Local and global gestalt laws: A neurally based spectral approach
LLocal and global gestalt laws: A neurally based spectral approach
Marta Favali ∗ , Giovanna Citti † and Alessandro Sarti ‡ Abstract
A mathematical model of figure-ground articulation is presented, which takes into account bothlocal and global gestalt laws and is compatible with the functional architecture of the primaryvisual cortex (V1). The local gestalt law of good continuation is described by means of suitableconnectivity kernels, that are derived from Lie group theory, and quantitatively compared with longrange connectivity in V1. Global gestalt constraints are then introduced in terms of spectral analysisof connectivity matrix derived from these kernels. This analysis performs grouping of local featuresand individuates perceptual units with the highest saliency. Numerical simulations are performedand results are obtained applying the technique to a number of stimuli. keywords:
Mathematical modelling, Quantitative gestalt, Figure-Ground Segmentation, PerceptualGrouping, Neural models, Cortical architecture.
Gestalt laws have been proposed to explain several phenomena of visual perception, such as grouping andfigure-ground segmentation ((Wertheimer, 1938; Kohler, 1929; Koflka, 1935) and for a recent review wequote: (Wagemans et al., 2012)). In particular, in order to individuate perceptual units, gestalt theoryhas introduced local and global laws. Among the local laws we recall the principle of proximity, similarityand good continuation. Particularly the local law of good continuation plays a central role in perceptualgrouping (see Figure 1, left).Regarding global laws, in the construction of percepts the feature of saliency is crucial and at thesame time it escapes to easy quantitative modelling. In the Berliner Gestaltheory the concept of saliencydenotes the relevance of a form with respect of a contextual frame, the power of an object to be presentin the visual field. The role of saliency is pivotal also in figure-ground articulation. Due to the perceptualgrouping process the scenes are perceived as constituted by a finite number of figures and the saliencyassigns a discrete value to each of them. In particular the most salient configuration pops up from theground and becomes a figure (Merleau-Ponty, 1945). Note that in case of continuous deformation of thevisual stimulus, the salient figures can change abruptly from one percept to a different one (Merleau-Ponty,1945). This happens for example in Figure 1 where a regular deformation is applied to the Kanizsasquare: we progressively perceive a more curved square, until it suddenly disappears and the 4 inducersare perceived as stand alone (see for example (Lee & Nguyen, 2001; Pillow & Nava, 2002; Petitot, 2008)). ∗ M. Favali, Center of Mathematics, CNRS - EHESS, Paris, France. [email protected] † G. Citti, Dipartimento di Matematica, Universit`a di Bologna, Bologna, Italy. [email protected] ‡ A. Sarti, Center of Mathematics, CNRS - EHESS, Paris, France. [email protected] a r X i v : . [ c s . C V ] A ug igure 1: Deformation of visual stimulus, represented by squares with different angles between theinducers: the angle regularly decreases and we perceive regular deformations of the subjective Kanizsasquare up to a certain value of curvature, when the square suddenly disappears and the inducers areperceived.A number of results have been provided in order to refine the principles of psychology of form andassess neural correlates of the good continuation law. In particular, Grossberg and Mingolla in (Grossberg& Mingolla, 1985) introduced a “cooperation field” to model illusory contour formation. Similar fields ofassociation and perceptual grouping have been produced by Parent and Zucker in (Parent & Zucker, 1989).In this contest, in the 1990s Kellman and Shipley provided a theory of object perception that specificallyadressed perception of partially occluded objects and illusory contours (Kellman & Shipley, 1991; Shipley& Kellman, 1992, 1994). Heitger and von der Heydt (Von Der Heydt et al., 1993) provided a theory offigural completion which can be applied to both illusory contour figures (as the Kanizsa triangle) and realimages. In the same years Field, Hayes and Hess (Field et al., 1993) introduced through psychophysicalexperiments the notion of association fields , describing the Gestalt principle of good continuation. Theystudied how the perceptual unit visualized in Figure 2 (b) pops up from a stimulus of Gabor patches (seeFigure 2 (a)). Through a series of similar experiments, they constructed an association field, that definesthe pattern of position-orientation elements of stimuli that can be associated to the same perceptual unit(see Figure 2 (c)). (a) (b) (c) Figure 2: The stimulus proposed by Field, Hayes and Hess (Field et al., 1993) (a) and the perceptualunit present in it (b). In (c) the field lines of the association field, that represents the elements in thepath which can be associated to the central point (Field et al., 1993).Starting from the classical results of Hubel and Wiesel (Hubel & Wiesel, 1977) it has been possible tojustify on neurophysiological bases these perceptual phenomena. The results of (Bosking et al., 1997)and (Fregnac & Shulz, 1999) confirmed that neurons sensitive to similar orientation are preferentiallyconnected. This suggests that the rules of proximity and good continuation are implemented in thehorizontal connectivity of low level visual cortices. A stochastic model which takes into account thestructure