A neural field model for color perception unifying assimilation and contrast
[['Color induction , which refers to a change in color appearance of a test stimulus under the influence of spatially neighboring stimuli in the field of view [1] , has been extensively studied in psychophysics [2] .', 'This effect has been observed for uniform inducing surrounds [3–7] and geometrically more complex ones as well [8–15] .', 'The geometry of spatial context , and especially the frequency of chromatic modulation , have been shown to play an important role in color induction .', 'Many works on this subject have been devoted to the study of two induction effects in particular , known as chromatic assimilation and simultaneous contrast ( see Fig 1 ) .', 'Chromatic assimilation is the fact that the chromatic appearance of a test stimulus changes towards the chromaticity of inducing stimuli .', 'Conversely , simultaneous contrast corresponds to the test chromatic appearance changing away from the chromaticity of inducing stimuli .', 'Contrast is interesting in that it involves the notion of color opponency: the change is often made towards an opponent or complementary color [5 , 7 , 16] .', 'These increased perceptual similarity or dissimilarity can be viewed as the results of attractive or repulsive effects respectively in color space .', 'While assimilation and contrast had previously been thought to occur separately , [15 , 17 , 18] suggested that they act simultaneously in a synergistic manner .', 'The idea that effects induced by context result from a balance between assimilation and contrast was also proposed in cognition [19] .', 'The experimental settings used by [18] to demonstrate the synergy relied on asymmetric color matching .', 'In psychophysics , this is a classical procedure to objectively measure the amount of color induction caused by spatial context .', 'A human observer views two still color images side by side , a test image Jtest whose pattern influences the perception of a test color ctest , and a comparison image Jcomp[ccomp] with a modifiable comparison color ccomp ( see Fig 2 ) .', 'In most experiments , the two images have the same geometric patterns and are composed of elementary shapes , such as rectangular , round or concentric patches , uniformly filled with different colors [3 , 7 , 20] .', 'The patches to be compared are filled with ctest and ccomp .', 'The observer is asked to change ccomp until color appearance between the test and comparison patches are the same , leading to a perceptual match .', 'The perceptual shift is then the difference between the final color cmatch and the test color ctest .', 'In Fig 3 , we illustrate this experiment with simple square patterns .', 'In [18] , color shifts were measured for patterns with concentric annuli as in Fig 1 ( Down ) , whose colors were distinguished by the stimulation of S cones only .', 'The matching surround was a neutral gray .', 'Stimuli were expressed in a variant of the cone-based chromaticity space proposed by [22] and specified by three coordinates s , l , Y , where s and l are defined as the ratios S L + M and L L + M respectively , and Y stands for luminance .', 'The definition of the S , M , L signals is recalled below .', 'Their results showed that the largest color shifts in s chromaticity were induced by patterns alternating between two distinct colors , such as purple and lime .', 'In particular , shifts induced by uniform backgrounds or patterns alternating between white and purple or lime were smaller .', 'Such large shifts could not be induced by optical factors ( spread light or chromatic aberration ) only , but implied some neural processing of the stimuli [23] .', 'To explain this , [18] suggested that stest was shifted towards the adjacent ring thanks to assimilation , while it was also repelled away from the second ring by contrast , resulting in the matching value smatch .', 'They proposed a S+/S- center-surround receptive field model to predict the shifts: at a point x in the test ring ,', 'shift at x ≔ s m a t c h - s t e s t = DOG * ( J t e s t - J c o m p [ s t e s t ] ) ( x ) ,', 'where Jtest and Jcomp , in s coordinates , are convolved with a Gaussian kernel beforehand to account for retinal blurring; the Difference of Gaussians DOG stands for the receptive field .', 'Color appearance would then be the combined contributions of chromaticity at the point of interest and influence of the surround filtered by the S+/S- receptive field .', 'After tuning the parameters , this simple linear model was capable of explaining data in the specific experimental setting of [18] , where the chromaticity s of the test ring was fixed while that of the surround was changed .', 'However , it was unable to explain the dependency of shifts upon stest reported in a later paper [17] where , conversely , stest was varied while the