statistics, vision, and the analysis of artistic...

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Opinion

Statistics, vision, and the analysisof artistic styleDaniel J. Graham,1 James M. Hughes,2 Helmut Leder1

and Daniel N. Rockmore2,3∗

In the field of literature, there is an established set of techniques that havebeen successfully leveraged in the analysis of literary style, most often to answerquestions of authenticity and attribution. With the digitization of huge troves of artimages come significant opportunities for the development of statistical techniquesfor the analysis of artistic style. In this article, we suggest that the progress madeand statistical techniques developed in understanding visual processing as itrelates to natural scenes can serve as a useful model and inspiration for visualstylometric analysis. © 2011 John Wiley & Sons, Inc. WIREs Comp Stat 2011 0 000–000

Keywords: stylometry; sparse coding; classification; logistic regression; imageprocessing

INTRODUCTION•

In 2011, a painting called ‘Salvator Mundi,’purportedly by Leonardo Da Vinci, surfaced under

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mysterious circumstances in New York. A paintingmatching its description was known to be part ofLeonardo’s œuvre, though at least 20 copies ofthe original work by Leonardo’s students and otherimitators were also produced, some of which were inthe past claimed by their owners to be the original DaVinci. Without a secure provenance—or the record ofthe chain of ownership of works from their creationuntil the present—the ‘determination’ of a work’sauthenticity is more akin to trying a court casethan administering a paternity test. While there isindeed a good amount of physical evidence (materialsanalysis of the pigments, canvas, and frame), atleast as much weight comes from the opinions ofexperts, or ‘connoisseurs,’ who, assuming that thephysical evidence agrees with their determination, acteffectively as judge, jury, and even counsel, formingan opinion that is shaped by a lifetime of looking at

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and thinking about the works of the artist in questionas well as the historical context in which these works

∗Correspondence to: [email protected] of Psychology, University• of Vienna2Department of Computer Science, Dartmouth College3Department of Mathematics, Dartmouth College

DOI: 10.1002/wics.197

were created.1 Ultimately, they try to answer questionsof the form, ‘Is this work in the style of the artist?’ or,‘Is the style of this work to be expected of the artistat this time in his or her career?’ The stature of theconnoisseur and the strength of his or her argumentsmake not only a significant historical impact, but afinancial one as well.

These questions of style comparison are at leastsuperficially ones that can easily be framed as ques-tions of statistics. Given the known work of the artistand the period, how likely is it that the artist wouldcreate this kind of work at that specific time? Atthis stage, of course, nothing more has been donethan to effectively translate the colloquial to the(vaguely) technical. The construction of a rigorousanalysis requires objective measurements and a meansof comparison performed in the service of articulatingand understanding visual style. This is the goal of thenascent field of visual stylometry. With the increasingavailability of large collections of high-resolution dig-ital images of works of art, as well as new advancesin image processing and machine learning and theunderstanding of the visual process, it is an area ofresearch poised to make great progress.

The general thesis put forward here is that ulti-mately, style is a perceived phenomenon, and shouldbe treated as such. In particular, our understandingof the human visual system and models of the wayin which we ‘see’ natural scenes can serve as a usefulmeans for visual stylometric analysis. In particular, the

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Opinion wires.wiley.com/compstats

two primary components of stylometry, selection andextraction of features and classification (or grouping)according to style, can benefit from this perspective.We show that going beyond mere metaphor and intro-ducing statistical ideas from the understanding ofhuman perception of natural scenes, while incorpo-rating high-level perceptual information into modelsof style, proves effective in meaningfully organizing artimages according to their style and in understandingsome aspects of the perceptual basis of style.