of the cortex, with position an orientation features, was proposed by Mumford (Mumford, 1994),and further exploited by Williams and Jacobs in (Williams & Jacobs, 1997) and (August & Zucker, 2000).They modelled the analogous of the association fields with Fokker-Planck equations, taking into accountdifferent geometric features, such as orientation or curvature. Petitot and Tondut (Petitot & Tondut,1999) introduced a model of the functional architecture of V1, compatible with the association field. Citti2nd Sarti in (Citti & Sarti, 2006) proposed the model of the functional architecture as a Lie groups,showing the relation between geometric integral curves, association fields, and cortical properties. Thismethod has been implemented in (Sanguinetti et al., 2008) and (Boscain et al., 2012). Exact solution ofthe Fokker-Planck equation has been provided by Duits and van Almsick (Duits & van Almsick, 2008),and their results have been applied by Duits and Franken (Duits & Franken, 2009) to image processing.The described local laws are insufficient to explain the constitution of a percept, since a perceivedform is characterized by a global consistency. Different authors qualitatively defined this consistencyas pregnancy or global saliency (Merleau-Ponty, 1945), but only a few quantitative models have beenproposed (Koch & Ullman, 1985). In particular spectral approach for image processing were proposed by(Perona & Freeman, 1998; Shi & Malik, 2000; Weiss, 1999; Coifman & Lafon, 2006). In (Sarti & Citti,2015) it is shown how this spectral mechanism is implemented in the neural morphodynamics, in terms ofsymmetry breaking of mean field neural equations. In that sense, (Sarti & Citti, 2015) can be consideredas an extension of (Bressloff et al., 2002).In this paper we further develop the approach introduced in (Sarti & Citti, 2015) and describean algorithm for the individuation of perceptual units, using both local and global constraints: localconstraints are modelled by suitable connectivity kernels, which represent neural connections, and theglobal percepts are computed by means of spectral analysis. The model is described in the geometricsetting of a Lie group equipped with a Sub-Riemannian metric introduced in (Petitot & Tondut, 1999;Citti & Sarti, 2006; Sarti et al., 2008). Despite the apparent mathematical difficulty, it helps to clarify ina rigorous way the gestalt law of good continuation.Here we introduce various substantial differences from the techniques in literature. While studyingthe local properties of the model, we focus on the properties of the connectivity kernels. The FokkerPlanck and the Laplacian kernel in the motion group are already largely used for the description of theconnectivity, since they qualitatively fit the experimental data (Sarti & Citti, 2015). Here we perform aquantitative fitting between the computed kernels and the experimental ones, in order to validate themodel. Moreover we propose to use also the Subelliptic Laplacian kernel, in order to account for thevariability of connectivity patterns. Secondly we accomplish grouping with a spectral analysis inspiredfrom the work of (Sarti & Citti, 2015), who proved the neurophysiological plausibility of this process.In the experiments we manipulate the stimuli to demonstrate the relation between the pop up of thefigure and the eigenvalue analysis. We will analyze in particular the swap between one solution and theother while smoothly changing the stimulus in many grouping experiments. Finally we enrich the model,exploiting the role of the polarity feature, which allows to work with two competing kernels.The plan of the paper is the following. The Section 2 is divided in two parts, in the first we describelocal constraints and in the second the global ones. We will first recall the neurogeometry of the visualcortex and see how the cortical connectivity is represented by the fundamental solution of Fokker Planck,Sub-Riemannian Laplacian and isotropic Laplacian equations. We propose a method for the individuationof perceptual units, first recalling the notions of spectral analysis of connectivity matrices, obtained bythe connectivity kernels. We will see how eigenvectors of this matrix represent perceptual units in theimage. In Section 3 we present numerical approximations of the kernels and we will compare kernelswith neurophysiological data of horizontal connectivity (Angelucci et al., 2002; Bosking et al., 1997). Wealso perform a quantitative validation of the kernel considering the experiment of (Gilbert et al., 1996),showing the link between the connectivity kernel and cell’s response. Finally in Section 4 we present theresults of simulations using the implemented connectivity kernels. We will identify perceptual units indifferent Kanizsa figures, highlighting the role of polarity, discussing and comparing the behavior of thedifferent kernels. 3
The mathematical model
In this section we identify a possible neural basis of local Gestalt laws in the functional architecture ofthe primary visual cortex, that is the first cortical structure that underlies the processing of the visualstimulus. We do not claim here that the process of grouping has to be attributed exclusively to V1, sinceseveral cortical areas are involved in segmentation of a figure. However neural evidence ensures that ittakes place already in V1 (see (Lee & Nguyen, 2001; Pillow & Nava, 2002)). Hence we focus on this areawhere the first elaboration is made and it is particular important for all the geometrical aspects of theprocess.