surround was left unchanged .', 'Indeed , in the above equality , the central ring of the difference image Jtest − Jcomp[stest] has chromaticity s = 0 , after cancellation of the test and comparison chromaticities .', 'The model is also conceptually difficult to justify , because it treats the central ring and the surround as fundamentally different spatial components , while handling single chromaticities and spatial integrations at the same conceptual level .', 'This prevents their computational framework from being extended to various patterns and other matching experiments .', 'Here , we build a framework dedicated to general color matching experiments .', 'It is able , in particular , to explain the nonlinear behavior of color shifts found by [17 , 18] .', 'As a starting point , we reformulate their fundamental observation into a Principle of Synergy which relies on the notion of color opponency introduced by Hering [16]: This viewpoint implies an appropriate change of vocabulary: in the sequel , chromatic assimilation and simultaneous contrast refer to local interactions , which may act at the same time but at two different local scales .', 'The global effect observed is then the integration of infinitesimal influences induced by spatially neighboring points .', 'Perception can in particular result in attraction ( assimilation wins over contrast ) , repulsion ( contrast wins over assimilation ) , or none of them .', 'Thus assimilation and contrast , which seem to be contradictory effects , can be described as concomitant local phenomena .', 'Beyond giving a merely computational model , we aim at designing a framework consistent with the physiological and anatomical observations currently available .', 'Light entering the eye stimulates L , M and S cones of the retina proportionally to quantal absorption rate [24] , in a “wavelength-blind” fashion according to the principle of univariance [25] .', 'At a point x of the retina R , the stimulus Lx of L cones can be approximated at first order as [26 , 27]', 'L x ≔ ∫ λ ∈ Λ C x ( λ ) S L x ( λ ) d λ = ⟨ C x , S L x ⟩ L 2 ( Λ ) , ( 1 )', 'where C is the spectral distribution of the light over the visible spectrum Λ , and the spectral sensitivity S L x of L cones located at x depends on their local density .', 'L , M , S signals are then relayed by the Lateral Geniculate Nucleus ( LGN ) and transmitted to the primary visual cortex ( V1 ) through axonal projections .', 'It has been established that chromatic input to the visual cortex from the LGN is encoded in an opponent fashion as Hering postulated [16] and as was later confirmed by neurophysiologists [28–30] .', 'This justifies the use of a color-opponent framework here .', 'LGN cells have been found to respond to linear combinations of L , M , S stimuli [23 , 30] .', 'L − M and S − ( L + M ) signals are transmitted to single-opponent cells in layers ( 4Cβ ) and ( 2/3 and 4A ) of the visual cortex respectively [2 , 23] .', 'In contrast to them , cells clustered inside or around Cytochrome Oxydase ( CO ) blobs in layer 2/3 of V1 are sensitive to a continuum of colors instead of three cardinal axes , nonlinearly with respect to cones [23 , 31 , 32] .', 'As such , they may have a prominent function in encoding color in the cortex , as proposed in [33 , 34] .', 'Most of them were found to be double-opponent cells [2 , 35 , 36] .', 'Single- and double-opponent cells are very likely to play a fundamental role in color processing in the brain [2 , 23] .', 'The former are important for analyzing color in large areas while the latter are sensitive to edges and orientation , implying that color assimilation would be due to single-opponent cells and color contrast to double-opponent cells [2] .', 'The visual cortex has the specificity to be organised into hypercolumns , i . e . , groups of neurons sharing the same receptive field and coding for a particular physical quantity at this position , such as orientation , spatial frequency , and temporal frequency [32 , 37–39] .', 'These signals are mapped from the retina to V1 following an approximately logarithmic retinotopy [40 , 41] .', 'Unlike in the case of orientation , for which the existence of such hypercolumns in V1 is now well established [32] , the anatomical and physiological bases for a functional architecture encoding color are still debated .', 'However , in light of the promising findings made by [31] , and as discussed in [23 , 32 , 42] , it is reasonable to assume in our work a hypercolumnar organisation of cells tuned to a continuum of colors , having double-opponent characteristics , and related to CO blobs in layer 2/3 of V1 .', 'Our work also supposes the presence of long-range lateral connections between hypercolumns , in agreement with observations of [43] where horizontal connections tend to link blobs to blobs .', 'Note that we do not use further