{"title": "Third-Order Edge Statistics: Contour Continuation, Curvature, and Cortical Connections", "book": "Advances in Neural Information Processing Systems", "page_first": 1763, "page_last": 1771, "abstract": "Association field models have been used to explain human contour grouping performance and to explain the mean frequency of long-range horizontal connections across cortical columns in V1. However, association fields essentially depend on pairwise statistics of edges in natural scenes. We develop a spectral test of the sufficiency of pairwise statistics and show that there is significant higher-order structure. An analysis using a probabilistic spectral embedding reveals curvature-dependent components to the association field, and reveals a challenge for biological learning algorithms.", "full_text": "Third-Order Edge Statistics: Contour Continuation,\n\nCurvature, and Cortical Connections\n\nMatthew Lawlor\nApplied Mathematics\n\nYale University\n\nNew Haven, CT 06520\n\nmatthew.lawlor@yale.edu\n\nSteven W. Zucker\nComputer Science\nYale University\n\nNew Haven, CT 06520\n\nzucker@cs.yale.edu\n\nAbstract\n\nAssociation \ufb01eld models have attempted to explain human contour grouping per-\nformance, and to explain the mean frequency of long-range horizontal connections\nacross cortical columns in V1. However, association \ufb01elds only depend on the\npairwise statistics of edges in natural scenes. We develop a spectral test of the suf-\n\ufb01ciency of pairwise statistics and show there is signi\ufb01cant higher order structure.\nAn analysis using a probabilistic spectral embedding reveals curvature-dependent\ncomponents.\n\n1\n\nIntroduction\n\nNatural scene statistics have been used to explain a variety of neural structures. Driven by the\nhypothesis that early layers of visual processing seek an ef\ufb01cient representation of natural scene\nstructure, decorrelating or reducing statistical dependencies between subunits provides insight into\nretinal ganglion cells [17], cortical simple cells [13, 2], and the \ufb01ring patterns of larger ensembles\n[18]. In contrast to these statistical models, the role of neural circuits can be characterized function-\nally [3, 14] by positing roles such as denoising, structure enhancement, and geometric computations.\nSuch models are based on evidence of excitatory connections among co-linear and co-circular neu-\nrons [5], as well as the presence of co-linearity and co-circularity of edges in natural images [8],\n[7]. The fact that statistical relationships have a geometric structure is not surprising: To the extent\nthat the natural world consists largely of piecewise smooth objects, the boundaries of those objects\nshould consist of piecewise smooth curves.\nCommon patterns between excitatory neural connections, co-occurrence statistics, and the geometry\nof smooth surfaces suggests that the functional and statistical approaches can be linked. Statistical\nquestions about edge distributions in natural images have differential geometric analogues, such as\nthe distribution of intrinsic derivatives in natural objects. From this perspective, previous studies\nof natural image statistics have primarily examined \u201csecond-order\u201d differential properties of curves;\ni.e., the average change in orientation along curve segments in natural scenes. The pairwise statistics\nsuggest that curves tend toward co-linearity, in that the (average) change in orientation is small.\nSimilarly, for long-range horizontal connections, cells with similar orientation preference tend to be\nconnected to each other.\nIs this all there is? From a geometric perspective, do curves in natural scenes exhibit continuity in\ncurvatures, or just in orientation? Are edge statistics well characterized at second-order? Does the\nsame hold for textures?\nTo answer these questions one needs to examine higher-order statistics of natural scenes, but this\nis extremely dif\ufb01cult computationally. One possibility is to design specialized patterns, such as in-\ntensity textures [16], but it is dif\ufb01cult to generalize such results into visual cortex. We make use\nof natural invariances in image statistics to develop a novel spectral technique based on preserving\n\n1\n\n\fa probabilistic distance. This distance characterizes what is beyond association \ufb01eld models (dis-\ncussed next) to reveal the \u201cthird-order\u201d structure in edge distributions. It has different implications\nfor contours and textures and, more generally, for learning.