VISUAL STYLOMETRY

The origins of visual stylometry can be traced back tothe work of Giovanni Morelli (1816–1891), an Italianstatesman who took it upon himself to both protectand rescue the reputations of his artist countrymenwhom he felt were suffering from misattributions.Morelli was trained as both a paleontologist and amedical doctor and brought his scientific outlook andvisual classification skills to bear on this difficult prob-lem. The solution that he arrived at, called ‘scientificconnoisseurship,’ was to compare specific details suchas ears or hands in an unknown work to collectionof exemplars (which he called a ‘schedule’), effec-tively asking whether the same kinds of details in theunknown work were consistent with the known exem-plars. The choice of a less prominent feature like ahand or ear was a purposeful one, for Morelli believedthat in these details the artist would be less driven bymarket and societal concerns and pressures.2–4

What Morelli did by eye and brain, researchersare now attempting to accomplish via statistical andimage processing techniques. The problem remains adifficult one. Analogous efforts in the study of litera-ture, where the notion of ‘stylometry’ was born, havebeen successful, and today there is a host of techniquesthat are used to answer questions of genre, authorship,and the dating of works—all of which are questionsthat we might hope to address in a quantifiable wayin visual art as well.5 Indeed, the following tasks areones for which a combination of statistical and imageprocessing tools might be able to provide a noveland compelling form of evidence that complementsevidence acquired using more traditional techniquessuch as chemical and materials analysis, historicalrecords, and, of course, human connoisseurship:

• Authentication: Given a set of ‘known’ worksby a given artist, the task is to determine if anunknown work is or is not similar to the knownworks.

• Attribution: Given a work that could be by morethan one artist, the task is to develop a model

of each artist’s style to see which model bestexplains the work in question.

• Cultural evolution: Many questions in this areacan be approached with image processing such asthe influence of a given artist on contemporaries,students, and later artists, the influence of oneartistic movement on a later movement, or thestylistic evolution of a particular type of work.

• Technical art history: Statistical research andsimulation has produced important insightsinto artists’ methods for viewing, lighting, andcapturing scenes on canvas.

• Conservation: Artwork from all eras—includingthe 20th century—shows degradation over timebecause of color fading, faulty previous attemptsat conservation, air pollution, and other factors.Stylometric investigations can reveal importantinformation used to conserve artworks andrestore them to their original condition.

To date, progress in visual stylometry has beenachieved in something of an ad hoc manner. Inliterature, or more generally, writing, we have theadvantage of at least being able to identify a basicatom of relevance: the word. In visual art, we are lessfortunate in that finding the basic elements of style isalready a significant challenge. One might expect thatstandard image processing techniques such as SIFT6

or color histogram statistics7 would prove useful inthe analysis of style in visual art, and indeed, many ofthese techniques have shown some success in sortingartworks with respect to general stylistic categories(e.g., broad stylistic groupings such as Impressionism,or sorting the works of famous artists by author-ship). Nevertheless, many of these techniques oftenfail when confronted with the subtleties inherent instylistic categorization.

For example, Gunsel and colleagues8 work atthe level of pixels and define six features derivedfrom the pixel intensities in images of works of art.Using an •SVM classifier, the authors are able to sep- AQ3arate Cubist, Classicist, and Impressionist works withgood accuracy. However, the addition of works in theExpressionist and Surrealist style causes performanceto drop dramatically. While Cubist, Classicist, andImpressionist works may each be relatively distinct interms of these low level features, it is clear that thisapproach does not scale up to finer grained distinc-tions. Other work9 confirms the potential utility ofthese low level statistics for genre classification.

Some efforts attempt to aggregate pixel informa-tion at the level of the brushstroke. For example, theproblem of brushstroke identification and extraction

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WIREs Computational Statistics Statistics, vision, and the analysis of artistic style

can be approached via filtering and subsequent model-ing of lines, for example, using splines.10,11 Sablatnigand colleagues12 have developed sophisticated line‘skeleton’ models for extracting brushstrokes frompaintings. These kinds of models can also be usedto extract strokes with respect to the order in whichthey were applied to the canvas. Subsequent pro-cessing and comparison can be accomplished by avariety of means including hidden Markov trees.13–15

Unfortunately, it is often unclear how much the sty-lometric algorithms are measuring differences in thetechnical aspects of a painting (i.e., how the paintis applied to the canvas), and how much they aremeasuring differences in the visual ‘space’ representedin the painting (i.e., the depicted objects, scenes, andforms).