assumptions about the anatomical organization of color-tuned cells with respect to blobs ( inside , around , independent ) , which is still unclear [2 , 23] .', 'In this context , our model relies on a neural field [44–47] .', 'It is worth noting that neural fields have been previously applied to simpler examples of sensory processing in visual cortex , in order to study the spontaneous formation of population tuning curves .', 'Orientation tuning has been addressed in an important paper by Ben-Yishai and colleagues [48] .', 'Their model of a single cortical hypercolumn did not take into account the spatial relations between these hypercolumns , which was done in the 2001 landmark paper by Bressloff and colleagues [49] .', 'The problem was then revisited by Bressloff [50–52] , and later by two of the authors of the present paper [53] .', 'Spatial frequency tuning has been addressed by Bressloff and Cowan [54 , 55] , and by Chossat and Faugeras [56] .', 'The combination of orientation tuning in binocular vision giving rise to rivalry waves has been studied in [57] .', 'These simpler models have the benefit of explicit knowledge regarding feature preference maps ( image orientations , image textures ) and connectivity in V1 ( orientation hypercolumns and their relations with CO blobs ) .', 'This color neural field framework allows us to study color-space interactions .', 'In recent years , the interactions between color , and orientation/form/space , have received increasing attention [2] .', 'Double-opponent cells may strongly contribute to the relationship between color and form processing , since the shape of their receptive fields determines their orientation tuning [23] .', 'This supports the hypothesis that functional architectures for color and orientation would be intermingled and realize color-form interactions .', 'A framework was proposed by [42] for modeling color and orientation processing in V1 .', 'They assume that two populations of neural masses , one color-insensitive but orientation-tuned , and the other sensitive to both , interact through an extended version of the ring model [48] .', 'Our work is related but complementary to theirs , because rather than considering only two hypercolumns of each kind and using uniform inputs , without introducing space , we study the interactions between color and space by the means of multiple color hypercolumns and patterned inputs , without introducing orientation .', 'The relation between orientation and color falls outside the scope of this work , and should be examined in the future .', 'The goal of our work is therefore to provide a neural field model unifying assimilation and contrast inside a color-opponent framework , consistent with psychophysical data , and compatible with the physiology of V1 .', 'We have successfully achieved this aim .'], ['A . S . conceived and designed the experiments herself and consented to participate .', 'The rigorous definition of color is thoroughly explained in S1 Appendix , where we also present the most common representations of the color space .', 'Here , for the sake of simplicity , we only briefly state the minimal definitions and properties to be used in the model .', 'Color is mathematically defined as an equivalence class of metameric lights [26 , 58–60] .', 'Two physical lights of spectral distributions C 1 , C 2 ∈ L 2 ( Λ ) are metameric if they produce exactly the same visual effect under the same viewing conditions , and this identification strongly depends on the observer .', 'In our framework , metamerism can be expressed as the equality of the triplets of scalar products characterizing C 1 and C 2 ( L , M , S ) = ( ⟨ C i , S L ⟩ L 2 ( Λ ) , ⟨ C i , S M ⟩ L 2 ( Λ ) , ⟨ C i , S S ⟩ L 2 ( Λ ) ) ,', 'with S L the spectral sensitivity of L cones ( and likewise for M and S cones ) , see ( 1 ) .', 'We dropped the exponent x ∈ R 2 by considering for simplicity that cone density is constant across the retina and that light is spatially uniform ( although we can define metamerism with respect to any x ) .', 'Color is hence naturally identified to a three-dimensional vector for trichromats , being specified by a triplet of cone stimuli ( L , M , S ) .', 'For a light of spectral distribution C , its color is denoted [ C ] .', 'Color space , denoted C , is then defined as the subset of physically realizable colors which are visible to the eye .', 'Since the cone signals ( L , M , S ) are non-negative and cannot reach all possible positive values [60] , C can be identified to a bounded subset of ( R + ) 3 through a choice ϕ L M S : C → R 3 of coordinates .', 'In this work , we suppose that an appropriate choice of coordinate system ϕ o p p : C → R 3 leads to an opponent representation of the color space', 'C o p p ≔ ϕ o p p ( C ) ( 2 )', 'satisfying the following important properties .', 'First , it is a bounded