In the 70s Hubel and Wiesel discovered that this cortical area is organized in the so called hypercolumnarstructure (see (Hubel & Wiesel, 1962, 1977)). This means that for each retinal point ( x, y ) there is anentire set of cells each one sensitive to a specific orientation θ of the stimulus.The first geometrical models of this structure are due to Hoffman (Hoffman, 1989), Koenderink(Koenderink & van Doorn, 1987), Williams and Jacobs (Williams & Jacobs, 1997) and Zucker (Zucker,2006). They described the cortical space as a fiber bundle, where the retinal plane ( x, y ) is the basis,while the fiber concides with the hypercolumnar variable θ . More recently Petitot and Tondut (Petitot &Tondut, 1999), Citti, Sarti (Citti & Sarti, 2006) and Sarti, Citti, Petitot (Sarti et al., 2008), proposed todescribe this structure as a Lie group with a Sub-Riemannian metric (see also the results of (Duits &Franken, 2009)). This expresses the fact that each filter can be recovered from a fixed one by translationof the point ( x, y ) and rotation of an angle θ . In particular the visual cortex can be described as thesubset of points of R × S . Every simple cell is characterized by its receptive field, classically definedas the domain of the retina to which the neuron is sensitive. The shape of the response of the cell inpresence of a visual input is called receptive profile (RP) and can be reconstructed by electrophysiologicalrecordings (Ringach, 2002). In particular simple cells of V1 are sensitive to orientation and are stronglyoriented. Hence their RPs are interpreted as Gabor patches (Daugman, 1985; Jones & Palmer , 1987).Precisely they are constituted by two coupled families of cells: an even and an odd-symmetric one.Via the retinotopy, the retinal plane can be identified with the 2-dimensional plane R . A visualstimulus at the retinal point ( x, y ) activates the whole hypercolumnar structure over that point. Allcells fire, but the cell with the same orientation of the stimulus is maximally activated, giving rise toorientation selectivity.Formally curves and edges are lifted to new cortical curves, identified by the variables ( x, y, θ ), where θ is the direction of the boundary at the point ( x, y ). In (Citti & Sarti, 2006) it has been shown thatthese curves are always tangent to the planes generated by the vector fields. These curves have beenmodelled by (Citti & Sarti, 2006) as integral curves of suitable vector fields in the SE (2) cortical structure.Precisely, the vector fields they considered are: (cid:126)X = (cos θ, sin θ, , (cid:126)X = (0 , , . (2.1)All lifted curves are integral curves of these two vector fields such that a curve in the cortical space is: c (cid:48) ( s ) = ( k ( s ) (cid:126)X + k ( s ) (cid:126)X )( c ( s )) , c (0) = 0 . (2.2)It has been noted in (Citti & Sarti, 2006) that these curves, projected on the 2D cortical plane are agood model of the association fields. 4 .2 A model of cortical connectivity From the neurophysiological point of view, there is experimental evidence of the existence of connectivitybetween simple cells belonging to different hypercolumns. It is the so called long range horizontalconnectivity. Combining optical imaging of intrinsic signals with small injections of biocytin in thecortex, Bosking et al. in (Bosking et al., 1997) led to clarify properties of horizontal connections on V1 ofthe tree shrew. The propagation of the tracer is strongly directional and the direction of propagationcoincides with the preferential direction of the activated cells. Hence connectivity can be summarized aspreferentially linking neurons with co-circularly aligned receptive fields.The propagation along the connectivity can be modeled as the stochastic counter part of the de-terministic curves defined in Eq.(2.2) for the description of the output of simple cells. If we assume adeterministic component in direction X (which describes the long range connectivity) and stochasticcomponent along X (the direction of intracolumnar connectivity), the equation can be written as follows:( x (cid:48) , y (cid:48) , θ (cid:48) ) = (cos θ, sin θ, N (0 , σ )) = (cid:126)X + N (0 , σ ) (cid:126)X (2.3)where N (0 , σ ) is a normally distributed variable with zero mean and variance equal to σ . The probabilitydensity of this process, denoted by v , was first used by Williams and Jacobs (Williams & Jacobs, 1997)to compute stochastic completion field, by August and Zucker (August & Zucker, 2000, 2003) to definethe curve indicator random field, and more recently by R. Duits and Franken in (Duits & van Almsick,2008; Duits & Franken, 2009) to perform contour completion, de-noising and contour enhancement. Thekernel obtained integrating in time the density v Γ ( x, y, θ ) = (cid:90) + ∞ v ( x, y, θ, t ) dt (2.4)is the fundamental solution of the Fokker Planck operator F P = X + σ X . The kernel Γ is strongly biased in direction X and not symmetric; a new symmetric kernel can beobtained as following: ω (( x, y, θ ) , ( x (cid:48) , y (cid:48) , θ (cid:48) )) = 12 (Γ (( x, y, θ ) , ( x (cid:48) , y (cid:48) , θ (cid:48) )) + Γ (( x (cid:48) , y (cid:48) , θ (cid:48) ) , ( x, y, θ )) . (2.5)In Figure 3 (a) it is visualized an isosurface of the simmetrized kernel ω , showing its typical twistedbutterfly shape. The kernel ω has been proposed in (Sanguinetti et al., 2008) as a model of the statisticaldistribution of edge co-occurrence in natural images, as described in (Sanguinetti et al., 2008). Thesimilarity between the two is proved both at a qualitative and at a quantitative level (see (Sanguinetti etal., 2008)) (see also Figure 3 (a) and (b)).If we assume that intracolumnar and long range connections have comparable strength, the stochasticequation Eq.