Several studies have examined artwork for reg-ularities in the spatial statistical properties of thecontours in art images. Perhaps the most well-knownearly example of work in the field of visual stylometryis that of Taylor,16,17 who showed that the fractaldimension of works by Jackson Pollock, obtainedusing box-counting statistics, follows a very particu-lar trend over the span of the artist’s career. This workhas not been without its critics.18 Keren19 suggesteda model to describe local stylistic characteristics thatuses the discrete cosine transform (DCT), along witha naıve Bayes classifier. The author’s suggestion ofa paradigm of ‘recognition by type,’ as opposed torecognition based on content, is a critical distinctionin the application of image processing methods to theevaluation of artistic style, particularly in the contextof image retrieval. Furthermore, Keren suggests thatthe obtained results are in accordance with humanperception of style, and that his approach could beused to synthesize images of a particular style.19

In the work of Lyu et al.,20 the authors appliedspatial statistical techniques to the problem of distin-guishing a set of authentic drawings by Pieter Bruegelthe Elder from a set of imitation drawings usingquadrature mirror filters to decompose the drawings,along with a hierarchical set of features derived fromthe filter coefficients, to demonstrate that the authenticdrawings grouped tightly together in stylistic space.Work by Hughes et al.21 demonstrated the techniqueof the Empirical Mode Decomposition (also known asthe Hilbert–Huang transform)22 to the same dataset.The authors showed that a newly created frameworkfor a two-dimensional extension of EMD is capable ofdistinguishing between Bruegel and Bruegel-imitationdrawings, as well as between a set of drawings byRembrandt and his pupils. Other examples include theuse of a combination of a wavelet decomposition andartificial neural networks capable of distinguishing

between the works of Matisse and stylistic imitationsof the artist’s work.23

Other work focuses on nonspatially localizedfeatures or ‘bags of features,’ which are not orderedor arranged in any meaningful way. One example ofthis is the work of Berezhnoy et al.,24 who analyzed

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the paintings of Van Gogh in terms of their use ofcomplementary colors. They examined both the con-sistency of the artist’s use of complementary colors,as well as the usefulness of this measure for orga-nizing Van Gogh’s works by the period in whichthey were painted. Another example is the workof Zujovic et al.25 In this study, the authors used acollection of gray scale edge detection and Gaborwavelet pyramid sub-band statistics, along with colorhistogram statistics, to sort images into several stylis-tic categories. The idea of this study was to searchfor a solution that would be robust to a variety ofdigital image degradations and manipulations (e.g.,downsampling, color space modification, compres-sion, etc.). Thus, no attempt was made to normalizethe images or to acquire high quality, distortion freeimages. While the authors show that this approachcan be successful and may thus be of use in consumer-level applications involving variable-quality images, itremains unclear whether this approach can scale upto more than five categories or whether image qualitybiases themselves contributed to performance.

STATISTICAL MODELS OF VISIONAND STYLE FEATURES

The use of wavelet and other multiscale techniquesin stylometric analysis has been driven more by theirutility in edge and orientation detection than any con-nection to the organization of the visual cortex.26–28

On the other hand, our recent work has involvedthe development of new stylometric techniques thattake as a starting point models of visual cortex. Thesemodels attempt to explain the organization of visualcortex and the encoding of visual information as anefficient, if not optimal, means of matching and cod-ing the information contained in natural scenes. Forexample, the sparse coding model of Olshausen andField seeks to explain the response properties of simplecells in primary visual cortex in terms of the statis-tical structure of natural images.29 That is, the brainitself builds a model of the visual world that is anoptimal representation of the statistical structure innatural scenes, where optimality is defined in termsof coding efficiency. According to their model, visualinputs should be represented using activations of asfew model neurons as possible, such that the overallresponse characteristics of neurons is sparse, while at

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the same time preserving accuracy of neural represen-tation by minimizing the error between inputs andtheir reconstructions.29 Using a linear image model

I(x, y) =∑

i

αiφi(x, y) + ε, (1)

for some pixel location x, y in image patch I withcoefficients α and Gaussian noise term ε, the goal isto learn the functions φ, assuming a sparse prior on α

