A neural field model for color perception unifying assimilation and contrast

We address the question of color-space interactions in the brain, by proposing a neural field model of color perception with spatial context for the visual area V1 of the cortex. Our framework reconciles two opposing perceptual phenomena, known as simultaneous contrast and chromatic assimilation. They have been previously shown to act synergistically, so that at some point in an image, the color seems perceptually more similar to that of adjacent neighbors, while being more dissimilar from that of remote ones. Thus, their combined effects are enhanced in the presence of a spatial pattern, and can be measured as larger shifts in color matching experiments. Our model supposes a hypercolumnar structure coding for colors in V1, and relies on the notion of color opponency introduced by Hering. The connectivity kernel of the neural field exploits the balance between attraction and repulsion in color and physical spaces, so as to reproduce the sign reversal in the influence of neighboring points. The color sensation at a point, defined from a steady state of the neural activities, is then extracted as a nonlinear percept conveyed by an assembly of neurons. It connects the cortical and perceptual levels, because we describe the search for a color match in asymmetric matching experiments as a mathematical projection on color sensations. We validate our color neural field alongside this color matching framework, by performing a multi-parameter regression to data produced by psychophysicists and ourselves. All the results show that we are able to explain the nonlinear behavior of shifts observed along one or two dimensions in color space, which cannot be done using a simple linear model.


Introduction
Assimilation and Contrast Color induction refers to a change in the color appearance of a test stimulus under the influence of spatially neighboring stimuli in the field of view [52].It has been well established that the spatial context heavily influences the color perception of a central point in an image, be it a uniform inducing surround ( [9], [43], [48], just to cite a few) or a geometrically more complex surround ( [30], [7], [50], [21], [8], [44], [24] and many others...).The geometry of the surrounding context is a major factor in chromatic induction, and in particular the spatial frequency of the chromatic modulation.
Chromatic assimilation occurs when the chromatic appearance of the test stimulus changes toward the chromaticity of the inducing stimulus: in other words, the test color tends to be "attracted" by the inducing one, so that it becomes perceptually more similar.Simultaneous contrast is the inverse phenomenon.It occurs when the chromatic appearance of the test color changes away from the chromaticity of the inducing one: hence, quite opponent colors tend to "repell" and become perceptually even more opponent when put one near another.For reviews on contrast, see [49] and [9]).Both phenomena are illustrated in the following figures.[31].a) Simultaneous contrast: the small patches on the left and the right are identical, but they tend to appear darker on the yellow backround, lighter on the dark background.b) Chromatic assimilation: the background tends to be blue or green depending on the color and spatial frequency of the grid.c) Appearance of both phenomena: the pink-orange changes perceptually when surrounded by concentric annuli with a purple/lime or lime/purple pattern.
Experimental settings in [32] and [31] Color appearance was measured specifically for colors distinguished by only S cones.The stimuli were expressed in a cone-based chromaticity space proposed by [20] with three coordinates s, l, Y .The s coordinate (respectively l) is defined as the ratio S L+M (respectively L L+M ) where the S, M, L account for the coordinates in LMS cone space (defined in the next section).The Y coordinate stands for luminance.Subjects were asked to perform asymmetric color matching between a test pattern I test and a comparison pattern I comp .The test pattern is made of concentric annuli (see Figure 1.c).It contains one central test ring, surrounded from inside and outside by other rings alternating between two colors (among 'purple','lime','white'), which are constrained to differ only in s chromaticity.The comparison pattern is a central comparison ring, surrounded by a uniform background (for instance equal-energy white), keeping the same geometry as the test image.The configuration of each experiment is hence determined by: the geometry of the pattern, the test color, two colors for the inducing test pattern, and a comparison background.Observers had to control the color filling of the comparison ring until color appearance between the test and comparison rings were the same.[32], [24], [31] have demonstrated that both phenomena (assimilation and contrast) act simultaneously in a synergistic manner.Their results showed that patterns which induced the largest color shifts in s chromaticity were those which alternated between two distinct colors such as lime/purple or purple/lime.In particular, those shifts were larger than for uniform backgrounds, or for patterns alternating between white and another color.They also observed that the matched s coordinate was shifted towards the adjacent ring (assimilation), while it was also repelled away from the second ring (contrast).

Synergy of attraction and repulsion
A linear model to explain the shifts Such large color shifts could not be accounted for by optical factors only (e.g.spread light, chromatic aberration), but may be produced by a neural process of the received stimuli.[32] therefore proposed a simple, but quite efficient linear model.It relies on a +S/ − S center-surround receptive field model.More precisely, the color shift at some point x located in the central ring is predicted as: shift at x = DOG * (I test − I comp [c test ])(x).
I test and I comp are the test and comparison images expressed in s coordinates, convolved with a gaussian kernel beforehand to account for retinal blurring.DOG designates a Difference of Gaussians kernel.I comp [c test ] means that the comparison ring is identical to the test ring.They fitted the model to their data, and resulting predictions were satisfying, provided that the s chromaticity of the test color was not changed (in the experiments of [32], only the l coordinate of the test ring was changed).
Towards a non-linear model However, a straightforward consequence of their model is that the predicted shift does not depend on the s coordinate of the test color itself.Indeed, the difference I test − I comp [c test ] is a ringish image whose central ring has chromaticity s = 0, as test and comparison rings are equal and cancelled out.This prediction has been shown not to hold in a later paper by [31] (see Figure 15, left), where the color shift depends on the test color as well.Because the previous model does not explain the nonlinear behavior of color shifts, we propose a model which could explain in a nonlinear way what is observed with the concentric annuli patterns.Furthermore, we want our model to be conceptually more consistent with the process of color matching, by involving the whole range of possible comparison images {I comp [c]} c instead of I comp [c test ] alone.To do so, let us start with the fundamental observation made by the aforecited authors.If we use the theory of opponent colors introduced by Hering [18], it can be formulated as such: Definition.

1.
Adjacent neighbors surrounding a spatial point x in an image I tend to perceptually attract towards their color, in the sense that they contribute to make the color appearance at x more similar to theirs.

2.
Remote neighbors tend to repel towards their respective opponent color.They are not immediately adjacent but at some short distance.
3. Far neighbors are too far from x to have any substantial influence on the color perception at x.
This point of view implies a change of vocabulary: from now on, chromatic assimilation and simultaneous contrast more appropriately designate local interactions, which may act at the same time but at two different local scales.The global effect observed is then the integration of all infinitesimal influences coming from spatially neighboring points, and can result in either attraction (assimilation wins over contrast) or repulsion (contrast wins over assimilation).This is why we say that assimilation and contrast seem to be antagonistic effects, but in reality they are concomitant phenomena.Our goal now is to provide a neural field model associating them inside a common framework, by taking advantage of an appropriate opponent space.Note that, as we discuss in Section 4, this paper is not directly related to color constancy.