\n\nFigure 1: Outline of paper: We construct edge maps from a large database of natural images, and\nestimate the distribution of edge triplets. To visualize this distribution, we construct an embedding\nwhich reveals likely triplets of edges. Clusters in this embedded space consist of curved lines\n\n2 Edge Co-occurrence Statistics\n\nEdge co-occurrence probabilities are well studied [1, 8, 6, 11]. Following them, we use random\nvariables indicating edges at given locations and orientations. More precisely, an edge at position,\norientation ri = (xi, yi, \u03b8i), denoted Xri, is a {0, 1} valued random variable. Co-occurrence statis-\ntics examine various aspects of pairwise marginal distributions, which we denote by P (Xri, Xrj ).\nThe image formation process endows scene statistics with a natural translation invariance. If the\ncamera were allowed to rotate randomly about the focal axis, natural scene statistics would also\nhave a rotational invariance. For computational convenience, we enforce this rotational invariance\nby randomly rotating our images. Thus,\n\nP (Xr1 , ..., Xrn ) = P (XT (r1), ..., XT (rn))\n\nwhere T is a roto-translation.\nWe can then estimate joint distributions of nearby edges by looking at patches of edges centered at\na (position, orientation) location rn and rotating the patch into a canonical orientation and position\nthat we denote r0. Let T (rn) = r0. Then\n\nP (Xr1 , ..., Xrn ) = P (XT (r1), ..., Xr0)\n\nSeveral examples of statistics derived from the distribution of P (Xri, Xr0) are shown in Fig. 2.\nThese are pairwise statistics of oriented edges in natural images. The most important visible feature\nof these pairwise statistics is that of good continuation: Conditioned on the presence of an edge at\nthe center, edges of similar orientation and horizontally aligned with the edge at the center have high\nprobability. Note that all of the above implicitly or explicit enforced rotation invariance, either by\n\n2\n\nNaturalImagesX-Y-\u0398EdgesConditionalCooccurrenceProbabilitiesEmbeddingsEdgeClusters(A)(B)(C)(D)(E)xy(cid:30)xyxy(cid:30)\u03d52\u03d53\u03d54xy(cid:30)Likely edge combinationsin natural images\u03a6\fAugust and Zucker, 2000\n\nGeisler et al, 2001\n\nElder & Goldberg, 2002\n\nFigure 2: Association \ufb01elds derive from image co-occurrence statistics. Here we show three attempts\nto characterize them. Different authors consider probabilities or likelihoods; Elder further conditions\non boundaries. We simply interpret them as illustrating the probability (likelihood) of an edge near\na horizontal edge at the center position.\n\nFigure 3: Two approximately equally likely triples of edges under the pairwise independence as-\nsumption of Elder et. al. Conditional independence is one of several possible pairwise distributional\nassumptions. Intuitively, however, the second triple is much more likely. We examine third-order\nstatistics to demonstrate that this is in fact the case.\n\nonly examining relative orientation with respect to a reference orientation or by explicit rotation of\nthe images.\nIt is critical to estimate the degree to which these pairwise statistics characterize the full joint dis-\ntribution of edges (Fig. 3). Many models for neural \ufb01ring patterns imply relatively low order joint\nstatistics. For example, spin-glass models [15] imply pairwise statistics are suf\ufb01cient, while Markov\nrandom \ufb01elds have an order determined by the size of neighborhood cliques.\n\n3 Contingency Table Analysis\n\nTo test whether the joint distribution of edges can be well described by pairwise statistics, we\nperformed a contingency table analysis of edge triples at two different threshold levels from im-\nages in the van Hataran database. We computed estimated joint distributions for each triple of\nedges in an 11 \u00d7 11 \u00d7 8 patch, not constructed to have an edge at the center. Using a \u03c72\ntest, we computed the probability that each edge triple distribution could occur under hypothesis\nH0 : {No three way interaction}. This is a test of the hypothesis that\n\nlog P (Xri , Xrj , Xrk ) = f (Xri, Xrj ) + g(Xrj , Xrk ) + h(Xri, Xrk )\n\nfor each triple (Xri, Xrj , Xrk ), and includes the cases of independent edges, conditionally indepen-\ndent edges, and other pairwise interactions. For almost all triples, this probability was extremely\nsmall. (The few edge triples for which the null hypothesis cannot be rejected consisted of edges that\nwere spaced very far apart, which are far more likely to be nearly statistically independent of one\nanother.)