A lower bound on the number of mechanisms for discriminating fourth and higher order spatial correlations

The human visual system must employ mechanisms to minimize informational redundancy whilst maximizing dynamic range and maintaining that which is behaviorally relevant [1,2]. Previous research has concentrated on two-point correlation properties, as captured by spatial frequency and orientation tuning. There has been less research into higher-order correlations although they may inform us about cortical functioning [3]. Isotrigon textures can be used to probe the sensitivity of the human visual system. The obvious structure in isotrigons is exclusively due to 4 and higher-order spatial correlations [4]. Thus, in order to discriminate isotrigons from noise, it is necessary to identify higher-order structure. Although artificially generated, the same structural features that give isotrigons salience also create salience in natural images [2]. Factor analysis can be used to infer the number of underlying independent neurological mechanisms which govern isotrigon discrimination. In this study, mean performance functions were calculated for two subjects using ten new isotrigons (VnL2) (Figure 1A). Two forms of factor analysis identified 3 principal factors (Figure 1B) [5]. Previous studies support that the number of mechanisms is less than 10 [6], and more likely 2-4 [7,8]. Such mechanisms may represent some combination of recursive or rectifying processes. Simple models of cortical processing, based on recursion, can discriminate isotrigons [9]. The formation of recursively applied products is physiologically plausible and can occur via dendritic back-propagation or dendritic spiking [10]. Published: 18 December 2015


Introduction
Natural images contain large amounts of structural information characterised by higher-order spatial correlations (Franz & Schölkopf, 2005). Neurons have limited information capacity and energy budgets, so this volume of information presents a considerable challenge. The human visual system must employ mechanisms to filter out redundant or non-salient information, but retain that which is behaviourally relevant (Barlow, 2001;Barlow, 1963).
The orientation and the spatial-frequency tuning of mammalian simple cells is suited to a ''sparse'' coding system which minimizes informational redundancy (Field, 1987). The antagonistic centresurround receptive field architecture exploits the degree of second-order correlation in images to remove redundant information. The information removed in such ''predictive encoding'' processes is related to second-order correlations (Srinivasan, Laughlin, & Dubs, 1982). In this manner, the dynamic range of a neuron can be devoted to encoding a small range of non-redundant contrasts, and the visual system can encode spatial detail in a manner that minimizes the effects of intrinsic noise. Interestingly, the visual system also appears to filter out the predictable higher-order structural information from of natural scenes, whilst retaining and transmitting that which is not predictable (and therefore behaviourally relevant) (Tkacik et al., 2010). End-stopping may be particularly important in filtering out redundant higher-order structural information (Zetzsche & Nuding, 2005).
Taken together, these studies highlight the importance of investigating higher-order correlations in images and how they are processed. Such investigations may also tell us a great deal about cortical functioning (Purpura, Victor, & Katz, 1994;Victor, 1995;Victor & Conte, 1996). Previous research into the responses of single striate cortical neurons has concentrated on two-point correlation properties, as captured by spatial frequency and orientation tuning. There has been less research into third-and higher-order correlations, although the visual system is sensitive these relationships in natural images (Tkacik et al., 2010).
Using artificially generated textures we can probe the sensitivities and limitations of the human visual system to higher-order correlations. These artificial images can have very well controlled statistical properties, and an important class of such test images are the original set of isotrigon textures (e.g. Fig. 1). The word ''isotrigon'' derives from the fact that the average first to third order correlation functions of these textures is zero, as is the case for uniform noise. The obvious structure in these textures is therefore exclusively due to fourth and higher-order spatial correlations (Gilbert, 1980;Julesz, Gilbert, & Victor, 1978;Maddess et al., 2004;Maddess et al., 2007;Victor, 1994). Thus, in order to discriminate a particular isotrigon texture from noise, it is necessary to identify http://dx.doi.org/10.1016/j.visres.2014.12.023 0042-6989/Ó 2015 Elsevier Ltd. All rights reserved.
its complex, higher-order structure; isotrigon textures cannot be discriminated based on luminance or other lower-order properties. Our discrimination ability seems to be pre-attentive and changes little for long presentation times (Taylor, Maddess, & Nagai, 2008).