Theoretical settings
Lights and Power input Before going further, let us first define precisely our theoretical framework.When a human sees a visual scene, light coming from visible objects enters his eye through the pupil, is reverted and projected by the lens onto the retina.The visual scene projects onto a piece of retina identifiable with a flat region of R 2 .The incoming light stimulates the L, M, S cones, which convert and transmit this information to the cortical area V 1, via the LGN .At a point x of the retina which receives light coming from a point ξ of the visual scene, the power input L x received by the L cones is at first order given by ( [28], [29]): P ξ denotes the spectral power distribution of the incident illuminant (daylight, lamp, etc.).R ξ stands for the spectral reflectance of the elementary material surface at position ξ, it is the proportion of luminous energy reflected by its surface (depending on the wavelength λ).S x L is the spectral sensitivity of the L cones located at the point x of the retina.Hence, S x L depends on the density of L cones at x.Given λ, R ξ (λ) and S x L (λ) are respectively the probability to reflect or absorb a unit quantity of energy contributed by wavelength λ in one second (but R ξ and S x L are not themselves probability densities over Λ).Note that it concerns a same quantity of energy for each wavelength, and not a quantum of energy hc/λ depending on the wavelength.Of course, analogous formulas define M x and S x .
We also recall that the reflected spectrum is the only visual information accessible to the eye at point x and wavelength λ ( [15], [4]).But cones are not themselves "aware" of this spectral distribution, only the total visual input matters.Indeed, the univariance principle stated by [40] assures that a single cone, whether excited by quanta of energy provided by photons at wavelengths λ 1 or λ 2 , is excited in the same way, provided that the absorbed energy is the same.Cone excitation is directly proportional to quantal absorption rate ( [3]).But what distinguishes cones of different types is that they absorb a given quantum of energy at wavelength λ with different probabilities That is why information given by a single cone type is less rich than for three, as is the case for night vision.More importantly, a single type of cone alone cannot give access to this richer information, because cones are "wavelength-blind".Instead, the brain has to compare the three outputs.
In this work we make the simplifying assumption that the cone density is considered constant across the retina, so that we can drop the x index in the spectral sensitivities S L .At a point x of the retina the power input becomes where we recognize the standard scalar product in the space L 2 (Λ) of square integrable functions.
Color space In this paragraph we define three different nested spaces: the color vector space VC, the set of physically realizable colors RC, and the color space C, the last being the most important to us.We have C ⊂ RC ⊂ VC.
It is usually accepted that a color is a point in a finite-dimensional vector space ( [47]), here VC.A nice and rigorous construction of this space is given in [13]; a more classical construction is given in [26], which we summarize in its simplest form.
Given two physical lights of spectral distributions C 1 , C 2 ∈ L 2 (Λ), we say that they are metameric and we note C 1 ∼ C 2 if they produce exactly the same visual effect under the same viewing conditions.Mathematically, this corresponds to an equivalence relation.Metamerism is strongly dependent on the observer, though we can talk about a "standard observer".In fact, C 1 and C 2 are metameric when the triplets of scalar products • We define the color vector space as the quotient space VC := L 2 (Λ)/ ∼ and one element denoted in brackets [C] is a metameric class containing C. Each element of this space is thus a set of metameric lights which all give equal information after being analyzed by the cones' sensitivities.According to the Grassmann's fundamental laws of additive color mixture postulated in 1853, this gives a space of dimension 3. VC is then canonically equipped with a euclidean structure, hence we say that it is the color vector space.We can identify it to R 3 .However, there are many ways to do so, by choosing a coordinate system, or chart φ : VC → R 3 , inducing what we call a representation (VC, φ).For two charts φ 1 and φ 2 , φ 2 • φ −1