\n\n3\n\n\fn = 150705016\npercentage of triples where pH0 > .05\n\nthreshold = .05\n0.0082%\n\nthreshold = .1\n0.0067%\n\n4 Counting Triple Probabilities\n\nWe chose a random sampling of black and white images from the van Hataren image dataset[10].\nThey were randomly rotated and then \ufb01ltered using oriented Gabor \ufb01lters covering 8 angles from\n[0, \u03c0). Each Gabor has a carrier period of 1.5 pixels per radian and an envelope standard deviation\nof 5 pixels. The \ufb01lters were convolved in near quadrature pairs, squared and summed.\n\n(a)\n\n(b)\n\nFigure 4: Example image (a) and edges (b) for statistical analysis. Note: color corresponds to\norientation\n\nTo restrict analysis to the statistics of curves, we applied local non-maxima suppression across ori-\nentation columns in a direction normal to the given orientation. This threshold is a heuristic attempt\nto exclude non-isolated curves due to dense textures. We note that previous studies in pairwise edge\nstatistics have used similar heuristics or hand labeling of edges to eliminate textures. The resulting\nedge maps were subsampled to eliminate statistical dependence due to overlapping \ufb01lters.\nThresholding the edge map yields X : U \u2192 {0, 1}, where U \u2282 R2 \u00d7S is a discretization of R2 \u00d7S.\nWe treat X as a function or a binary vector as convenient. We randomly select 21 \u00d7 21 \u00d7 8 image\npatches with an oriented edge at the center, and denote these characteristic patches by Vi\nSince edges are signi\ufb01cantly less frequent than their absence, we focus on (positive) edge co-\noccurrence statistics. For simplicity, we denote P (Xri = 1, Xrj = 1, Xrk = 1) by E[XriXrj Xrk ].\nIn addition, we will denote the event Xri = 1 by Yri. (A small orientation anisotropy has been\nreported in natural scenes (e.g., [9]), but does not appear in our data because we effectively averaged\nover orientations by randomly rotating the images.)\nWe compute the matrix M + where\n\nij = E[Xri Xrj|Yr0]\nM +\n\nn(cid:88)\n\n\u223c 1\nn\n\nViV T\ni\n\ni=1\n\nFigure 5: Histogram of edge probabilities. The\nthreshold to include an edge in M + is p > 0.2,\nand is marked in red.\n\nwhere Vi is a (vectorized) random patch of edges centered around an edge with orientation \u03b8i = 0.\nIn addition, we only compute pairwise probabilities for edges of high marginal probability (Fig. 5)\n\n4\n\n\f5 Visualizing Triples of Edges\n\nBy analogy with the pairwise analysis above, we seek to \ufb01nd those edge triples that frequently co-\noccur. But this is signi\ufb01cantly more challenging. For pairwise statistics, one simply \ufb01xes an edge to\nlie in the center and \u201ccolors\u201d the other edge by the joint probability of the co-occurring pair (Fig. 2).\nNo such plot exists for triples of edges. Even after conditioning, there are over 12 million edge\ntriples to consider.\nOur trick: Embed edges in a low dimensional space such that the distance between the edges rep-\nresents the relative likelihood of co-occurrence. We shall do this in a manner such that distance in\nEmbedded Space \u223c Relative Probability.\nAs before, let Xri be a binary random variable, where Xri = 1 means there is an edge at location\nri = (xi, yi, \u03b8i). We de\ufb01ne a distance between edges\n\nD2\n\n+(ri, rj) = E[X 2\nri\n\n|Yr0 ] \u2212 2E[XriXrj|Yr0 ] + E[X 2\n\n|Yr0]\n\nrj\n\n= M +\n\nii \u2212 2M +\n\nij + M +\n\njj\n\nThe \ufb01rst and the last terms represent pairwise co-occurrence probabilities; i.e., these are the asso-\nciation \ufb01eld. The middle term represents the interaction between Xri and Xrj conditioned on the\npresence of X0. Thus this distance is zero if the edges always co-occur in images, given the hori-\nzontal edge at the origin, and is large if the pair of edges frequently occur with the horizontal edge\nbut rarely together. (The relevance to learning is discussed below.)\nWe will now show how, for natural images, edges can be placed in a low dimensional space where\nthe distance in that space will be proportional to this probabilistic distance.