(2.3) reduces to: ( x (cid:48) , y (cid:48) , θ (cid:48) ) = N (0 , σ ) (cid:126)X + N (0 , σ ) (cid:126)X (2.6)where N (0 , σ i ) are normally distributed variables with zero mean and variance equal to σ i . In this casethe speed of propagation in directions X and X is comparable. The associated probability density isthe fundamental solution of the Sub-Riemannian Heat equation (Jerison & Sanchez-Calle, 1986). Theintegral in time of this probability densityΓ ( x, y, θ ) = (cid:90) + ∞ v ( x, y, θ, t ) dt (2.7)5s the fundamental solution of the Sub-Riemannian Laplacian (SRL): SRL = σ X + σ X . It is a symmetric kernel, so that we do not need to symmetrize it and we use it as a model of theconnectivity kernel: ω (( x, y, θ ) , ( x (cid:48) , y (cid:48) , θ (cid:48) )) = Γ (( x, y, θ ) , ( x (cid:48) , y (cid:48) , θ (cid:48) )) . (2.8)In Figure 3 (c) it is shown an isosurface of the connectivity kernel ω .We will see in Section 3.2 that a combination of Fokker-Planck and Sub-Riemannian Laplacian fits theconnectivity map measured by Bosking in (Bosking et al., 1997), where the Fokker-Planck fundamentalsolution represents well the long distances of the trajectory, while the Sub-Riemannian Laplacian theshort ones. Combination of different Fokker-Planck fundamental solutions can also be used to model thefunctional architecture of primates experimentally measured by Angelucci in (Angelucci et al., 2002).While validating the model, we will see that a standard Riemannian kernel does not provide the sameaccurate results. In order to show this we will introduce an isotropic version of the previous model whichis a standard Riemannian kernel. To constuct it, we complete the family of vector fields in Eq.(2.1) withan orthonormal one: (cid:126)X = ( − sin( θ ) , cos( θ ) ,
0) (2.9)choosing stochastic propagation in any direction, in such a way that equation Eq.(2.3) becomes:( x (cid:48) , y (cid:48) , θ (cid:48) ) = N (0 , σ ) (cid:126)X + N (0 , ρ ) (cid:126)X + N (0 , σ ) (cid:126)X , (2.10)where N (0 , σ i ) are normally distributed variables with zero mean and variance equal to σ i . Itsprobability density will be denoted v and the associated time independent kernelΓ ( x, y, θ ) = (cid:90) + ∞ v ( x, y, θ, t ) dt (2.11)will be the fundamental solution of the standard Laplacian operator: L = σ X + ρ X + σ X = σ ( ∂ xx + ∂ yy ) + ρ ∂ θθ . One of its level sets is represented in Figure 3 (d). (a) (b) (c) (d)
Figure 3: An isosurface of the connectivity kernel ω obtained by symmetrization of the Fokker Planckfundamental solution Eq.(2.4) (a). The distribution of co-occurrence of edges in natual images (from(Sanguinetti et al., 2008)) (b). Isosurface of the connectivity kernel ω obtained from the fundamentalsolution Γ of the Sub-Riemannian Laplacian equation Eq.(2.7) (c). An isosurface of the fundamentalsolution of the isotropic Laplacian Eq.(2.11) (d).In section 3.1 we will describe a numerical technique to construct the 3 kernels described above.6 .3 Global integration Since the beginning of the last century perception has been considered by gestaltist as a global process.Moreover, following Koch-Ullman and Merleau-Ponty, visual perception is a process of the visual field,that individuates figure and background at the same time (Koch & Ullman, 1985; Merleau-Ponty, 1945).Then it continues in segmentation of the structures by succeeding differentiations.A cortical mechanism responsible for this analysis has been outlined by (Sarti & Citti, 2015), startingfrom the classical mean field equation of Ermentrout and Cowan (Ermentrout & Cowan, 1980) andBressloff and Cowan (Bressloff et al., 2002; Bressloff & Cowan, 2003). This equation describes the evolutionof the cortical activity and depends on the connectivity kernels. The discrete output h of the simple cells,selects in the cortical space ( x, y, θ ) the set of active cells and the cortical connectivity, restricted on thisset, defines a neural affinity matrix. The eigenvectors of this matrix describe the stationary states ofthe mean field equation hence the emergent perceptual units. The system will tend to the eigenvectorassociated to the highest eigenvalue, which corresponds to the most important object in that scene.Mathematically the approach is strongly linked to spectral analysis techniques for locality-preservingembeddings of large data sets (Coifman & Lafon, 2006; Belkin & Niyogi, 2003; Roweis & Saul, 2000),for data segregation and partitioning (Perona & Freeman, 1998; Meila & Shi, 2001; Shi & Malik, 2000),grouping process in real images (Weiss, 1999). We have seen that in presence of a visual stimulus cells aligned to its boundary give the maximal response.We will assume that a discrete number of cells N are maximally activated and we will denote them( x i , y i , θ i ) for i = 1 , ..., N . In Figure 9 (b) we show as an example the cells responding to a Kanizsafigure, represented with their Gabor-like receptive profiles. Following (Sarti & Citti, 2015) the corticalconnectivity is restricted to this discrete set and reduces to a matrix A : A i,j = ω (( x i , y i , θ i ) , ( x j , y j , θ j )) . (2.12)In this discrete setting the mean field equation for the cortical activity reduces to: dudt = − λu ( i ) + s (cid:16) N (cid:88) j =1 A ( i, j ) u ( j ) (cid:17) (2.13)where s is a sigmoidal function and λ is a physiological parameter. The solution tends to its stationarystates, which are the eigenvectors of the associated linearized equation: N (cid:88) j =1 A i,j u j = λs (cid:48) (0) u i (2.14)Hence these are the emergent states of the cortical activity, that individuate the coherent perceptualunit in the scene and allow to segment it. This is why we will assign to the eigenvalues of the affinitymatrix the meaning of a saliency index of the objects. Since we have defined three different kernelsdifferent affinity matrices will be defined. However all kernels are real and symmetric, so that the matrix A is a real symmetric matrix A i,j = A j,i . Their eigenvalues are real and the highest eigenvalue is defined.The associated principal eigenvectors emerge as symmetry breaking of the stationary solutions of meanfields equations and they pop up abruptly as emergent solutions. The first eigenvalue will correspond tothe most salient object in the image. 