1
: R 3 → R 3 is called a change of representation, and can be linear or nonlinear.The most natural setting is the LMS representation, in which the L, M, S coordinates are exactly the triplets given by the isomorphism φ LM S : The color vector space VC is thus entirely characterized by the spectral sensitivies {S L , S M , S S }.These functions are an example of color matching functions and denoted { l, m, s} in the literature. 1 Other similar representations exist, such as {r, ḡ, b} and {x, ȳ, z} which are derived from the former.Color coordinates are scalar products with the specific color matching functions: Both representations are linearly related to LMS representation, being just different choices for the basis (hence, φ RGB and φ XY Z are also isomorphisms).
The linear relation (which is the change of representations) converting from LMS to RGB or XYZ depends on the observer, but for universality the CIE commission has fixed it to a reference value, relative to the "standard observer".However, "the advantage of using [LM S cone space] is that cones represent the initial encoding of light by the visual system", as stated by [3].
In fact, {r, ḡ, b} and {x, ȳ, z} are entirely derived from {S L , S M , S S }.Moreover, the matching functions {r, ḡ, b} also involve a notion of spatial extent, whereas { l, m, s} are (at least theoretically) intrinsically well-defined and context-independent (see more details about {r, ḡ, b} color matching functions in the Appendix).
• Now, we introduce the set of physically realizable colors RC.Notice that VC includes non physically realizable lights, with negative spectral distributions for instance.We have to restrict to real lights.We define RC as the set of metameric classes containing at least one physical light: From [13] (Theorems 3.13 and 3.15), we know that RC is a convex (mathematical) cone.It is also the convex hull of the set of monochromatic lights.2 • Finally, we introduce the color space C, which is the subset of realizable colors which are visible by the eye.It is not true that all physically realizable colors can be seen by the eye, as remarked by [38].Indeed, if [C] ∈ RC, for extreme scaling values 0 < α << 1 or α >> 1, α[C] does not belong to RC; because for low light power, cones are no more enough excited and scotopic vision is ensured by rods, while on the other hand, cones and rods are saturated by excessive light energy, so that defining such a color does not make any sense.Therefore we suppose that C is a bounded and convex subset of RC.
What is a good opponent representation C opp ?Let us extend the definition of a chart by allowing it to be defined on the subset C ⊂ VC only: φ : C → R3 , and a change of representation φ 2 • φ −1 1 : φ 1 (C) → φ 2 (C).Our model relies on a "good" opponent representation (C, φ opp ).We then set We can obtain φ opp := T LM S→opp • φ LM S if we are given a change of representation T LM S→opp : φ LM S (C) → R 3 .C opp will be abusively called an "opponent color space" or "opponent representation".From now on, when we write c 1 + c 2 ∈ C opp , we use the own additive structure of C opp ⊂ R 3 . 3We implicitly refer to the corresponding color [C] = φ −1 opp (c).We now state the properties that a good opponent representation C opp should enjoy: • and −c is the opponent color of c in the sense of [18].Hence, all color regions of C opp come into opposed pairs, such as Yellow and Blue or Red and Green regions.
• The zero color, noted 0 ∈ C opp , is not black, but is a color which is its own opponent, such as some gray midway between any color and its opponent.
• Ideally, T LM S→opp should be an affine (or projective) change of coordinates.Indeed, as specified by [2], the only intrinsic structure of C is its affine structure, so we should preserve it.
Which opponent representation do we choose?It is believed that the L, M, S signals are recombined in the brain into Yellow-Blue, Red-Green and Achromatic independent channels, as Hering postulated [18].Therefore a possible choice is to consider Hering's opponent space.Suppose a color [C] is specified by coordinates denoted L, M, S and Y B, RG, Lum, respectively.We can set T LM S→opp to be a simple linear change of coordinates, as in ( [42]): or consider instead a nonlinear transformation, such as in ( [25]): However in our work we will either use the (l, s, Y ) representation or the (H, S, L) representation, that we define below.We will also restrict to a lowerdimensional color subspace: a one-dimensional subspace based on chromaticity s, defined by c := s − 1, and the chromatic disk of constant Luminance L = 1/2 respectively with (c 1 , c 2 ) := (S cos(H), S sin(H)) and the color specified by coordinates (H, S, 1/2).
• (l, s, Y ) representation.This representation is a variant of the one proposed in [20], and is used in [32]: We will then use the one-dimensional color space based on the change of coordinates c where the number 2 is arbitrary, but [0, 2] is the typical range of s values, (for purple, s = 2.00, and for lime, s = 0.16), so that C opp contains it.
• (H, S, L) representation.In this context, the letter 'S' of HSL refers to Saturation (and not Short cones).We made our experiments using a standard computer screen.We claim that the details of the display, such as color gamut and screen specificities, are not important in themselves for our methods and results, provided that all experiments are made in the same conditions, with the same device.Experiments explore the sRGB unit cube subspace [13], or gamut, that can be generated by our device.Given an observer with color space C, this sRGB cube is in pointwise correspondance with a subspace C dev ⊂ C, thanks to a chart φ sRGB : C dev → [0, 1] 3 .C dev depends on the device and the observer, and is strictly smaller than C which contains visible colors not reproducible by the screen. 4The conversion T sRGB→HSL is then easily made with standard formulas (see [22] or on the web).We apply our model to the chromatic disk defined as the section of the plane of Luminance L = 1/2 with the HSL cylin- Cortex Cones in the retina convert power inputs into electrical and chemical information.It is sent to cortical layer 4C of the visual area V 1 by the means of axonal projections.The visual cortex is organised into hypercolumns, i.e. groups of neurons sharing the same receptive field around a point x in the retina R. A hypercolumn is subdivised into microcolumns or neural masses, each coding for a particular physical quantity at position x, such as orientation, spatial frequency, temporal frequency ( [19], [12], [33]).
In this work, we assume such a columnar organisation encoding for colors in C. The encoding of colors could be due to the presence of blobs ( [17], [25]).The axonal projections from the retina to the cortex ensure a correspondence, called retinotopic mapping, between receptive fields of the retina and the hypercolumns of V 1, so that close neighboring elements in the retina are kept close after projecting in the cortex, and reciprocally.Let us identify the pieces of retina and cortex to flat regions of the plane R 2 , respectively denoted R and Ω, hence considering the receptive fields and hypercolumns not to be discrete but rather continuous entities.Approximations of the retinotopic mapping have been proposed, such as a logarithmic transformation χ : R → Ω ( [41], [36]): Notations The point (r, c) designates the neural mass selective for cortical position r ∈ Ω and color c ∈ C opp .We say (r , c ) is a neighboring neural mass when r is in the first two categories of neighbors with respect to r: adjacent or remote (see Definition 1).We use I : Ω → C opp ⊂ R 3 to denote the image of the scene projected onto the cortex.I takes values in the opponent color space C opp defined in (2).The image I can be viewed as a triplet of three scalar images.The numerical computation of the input image I uses an approximation of the retinotopic mapping, as later exposed.A summary of notations used throughout the paper is given in the tables below.
neural activity at time t of neural mass (r, c) is a convex bounded set in VC C ⊂ RC ⊂ VC not universal, they all depend on the observer

Structure of the model
To each hypercolumn r, we associate a neural activity denoted a, which is a function of position r ∈ Ω, color c ∈ C opp and time t ∈ R. The main thrust of this article is twofold : first, we put forward an evolution model for the dynamics of neural activities, considered as a spatial and color neural field [1,14,5,11].Second, we propose a formal definition for color matching experiments in the context of color perception.

Color Neural Field
We now describe our model.We assume that the neural activity a : Ω × C opp × I → [0, 1] is solution to an integrodifferential equation of Wilson-Cowan type ( [51]): where I ⊂ R is a time interval containing 0, and at each time a(t) takes values in [0, 1] so that it can represent a firing rate, or any physical activity. 5he connectivity kernel ω is designed to encode the antagonistic actions of contrast and assimilation.The operation in fact depends on the opponent representation C opp .We set: and the different functions are such that: • g is a classical difference of gaussians or "mexican hat", parameterized by the weights µ,ν and variances α,β: To have local excitation, we suppose g(0) > 0 i.e. µ > ν.The kernel g weights the influence of spatially neighboring hypercolumns.
) is a difference of gaussians, one which is centered at c , the other one at its opponent −c : Introducing the gaussian kernels f measures the influence of column c and its opponent −c over column c.
• The activation function F to be the sigmoid converging to 0 and 1 at ±∞: where γ is a parameter proportional to the highest slope of the sigmoid • H is the color input sent by the LGN and related to the cortical image I(r, t) in opponent coordinates: H measures through a gaussian kernel how far is the color c encoded by (r, c), from the real input I(r, t).It has a greater value when c is closer to the input.
In this dynamics H represents the "forcing term" imposed by the image seen by the subject.
• τ is the typical speed of the dynamics, but in this work it will be of less importance than other parameters.Without loss of generality we can take τ = 1 up to rescaling of the time axis.
In the sequel, the connectivity kernel also refers to The neural field dynamics can then be written as: for all (r, c, t) In this work, we only consider fixed images and not movies, so H(r, c, t) = H(r, c) is kept constant.

Interpretation of the connectivity kernel
The connectivity kernel ω is the most important item in the neural field model, because it expresses the antagonism between contrast and assimilation.For a fixed neural mass (r, c), it describes how neighboring neural masses (r , c ) excite or inhibit (r, c) through the operation ω a (r, c, t).The "lateral" connection from (r , c ) to (r, c) is excitatory if ω(r, c, r , c ) is positive, inhibitory if it is negative; the strength of the connection corresponds to its absolute value.In the double integral involved in ω a, the contribution of neighbors is their level of activity a(r , c , t) > 0, positively or negatively weighted by ω(r, c, r , c ).The sign depends on the relative positions of r and r, c and ±c.There are four situations according to the respective signs of g(r − r ) and f (c, c ), summarized in table 3: The function f (c, •) for fixed c.The function g(r − •) for fixed r. r 1 and r 2 are resp.adjacent and remote neighbors.Hence, the only way for (r , c ) to excite (r, c) is when r is adjacent to r (i.e.g(r − r ) > 0) and c close to c, or r is a remote neighbor (i.e.g(r − r ) < 0) and c is close to the opponent color −c (see Figure 4).Otherwise the connection is inhibitory.This behavior typically models assimilation and contrast.In the computational results, we illustrate the roles of the connectivity kernel ω and cortical input H.