Humans and primates have been demonstrated to be sensitive to the higher order correlations in isotrigon textures in a number of psychophysical (Maddess & Nagai, 2001;Maddess et al., 2007;, VEP (Victor & Conte, 1991), fMRI (Beason-Held et al., 2000, Beason-Held et al., 1998a, PET (Beason-Held et al., 1998b) studies, and also single cell recordings (Purpura, Victor, & Katz, 1994). Although they are artificially generated, the same structural features that give isotrigon textures visual salience also create visual salience in natural images (Tkacik et al., 2010). Therefore, they are an ideal tool for evaluating human texture perception. This paper investigates a lower bound on how many mechanisms might operate to discriminate correlations at fourth and higher orders.
Most of the isotrigon textures used to date have been created using a recursion procedure in which a combinatorial rule operates on n pixels selected by a gliding template, or glider, determines the value of an output pixel (Victor & Conte, 1991). For details see the caption of Fig. 1. Here we introduce the VnL2 isotrigon textures where n can be 4, 5 or 6 and the gliders are 4 Â 4 pixels (Fig. 2), the L2 refers to the number of grey levels. This is the first paper to investigate these particular isotrigon textures.
Humans use a number of neurophysiological mechanisms to capture visually salient higher order structure. Evidence has been presented that the number of independent mechanisms is less than 10 (Taylor, Maddess, & Nagai, 2008), and is more likely 3 to 4 (Maddess & Nagai, 2001;Maddess et al., 2007). One statistically Even Odd Binary Isotrigon Textures Fig. 1. Examples of the first set of binary isotrigon textures to be published (middle and left column). The textures are created by a recursive rule-based process where the rule is operated in concert with glider patterns (Victor & Conte, 1991). To begin the process a matrix is set randomly dark (À1) or bright (1) with equal probability, each entry representing a pixel. A 3 Â 3 pixel glider is then moved over the matrix. The gliders are shown in the left column, where they are greatly magnified, their pixels actually match those of the textures. How the glider affects the underlying pixels is determined by the shape of the glider's active regions. Glider pixels marked a, b, c are the input pixels, and f is the position of the output pixel. Two rules termed Even and Odd were used to create all the textures. The Even and Odd rules are f = abc and f = Àabc. The glider is moved in steps of one pixel across the matrix modifying the entries using the selected rule until the texture is complete. Since former outputs become inputs the process is recursive. In fact only the first few rows and columns need to be seeded with random À1 and 1 values and those will determine the outcome, hence these methods are said to be deterministic. well principled method is to use factor analysis of human performance functions from many individuals, or many repeats for single individuals, to infer the number and form of underlying independent mechanisms (Maddess & Kulikowski, 1999;Rosli, Bedford, & Maddess, 2009;Sekuler, Wilson, & Owsley, 1984;Simpson & McFadden, 2005). For example, the technique has been used to evaluate the mechanisms which underpin contrast sensitivity (Peterzell & Teller, 1996;Peterzell, Werner, & Kaplan, 1993). In this study, we apply that approach to performance functions for discriminating the new VnL2 textures from random textures. Additionally, given that many repeats were involved, we also examined learning effects.

Materials and methods
Two healthy male subjects (JWS and ALB) were recruited to the study (age 34 and 22). They had corrected to normal vision. Four other normal subjects aged 22, 33, 40, and 56 (one female) also participated to make a third, combined data set. In this data set, only early data from JWS and ALB was used, to equate the level of experience of the six participants. Informed consent was obtained under ANU Human Experimentation Ethics Committee protocol 238/04. All research adhered to the tenets of the Declaration of Helsinki.
The VnL2 textures ( Fig. 2) were created by the procedure discussed above (Maddess & Nagai, 2001;Victor & Conte, 1991). Several hundred VnL2 texture types were created and screened for being isotrigon by examining their mean second-and third-order correlation functions (Maddess et al., 2004;Maddess et al., 2007). All texture production, stimulus presentation, and analysis was done using MatLab (The Mathworks, Natick, MA).
Ten VnL2 textures were selected because they exhibit a good variety of image structure, from quite salient to almost random in appearance. The set presumably modulates the human texture mechanisms well. Importantly, no texture was so salient that the performance came close to saturating at 100%: this would remove the variability that factor analysis is based upon.