and convex subset of R 3 which enjoys symmetry: if c ∈ C o p p , then - c ∈ C o p p .', 'Second , the symmetry operation c ↦ −c must pair any color to its opponent one , in the sense of Hering [16] .', 'Hence , color regions of C o p p come into opposed pairs , for example Yellow and Blue or Red and Green regions , or likewise .', 'Third , C o p p must contain the neutral or zero color 0 , opponent to itself , which would correspond to some neutral gray with no hue ( for a fourth condition , see details in S1 Appendix ) .', 'Following up on this , we consider in this work an opponent representation in the style of Hering’s theory .', 'Indeed , in view of the physiological results exposed in the Introduction , it is now accepted that L , M , S signals are recombined in area V1 into Yellow-Blue , Red-Green and Achromatic independent channels [61] , although the right choice of the opposite axes has been debated [62] .', 'Our model does not depend in a decisive manner upon the choice of a specific color opponent space .', 'In fact our particular choice of Hering’s coordinates is not very different from the cone opponency coordinates defined in [30] .', 'Also , we do not require that the opponent axes point towards perceptually unique hues , unlike in the original theory of Hering .', 'Supposing such a simple relationship between physiology of the cells and psychology related to hue pureness was indeed criticized [23 , 30 , 62] .', 'Here , we rely on the ( l , s , Y ) and ( H , S , L ) representations , and restrict ourselves to a lower-dimensional subspace of the original color opponent space , also denoted C o p p for convenience ( in this context , the letters ‘S’ and ‘L’ of HSL stand for Saturation and Luminance respectively , not Short or Long cones , while ‘H’ stands for Hue ) .', 'In the case of the ( l , s , Y ) representation , we apply our model to the one-dimensional subspace based on the chromaticity s and defined by c ≔ s − 1 .', 'The ( l , s , Y ) representation used in [15 , 17 , 18] , a variant of the one proposed by [22] , is defined as', '{ s = S L + M l = L L + M Y = L + M + S .', 'We then define C o p p to be the one-dimensional color subspace based on the change of coordinates c ≔ s - 1 ∈ C o p p ≔ [ - 2 , 2 ] , where the number 2 is arbitrary , but covers the typical range of c values used in experiments ( purple , lime and white correspond to ( l , s , Y ) = ( 0 . 66 , 2 . 0 , 15cd/m2 ) , ( 0 . 66 , 0 . 16 , 15cd/m2 ) and ( 0 . 66 , 0 . 98 , 15cd/m2 ) respectively .', ') .', 'The Hue , Saturation and Luminance or ( H , S , L ) representation ( note that the letters ‘S’ and ‘L’ are not referring to Short or Long cones ) , often used in computer graphics , maps the sRGB unit cube or gamut of a device to a cylinder whose central axis is achromatic and perpendicular to a chromatic disk [58 , 63] .', 'Standard formulas provide the change of coordinates T s R G B → H S L [21] .', 'We use the two-dimensional chromatic disk C o p p , defined as the intersection of the constant luminance plane L = 1/2 with the HSL cylinder , and identified to the unit disk , so that ( c 1 , c 2 ) ≔ ( S cos ( H ) , S sin ( H ) ) ∈ C o p p .', 'The gamut corresponds in fact to a subject- and device-dependent subspace C d e v strictly smaller than the subspace of chromatic colors visible by the observer , since they are not all reproducible by a screen .', 'The gamut of standard devices however covers a large part of visible colors , hence justifying its use , and the HSL representation has already proven its efficiency in computer graphics .', 'We claim that the specific details of the display , such as gamut and screen characteristics , do not play a major role in our methods and results , provided that all experiments are consistently made in the same conditions .', 'The main thrust of our model is twofold: first , we put forward an evolution equation for the dynamics of neural activities , considered as a spatial and color neural field [44–47] .', 'Second , we propose a theoretical framework for color matching experiments in the context of color perception , and introduce a formal definition of color sensation .', 'The Color Neural Field Eq ( 3 ) describing ( at least , theoretically ) how the visual cortex reacts to a color image , we now link our model to psychophysical data .', 'This subsection introduces the central notion produced by our model , i . e . , that of color sensation .', 'Basically , it corresponds to some feeling produced in the brain when observing a still image .', 'We define it to be a steady state of the neural field dynamics of Eq ( 3 ) , then restricted to the hypercolumn in correspondance to the test point .', 'This concept allows us to propose a mathematical description of color matching experiments ( see Introduction ) , where matching is considered as the projection of the test color