7 .5 Individuation of perceptual units Since the three different kernels assign different role to different direction of connectivity, the differentaffinity matrices and their spectrum will reflect these different behavior. Consequently the resulting dataset partitioning will be stronger in the straight direction using the Fokker Planck ω kernel, or will allowrotation using the ω kernel (see also (Cocci et al., 2015) for a deeper analysis). Using the kernel ω weexpect an equal grouping capability in the collinear direction and in the ladder direction.In Figure 4 we visualize the affinity matrix of the image presented in Figure 9 (a). It is a square matrixwith dimensions NxN, where N is exactly the number of active patches. It represents the affinity of eachpatch with respect to all the others. The structure of the affinity matrix is composed by blocks and theprincipal ones corresponds to coherent objects. On the right we visualize the complete set of eigenvaluesin a graph (eigenvalue number, eigenvalues). Let us explicitly note that the first eigenvector will have themeaning of emergent perceptual unit. The other eigenvectors do not describe an ordered sequence offigures with different rank. However, their presence is important, above all when two eigenvalue havesimilar values. In this case, small deformation of the stimulus can induce a change in the order of theeigenvalues and produce a sudden emergence of the correspondent eigenvector with an abroupt change inthe perceived image.This is in good agreement with the perceptual characteristics of salient figures of temporal and spatialdiscontinuity, since they pop up abruptly from the background, while the background is perceived asindifferentiated (Merleau-Ponty, 1945). Spectral approaches give reason to the discontinuous characterof figure-ground articulation better than continuous models, who instead introduce a graduality in theperception of figure and background (Lorenceau & Alais, 2001).To find the remainig objects in the image, the process is then repeated on the vector space orthogonal to p , the second and the following eigenvectors can be found, until the associated eigenvalue is sufficientlysmall. In this way only n eigenvectors are selected, with n < N , this procedure reduces the dimensionalityof the description. This procedure neurally reinterprets the process introduced by Perona and Freeman in(Perona & Freeman, 1998).Figure 4: On the left it is visualized the affinity matrix that contains informations about the affinity ofan active patch with respect to all the others. On the right the set of its sorted eigenvalues.8 Quantitative kernel validations
In this section we numerically approximate the connectivity kernels ω i , defined in Section 2.We obtain the discrete fundamental solution Γ of Eq.(2.4) by developing random paths from the numericalsolution of the system (2.3), that can be approximated by: x s +∆ s − x s = ∆ s cos( θ ) y s +∆ s − y s = ∆ s sin( θ ) , s ∈ , ..., Hθ s +∆ s − θ s = ∆ sN ( σ,
0) (3.1)where H is the number of steps of the random path and N ( σ,
0) is a generator of numbers takenfrom a normal distribution with mean 0 and variance σ . In that way, the kernel is numerically estimatedwith Markov Chain Monte Carlo methods (MCMC) (Robert & Casella, 2013). Various realizations n ofthe stochastic path will be given solving this finite difference equation n times; the estimated kernel isobtained averaging their passages over discrete volume elements, as described in detail in (Higham, 2001;Sarti & Citti, 2015). Proceeding with the same methodology the numerical evaluation of fundamentalsolution Γ of the hypoelliptic Laplacian (Eq.(2.7)) is obtained and the system (2.6) discretized: x s +∆ s − x s = ∆ sR cos( θ ) y s +∆ s − y s = ∆ sR sin( θ ) , s ∈ , ..., Hθ s +∆ s − θ s = ∆ sN ( σ ,
0) (3.2)where R = N ( σ ,
0) and σ is the variance in the θ direction. The kernel represented in Figure 3 (c) isobtained by the numerical integration of that system and averaging as before the resulting paths.Finally, the system (2.10), that is a model for isotropic diffusion equation (Eq. 2.11), is approximated by: x s +∆ s − x s = ∆ sN ( σ, y s +∆ s − y s = ∆ sN ( σ, , s ∈ , ..., Hθ s +∆ s − θ s = ∆ sN ( ρ,
0) (3.3)where σ , ρ are the variances in the x , y , θ directions. In order to obtain the approximation of the kernel ω , visualized in Figure 3 (d), the system is integrated with the same technique used before.These kernels will be used to construct the affinity matrices in Eq.(2.12). We will now study in which extent kernels ω i , i = 1 , ω isused for comparison and to show that an uniform Euclidean kernel does not capture the anysotropicstructure of the cortex. Random paths that we compute through MCM are implemented in the functionalarchitectures in terms of horizontal connectivity of a single cell. On the other hand the connectivity ofan entire population of cells corresponds to the set of all single cells connectivities, then to the FokkerPlanck fundamental solution.A first qualitative comparison between the kernels ω , ω and the connectivity pattern has beenprovided in (Sarti & Citti, 2015). Here we follow the same framework, but we propose a more accurate,quantitative comparison. 9t is well known that the 3D cortical structure is implemented in the 2D cortical layer as a pinwheelstructure, which codes for position and orientations (see Figure 5 (b)). The pinwheel structure has alarge variability from one subject to one other, but within each species common statistical propertieshave been obtained. Cortico-cortical connectivity has been measured by Bosking in (Bosking et al.,1997) by injecting a tracer in a simple cell and recording the trajectory of the tracer. In Figure 5 (a)the propagation through the lateral connections is represented by black points. Bosking found a largevariability of injections, which however have common stochastic properties as the direction of propagation,a patchy structure with small blobs at approximately fixed distance and the decay of the density of traceralong the injection site.We model each injection with stochastic paths solutions of Eq. (2.3). Then we evaluate the stochasticpaths on the pinwheel structure.Due to the stochastic nature of the problem, we do not compare pointwise the image of the tracer andthe stochastic paths but we average them on the pinwheels. We partition both the images of the tracerand of the stochastic paths in M regions corresponding to the pinwheels: I = ∪ i R i (3.4)and for every R i we compute the density of tracer DT i and the density of the stochastic paths DP i . Thetwo vectors DT i and DP i are then rescaled in such a way to have unitary L -norm and the mean squareerror is computed: E = (cid:118)(cid:117)(cid:117)(cid:116) M M (cid:88) i =1 (cid:16) DP i − DT i (cid:17) (3.5)The free parameters of the model are the value of the standard deviation, the number of paths, thenumber of steps, appearing in Eq.