The achromatic (colour temperature 6500 K) texture patterns were displayed at mean luminance of 395 cd/m 2 on a Phillips MGD 403 ultra-high brightness monitor at a pixel resolution of 512 Â 420 square pixels, and a refresh rate of 101 Hz. The ambient illumination was that provided by the monitor in an otherwise darkened room. Subjects viewed the patterns binocularly from a fixed distance of 46 cm, and head position of the subjects was maintained using a chin rest. The angular size of the stimuli is given in Fig. 3 and they were 11.3 and 22.6 degrees square.
The experiment was designed as a series of classification tasks in which subjects attempted to discriminate between VnL2 textures and random binary noise. Two sizes of textures were employed (16 Â 16 and 32 Â 32 pixels), providing two difficulty levels. Subjects practiced using the larger size before being tested with the smaller stimuli (Fig. 3). The sizes were similar to previous studies (Maddess & Nagai, 2001;Victor & Conte, 1991), which had shown that the relevant range of salient correlations is less than ±6 pixels.
The stimuli were presented for 297 ms within a smooth temporal window. During each texture presentation, the change in contrast of the textures incorporated a sigmoidal onset and offset as determined by a Blackman function and details of the exact function are given elsewhere (Maddess & Nagai, 2001). This was designed to mitigate the nonlinear effects associated with abrupt changes in contrast. The textures were at full contrast for 204 ms which constitutes pre-attentive viewing (Julesz et al., 1973, Maddess & Nagai, 2001. Subjects used a mouse button to indicate whether the texture was random or non-random. To facilitate learning, incorrect choices were indicated by a tone (Maddess & Nagai, 2001).
Test sessions proceeded as follows. One of the 10 VnL2 texture types was selected at random for testing. Before testing the subject inspected a sheet depicting 15 32 Â 32 pixel examples of the texture type to be tested. For example, a subject was shown printed examples of texture 048. The subject was then presented with a sequence of 20 images: either an example of texture 048 or a random binary noise pattern. The subject was asked to determine whether the image they were presented was random or textured. First twenty 32 Â 32 pixel examples were tested, and then twenty 16 Â 16 pixel examples, the larger textures providing some training for the smaller stimuli. We will refer of the testing of 40 examples as a test ''block''. After the last of the images was presented, a new texture was selected at random from the remaining 9 VnL2s. Once again, printed examples of the texture were provided before the trial commenced. This pattern was repeated until blocks for all 10 texture types had been tested. In later trials of subjects ABL and JWS the 32 Â 32 examples were replaced by 16 Â 16 meaning that 40 examples were shown in each block. For each texture size the mean performance, i.e. probability correct, was calculated for each texture type providing a performance function comprising ten points, one function for each repeat.
Subjects JWS and ABL each completed 16 blocks as described above (6400 individual presentations). The test blocks were not carried out in one sitting, but were spread out over a period of days. Data from four other subjects was combined (minimum of 40 presentations each). The data from the first few presentations to JWS and ABL were included to provide two further data sets . Two sizes of textures were employed (32 Â 32 and 16 Â 16 pixels), which thereby provided two difficulty levels. In the first phase of testing, subjects made 20 discriminations between 32 Â 32 pixel textures. Then, in the second phase, 20 discriminations were made between 16 Â 16 pixel textures. The same isotrigon texture was compared with random textures in each of these blocks of 40 discriminations. The order of these texture test blocks was randomised. In some subjects this was repeated several times, and in later blocks only the smaller textures were shown in further repeats. Each texture pixel was actually 8 display pixels square. The textures were displayed on a neutral grey background at the middle luminance of the texture checks. from relatively inexperienced subjects for 16 Â 16 and 32 Â 32 textures. These data sets are labelled All16 and All32.
A further data set from a previous study was added (Maddess & Nagai, 2001). The spatial and temporal presentation aspects were identical except that the two subjects saw blocks of 12 texture/random alternates for each of 32 Â 32, 16 Â 16 and 8 Â 8 pixels in block. The 18 V3L2 textures were the same as in Fig. 1 except that Even and Odd versions of texture called Triangle were added. The two subjects each repeated 5 sets of blocks, and were very experienced viewers (TM and YN).