sensation onto a family of color sensations elicited by comparison images .', 'Such a “matching as a projection” framework allows the model to predict color shifts .', 'It could be generally applied to other dynamics than Eq ( 3 ) , as well as other definitions of sensation relatively to these dynamics .'], ['In Fig 6 , we set W ( ⋅ , ⋅ ) ≔ ω ( r0 , c0 , ⋅ , ⋅ ) , and display it for two values of ( r0 , c0 ) .', 'The two configurations of Table 2 ( excitatory or inhibitory connections ) can be clearly seen in the figure .', 'In fact , most of the neural masses have negligible influence on ( r0 , c0 ) .', 'This occurs when c′ is neither close to c0 nor to −c0 , or when r′ is too far from r0 ( not shown on the figure , because the outer variance of the DOG g has a great value compared to the extent of Ω ) .', 'The properties of the connectivity values are analogous to double-opponent cells’ center-surround behavior .', 'In Fig 7 , we show the cortical input H ( r , c ) = h ( c − I ( r ) ) ) , where the purple/lime patterned image I is as in Fig B in S2 Appendix and corresponds to the cortical counterpart of one typical test pattern of [18] .', 'We also show the color sensations a∞ after convergence .', 'The input H is obtained by “lifting” the image I inside Ω × C o p p .', 'Altitudes of maximal values hence alternate between lime ( c ≃ −1 ) , purple ( c ≃ 1 ) , and white ( c ≃ 0 ) .', 'The shape of H heavily determines that of the final activities a∞ , since it has the role of cortical input .', 'It can be noticed that a∞ reaches values lower than 1/2 in the bottom part of the heatmap .', 'In order to emulate a color matching experiment , the first building block of our algorithm is to simulate the Color Neural Field dynamics in Eq ( 3 ) , as shown in Fig 8 , and accordingly to Algo 1 in S2 Appendix ( with dt = 1 ) .', 'Once again , the cortical image is given by Fig B in S2 Appendix .', 'Color matching can then be emulated by applying Algo 2 in S2 Appendix .', 'The parameters have to be regressed to the experimental data in order to reproduce the color shifts .', 'Fig 9 shows that our model is able to explain the shifts measured for the observers called ‘MC’ and ‘AZ’ in [18] , by using the regressed values qMC = ( 0 . 60 , 0 . 69 , 0 . 30 , 0 . 40 , 4 . 42 , 1 . 82 , 0 . 58 , 8 . 35 , 0 . 47 , 0 . 30 , 1 . 80 ) and qAZ = ( 0 . 60 , 0 . 69 , 0 . 31 , 0 . 40 , 4 . 42 , 1 . 81 , 0 . 60 , 8 . 35 , 0 . 47 , 0 . 30 , 1 . 80 ) , respectively .', 'The slight difference observed between the parameter values could partly account for subject differences .', 'We compare in Fig 10 ( left ) the color sensations a q t e s t , a q c o m p [ c m a t c h ] , and a q c o m p [ c q p r e d ] ( refer to S2 Appendix for the notations ) , taken at the point of interest r0 = ( 0 , 0 ) , and with q = qMC .', 'They are generated by the purple/lime test image ( as before ) , the comparison image filled with the experimental value cmatch , or with the predicted matching value c q p r e d , respectively .', 'After regression , c q p r e d becomes close to the experimental value cmatch , so that the two corresponding curves nearly coincide .', 'Among all curves { a q c o m p [ c ] } c , a q c o m p [ c m a t c h ] should be the nearest one to atest with respect to the L ∞ norm , and we illustrate this in Fig', '11 . The qualitative difference between the test and comparison curves mainly comes from the difference of complexity of their respective inputs: the test image has a complicated pattern , while the comparison images are simpler .', 'We confront our model to the nonlinear shifts observed by [17] in Fig', '12 . We find that it is able to reproduce the data after regression , with the fitting parameter value qnonlin = ( 0 . 42 , 0 . 71 , 0 . 63 , 1 . 16 , 4 . 43 , 1 . 72 , 0 . 56 , 6 . 35 , 0 . 47 , 0 . 30 , 1 . 80 ) .', 'This non-trivial result , alongside the results in Fig 9 , provides a strong justification to our framework .', 'In Fig 13 , instead of qnonlin , we used the value qAZ that explained the data on Fig 9 ( right ) observed by [18] .', 'Let us recall that in the experimental settings of [18] and Fig 9 , stest was fixed while the surround varied , unlike in the settings of [17] and Fig 12 , where conversely for some fixed test patterns stest was changed .', 'The predictions in Fig 13 are hence not close to the ground truth data , especially at the endpoints where the orange and blue curves are crossing .', 'Shifts are also smaller in magnitude .', 'However , it is remarkable that the model predicts a similar trend with respect to stest .', 'Crossings are also quite expected at stest of magnitude great enough , for which a reversing of the shift direction is plausible .', 'For too great magnitudes , the shifts are likely to become negligible .', 'We obtain similar results by using qMC .', 'Finally , as a further important validation , we show in Fig 14 that our framework is also efficient in explaining the vector field of shifts in the chromatic disk ( Fig 5 ) .', 'The convergence towards the opposite blue is made obvious , with the regressed parameter value qHSL = ( 0 . 