(2.3) and in the system (3.1). The best fit between the experimentaland simulated distributions has been accomplished by minimizing the mean square error by varying theseparameters.Due to the different role of the directions X and X in the definition of these kernels, the Sub-Riemannian Laplacian paths and the Fokker Planck paths have different structure.The Subriemannian Laplacian allows diffusion in direction X , favors the changement of the angle andit can be used to describe short range connectivity as described in Section 4.4. Hence it is responsible forthe central blob, in a neighborhood of the injection points (see Figure 5 (c)). The Fokker Planck kernelproduces an elogated, patchy structure and seems responsible for the long range connection (see Figure 5(d)). We apply our quantitative fit only to the long range connectivity, hence discarding the tracer ina neighborhood of the injection. For this reason the Sub-Riemannian Laplacian is not involved in thevalidation of the model.The method is first applied to fit the image of the tracer taken by Bosking (Bosking et al., 1997) (seeFigure 5 (a)). All the kernels are evaluated on the pinwheels provided in the same paper (see figure 5(b)), to obtain a patchy structure. In order to apply the formula (3.4), we cover both the image of thetracer and the Fokker Planck with a regular distribution of rectangles, with edges equal to the meandistances between pinwheels (see Figure 5 (c),(d)) (clearly we do not cover the central zone, where we cannot fit the Fokker Planck kernel). The resulting error value is E <
E <
E < (a) (b) (c) (d)(e) (f) (g) (h)(i) (j)
Figure 5: The connectivity map measured by Bosking (Bosking et al., 1997) (a) and Angelucci (Angelucciet al., 2002) (e), the pinwheel structure used for the estimate ((b) and (f)), the tracer partitioned accordingto rectangles with sides equal to the distance between pinwheels ((c) and (g)) and the best fit results ((d)and (h)). In (i) the tracer superimposed to the piwheel structure found by (Bosking et al., 1997) and in(j) the isocontours obtained from a combination of Fokker Planck.11 .3 Perceptual facilitation and density kernels
In order to obtain a stable and deterministic estimate of this stochastic model, we used the density kernel,which is a regular deterministic function, coding the main properties of the process. We perform here aquantitative validation of these regular kernel comparing to an experiment of (Gilbert et al., 1996).This work studies the capability of cells to integrate information out of the single receptive field of thecells. This integration process is due to the long-range horizontal connections, hence it can be used tovalidate our model of long range connectivity. As we have recognized in the previous section it is theFokker Planck kernel which can be considered as a model for long range connectivity, hence we use herethis kernel.Figure 6: On the left the experiment of (Gilbert et al., 1996), with the stimulus composed by randomlyplaced and oriented lines and the black histogram of cell’s response. On the right the histogram evaluatedfrom the probability density in response to the same distribution of lines.In Figure 6 (left) it is shown the results of (Gilbert et al., 1996), where it is visualized the cell’sresponse to randomly placed and oriented lines in a black histogram. A vertical line is present in thereceptive field of a cell selective to this orientation and the intensity of its response is represented inthe first column of the histograms. If the stimulus is surrounded by random elements aligned withthe first one, the cell’s response increases (respectively in the second, third and the last column of thehistograms). When the other random elements are not aligned with the fixed one (as in the fifth, sixth,seventh columns), the cell’s response decreases because it reflects an inhibitory effect.On the right in the blue histogram we evaluate the probability density modelled by the kernel inEq.(2.5) in presence of the same configuration of elements. The same trend is obtained considering theprobability density distribution, as visualized in Figure 6 (right). In order to consider the inhibitory effectwe evaluate the kernel with 0 mean. A quantitative analysis of the differences between them have beenevaluated considering the mean square error between the two normalized histograms. The error of 8%underlines how this connectivity kernel well represents neural connections.
In the following experiments some numerical simulations will be performed in order to test the reliability ofthe method for performing grouping and detection of perceptual units in an image. The kernel consideredhere only depends on orientation. Hence it can be applied to detect the saliency of geometrical figureswhich can be very well described using this feature.The purpose is to select the perceptual units in these images, using the following algorithm:
1. Define the affinity matrix A i,j from the connectivity kernel.2. Solve the eigenvalue problem A i,j u i = λ i u i , where the order of i is such that λ i is decreasing. . Find and project on the segments the eigenvector u associated to the largest eigenvalue. The parameters used are: 1000000 random paths with σ = 0 .
15 in the system (3.1), σ = 1 . σ = 0 . σ , ρ = 0 .