We applied factor analysis to estimate how many neural mechanisms might contribute to the performance functions, and their possible weightings for each texture type. This exploratory statistical method has been used by ourselves (Maddess & Kulikowski, 1999;Rosli, Bedford, & Maddess, 2009) and others (Peterzell & Teller, 1996;Peterzell, Werner, & Kaplan, 1993;Sekuler, Wilson, & Owsley, 1984;Simpson & McFadden, 2005) to address similar questions. Factor analysis exploits covariance across performance functions to provide information about any underlying mechanisms. Simply stated, factor models can present simplified representations of the many data variables in terms of a small number of unobservable variables: in the present case putative texture discrimination mechanisms that determine the form of the measured performance functions. The basic equation of a Factor analysis is: where X is the data matrix, in the present case the rows were observations (performance functions) for the 10 textures (columns), one row for each repeat or subject depending on the data set. S is the matrix of factor scores, i.e. the estimated responses of the mechanisms to each texture over the repeats. L is the matrix of factor loadings: the loadings are regression weights that are used to combine the factor scores into the performance variables (X), here representing the influences of the independent putative mechanisms (factors) upon the discrimination of each texture type. Importantly, if these loadings are similar for different subjects then the factors are more likely to be real. S has size N-repeats x k-factors (see below), and L is N-textures x k-factors. The loadings were computed from R, the correlation matrix of X by singular value decomposition (SVD), such that R = F Â D Â G providing: and the factor scores were given by, The columns of F (or G, they are equal in this case) are the eigenvectors of R and the elements of the diagonal matrix D are the eigenvalues. The eigenvalues provide the proportion of the variance in R provided by each eigenvector. Commonly the eigenvalues are sorted and plotted to form a so called scree plot. If some small number of factors determines the observed variables then a sharp step can occur in this scree plot the number of eigenvalues before the step providing an indication of the number (k) of factors to consider, the remaining small components being discarded as noise. Note that Eqs. (2) and (3) are computed for the k columns of F and D, which provide the k highest eigenvalues to provide a k-factor model, the selected eigenvectors being the estimated factors. We will refer to this SVD based method as a Principal Components (PC) factor analysis. The overall objective is to find the smallest set of factors that account for the original data well. Here this would provide a lower bound on the number of mechanisms for discriminating higher order structure.
One measure of the adequacy of these models is provided by the communalities (Reyment & Joreskog, 1996), which are akin to r 2values and indicate what proportion of each of the original vari-ables is accounted for by the k-factor model, hence the communalities for each of the points in the performance functions examined should be uniformly high if the reduced factor model accounts well for all the input variables (e.g. Fig. 7D,F). Before forming L (Eq. (2)) the loadings were rotated according the Varimax method (Reyment & Joreskog, 1996). Through the Varimax method each rotated factor is a combination of only the highly interdependent input variables. That is the factors tend to be loaded onto (i.e. reflect) the input variables.

Mean performance
The mean performance functions for all subjects are shown in Fig. 4. The upper two panels show the means for the first and last 10 repeats for JWS and ABL. For relatively naïve subjects the performance data is split according to texture size (All16 and All32). Although ABL and JWS differ in their performance on texture 042 all subjects agree quite well on all other textures in terms of the form of their mean performance functions. The performance of last 10 repeats of JWS was on average significantly higher than his first 10 (p = 0.04), but that of ABL was not.  The performance data (probability correct) is presented separately according to subjects and texture size. For All16 (16 Â 16 textures) and All32 (32 Â 32 textures) the error bars are SE for N = 6 subjects. The JWS and ABL the data use the first and last 10 repeats of 26 and 21 total for 16 Â 16 textures. The glider shapes have been added in the bottom panel to improve the link between the performance functions and Fig. 2. Fig. 5 shows scree plots which display the proportion of variance accounted for the 10 possible factors from each of the 4 data sets (JWS, ABL, All16 and All32). The proportions are plotted in order of their relative influence, from the largest to smallest. If the data being evaluated is governed by a small number of principal mechanisms, we may see a few large eigenvalues followed by a significant drop-off: the lesser factors exert less influence on our data and trail off exponentially into noise.

