basis Figure 3 The rows of a single 1 slice are placed into a new mea. As a side note the output of this procedure differs from surement tensor rows are laid horizontally above with all other. the best rank 3 3 3 approximation given by HOSVD 11 entries set to zero The marks the nonzero entries. HOSVD only gives orthogonal bases as output and the rank. 3 3 3 truncation does not in general yield a closest rank 3. tensor HOSVD may however provide a good initial guess section as these methods propose more intuitive guidelines. The following details on TALS mimic the discussion in rather than formal relationships. 14 For further information see 14 8 and the references Previous analyses treat the von Kries compatibility con. therein The Khatri Rao product of two matrices A and B ditions as constraints on spectra whereas the analysis here. with N columns each is given by treats them as constraints on color measurements In this. h i section we translate between the two perspectives To. A B a1 b1 a2 b2 aN bN 4 go from spectra to measurement tensors is straightforward. To go the other way is a bit more tricky In particular. where is the Kronecker product given a measurement tensor with rank 1 k there is not. Denote the flattening of the measurement tensor M by a unique world generating this data Any set of illumi. M IJ 3 if the elements of M are unrolled such that the nants Ei i 1 I and reflectances Rj j 1 J satisfying. rows of matrix M IJ 3 loop over the i j indices with Equation 1 with M and k fixed will be consistent with. i 1 I as the outer loop and j 1 J as the inner the data Many constructions of worlds are thus possible. loop The column index of M IJ 3 corresponds with the di But if one first selects particular illuminant or material spec. mension of the measurement tensor that is not unrolled in tra as mandatory inclusions in the world then one can state. this case k 1 2 3 The notation for other flattenings is more specific conditions on the remaining spectral choices. defined symmetrically We can then write In 5 4 it is shown that if the illuminant space is 3 di. mensional and the material space is 2 dimensional or vice. M JI 3 B A C T 5 versa then the resulting world is generalized von Kries. compatible As a measurement tensor this translates into. By symmetry of equation 5 we can write out the least stating that any 3x3x2 measurement tensor is complex. squares solutions for each of the matrices with the other rank 3 However this 3 2 condition is clearly not nec. two fixed essary as almost every rank 3 tensor is not reducible via. change of bases to size 3x3x2 In fact one can always ex. h iT tend a 3x3x2 tensor to a 3x3x3 core tensor such that the. B C M J3 I 6 k are still rank 1 The illuminant added by this exten. h iT sion is neither black with respect to the materials nor in the. C A M 3I J 7 linear span of the first two illuminants. h iT The necessary and sufficient conditions provided in 16. can be seen as special cases of Lemma 1 The focus on. C B A M JI 3 8, spectra leads to a case by case analysis with arbitrary spec. tral preferences However the essential property these con. 4 Relationship to Previous Characterizations ditions point to is that the 2x2 minors of k must be zero. As mentioned in the introduction there are two main i e k must be rank 1. sets of theoretical results There are the works of 7 16 One case from 16 is explained in detail in 7 They fix. that give necessary and sufficient conditions for von Kries a color space a space of material spectra and a single ref. compatibility under a predetermined choice of color space erence illumination spectrum They can then solve for the. and are able to build infinite dimensional von Kries com unique space of illumination spectra that includes the ref. The von Kries Hypothesis and a Basis for Color Constancy Hamilton Y Chong Harvard University hchong fas harvard edu Steven J Gortler Harvard University sjg cs harvard edu Todd Zickler Harvard University zickler seas harvard edu Abstract Color constancy is almost exclusively modeled with di agonal transforms However the choice of basis under which diagonal transforms are taken is

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