73 , 0 . 15 , 0 . 52 , 0 . 68 , 4 . 41 , 1 . 84 , 0 . 51 , 8 . 35 , 0 . 47 , 0 . 30 , 1 . 80 ) .', 'As a remark , the sampling resolution of the 2D color space is quite low for computational reasons , so that the meaningfulness of parameter values has to be carefully considered .', 'We also obtain a smoother result than in the experimental data , as a SoftMin method is used to search for optimally matching colors ( see associated code ) .'], ['The present work has to be distinguished from those dealing with color constancy problems .', 'Color constancy is the ability of humans to guess the reflectance of objects despite very different illumination conditions [69] .', 'Reflectance is a property characterizing matter , defined as the proportion of luminous energy reflected by its surface .', 'In Eq ( 1 ) , it is linked to the spectral power distribution P of the incident illuminant ( daylight , lamp ) through the relation', 'C x ( λ ) ≔ P x ( λ ) R x ( λ ) ,', 'where reflectance and illuminant power are taken at the point of the scene which sends light to x in the retina .', 'The phenomenon of color constancy , first studied by E . Land who proposed the Retinex algorithm [70 , 71] , has since been the subject of much research .', 'Given cone inputs ( L , M , S ) , how does the brain retrieve the spectral reflectance R of an object ?', 'By contrast , we are not interested in how color is linked to reflectance , but rather in how identical stimulation of the cones can lead to different sensations because of spatial context .', 'In view of the previous discussion , to the naive question “What is the color of this object ? ” , we see that at least three types of answers are theoretically possible .', 'First , identify the colored material likely to produce such a visual effect , i . e . , guess its reflectance R; second , report the color [ C ] produced by cone stimulation; and third , describe the color sensation , as introduced in this work .', 'In practice , none of these tasks is trivial , even regarding color naming , since thought and language related to color involve complex cognition [72] .', 'Summarizing the previous ideas , the three punctual relationships', 'R 1 x 1 = R 2 x 2 [ C 1 x 1 ] = [ C 2 x 2 ] S 1 x 1 = S 2 x 2', 'are independent , because there are situations where any of them can hold or not .', 'The reader can be convinced by simple examples .', 'For instance , the same object seen in daylight or under shadow has different colors because [ C d a y l i g h t ] ≠ [ C s h a d o w ] , but we can easily recognize it and guess that 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 , [ C 1 ] = [ C 2 ] but R 1 ≠ R 2 .', 'Another example is given by color induction , as largely discussed here .', 'Because of the surrounding context , two colors [ C 1 ] = [ C 2 ] deriving from identical L , M , S cones stimulation can be perceived as different color sensations S 1 ≠ S 2 .', 'The reverse situation can occur , where different colors are perceived as similar with different contexts .', 'However , if the whole retinal space is taken into account , then the sets', '{ R x } x ∈ R ( 10 ) { [ C x ] } x ∈ R ( 11 ) { S x } x ∈ R ( 12 )', 'are linked to one another .', 'While papers on color constancy are devoted to the link between Eqs ( 10 ) and ( 11 ) [69 , 73] , ours focuses on the relationship between Eqs ( 11 ) and ( 12 ) .', 'Our framework supposes that { S x } x ∈ R depends on the sole knowledge of { [ C x ] } x ∈ R . Our model should not be considered as an image processing algorithm .', 'First , our work involves a psychophysically and physiologically relevant neural structure , while image processing models often lie at the conceptual level of an image .', 'In the literature , many algorithms are designed to make the image perceptually better ( contrast enhancement , histogram equalization , etc . ) and allow for image compression , in accordance with various perceptual criteria , an early work being [74] for example .', 'Some of them are computational models simulating perception , and inspired by neural mechanisms underlying vision , as in [75] , and more recently [76] .', 'Taking an image as input , their algorithms produce another image as output , which represents “the” perceived color image , in order to reproduce color induction effects .', 'In fact , the simple convolutional receptive-field model of [18] is already a basic version in this family of algorithms .', 'In the more sophisticated approach of [75] , the image evolves according to a Wilson-Cowan dynamic , which is akin to descending the gradient of some energy , leading to