Factor analysis
There is a drop-off in the accounted proportion of variance after the first two factors for All16 (Fig. 5A). The last 10 visits of ABL also seem to short drop off after 2 factors. Fig. 5D shows the results for the last 10 repeats when the data were transformed to d' units. The results are very similar with a possible drop off at 3 factors for JWS. This may suggest that two, or perhaps three, independent neurophysiological mechanisms govern VnL2 texture discrimination. Better evidence for this idea comes from the communalities and the factor loadings. Fig. 6 shows an evaluation of the factor model validity for N factors using communalities. The communalities indicate the proportion of variance accounted for within the data for each texture. A robust model will provide a balanced account of the data for each texture, i.e. communalities of even size. Communalities obtained for the largest 5 factors are shown for the All16 data set, which is the most representative of typical performance among the available data sets. It is clear that by 3 to 4 factors a balanced account of the data is obtained. A three factor model is sufficient to describe the majority of the data points moderately well. Texture 042 was the least well described by the three factor model in Fig. 6. It was also the least well described for 2 of the other 3 data sets (not shown). Across the 4 sets the mean (±SD) communality for 042 was 0.59 ± 0.17. For the other textures taken together the three factor model provided an overall mean communality of 0.60 ± 0.09, the small SD indicating a rather even distribution of communalities. Including 042 the four data sets the 3-factor models accounted for 0.54 ± 0.19, 0.58 ± 0.18, 0.63 ± 0.14, and 0.66 ± 0.16 of the variance, for JWS, ABL, ALL16 and All32 respectively.   The communalities for five different factor models beginning with a model including only the largest single factor (nf = 1), to a model containing the five largest factors (nf = 5) for the All 16 data set. As the number of factors grows the profile of bars becomes flatter indicating that the models progressively account for the data in a more balanced way. The other three data sets were examined and showed many common features such as texture 042 being less well described until 4 or more factors are included.

Factor loadings
We now address the contributions of the top three factors to the performance of subjects on a given texture (Fig. 7). This can be done by examining the so-called factor loadings, which are like regression weights between the factors and performance for each texture. Loadings with large non-zero weights indicate the possible form of the putative texture discrimination mechanisms. To highlight the differences and similarities between the four data sets, we compared their raw correlation coefficients. Given that the functions describing the loadings for each factor can be a 10-point function of any shape, significant correlation between those functions strongly indicates that they are real.
In this analysis, factor loadings FL1 to FL3 of dataset 1 (JWS) were compared to those of datasets ABL, All16, and All32. A problem with Factor analysis is that the ordering of factors used here is based on the variance accounted for by each factor in each data set. Therefore a small amount of noise might cause the order of factors to be permuted. Furthermore, the sign of the factor loadings is arbitrary. So we used correlation analysis as a way of determining which factors were most similar between data sets. Thus in Fig. 7 we grouped the factor loadings from each data set that had the highest correlations. As expected this only permuted a few factors by one step. For Fig. 7A the best correlation was between JWS and All16 (r = 0.84, p = 0.002). For Fig. 7B the best two correlations were between JWS and ABL or All16 (r = 0.54 and 0.57; p = 0.11 and 0.09). For Fig. 7C the best correlation was again between JWS and All16 (r = 0.75, p = 0.01), and JWS and ABL were marginally correlated (r = 0.60, p = 0.06).

Other textures
Data from a previous study (Maddess & Nagai, 2001) was added (Methods). These included the V3L2 textures of Fig. 1 but the data for the Box Even texture had low variance so was dropped from the factor analysis. Fig. 8 shows the results for 16 Â 16 pixel data from the 17 textures. Fig. 8A shows a marked drop in the variance explained after 2 factors, and perhaps a second at 4 factors. The communalities for a three factor model were quite flat (Fig. 8B) and explained 0.85 ± 0.11 of the variance (mean ± SD). A four factor model accounted for 0.94 ± 0.06 of the variance for each texture. Results for 32 Â 32 pixel versions were similar.

Discussion
Subjects differed somewhat in their ability to detect image structure defined by 4th and higher order spatial correlations (Fig. 4). This study supports the possibility of about three mechanisms underlying this higher order texture discrimination. This is similar to results reported for other isotrigon textures using different methods (Maddess & Nagai, 2001;Maddess et al., 2007). Three factor models gave a relatively balanced set of communalities for the VnL2 textures (Fig. 6), and accounted for about 85% of the variance for a separate V3L2 17-texture data set from two subjects (Fig. 8). For those V3L2 textures the lowest communalities were for the Even and Odd El textures, which interestingly are among the least visually salient of these textures (Maddess & Nagai, 2001;Tkacik et al., 2010;Victor & Conte, 1989;Victor & Conte, 1991) and that type of 4th order correlation rarely occurs in natural scenes (Tkacik et al., 2010).