(e.g., a Cauchy distribution). The resultant functionsφ, which are treated as a basis for representing inputs,possess a structure very similar to the response prop-erties of cortical simple cells when trained on naturalimage inputs, in that they are spatially localized andhave orientation and spatial frequency selectivity.29,30

Figure 1a shows a set of sparse coding basis functionstrained on the set of natural images used in Ref 29.Furthermore, although the sparse coding model buildsa basis for the space R

n, where n is the number ofpixels in an input image patch, the basis functionsare merely linearly independent and not guaranteed tobe orthogonal. Such a representation results from theconstraint that the patch coefficient distributions besparse, that is, that only a few basis functions shouldhave significantly nonzero weight for representing anyparticular patch. This stands in contrast to an orthogo-nal basis for the inputs such as a Fourier basis or a PCAbasis created from the input patches, each of whichwould potentially utilize all basis functions in creatinga minimum-error reconstruction for each patch.

The sparse coding model was first used for sty-lometry to capture the structure of a set of authenticdrawings by Pieter Bruegel the Elder31 where thebasis functions were then used to distinguish authen-tic and imitation drawings based on how efficiently

(a) (b)

FIGURE 1 | Two set of basis functions trained using the sparsecoding model. The set (a) was trained on the natural image set used inthe original work of Olshausen and Field.29 The set (b) corresponds tothe art image (shown in Figure 2) that produced a set of basis functionsthat had minimal average histogram intersection using the distributionsof spatial frequency and orientation bandwidth with respect to the basisfunctions (a).

they represented the statistical structure in a set oftest images. A further study32 explored the extentto which various statistics derived from the learnedsparse coding basis functions were useful for classifi-cation according to style, with some promising initialresults. In particular, the authors trained a set of basisfunctions to represent each work in a set of art imagesby several artists. The learned functions were thenexamined for features such as spatial frequency andorientation selectivity and spatial frequency and ori-entation bandwidth that were capable of grouping theimages by artist.32 Additionally, statistics derived fromsparse coding basis functions have proven useful fororganizing artwork according to its painterliness.33

The efficiency criterion of the sparse codingmodel suggests that comparing sets of learnedfilters—and the statistical characteristics of coefficientdistributions for inputs—may highlight the underlyingstatistical differences in potentially different sets ofinputs. Furthermore, because the sparse coding modelcaptures higher order statistical characteristics asopposed to two-point correlations between spatialfrequencies and orientations, a learned sparse codingmodel is a window into the important structurepresent in images, and in particular a window thatallows insight into structure at a scale that is likely tobe relevant for human perception.

Previous work has shown that artwork pos-sesses a statistical structure similar to that of naturalscene images, but with important distinctions. Workby Graham and Field34 and Redies et al.35 estab-lished that natural scenes and visual art share similarFourier spatial frequency amplitude spectra, which arefound to be roughly scale invariant (amplitude scalesapproximately as 1/f k where k ≈ 1, for frequency f ).Spatial frequency and orientation information can beindicative of some of the higher order statistical prop-erties that differentiate art from natural images andhighlight the effectiveness of the sparse coding modelat articulating these differences, albeit in a numericalfashion. In order to see this, we computed the fol-lowing features on the set of basis functions shown inFigure 1a and on a set of 308 high-resolution imagesof works of art (the same set used in Ref 32):

1. Distribution of spatial frequency bandwidths:given the two-dimensional Fourier transformof each basis function, compute the band-width in octaves (measured by full width athalf-maximum) of the function, averaged acrossall orientations, centered around its peak spa-tial frequency ω∗. This quantity measures howselective the basis functions are for their pre-ferred spatial frequencies.

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2. Distribution of orientation bandwidths: giventhe two-dimensional Fourier transform of eachbasis function, compute the bandwidth inoctaves (measured by full width at half-maximum) of the function, averaged across allspatial frequencies, centered around its peakorientation θ∗. This quantity measures howselective the basis functions are for their pre-ferred orientations.

Here, ω∗ and θ∗ are defined as follows: given thetwo-dimensional Fourier transform F(ω, θ) of a basisfunction (viewed as a function of frequency ω andangle θ),

ω∗ = arg maxω

1|�|

∑θ∈�

|F(ω, θ)|,

θ∗ = arg maxθ

1|�|

∑ω∈�

|F(ω, θ)|.