15 in the system (3.3). The value of H is defined as follows: H = d max ,where d max is the maximum distance between the inducers of the stimulus. Similar parameters have beenused for all the experiments. In this section we consider some experiments similar to the ones of Field, Hayes and Hess (Field et al.,1993), where a subset of elements organized in a coherent way is presented out of a ground formed by arandom distribution of elements. A first stimulus of this type is represented in Figure 7 (first row). Theconnectivity among these elements is defined as in equations (2.4) and (2.7).After the affinity matrix and its eigenvalues, the eigenvector corresponding to the highest eigenvalueis visualized in red. The results show that the stimulus is well segmented with the fundamental solutionsof Fokker Planck and Sub-Riemannian Laplacian equations (Figure 7 (b)).13 a) (b)(c) (d) (e) (f) (g)(h) (i) (j) (k)
Figure 7: First row. Example of stimulus (a) similar to the experiments of (Field et al., 1993). Thestimulus containing a perceptual unit is segmented with Fokker Planck and Sub-Riemannian Laplacian(b), using the first eigenvector of the affinity matrix.Second row. In red the first eigenvector of the affinity matrix considering images containing paths inwhich the orientation of successive elements differs by 15 (c), 30 (d), 45 (e), 60 (f) and 90 (g) degrees.Third and fourth rows. Examples (h) with two units in the scene, where a change in the agle leads to achange in the order of the eigenvalues (i),(j),(k).Now we consider a similar experiment proposed in (Field et al., 1993), where the orientation ofsuccessive elements differs by 15, 30, 45, 60 and 90 degrees and the ability of the observer to detect thepath was measured experimentally. It was proved that the path can be identified when the successiveelements differ by 60 deg or less. With our method, we obtain similar results: if the angle betweensuccessive elements is less than 60 degree (Figure 7 (c), (d),(e)), the identification of the unit is correctlyperformed. With an angle equal to 60 degree (Figure 7 (f)) only a part of the curve is correctly detected:this can be interpreted as the increasing observer’s difficulty to detect the path. Finally with higher angles(Figure 7 (g)) the first eigenvector of the affinity matrix corresponds to random inducers, confirming theresults. 14inally we present an example where there are two units in the scene with roughly-equal salience, theyhave roughly-equal eigenvalues. In the third and in the fourth row of Figure 7 the stimuli are composedby a curve and a line in a background of random elements. In the stimulus (h) represented in the thirdrow, the elements composing the curve are perfectly aligned and very nearby, so that this has the highestsaliency and it represents the eigenvector associated to the first eigenvalue (as shown in red in Figure 7(i)). The second eigenvalue in this case is sligtly smaller. After the computation of the first eigenvector,the stimulus is updated (Figure 7 (j)), the first eigenvector of the new affinity matrix is computed and itcorresponds to the inducers of the line (Figure 7 (k)).In the fourth row we slightly modify the stimuli, in particular the alignement of the element formingthe curve (e.g. an angle of pi/18). As a consequence, the line becomes the most salient perceptual unitand the first eigenvector (Figure 7 (i), fourth row). The stimulus is updated (Figure 7 (j), fourth row)and the first eigenvector of the new affinity matrix corresponds to the inducers of the curve (Figure 7 (k),fourth row). It is notable that in this case a small changement of the eigenvalues corresponds to smallchangement of the eigenvectors, but the first eigenvalue swaps with the second one and consequently weobtain an abrupt change in the perceved object.
The term of polarity leads to insert in the model the feature of contrast: contours with the same orientationbut opposite contrast are referred to opposite angles. For this reason we assume that the orientation θ takes values in [0 , π ) when we consider the odd filters and in [0 , π ) while studying the even ones. (a) (b) (c) (d) Figure 8: In the first row schematic description of the whole hypercolumn of odd simple cells centeredin a point ( x, y ). The maximal activity is observed for the simple cell sensitive to the direction of theboundary of the visual stimulus. The set of maximally firing cells are visualized in the last image.In the second row: a cartoon image (a), the first eigenvector of the affinity matrix without polarity (b),its representation with polarity dependent Gabor patches (b) and the corresponding first eigenvector (d).The response of the odd filters in presence of a cartoon image is schematically represented in Figure 8.At every boundary point the maximally activated cell is the one with the same direction of the boundary.Then the maximally firing cells are aligned with the boundary (Figure 8, top right).In order to clarify the role of polarity we consider an image in Figure 8 (a), that has been studied by(Kanizsa, 1980), in the contest of a study of convexity in perception. In this case, if we consider onlyorientation of the boundaries without polarity, we completely loose any contrast information and we15btain the grouping in figure 8 (b). Here the upper edge of the square is grouped as an unique perceptualunit. On the other side, while inserting polarity, the Gabor patches on the upper edge boundary of theblack or white region have opposite contrast and detect values of θ which differs of π (see Figure 8 (c)).In this way, there is no affinity between these patches, and the first eigenvector of the affinity matrixrepresented in red correctly detects the unit present in the image and corresponds to the inducers of thesemicircle (see Figure 8 (d)). This underlines the important role of polarity in perceptual individuationand segmentation. We also note that the fist perceptual unit detected is the convex one, as predicted bythe gestalt law (see (Kanizsa, 1980)). We consider here stimuli formed by Kanizsa figures, represented by oriented segments that simulate theoutput of simple cells. In (Lee & Nguyen, 2001) it is described that completion of Kanizsa figures takesplace in V1.We first consider the stimulus of Figure 9 (a). The connectivity among its elements will be analysiedwith the kernels defined in equations Eq. (2.4),(2.7).The results of simulations with the fundamental solutions of Fokker Planck and Sub-RiemannianLaplacian equations are shown in Figure 9. The first eigenvector is visualized in red and it corresponds tothe inducers of the Kanizsa triangle (Figure 9 (c)). In this example, after the computation of the firsteigenvector of the affinity matrix, this matrix is updated removing the identified perceptual unit and thenthe first eigenvector of the new matrix is computed (Figure 9 (d))): these simulations show that circlesare associated to the less salient eigenvectors. In that way, the first eigenvalue can be considered as aquantitative measure of saliency, because it allows to segment the most important object in the scene andthe results of simulations confirm the visual grouping. (a) (b) (c) (d)