Convolution form The kernel
allows us to rewrite the double integral into a form similar to a convolution: where the convolutions are computed in their respective spaces, and S = −Id C operates as the symmetry in color space, which we define as Copp (S • a) thanks to the symmetry of f 2 and C opp .

Color sensation and color matching experiments
In this paragraph we introduce the central notion produced by our model, i.e. that of color sensation.This allows us to propose a mathematical description of a color matching experiment.Now that we (theoretically) know how the visual cortex reacts to a color image, we describe how to link the model to psychophysical data.
Setting A human viewer is presented with two still color images, one of them being a reference image called test image and the other one a comparison image which can be modified by the subject.In most experiments, the images generally have a simple geometry which is composed of elementary shapes, such as squares, rectangles, disks and annuli.These elementary patches are usually uniformly filled with colors.The geometric frames of the two images are generally the same.Typically, the color to be tested uniformly fills a patch (test patch), and similarly for the comparison color (comparison patch).
Task.The subject is asked to modify the comparison color until it perceptually matches the test color.Then, the final value is saved and the difference with the initial value defines a shift.There is a perceptual match when the test and comparison color, along with their respective surroundings, (approximately) create the same effect from the subject's viewpoint, in other words, they appear to be "the same color".During the process, the observer alternatively directs her/his gaze towards the test and comparison patches.The observer is asked to look at a point x 1 in image J test , and to manually adjust the parameterizing color c of J comp [c].We denote c match the specific color c which makes J comp [c] perceptually equivalent to J test at the respective points x 2 and x 1 .The comparison is relative to specific locations, and so there is no reason for the final comparison image to be everywhere perceptually equivalent to the test image.See Figure 5.There is probably no sense to define "the" perceived color, which would be an element in C. For instance, take the test image and the matching comparison image of Figure 5.We would say, "the" perceived color in J test is some green c match .However, if we had chosen a lighter or darker gray for the comparison image, the respective matching colors would be different, because they depend on the comparison background.Yet, the perception of a test color should not depend on the comparison image.Instead, we find more appropriate to talk about a color sensation elicited at point x in image J.

Color sensation and not perceived color
Indeed, a matching process corresponds to matching two brain states, whatever the perception being matched, such as color, texture, touch, pitch, timbre, or any other feeling.
Definition (Color sensation).Let J be a fixed image inducing the cortical image I(r) := φ opp (J(x)), r = χ(x).Suppose that there exists a unique stationary solution to which the dynamics of Equation ( 7) converges 6 , denoted a J (•, •, ∞) := lim t→∞ a J (•, •, t).Then the color sensation perceived at a cortical point r 0 generated by J is the element of L ∞ (C opp ) Note that in our framework, a color sensation at r 0 is a function on C opp which represents how the hypercolumn at r 0 responds to any color in the whole space C. We can extend this concept to a group of hypercolumns and consider the collection of corresponding color sensations.Our representation is much richer than defining "the" perceived color, since we have instead a whole set of functions attached to each hypercolumn, as an echo of the complexity of the color perception problem.
Color matching as a mathematical projection Proposition (Color matching experiments are projections).Suppose we are given a test image J test , a comparison image J comp [c] parameterized by c ∈ C, and two interesting locations r test and r comp to look at (for instance, the respective centers).Denote a test and a comp [c] the associated color sensations.A color matching experiment consists in choosing c match ∈ C so that a comp [c match ] is the closest to a test , in the sense that c match satisfies where dist(a test , a Hence, color matching is formally the projection of a test onto a nonlinear manifold whose elements are {a comp [c]} c∈C .In the Appendix (Section 7), we show that under conditions 19, we can smoothly parameterize color sensations a comp [c] with respect to c, provided that J comp [c] is also smooth with respect to c.

Simulation and Regression
In this part, we simulate our model and confront it to 1) experimental data coming from [32], [31], and 2) our data.For 1), as they showed that color shifts mainly concern the s chromaticity coordinate, we apply the model in its one-dimensional version.
The algorithm which computes the discretized dynamics of the neural activities (7) has the structure of what is called a "neural network" in the machine learning community [34].This has nothing to do with our perceptual model itself.However, the analogy will turn out to be of great importance when undertaking the regression, because we can take advantage of the automatic differentiation tool PyTorch provided by [35].In practice the regression could be compared to a learning process, although conceptually we do not mean to use this analogy.
The model is parameterized by a tuple q of several scalar parameters involved in the gaussians (weights and variances): In the regression part, we aim at finding an appropriate q which best predicts the color matching results.
Discrete setting Here, a vector of length N is indexed by i = 0, ..., N − 1.
For the sake of simplicity, we consider that Ω = [−1, 1] × [−1, 1] is a square domain.Each side is equally sampled by 2 • N x + 1 points.Hence, zero is indexed by N x .Ω n = {(x i , y j )} i,j=0,...,2•Nx denotes the discretized domain, where Nx .The point of interest at which color matching is performed is r 0 = (0, 0). 1) For the data from [32] and [31], as explained before, we make use of an interval to represent the opponent color space : C opp = [−2, 2] with the conversion rule c := s − 1, so that the "neutral" point c = 0 corresponds to s = 1 (so that purple and lime become c = 1.00 and c = −0.84).C n = {c i } i=0,...,2•Nc denotes the discretized color space, with Nc .We used N x = 10 and N c = 20, which correspond to 21 and 41 sampling points respectively, along spatial and color axes.
2) For our personal data, we work in HSL space using the X, Y cartesian coordinates (X, Y ∈ [−1, 1]) and restrict the color space to the chromatic disk.Here, the "neutral" color is the gray defined by (H, S, L) = (0, 0, 1/2).This choice is arbitrary, but computationally efficient.We used N x = 4 and N c = 4 for a total of 9 sampling points along each axis, which amounts to 9 4 = 6561 sampling points in Ω n × C n .The sampling resolution is quite small, but this drastically increases the speed of computation.
Convolutions An image I and a neural activity a are all functions defined over a finite set.We use the brackets [...] to denote a discrete variable, as in The neural activity a is a 3D array containing (2 The computations become increasingly involved when the dimension is 2 or 3, especially when simulating the color matching experiments (because we have to look for the best c match in the whole color space.)In discrete form, the operation (8) is written where dx := 1  Nx and dc := 2 Nc are the discretization steps.In practice, we computed successive discrete convolutions, as in (11), where the symmetry S boils down to a flip along the C n axis.We exploit the separability of gaussians as much as possible in order to speed up computations (a convolution with a multi-dimensional gaussian is equivalent to several 1D convolutions along each axis).In particular, the connectivity kernel ω is separable in physical and color space, and in respective spaces we can split gaussians into 1D kernels. 7nput image Ideally, the cortical image I in Equation ( 9) should be computed from the original retinal image J (an example is shown in Figure 6), by applying a retinotopic map such as in Equation ( 6).In particular, the visual field width should be in relation to R. Also, the visual angles, the spacing and spatial frequency of the bands (of 3.3 cycles per degree for instance) should be taken into account before applying the retinotopic map.However, to keep computations simple, we make the drastic and simplistic assumption that the gaze is on average directed towards the center of the concentric annuli, so that the annuli after transformation are approximately parallel bands as in Figure 7.    Dynamics The neural dynamics (7) are simulated with a forward Euler scheme: where dt is a small time step.To search for the stationary state of the dynamics, we use a fixed point iteration method (equivalent to fixing dt = 1 in the previous formula): In practice, with our discretization steps, the activity converges after 15 or 20 iterations.It is the unique fixed point of the operator Φ q (a; J) := F (ω a + H), given the parameters q on which the functions F , ω, H depend.
The algorithm can be stated as follows: We start with a set of N exp test images, denoted J test [i], where 1 ≤ i ≤ N exp .For each index i, let a test q [i] denote the corresponding color sensation at r 0 = (0, 0).It is the result of the computation through Algorithm 1. Now, we also have a set of comparison patterns {J comp [i], i = 1, ..., N exp parameters : q := (µ c , ν c , α c , β c , µ, ν, α, β, µ h , σ h , γ) initialization: for all input images, compute H and set a = 0 for c ∈ C n do compute a comp q [c] using Algorithm 1 and save it end for i = 1, ..., N exp do compute a test q [i] using Algorithm 1 and save it ; Regression Now that we are able to emulate a person realizing several color matching experiments, our model parameterized by q has to be regressed to experimental data.