We can compare the distributions of these quantitiesbetween images using, for example, the symmetrizedhistogram intersection statistic,36

HI(H1, H2) = 12

[∑b min(H1[b], H2[b])∑

b H1[b]

+∑

b min(H1[b], H2[b])∑b H2[b]

]

for histograms H1, H2, and bins b. Note that his-togram intersection is a similarity metric, rather thana dissimilarity metric, so that large values of HI implygreater similarity between images. For the experi-ments presented here, we considered both the spatialfrequency bandwidth and orientation bandwidth his-togram intersections.

Figure 1b shows a set of basis functions trainedon an art image from our dataset; in particular, thesebasis functions correspond to the art image that pro-duced the set of basis functions with smallest averagehistogram intersection for spatial frequency and ori-entation information. This set of basis functions wastrained using the image in Figure 2a. This image, whiledepicting a ‘natural scene,’ that is, a village, is itselfnot particularly realistic. On the other hand, the imagein Figure 2b, which is the image corresponding to theset of basis functions that had the largest averagehistogram intersection with the natural image bases,depicts a natural scene not unlike the ones used to trainthe natural image bases, albeit in a painterly style.

If on the other hand we consider only spatialfrequency information, the result is somewhat differ-ent. The closest image, shown in Figure 2d, is indeed

(a)

(b)

(c) (d)

FIGURE 2 | Four art images from the database used in our analysis.Image (a) corresponds to the image that had the weakest similarity tothe natural images, using the histogram intersection statistic on thespatial frequency and orientation bandwidth distributions. Image (b) isthe art image that was maximally similar to the natural images. Notethat it depicts a common natural scene (albeit in a painterly manner).Image (c) is the art image that produced the basis functions mostdifferent from the natural image basis functions, according to thehistogram images between the spatial frequency bandwidthdistributions only. Image (d) is a detail of the art image that wasmaximally similar under the same analysis (i.e., considering only spatialfrequency information). Images (a)–(c) are courtesy of the Herbert F.Johnson Museum of Art, Cornell University, and image (d) is courtesy ofthe Metropolitan Museum of Art, New York.

a ‘natural scene’ (of a village in a drawing by animitator of Bruegel), thought it is quite different fromthe painterly one in Figure 2b. The furthest-away artimage is an abstract rendering with little in com-mon with natural scenes, and is shown in Figure 2c.Figure 3 shows the histograms of spatial frequencybandwidths for the basis functions corresponding tothe images in Figure 2c and d, along with the spa-tial frequency bandwidth histogram for the naturalimage basis. Clearly, the bandwidth distributions forthe natural image basis and the basis for the artimage in Figure 2d and quite different from the basistrained on the image shown in Figure 2c, suggestingthat this approach is capable of meaningfully relatingthe statistical structure of natural scenes and art.

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0

1 2 3

Spatial frequency bandwidth

4 5

10

20

Cou

nt

30

40

50

ImageNatural image

Closest art image

Furthest art image

Histogram of spatial frequency bandwidths of learned basis functions

FIGURE 3 | Histogram of basis function spatial frequency bandwidths for the natural image basis (blue) and the images shown in Figure 2c and d(orange and green, respectively).

PERCEPTUALLY DRIVEN FEATURESELECTION

Historically, psychologists and art historians havetaken an interest in the perception of style in visualart, at least from a theoretical perspective.37–41 Morerecent work by Leder and colleagues has elucidateda number of basic properties of style perception. Intheir work on empirical aesthetics, style as a prop-erty of art has been treated as a candidate for aspecial mode of visual processing. The model ofaesthetic experiences by Leder and colleagues,42 isbuilt around the central assumption that process-ing of style—the manner of depiction as opposedto content of depictions—distinguishes relevant waysof approaching artworks and yields style-related pro-cessing as a distinctive, art-typical experience.

Beyond the process of generating features, it canbe useful to take advantage of perceptual informationfor classifying art images according to their style.