histogram equalization .', 'M . Bertalmío also proposes a method to reproduce the lightness matching data of [77] , in particular some assimilation or contrast effects ( in the usual global sense , not at the local scale ) .', 'His attempt to match psychophysical data based on neural-like dynamics can be considered similar to ours .', 'However , shifts are estimated through a direct algebraic computation of the values found at different locations of the disks after convergence .', 'This prevents any straightforward generalization of the method to spatially more complicated patterns , an impediment which we already mentioned in the Introduction for the model of [18] .', 'More importantly , a major conceptual problem in this image-based approach is that , it seems inappropriate to simulate perception by producing an output image of the same nature as the original one , that represents “the” perceived color image .', 'Indeed , the original and final images do look different: the final image hardly represents our perception of the original one , since it is also processed by the brain .', 'This ambiguity should be clearly discussed when dealing with the output image .', 'For instance , [74] proposes to map color images into an opponent perceptual space , where they are processed and compared through a perceptual metric , that has a role clearly different from the usual RGB color space of images .', 'It is the fate of any model to meet its own limitations .', 'Below , we depict non-exhaustively some questionable features or drawbacks encountered by ours .', 'Just as for neural field models modeling orientation vision , we can study bifurcations of the solutions of Eq ( 3 ) around stationary states [49 , 53 , 56 , 82] .', '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 [49] 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 .', 'Our work addresses the question of color-space interactions , by providing a color neural field model alongside a general framework to account for matching experiments .', 'We propose to consider color matching as a mathematical projection , in agreement with the principles of psychophysics , where subjective notions are assessed by means of objective procedures .', 'Our neural field unifies assimilation and contrast at the cortical level , and relies on the idea of color opponency .', 'The notion of color sensation that we introduce bridges the gap between these cortical and perceptual levels , and is a nonlinear percept involving a whole distribution of neurons .', 'This framework allows the study of psychophysical phenomena such as color induction , by taking advantage of a classical computational neuroscience tool .', 'To our knowledge , this is the first color neural field model consistent with psychophysical data and compatible with physiological findings .', 'The assumption that V1 is organized into a structure similar to color hypercolumns has still to be experimentally proved though .', 'We believe that the prop
['Introduction', 'Materials and methods', 'Results', 'Discussion']
['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 .']
['The color perception produced by an image heavily depends on the spatial distribution of its colors .', 'From this “color in context” phenomenon , extensively studied in psychophysics for decades , has arisen the question in neuroscience of how color and space interact in the brain .', 'Visual signals are indeed processed in such a way that neighboring pixels make the perception at some point different from its real color , inducing a color shift .', 'In this work , we propose to emulate perception in context by modeling the activity of color sensitive neurons with a neural field .', 'Our framework unifies two antagonistic effects , assimilation and contrast , which have been suggested to occur simultaneously but at different scales .', 'We use the notion of color opponency inspired by the work of Hering , so as to express these effects as a combination of attraction and repulsion in physical and color spaces .', 'We introduce the concept of “color sensation” , and show how to rigorously link the neural field model to perceptual shifts , by considering color matching as a mathematical projection on color sensations .', 'The results show that our model is able to reproduce some nontrivial behaviors of the color shifts observed in experiments .']
['medicine and health sciences', 'ocular anatomy', 'brain', 'social sciences', 'light', 'neuroscience', 'electromagnetic radiation', 'perception', 'cognitive psychology', 'luminance', 'visible light', 'vision', 'animal cells', 'visual cortex', 'physics', 'psychology', 'cellular neuroscience', 'retina', 'cell biology', 'anatomy', 'color vision', 'neurons', 'psychophysics', 'biology and life sciences', 'cellular types', 'sensory perception', 'physical sciences', 'ocular system', 'cognitive science']