Problem (Regression). Assume we have the following data:
• a finite set of test images J test [i], i = 1, ..., N exp ; • a family of comparison images {J comp [c]} c∈Cn ; • and the corresponding experimental color matching results c match [i].
We aim to minimize the squared error: where for each experiment i = 1, ..., N exp , c pred q [i] is the minimizer of: produced by Algorithm 2.
8 Remember that a test q [i] and a comp q [c] are color sensations, hence functions from C to [0, 1].
As said before, the gradient is computed using the PyTorch library which provides automatic differentiation ( [35]).With this tool, we are able to minimize E(q), using L-BFGS-B optimisation. 9

Data
Reference data We used the data in the figures of [32] and [31].Usage of the data was kindly approved by the corresponding authors.
Personal Data We fixed two particular backgrounds to serve as test and comparison backgrounds, Yellow (HSL = (60 • , 50%, 50%) and Gray HSL = (0 • , 0%, 50%).One of the authors (A.S.) performed color matching experiments as in Figures 2 and  3. Experiments were made on the same device, in a dark room.Squares were presented against a black background.Several test colors were presented surrounded by Yellow, giving different comparison colors surrounded by Gray, which lead to several couples (test color, comparison color).We realized the experiments for all three dimensions in color space, at regularly spaced locations.We obtain a vector field of color shifts in HSL space.Our data was obtained by averaging the shift along the third Luminance coordinate, to obtain a field of shifts in the chromatic disk only (see Figure 17, left).We are indeed only interested in chromatic shifts.The treatment of Luminance seems to be more complex, and has to be separate from that of chromaticity.We clearly see that, neighboring masses (r , c ) such that r is an adjacent neighbor of r 0 and c is close to c 0 , or such that r is a remote neighbor and c is close to −c 0 , have excitatory connections with (r 0 , c 0 ) (see Table 3).The two other cases are also represented in the figure.In fact, most of the neural masses have negligible influence on (r 0 , c 0 ).This occurs when c is neither close to c 0 nor to −c 0 , or when r is too far from r 0 (which we cannot see on the figure, since the outer variance of the DOG g has a great value compared to the extent of Ω n ).
We provide an interactive 3D animation of the connectivity kernels ω(r 0 , c 0 , •, •) for all values of c 0 ∈ C opp , in the Supplementary Materials.
For varying r 0 , the kernel is just spatially translated along r 0 .However, for varying c 0 , the positive and negative gaussian kernels in f follow c 0 and −c 0 , collide when c 0 goes through zero, then exchange of position when |c 0 | grows again.The color bar extends between 0 and 1, and is set so that small variations are easily seen.H is obtained by 'lifting' the cortical image I inside Ω × C. Hence, the altitudes of the extremal values alternate between lime (c −1) and purple (c 1).Its shape heavily determines that of the final activities a ∞ , since it has the role of cortical input.Notice how in a ∞ , the activities of the neural masses are lower than 1/2 for small values c.We provide an interactive 3D animation of the evolving activities a(•, •, t) along the iterations of the fixed point algorithm in the Supplementary Materials.We can see that the convergence is quite fast and 20 iterations are sufficient.7), with input image I given by the purple/lime pattern as in Figure 7. Plotted after a. one iteration, b. two iterations, c. thirty iterations (convergence reached after fifteen iterations).The red curve is the activity a test q (r 0 , •) corresponding to the spatial point r 0 .The four other blue curves correspond to spatial points r i located on the other rays.Notice that we only show four and not eight different curves, because of the axial symmetry that we artificially introduced in the numerical computations (see the simplification exposed before, as in Figure 7).A video of the evolution is provided in the Supplementary Materials.Red dots indicate the experimental data while blue crosses are the predicted matching comparison colors.The data is an average over three sets of experiments (as detailed in the original article).The ordinate corresponds to the color shift expressed in C opp coordinates c = s − 1.The abscissa i = 0, ..., 7 refers to the test pattern: p/p, l/l, p/w, l/w, w/p, w/l, p/l, l/p (p stands for purple, l for lime, w for white).The regressed value for q is resp.q M C = (0.60, 0.69, 0.30, 0.40, 4.42, 1.82, 0.58, 8.35, 0.47, 0.30, 1.80) and q AZ = (0.60, 0.69, 0.31, 0.40, 4.42, 1.81, 0.60, 8.35, 0.47, 0.30, 1.80).We note that there is only a slight difference between the parameters, which can partly account for the differences between subjects.Figure 13: Left.Converged neural activities after 40 iterations, computed for the purple/lime configuration with the regressed parameter q = q M C (the same as in Figure 12, left).a test q ,a comp q [c match ] and a comp q [c pred ] are plotted as functions over C opp , where c match is the data point and c pred numerically predicted.The curves of a comp q [c match ] and a comp q [c pred ] coincide because c match and c pred are practically the same.Among all curves {a comp q [c]} c it is the nearest to a test for the L ∞ norm.Right.Functions f and g for the regressed parameter value q = q M C .Up: heatmaps for g and f .Down: corresponding side views.15.Left but instead of using the data to find the best vector q we use the regressed value q AZ for observer 'AZ' [32] where s test is not changed, to simulate the setting where s test changes [31].Our predictions are of course not close to those of Figure 15 (especially at the endpoints where the red and blue curves cross10 , and shifts are smaller).However it is remarkable that the model predicts a similar trend with respect to the s test chromaticity values of the test color.We have obtained similar results for the observer 'MC'.8).The test colors sample reasonable well the HSL space.b.Predicted results, after regression.Note that the sampling resolution of HSL space is quite low for computational reasons, so the meaningfulness of the parameter values has to be carefully considered.We obtain a smoother result than in the experimental data, and the convergence towards the opposite blue becomes more obvious.