Classification schemes can take advantage ofperceptual information in several ways, but perhapsthe most obvious is to consider perceptual similar-ities between works, derived using psychophysicalexperiments in which human subjects rate the simi-larity between two works on some predefined scale.Although such an approach is certainly laborious, ithas the potential to yield a significant amount of fruit-ful data. Once these similarities are obtained, they canbe used to train classification models in a supervisedfashion or to ‘discover’ latent stylistic dimensions inan unsupervised learning model. As an example ofthe former, weighting statistical features based onperceptual information leads to automatic stylisticdistinctions that agree well with human perceptualjudgments.32 In essence, stylistic perception is not

random, and taking advantage of perceptual informa-tion can guide the use of image features in explainingvariations in artistic style perception.

Another possible approach would be to use per-ceptual data to learn feature weights in the context oflogistic regression.43 Such a model is a natural can-didate as it transforms an inner product that reflectssimilarity between two images into a scale (i.e., a prob-ability in [0, 1]) that better reflects the intuitive notionof similarity. For example, if pairs of art images werelabeled ‘similar’ and ‘dissimilar,’ these binary labelswould serve as natural class labels in a logistic regres-sion setting. For example, given two images I1, I2, avalue L that equals 1 if the two images are similar and0 if they are not, and some set of weights β, we let

P(L|I1, I2) = σ

(β0 +

M∑j=1

βjκj(φI1j , φI2

j ))L

×[1 − σ

(β0 +

M∑j=1

βjκj(φI1j , φI2

j ))]1−L

.

In this case, κj is some similarity function between fea-tures φ

I1j and φ

I2j , which could be scalar, vector-valued,

etc., and σ is the logistic sigmoid function. More com-plex models could be utilized to take advantage ofricher perceptual information, such as the similarityratings mentioned above, rather than binary inputs.

Another possible avenue for taking advantageof perceptual information in classifying art imagesaccording to style is to utilize this information for fea-ture elimination, for example, as in recursive featureelimination,44 or by performing a biased dimension-ality reduction that takes advantage of perceptualinformation. Some previous work has also been done

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to relate statistical features to the axes of embed-dings of art images obtained using multidimensionalscaling.43,45

CONCLUSIONAlthough a great deal of progress has been made inunderstanding and organizing art images accordingto their style, we argue that a significant majorityof the research conducted in this field does not takeadvantage of the fundamental connection betweenvisual perception and artistic style—and the waysin which style should be measured. Because style isultimately perceived, it is important to concentrateon developing features and classification methods fororganizing and understanding art images that utilizethe information vision science and visual psychol-ogy can offer about human perception. Moreover,because our visual system is well adapted to the sta-tistical structure of the natural world, we believe thatthe key to understanding the salient characteristicsof artistic style is to quantify the intrinsic differences

between art and natural images. Understanding thisfundamental distinction will also allow us to refine ourknowledge of human perception. In a sense, the way inwhich natural scene statistics are manipulated to cre-ate visual art will provide insight into the ‘allowabledeformations’ of visual content that are important forhuman perception. Because we are dealing with digitalrepresentations of art, image processing techniqueswill remain critical in enabling us to quantify the stylepresent in works of art. Nevertheless, these techniquesshould be shaped and utilized in accordance with ourunderstanding of the fundamental processes in humanvision. Such an approach will also allow us to extendthe scientific reach of stylometric investigations from‘canned’ problems such as authentication or attribu-tion of works whose attribution is already known tosubtler and more complex descriptions of art. Withthe growing availability of digital representations ofmany types of cultural heritage, the ability to mean-ingfully organize these objects is of ever-increasingimportance.

ACKNOWLEDGMENTS

D.J.G., J.M.H., and D.N.R. gratefully acknowledge the support of the Kress Foundation for this work, underthe grant ‘The Workshop Practices of Botticelli Before Rome: Collaboration with Filippino Lippi in ‘The StoryOf Esther.’

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FURTHER READINGAtick JJ, Redlich AN. What does the• retina know about natural scenes?

AQ11

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Graham DJ, Chandler DM, Field DJ. Can the theory of ‘‘whitening’’ explain the center-surround properties of retinalganglion cell receptive fields? Vision Res 2006, 46:2901–2913.

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