We presented some data for the case of transforming the raw performance probabilities to d 0 values (Fig. 5D). This was suggested Factor loadings are regression weights between the factors and performance for each texture. These weights can be positive or negative, thus weights near 0 contribute little. This allows us to evaluate what proportion each factor contributes to performance functions (Fig. 4) for each texture. The loadings are thus an indication of the relative influences of the putative texture discrimination mechanisms.  Fig. 8. Results from the seventeen V3L2 textures. They are the same textures as those of Fig. 1 plus 2 more, but not the Box Even texture (see Methods). (A) Scree plot showing an abrupt drop between the 2nd and 3rd factors. (B) the communalities for a 3 factor model account for an average of 0.85 ± 0.11 of the variance (mean ± SD) for discrimination of each texture. On the abscissa labels '+' indicates the Even rule ( Fig. 1 caption), and 'À' the Odd rule; e.g. Cr+ is Cross Even, and CrÀ is Cross Odd. The order of texture types thus follows Fig. 1. by a reviewer who pointed out that the restriction of the probabilities to a maximum of 1 was somewhat artificial and may have affected the factor analysis. For that data set the d 0 -transformation increased the goodness of the model fit as expressed by an increase in the mean communalities for 3 factors from 0.64 to 0.68, with no change in uniformity of the communalities as measured by the standard deviations (0.20 cf. 0.21). Similarly there was little change in the communalities for the early phase data (i.e. First 10 repeats Fig 5B): the original analysis provided communalities of 0.68 ± 0.21 and the d 0 case 0.68 ± 0.24 (mean ± SD). We also examined d 0transformed data for the data set of Fig. 8 where discrimination performance was higher. The communalities for the 3-factor model dropped slightly from 0.85 ± 0.11 to 0.81 ± 0.16.
Texture 042 was the least well explained by the 3 factor model showing the lowest communalities in 3 of the 4 data sets (e.g. Fig. 6). As a way of possibly addressing this issue the Minkowski functionals for the textures were also investigated. These are three combinations of low to 4th order correlations computed for 2 Â 2 pixel blocks of the texture: the Area, Perimeter and Euler Number (porosity). On a square lattice there are actually two Euler numbers based on 4-and 8-way connectedness, v 4 and v 8 . Linear combinations of the functionals have been suggested to explain the performance data for discrimination of isotrigons (Barbosa, Bubna-Litic, & Maddess, 2013). In this case, there are some notable differences in the Minkowski functional data across the textures used here, but there was nothing that uniquely separated texture 042 from the others.
The study of Barbosa, Bubna-Litic, and Maddess (2013) asked if linear combinations of various moments of the Minkowski functionals could explain human performance functions for 33 types of isotrigon texture, including 25 of those used here. The model with lowest deviance contained combinations of some of the mean, variance, kurtosis and skew of v 4 and v 8 . Thus it is possible that the mechanisms for V2LN and VnL3 are related, but we have no evidence for that here.
An interesting feature of the Minkowski functionals is that they can be extended to describe 3D materials composed of two components, either two materials (like fibreglass) or a material and voids (like sponge), and in these cases are related to the strength of the materials. Thus, the Minkowski functionals may be related to the surface properties of some real surface textures and how they inform us about the mechanical properties of the materials (Schroder-Turk et al., 2011, Schroder-Turk, 2010. In terms of the factor loadings ( Fig. 7) the consistent correlation between JWS and the 6-person All16 data set was encouraging in that they were two of the largest data sets, having 26 and 16 repeats respectively, and so were the less likely to be affected by noise. Overall the largest correlations between the loading functions were for the data sets for presentations of 16 Â 16 pixel textures. Whether this implies something different happening for the larger 32 Â 32 pixel examples will require more work. Taken together, the fact that a 3 factor model could produce quite evenly distributed communalities, and that the resulting factor loadings showed significant correlation, indicates that a lower bound of about 3 mechanisms is reasonable. Experiments on a larger set of different textures would be required to verify this preliminary result.
It should be noted that PCA analysis extracts orthogonal (uncorrelated) components. However, it is possible that the true mechanisms for extracting information about 4th and higher order textures may not be orthogonal, and therefore the number of mechanisms may be larger. That is a larger number of correlated mechanisms could span the same 3D space. Such non-orthogonal mechanisms could possibly be picked up by methods like nonlinear PCA. Alternatively our 10 textures may not probe all the dimensions of the true set of mechanisms. Another possibility is that independent texture mechanisms have been shown to be able to produce mutually antagonistic outputs (Victor, Thengone, & Conte, 2013). The relevance to the present case is that some of our textures might contain correlation types that stimulate independent mechanisms but when presented together that activity could cancel, thus masking the mechanisms to a factor analysis. That being said it is perhaps unlikely that most of our texture contained the same cancelations, so across the 10 new texture types one might still expect to see evidence of most of the mechanisms.