Difference with color constancy problems
If one is asked: "What is the color of this object ?", at least two answers come to mind.A first possibility is to report what underlying colored matter has produced such a visual effet, in other words, to identify the reflectance R ξ (which is a property of the material).The second one is to report what is the physical light sent by the object to the sensor (device, eye...), after being reflected; in this case, we are trying to retrieve the visual information C of Equation ( 1) or rather the color equivalence class [C] in which it falls, uniquely specified by the (L, M, S) coordinates.This is the main difference between color constancy-related problems, and ours.In the next subsection we state that a third answer is now possible, thanks to our model, and is based on color sensation.
The problem of color constancy Color constancy is the ability of the human observer to guess objects reflectance despite very different illumination conditions.This phenomenon, first studied by E. Land who proposed the Retinex algorithm ( [27], [39]), has since produced a lot of research.For a good review on color constancy, see [15].The problem can be formulated as such: given the cone inputs (L, M, S), how does the brain retrieve the spectral reflectance R ?For instance, the same object seen in daylight or under shadow has different colors because [C daylight ] = [C shadow ], but we can easily recognize it and guess R = R daylight = R shadow .On the reverse, two objects having the same color can be guessed as being made of different materials, if their surroundings are not the same, that is, Color sensation: subjectivity vs objectivity Now, let us forget about the substance which reflects the lights and concentrate on the visual information C only.Even though the colors [C 1 ] = [C 2 ] are the same (identical LMS coordinates), they can be perceived as different, in other words, induce different color sensations S 1 = S 2 .This is possible under the influence of different surrounding stimuli.The reverse situation can occur, where two different colors are perceived as identical because of the context.Unlike the first two physical and objective quantities, the concept of color sensation is completely subjective.However, we are able to compare two color sensations in a quite objective manner, through color matching experiments.Thus we have the means to indirectly, but objectively, gain access to this subjective information.In other words, if not an absolute concept, perception is certainly above all a relative concept that one can assess through comparison.

What is color? Three levels of definition
In fact, a third answer is now possible.To summarize, if we use punctual notations again, where x 1 , x 2 ∈ R are retinal points corresponding to ξ 1 , ξ 2 two points in the scene, the three relationships are independent, in the sense that we can find situations where any of them can hold or not.One can convince himself of this with simple examples, because punctual informations without context are independent.However, if we take into account all the scene with all points x, then the sets {S x } x∈R (18) are linked one to another.While papers on color constancy work on the link between Equations ( 16) and ( 17) ( [15], [4]), our paper concentrates on the relationship between Equations ( 17) and (18).We believe that {S x } x∈R entirely depends on the sole knowledge of {[C x ]} x∈R .

The need for a universal model
As any model, ours obvioulsy has its own limitations, that we list below.Some standard features of the human visual system are not reproduced correctly, which call for improvements, or extensions.
• Difficulty to have a perfectly symmetrical opponent space.An ideal mathematical setting eases the analysis, but the geometry of the color space C is far from being known, and that an opponent representation gives it perfect symmetry is quite improbable.However, it is still possible to talk about a nearly symmetric space and work in the biggest symmetric opponent space we can define inside C.
• What about luminance?We did not compare our model to data varying along the Luminance axis, because it certainly needs a special treatment, separately from the chromatic plane.An easy adaptation of our model would be to consider anisotropic gaussian kernels in f and g, and consider a nonseparable connectivity kernel ω.It could be for instance a finite sum of separable kernels ω i = f i ⊗ g i .This should already allow different treatment of the Luminance axis and chromatic plane.
• Perceived or not perceived color We insisted on using color sensations (functions on C) and not perceived colors (an element in C), to describe the perception of a color.We already argued in the text that, the concept of 'the' perceived color fails to be appropriate, as soon as it depends on the comparison image.However, a way to define it only based on the test image would be to take c max ∈ C maximizing the neural activity of the hypercolumn r 0 .As in the literature dealing with the perception of visual orientations, the 'winner takes all' law assumes that the perceived quantity is the one for which the firing rate is maximal.
• Positive / negative afterimage However, if 'the' perceived color is indeed c max defined above, then our model can only predict positive afterimages and not negative ones.Indeed, our model positively adds the contribution of the input image.When the image is dynamic and goes back to some static white or black image, just after fading, the activity is still close to the one just before, and so is the argmaximum which defines 'the' perceived color: we would still perceive 'the' same color.
• Watercolor effect The model cannot explain the spread of a local feature (two thin borders of dissimilar colors) to a global impression (one of the delimited area faintly seems to be colored by the inner border) (see [37]).
For explaining such an effect, the contribution of edges in the influence of context over perception has to be taken into account.
• The role of edges More generally, as for explaining watercolor effects, a better model should also integrate edges in the discussion.Indeed, 'the perception of the color of a surface depends strongly on the color difference across the boundary of the surface, on visual edge contrast' ( [16]) and 'V1 retains the color sensitivity provided by the LGN, and adds spatial selectivity for color boundaries' ( [23]).A simple integration over the spatial domain cannot sufficiently account for the effect of a color border over the entire perception.

Towards color hallucinations
Just as neural field models for the vision of orientations, we can study the bifurcations of the solutions to equation ( 7) around stationary states ([6], [10], [46], [45]).Under some hypotheses of symmetry and periodicity, we can predict, using equivariant bifurcation theory, the emergence of visual patterns or "planforms".In the same fashion as [6] who explained orientation-based geometric hallucinations, a color neural field model can predict patterned color hallucinations.Future psychophysical experiments, may confirm this and support the relevance of this kind of model for color vision.