Figure 9: The Kanizsa triangle (a) and the maximally responding odd filters (b). In (c) it is shown thefirst eigenvector of the affinity matrix, using the fundamental solutions of Fokker Planck (2.4,3.1) andSub-Riemannian Laplacian equations (2.7,3.2). After this computation, the affinity matrix is updatedremoving the detected perceptual unit; the first eigenvector of the new matrix is visualized (d).When the affinity matrix is formed by different eigenvectors with almost the same eigenvalues, asin Figure 9 (d), it is not possibile to recognize a most salient object, because they all have the sameinfluence. We choose here to show just one inducer in red. The other two have the same eigenvalue. Thatalso happens, for example, when the inducers are not co-circularly aligned or they are rotated.Now we consider as stimulus the Kanizsa square and then we change the angle between the inducers, sothat the subjective contours become curved (Figure 10 (a), (b), (c), (d), first row). The fact that illusoryfigures are perceived depends on a limit curvature. Indeed we perceive a square in the first three cases,but not in the last one. The results of simulations with the fundamental solutions of Fokker Planck andSub-Riemannian Laplacian equations confirm the visual grouping (Figure 10 (a), (b), (c), (d), secondrow): when the angle between the inducers is not too high (cases (a), (b), (c)) the first eigenvectorcorresponds to the inducers that form the square, otherwise (case (d)) the pacman becomes the most16alient objects in the image. In this case, we obtain 4 eigenvectors with almost the same eigenvalue.Now we consider a Kanizsa bar (Figure 10 (e), first row), that is perceived only if the inducers are aligned.Also in that case, the result of simulation confirms the visual perception if we use the fundamentalsolutions of the Fokker-Planck and the Sub-Riemannian Laplacian equations. When the inducers are notaligned, all the kernels confirm the visual perception, showing two different perceptual units (Figure 10(f)). (a) (b) (c) (d) (e)(f) (g) (h)
Figure 10: Examples of stimulus (first row) with aligned and not-aligned inducers. Stimulus with rotated(g) and not-aligned (f),(h) inducers (third row). The first eigenvectors of the affinity matrix using thefundamental solutions of Fokker Planck and Sub-Riemannian Laplacian are visualized in red (second andfourth row).Considering a stimulus composed of rotated or not-aligned inducers, as in Figure 10 (g), (h) it is notpossible to perceive it and the results of simulations, using all the connectivity kernels described, confirmthe visual grouping. In that case, the affinity matrix is decomposed in 3 eigenvectors with almost thesame eigenvalues, which represent the 3 perceptual units in the scene.17 .4 Sub-Riemannian Fokker Planck versus Sub-Riemannian Laplacian
We have outlined in Section 2.2 and 3.2 that the Fokker Planck kernel accounts for long range connectivity,while Sub-Riemannian Laplacian for short range. In the previous examples we obtain good results withboth kernels, but this difference emerges while we suitable change the parameters. In Figure 11 wecompare the action of these two kernels. (a) (b) (c) (d)(e) (f)
Figure 11: In the first row a few aligned segments, which are correctly grouped by the Fokker Planck andthe Sub-Riemannian Laplacian (a), (b). When we separate the inducers, the perceptual unit is correctlydetected using the Fokker Planck kernel (c), while the Sub-Riemannian Laplacian is not able to performthe grouping (d). In the second row we consider an angle. In this case the Fokker Planck is unable toperform the grouping (e), while the Sub-Riemannian Laplacian can correctly perform the grouping (f).In the first row we see some segments, which form an unique perceptual unit. If they are not too far,the grouping is correctly performed both by the Fokker Planck and the Sub-Riemannian Laplacian (Figure11 (a),(b)). When we separate the inducers, the perceptual unit is correctly detected by the FokkerPlanck kernel (Figure 11 (c)), while the Sub-Riemannian Laplacian is not able to perform the grouping(Figure 11 (d)). This confirms that the Fokker Planck kernel is responsible for long range connectivity.In the second row we consider an angle. When the angle is sufficiently big, the Fokker Planck becomesunable to perform the grouping (Figure 11 (e)), while the Sub-Riemannian Laplacian, correctly performsthe grouping of the perceptual unit (Figure 11 (f)). This confirms that the Sub-Riemannian Laplaciancan be used as a model of short range connectivity.
In order to further validate the Sub-Riemannian model we show that the model applied with the isotropicLaplacian kernel does not perform correctly. As shown in Figure 12 (first row) the visual perception isnot correctly modeled: the first eigenvectors coincide with one of the inducers and the squares are notrecognized. That also happens for the stimulus of Figure 9 (a) and when the inducers are not co-circularlyaligned or they are rotated. 18igure 12: Stimulus of Figure 9 and 10. The results do not fit the visual perception if we use the isotropicLaplacian equation (2.11 , 3.3) and confirm the necessity to use a Sub-Riemannian kernel to model thecortical connectivity.
Conclusions
In this work we have presented a neurally based model for figure-ground segmentation using spectralmethods, where segmentation has been performed by computing eigenvectors of affinity matrices.Different connectivity kernels that are compatible with the functional architecture of the primary visualcortex have been used. We have modelled them as fundamental solution of Fokker-Planck, Sub-RiemannianLaplacian and isotropic Laplacian equations and compared their properties.With this model we have identified perceptual units of different Kanizsa figures, showing that this can beconsidered a good quantitative model for the constitution of perceptual units equipped by their saliency.We have also shown that the fundamental solutions of Fokker-Planck and Sub-Riemannian Laplacianequations are good models for the good continuation law, while the isotropic Laplacian equation is lessrepresentative for this gestalt law. However it retrieves information about ladder parallelism, a featurethat can be analysed in the future. All the three kernels are able to accomplish boundary completionwith a preference for the operators Fokker Planck and the Sub-Riemannian Laplacian.The proposed mathematical model is then able to integrate local and global gestalt laws as a processimplemented in the functional architecture of the visual cortex. The kernel considered here only dependson orientation. Hence it can be applied to detect the saliency of geometrical figures which can be very welldescribed using this feature. The same method can be applied to natural images if their main featuresare related to orientations, as in retinal images (see (Favali, 2015)). The ideas presented here could beextended to more general kernels able to detect geometrical features different from orientation and we areconfident that there is a relation between the highest eigenvector and the salient object. However forgeneral images we can not rely on this simple geometric method, since different cortical areas can beinvolved in the definition of the saliency, with a modulatory effect on the connectivity of V1.
Acknowledgements
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