The work of Victor and Conte (2012) indicates that for correlations of order 1 to 4, computed within 2 Â 2 blocks of pixels that there is a possibly a 10-dimensional space. This can be reduced to 5-dimensions or fewer if rotational symmetry is considered and some of the dimensions show some correlation between them (Victor & Conte, 2012;Victor, Thengone, & Conte, 2013). That set of correlations only includes a single measure for 4th order correlations, and larger patch sizes, e.g. 3 Â 3 pixels, would be needed to be considered to survey more complex 4th order interactions, which could open up a large number of possible dimensions. Certainly there is evidence for salient and non-salient higher order correlations at such scales in isotrigons and natural textures (Tkacik et al., 2010). Previous studies have suggested that the range of salient correlations is probably less than 6 pixels, and a self-similar analysis occurs for a wide range of pixel sizes (Maddess & Nagai, 2001;Victor & Conte, 1989). We have previously provided evidence that the number of independent mechanisms is less than 10 (Taylor, Maddess, & Nagai, 2008), and is perhaps as small as 3 to 4 (Maddess & Nagai, 2001;Maddess et al., 2007) including different types of isotrigon textures than those used here.
Although we address mechanisms sensitive to 4th and higher order correlations here it is worth mentioning that elegant studies of interactions between 1st and 4th order (Victor, Chubb, & Conte, 2005), and 2nd and 3rd order correlations (Victor, Thengone, & Conte, 2013), have been reported. These have implications for the total number of mechanisms involved in texture discrimination more generally. See also related work on the black shot mechanism (Chubb, Landy, & Econopouly, 2004).
Other than JWS none of the subjects showed significant evidence of learning. One possible feature was a change in the scree plot for ABL with learning (Fig. 5). In a previous study some subjects markedly improved their ability to discriminate V3L2 textures; the increase in probability correct for 16 Â 16 textures was 20% for one and 30% for another texture type in just 5 repeats (Maddess & Nagai, 2001). Learning has also been studied for isotrigon textures that had 3 grey levels (Taylor, Maddess, & Nagai, 2008). The effects were moderate however, with discrimination performance improving little, either after experience or when the textures were presented for several seconds. A recent study by our group, which used a set of 17 V3L2 textures, found strong evidence of learning effects described by an exponential rise with a time constant of about 5 days (Coy, Barbosa, & Maddess, 2014). That data also showed that a three factor model was reasonable.
How might fourth-order correlations be computed by the brain? Maddess and Nagai (2001) have proposed cascading two motion computation-like mechanisms to yield the quadruple products required for fourth-order correlations. Nagai and Maddess have shown that simple models of cortical processing, based on recursion, can discriminate isotrigon textures (Nagai et al., 2009). They found that oscillator networks, containing as few as 3 oscillators, were able to emulate human isotrigon discrimination performance for 53 binary and ternary isotrigon texture types.
The formation of recursively applied products is physiologically plausible and may occur via dendritic back-propagation (Buzsaki & Kandel, 1998;Stuart, Schiller, & Sakmann, 1997) or dendritic spiking (Hausser, Spruston & Stuart, 2000, Mel, 1993Stuart, Schiller, & Sakmann, 1997). An unpublished modelling study from our group has also shown that dendritic back propagation in one pyramid cell is sufficient to discriminate some isotrigon textures from random ones (Taylor, 2013). Overall the mechanisms identified in this study, and previous studies (Maddess & Nagai, 2001;Maddess et al., 2007), may represent some combination of recursive or rectifying processes.

Conclusions
In this study, a novel set of isotrigon textures (VnL2) were used to evaluate the underlying mechanisms of texture discrimination. Subjects discriminated random binary noise patterns from the ten isotrigon textures. Factor analysis revealed that as few as three mechanisms govern the detection of fourth and higher order image structure. The mechanisms identified in this study may represent some nonlinear combination of recursive and/or rectifying processes computed over relatively short ranges.