Conclusion
We have proposed an application of a classical neuroscience framework -that of neural fields -to the study of psychophysical phenomena for color in context.To our knowledge, this is the first one.Our model accounts for some nonlinear behaviors observed from experimental data, by combining locally opposing influences in physical space at two different scales, and in color space.The assumption that V1 is organized into a structure similar to color hypercolumns has to be experimentally proved though.
Our work also proposes two natural ideas.First, we propose to consider color matching as a mathematical projection, which is a quite intuitive idea.This satisfies the principles of Psychophysics, which seeks to access subjective notions, such as perception, through the means of objective procedures, such as matching.Second, we do not push forward the concept of the perceived color, but rather that of a color sensation, which is a much richer object, since it already takes into account the spatial complexity of the problem.Indeed, the final effect is produced after integrating the pointwise influence of neighboring neural masses.These two ideas could be adapted to other perceptual situations, such as hearing or touch.

Appendix
Color matching functions and cone sensitivities In this paragraph we explain how the {r, ḡ, b} matching functions fit inside our theoretical framework.Classically, to test metamerism between lights of density C 1 and C 2 , an observer is shown a 2 • or 10 • diameter disk, where spectral lights C 1 and C 2 are displayed on the two vertical halves.If the disk is perceptually uniform then they are declared to be metameric.Now, we fix three independent primaries C 1 , C 2 , C 3 ∈ L 2 (Λ) that we call R, G, B. Usually, they are three monochromatic lights.For any light C, we can assess how much of each primary light is necessary to make C metameric to aR + bG + cB, such as in Wright's or Guild's historical experiments.We then obtain functions r, ḡ, b ∈ L 2 (Λ) so that the coefficients are a = r, C , b = ḡ, C , c = b, C .In practice, they are obtained by taking C as a monochromatic light concentrated around some wavelength λ, and r(λ) is then the scalar coefficient in front of R.
Metamerism implies in our settings that, for any light C and corresponding coefficients a, b, c, which can be inverted.Thus, the matching functions of an observer defined with three fixed primaries are always linear combinations of his/her cone sensitivities.Note that, the matching functions as well as the sensitivities and the conversion matrix depend on the subject.Besides, the matching functions depend on the introduction of a spatial extent, which is not the case for sensitivities.Indeed, the radius of the disk impacts on the metamerism relationship (it is not apparent in the computations above, since we considered a constant cone sensitivity S x L = S L at any point x of the retina, for simplicity).Some properties of the dynamics In all the following, we use the notation x ∈ L ∞ (Ω×C opp ) instead of letter a as in Equation (7), because it is mathematically more rigorous to speak of a functional equation on the Banach space L ∞ (Ω × C opp ), dx dt = −x(t) + F (ω x(t) + H) =: Θ(x(t)) and a(r, c, t) := x(t)(r, c).H is also supposed to be constant w.r.t.time.
Lemma 1 (Condition for the existence of a unique stationary solution).Let d denote the dimension of the color space (d ∈ {1, 2, 3}).Suppose that Then there exists a unique stationary solution to (7) in L ∞ .More precisely, for H ∈ L ∞ (Ω × C opp ), the map is Lipschitz continuous with the Lipschitz constant given in the left hand side of (19), so that the inequality ensures Φ H to be a contraction for the L ∞ norm.
Proof.For any x ∈ L ∞ (Ω × C opp ), Thus, for x, y, In fact, the same conditions ensure linear stability of the solution, which is the object of the next lemma.Let E denote L ∞ (Ω × C opp ).
Lemma 2 (Stability).Under the conditions of Lemma 1, the unique stationary solution is linearly stable.
Let L := DΘ(x 0 ) denote the linear part.Then, L = −Id + T where T := F (ω x 0 + H) ω is a linear operator such that T < 1 thanks to condition (19).Note that T takes values in C 0 (Ω × C opp ) the set of continuous functions defined on the domain.The spectrum of L, denoted Σ(L) := {σ ∈ C | L − σId not bijective}, is then equal to −1 + Σ(T ) which is a compact contained in a disk centered on −1 and of radius T .Thus, for any σ ∈ Σ(L) we get that σ < 0, which ensures linear stability.
Notice that this does not imply global convergence of the dynamics to the unique stationary solution.
Lemma 3. Let d denote the dimension of the color space, g 1 and g 2 the two gaussians such that g = g 1 − g 2 and D the closed disk on which g 1 ≥ g 2 .The radius of the disk is given by The contraction condition (19) is equivalent to where and where the bracket is equal to Color matching as a projection where the map Q is defined as

Figure 1 :
Figure 1: Figure from[31].a) Simultaneous contrast: the small patches on the left and the right are identical, but they tend to appear darker on the yellow backround, lighter on the dark background.b) Chromatic assimilation: the background tends to be blue or green depending on the color and spatial frequency of the grid.c) Appearance of both phenomena: the pink-orange changes perceptually when surrounded by concentric annuli with a purple/lime or lime/purple pattern.

Figure 4 :
Figure 4: Displayed on a 1D axis for illustration purpose.
In this work, we use the most general framework to define color matching experiments in a mathematical sense.In particular, we do neither suppose that the test and comparison images have simple geometric frames, nor that the latter are the same.Formally, let J test , J comp ∈ C R denote the test and comparison images projected onto the retina R ⊂ R 2 .J comp = J comp [c] is an image parameterized by a color c, and {J comp [c]} c∈C forms a family of images living in C R (we no more use [C] but c to denote an abstract color).

Figure 5 :
Figure 5: The test image J test has a yellow surround and we are testing the green color c test .The comparison image J comp has a gray surround.We change the comparison patch until a match with the test patch.We save the final value c match and the shift.

Figure 6 :
Figure 6: Physical images assimilated to the retinal images J. a. purple/lime annuli pattern surrounding a test color, with modified coordinates s − 1.In our simplified framework J corresponds in fact to the left or right half of the image.b.Comparison color surrounded by equal energy white s − 1 = 0. Colors do not reproduce experimental settings, but correspond to the color scale.

Figure 7 :
Figure 7: Cortical images I corresponding to Figure 6: a. the purple/lime pattern shows the geometry of the inputs to our algorithms (following the simplification exposed before).b.Comparison color surrounded by equal energy white s − 1 = 0. Colors do not reproduce experimental settings, but correspond to the color scale.
[c]} c∈Cn , parameterized by colors.A second application of Algorithm 1 to all comparison images allows us to compute the corresponding color sensations a comp q [c] induced at r 0 by J comp [c].Then, for each experiment i, we use the procedure explained in Paragraph 2.2.4 to determine which comparison pattern gives the most similar color sensation to the test image.This gives the corresponding predicted matching colors c pred q [i] ∈ C n . 8Algorithm 1 emulates the color neural field dynamics.It is the building block in Algorithm 2 below, which emulates the color matching experiment by selecting c pred q .input : test images I test [i], i = 1, ..., N exp comparison patterns {I comp [c]} c∈Copp output : predicted matching colors c pred q

Figure 8 :
Figure 8: Given fixed test and comparison surrounds (resp.Yellow and Gray), we obtain a vector field of shifts (test color → comparison color) in HSL space.There is clearly a 'yellow pushes towards blue' phenomenon (contrast wins).