\n\n6 Dimensionality Reduction via Spectral Theorem\n\nWe exploit the fact that M + is symmetric and introduce the spectral expansion\n\nM + =\n\n\u03bbl\u03c6l(i)\u03c6l(j)\n\nn(cid:88)\n\nl=1\n\n\u2192 Rn\n\n(cid:112)\n\nwhere \u03c6l is an eigenvector of M +.\n\nDe\ufb01ne the spectral embedding \u03a6 :\n\n(cid:33)\n\n(cid:32) xi\n\u03a6(ri) = {(cid:112)\n\nyi\n\u03b8i\n\n(cid:112)\n\n\u03bbn\u03c6n(i)}\n\n(1)\n\n\u03bb1\u03c61(i),\n\n\u03bb2\u03c62(i), ...,\n\nThe Euclidean distance between embedded points is then\n\n(cid:107)\u03a6(ri) \u2212 \u03a6(rj)(cid:107)2 = (cid:104)\u03a6(ri), \u03a6(ri)(cid:105) \u2212 2(cid:104)\u03a6(ri), \u03a6(rj)(cid:105) + (cid:104)\u03a6(rj), \u03a6(rj)(cid:105)\n\n= M +\n= D2\n\nii \u2212 2M +\n+(ri, rj)\n\nij + M +\n\njj\n\n\u03a6 maps edges to points in an embedded space where squared distance is equal to relative probability.\nThe usefulness of this embedding comes from the fact that the spectrum of M + decays rapidly\n(Fig. 6). Therefore we truncate \u03a6, including only dimensions with high eigenvalues. This gives a\ndramatic reduction in dimensionality, and allows us to visualize the relationship between triples of\nedges (Fig. 7). In particular, a cluster, say, C, of edges in embedding space all have high probability\nof co-occurring, and the diameter of the cluster\n\nd = max\ni,j\u2208C\n\nD2(ri, rj)\n\nbounds the conditional co-occurrence probability of all edges in the cluster.\n\nE[Xri, Xrj|Yr0] \u2265 2p \u2212 d\n\n2\n\n5\n\n\fFigure 6: Spectrum of M +. Other spectra are similar. Note rapid decay of the spectrum indicating\nthe diffusion distance is well captured by embedding using only the \ufb01rst few eigenfunctions.\n\nSpectral embedding colored by embedding coordinates\n\nEdge map colored by embedding coordinates\n\n\u03c62\n\n\u03c63\n\n\u03c64\n\nFigure 7: Display of third-order edge structure showing how oriented edges are related to their\nspectral embeddings. (top) Spectral embeddings. Note clusters of co-occurring edges. (bottom)\nEdge distributions. The eigenvectors of M + are used to color both the edges and the embedding.\nThe color in each \ufb01gure can be interpreted as a coordinate given by one of the \u03c6 vectors. Edges that\nshare colors (coordinates) in all dimensions (\u03c62, \u03c63, \u03c64) are close in probabilistic distance, which\nimplies they have a high probability of co-occurring along with the edge in the center. Compare with\nFig. 2 where red edges all have high probability of occurring with the center, but no information is\nknown about their co-occurrence probability.\n\nwhere p = mini E(Xri|Yr0). For our embeddings p > .2 see Fig. 5.\nTo highlight information not contained in the association \ufb01eld, we normalized our probability matrix\nby its row sums, and removed all low-probability edges. Embedding the mapping from R2 \u00d7 S \u2192\nRm reveals the cocircular structure of edge triples in the image data (Fig. 7). The colors along each\ncolumn correspond, so similar colors map to nearby points along the dimension corresponding to\nthe row. Under this dimensionality reduction, each small cluster in diffusion space corresponds to\nhalf of a cocircular \ufb01eld. In effect, the coloring by \u03c62 shows good continuation in orientation (with\nour crude quantization) while the coloring by \u03c64 shows co-circular connections. In effect, then, the\n\n6\n\n051015202530354000.20.40.60.811.2lambdaSpectrum of co\u2212occurance kernel\u22120.1\u22120.0500.050.1\u22120.2\u22120.15\u22120.1\u22120.0500.050.1\u22120.1\u22120.0500.050.1\u22120.2\u22120.15\u22120.1\u22120.0500.050.1\u22120.1\u22120.0500.050.1\u22120.2\u22120.15\u22120.1\u22120.0500.050.1\fassociation \ufb01eld is the union of co-circular connections, which also follows from marginalizing the\nthird-order structure away. We used 40,000 (21 \u00d7 21 \u00d7 8) patches.\nShown in Fig. 7 are low dimensional projections of the diffusion map and their corresponding color-\nings in R2 \u00d7 S. To provide a neural interpretation of these results, let each point in R2 \u00d7 S represent\na neuron with a receptive \ufb01eld centered at the point (x, y) with preferred orientation \u03b8. Each cluster\nthen signi\ufb01es those neurons that have a high probability of co-\ufb01ring given that the central neuron\n\ufb01res, so clusters in diffusion coordinates should be \u201cwired\u201d together by the Hebbian postulate. Such\ncurvature-based facilitation can explain the non-monotonic variance in excitatory long-range hori-\nzontal connections in V1 [3, 4]. It may also have implications for the receptive \ufb01elds of V2 neurons.