Figure 9 :
Figure 9: The connectivity values W (•, •) := ω(r 0 , c 0 , •, •) are shown as a function defined over the product of the 2D physical space by the 1D color space.The parameter vector q is equal to q M C .The color bar extends between −1 and 1.It goes from dark green (negative values, strong inhibition) to dark orange (positive values, strong excitation).Left: r 0 = 0 and c 0 = 1 defining W 1 .Right: r 0 = 0 and c 0 = .5defining W 2 .We clearly see that, neighboring masses (r , c ) such that r is an adjacent neighbor of r 0 and c is close to c 0 , or such that r is a remote neighbor and c is close to −c 0 , have excitatory connections with (r 0 , c 0 ) (see Table3).The two other cases are also represented in the figure.In fact, most of the neural masses have negligible influence on (r 0 , c 0 ).This occurs when c is neither close to c 0 nor to −c 0 , or when r is too far from r 0 (which we cannot see on the figure, since the outer variance of the DOG g has a great value compared to the extent of Ω n ).We provide an interactive 3D animation of the connectivity kernels ω(r 0 , c 0 , •, •) for all values of c 0 ∈ C opp , in the Supplementary Materials.For varying r 0 , the kernel is just spatially translated along r 0 .However, for varying c 0 , the positive and negative gaussian kernels in f follow c 0 and −c 0 , collide when c 0 goes through zero, then exchange of position when |c 0 | grows again.

Figure 10 :
Figure 10: Left: cortical input H(r, c) := h(c − I(r))) where the cortical image is I, shown on Figure 7. Right: final activities (or color sensations) a ∞ .The color bar extends between 0 and 1, and is set so that small variations are easily seen.H is obtained by 'lifting' the cortical image I inside Ω × C. Hence, the altitudes of the extremal values alternate between lime (c −1) and purple (c 1).Its shape heavily determines that of the final activities a ∞ , since it has the role of cortical input.Notice how in a ∞ , the activities of the neural masses are lower than 1/2 for small values c.We provide an interactive 3D animation of the evolving activities a(•, •, t) along the iterations of the fixed point algorithm in the Supplementary Materials.We can see that the convergence is quite fast and 20 iterations are sufficient.

Figure 11 :
Figure11: Evolution of neural activities according to Equation(7), with input image I given by the purple/lime pattern as in Figure7.Plotted after a. one iteration, b. two iterations, c. thirty iterations (convergence reached after fifteen iterations).The red curve is the activity a test q (r 0 , •) corresponding to the spatial point r 0 .The four other blue curves correspond to spatial points r i located on the other rays.Notice that we only show four and not eight different curves, because of the axial symmetry that we artificially introduced in the numerical computations (see the simplification exposed before, as in Figure7).A video of the evolution is provided in the Supplementary Materials.

Figure 12 :
Figure 12: Results of the prediction of our model after regressing over the data of 'MC' observer (Left) and 'AZ' observer (Right), respectively ([32]).Red dots indicate the experimental data while blue crosses are the predicted matching comparison colors.The data is an average over three sets of experiments (as detailed in the original article).The ordinate corresponds to the color shift expressed in C opp coordinates c = s − 1.The abscissa i = 0, ..., 7 refers to the test pattern: p/p, l/l, p/w, l/w, w/p, w/l, p/l, l/p (p stands for purple, l for lime, w for white).The regressed value for q is resp.q M C = (0.60, 0.69, 0.30, 0.40, 4.42, 1.82, 0.58, 8.35, 0.47, 0.30, 1.80) and q AZ = (0.60, 0.69, 0.31, 0.40, 4.42, 1.81, 0.60, 8.35, 0.47, 0.30, 1.80).We note that there is only a slight difference between the parameters, which can partly account for the differences between subjects.

Figure 14 :
Figure14: The blue and green curves are as in Figure13and correspond to a test q

Figure 15 :
Figure 15: Left.Data from[31].Dots indicate the means for four different subjects, and the thick black lines best approximate the means of the four results for seven experimental settings, where the test background is fixed (as in our experiments), but the test color is varied.The straight thin horizontal black lines show the predictions made by their model.However, the experiments indicate a color shift which depends in a non trivial way upon the s coordinate of the test ring.Right.Predictions of our model.The best fitted value of q is q = (0.42, 0.71, 0.63, 1.16, 4.43, 1.72, 0.56, 6.35, 0.47, 0.30, 1.80).Red crosses are the means of the experimental data which stand as ground truth.

Figure 17 :
Figure17: a. 36 pairs of experimental data points (test, comparison colors) in the HSL chromatic disk (constant Luminance), which are the averages of measured shifts (refer to Figure8).The test colors sample reasonable well the HSL space.b.Predicted results, after regression.Note that the sampling resolution of HSL space is quite low for computational reasons, so the meaningfulness of the parameter values has to be carefully considered.We obtain a smoother result than in the experimental data, and the convergence towards the opposite blue becomes more obvious.

S
L , C = S L , aR + bG + cB S M , C = S M , aR + bG + cB S S , C = S S , aR + bG + cBThe first equality can be reformulated asS L , C = S L , R r, C + S L , G ḡ, C + S L , B b, C S L , C = S L , R r + S L , G ḡ + S L , B b , Cand so S L = S L , R r + S L , G ḡ + S L , B b.We obtain similar results for the other sensitivities, resulting in a linear relationship Indeed, for any (r, c)∈ Ω × C opp , |ω x|(r, c) ≤ Ω |g(r − r )| dr Copp |f 1

Lemma 4 .
Suppose that J comp [c] is smooth function of c, and that condition (19) holds.Then the unique stationary solution a[c] to the dynamics with input H[c] related to J comp [c] is smoothly parameterized by c. Hence under these assumptions, color matching consists in projecting a test on the image set of the parameterization {a comp [c]}.Proof.For any c ∈ C, the unique stationary solution a[c] satisfies 0 = Q(a[c], c) and the partial differential D a Q(a, c) defined below is invertible:D a Q(a, c) • da = −da + F (ω a + H[c]) ω da because for any b ∈ L ∞ , da → F (ω a + H[c]) ω da − b defines a contraction mapping in L ∞ under condition(19) (we used the fact that |F | ≤ F (0)), and we can apply Picard's theorem.Then, in a neighborhood of each c 0 and a[c 0 ] the map c → a[c] is C k thanks to the Implicit Function Theorem.We thus obtain a smoothly parameterized family of elements in L ∞ (C) {a[c]} c∈C .
Algorithm 1: Color Neural Field NetworkColor matching We now describe how our model can be used to simulate a set of virtual color matching experiments whose results can in turn be used to estimate the model parameters q.