\nAs clusters of co-circular V1 cells are correlated in their \ufb01ring, it may be ef\ufb01cient to represent them\nwith a single cell with excitatory feedforward connections. This predicts that ef\ufb01cient coding models\nthat take high order interactions into account should exhibit cells tuned to curved boundaries.\n\n7\n\nImplications for Inhibition and Texture\n\nOur approach also has implications beyond excitatory connections for boundary facilitation. We\nrepeated our conditional spectral embedding, but now conditioned on the absence of an edge at the\ncenter (Fig. 8). This could provide a model for inhibition, as clusters of edges in this embedding\nare likely to co-occur conditioned on the absence of an edge at the center. We \ufb01nd that the embed-\nding has no natural clustering. Compared to excitatory connections, this suggests that inhibition is\nrelatively unstructured, and agrees with many neurobiological studies.\n\nFigure 8: Embeddings conditioned on the absence of an edge at the center location. Note how\nless structured it is, compared to the positive embeddings. As such it could serve as a model for\ninhibitory connections, which span many orientations.\n\nFinally, we repeated this third-order analysis (but without local non-maxima suppression) on a struc-\ntured model for isotropic textures on 3D surfaces and again found a curvature dependency (Fig. 9).\nEvery 3-D surface has a pair of associated dense texture \ufb02ows in the image plane that correspond to\nthe slant and tilt directions of the surface. For isotropic textures, the slant direction corresponds to\nthe most likely orientation signaled by oriented \ufb01lters.\nAs this is a representation of a dense vector \ufb01eld, it is more dif\ufb01cult to interpret than the edge map.\nWe therefore applied k-means clustering in the embedded space and segmented the resulting vector\n\ufb01eld. The resulting clusters show two-sided continuation of the texture \ufb02ow with a \ufb01xed tangential\ncurvature (Fig. 10).\nIn summary, then, we have developed a method for revealing third-order orientation structure by\nspectral methods. It is based on a diffusion metric that makes third-order terms explicit, and yields\na Euclidean distance measure by which edges can be clustered. Given that long-range horizontal\nconnections are consistent with these clusters, how biological learning algorithms converge to them\nremains an open question. Given that research in computational neuroscience is turning to third-\norder [12] and specialized interactions, this question now becomes more pressing.\n\n7\n\n(cid:239)10(cid:239)50510(cid:239)1001012345678\u22120.2\u22120.15\u22120.1\u22120.0500.050.10.15\u22120.2\u22120.15\u22120.1\u22120.0500.050.10.15\f(a)\n\n(b)\n\n\u03c62\n\n\u03c63\n\n\u03c64\n\nFigure 9: (top) Oriented textures provide information about surface shape.\n(bottom) As before,\nwe looked at the conditional co-occurrence matrices of edge orientations over a series of randomly\ngenerated shapes. Slant orientations and embedding colored by each eigenvector. The edge map is\nthresholded to contain only orientations of high probability. The resulting embedding \u03c6(vi) of those\norientations is shown below. The eigenvectors of M + are used to color both the orientations and\nthe embedding. Clusters of orientations in this embedding have a high probability of co-occurring\nalong with the edge in the center.\n\nFigure 10: Clustering of dense texture \ufb02ows. Color corresponds to the cluster index. Clusters were\nseparated into different \ufb01gures so as to minimize the x, y overlap of the orientations. Embedding on\nthe right is identical to the embeddings above, but viewed along the \u03c63, \u03c64 axes.\n\nReferences\n\n[1] Jonas August and Steven W Zucker. The curve indicator random \ufb01eld: Curve organization\nvia edge correlation. In Perceptual organization for arti\ufb01cial vision systems, pages 265\u2013288.\nSpringer, 2000.\n\n[2] A.J. Bell and T.J. Sejnowski. 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Science, 287(5456):1273\u20131276, 2000.\n\n9\n\n\f", "award": [], "sourceid": 890, "authors": [{"given_name": "Matthew", "family_name": "Lawlor", "institution": "Yale University"}, {"given_name": "Steven", "family_name": "Zucker", "institution": "Yale University"}]}