\documentclass[twocolumn,global]{svjour}\usepackage{times,bibay2e}\include{figmacros}\usepackage{psfig}\usepackage{subfigure}\usepackage{amsmath}\journalname{Biological Cybernetics}%------------------------------------\def\ackname{{\it Acknowledgements.}}\def\bea{\begin{eqnarray}}\def\eea{\end{eqnarray}}\def\beao{\begin{eqnarray*}}\def\eeao{\end{eqnarray*}}\def\nn{\nonumber}\def\arraystretch{1.1}\def\floatcounterend{.~}\def\abstractname{{\bf Abstract.}}%\makeatletter%\def\@biblabel#1{}%\def\@openbib@code{\leftmargin\parindent\itemindent-\parindent}\makeatother\begin{document}%\documentclass[global,twocolumn]{svjour}%\usepackage{times,mathptm}%\newcommand{\SJour}{\textsc{SVJour}}

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\title{A rotation and translation invariant discrete saliency network}\author{Lance R. Williams\inst{1}, John W. Zweck\inst{2}}\institute{Department of Computer Science, University of New Mexico,Albuquerque, NM 87110, USA \andDepartment of CS and EE, University of Maryland, Baltimore County, MD21250, USA}\date{}\mail{L.R. Williams (e-mail: Williams@cs.unm.edu)}%\begin{document}\maketitle\markboth{}{}\begin{abstract} We describe a neural network which that enhances and completes salient closed contours in images. Our work is different from all previous work in three important ways. First, like the input provided to primary visual cortex (V1) by the lateral geniculate nucleus (LGN), the input to our computation is isotropic. That is, it is composed of spots, not edges. Second, our network computes a well- defined function of the input based on a distribution of closed contours characterized by a random process. Third, even though our computation is implemented in a discrete network, its output is invariant to continuous rotations and translations of the input image.\end{abstract}\section{Introduction}Vision, whether that of man humans or machines, must solve certain fundamentalIinformation- processing problems. One such problem is robustness torotations and translations of the input image. Ideally, the output of

a visual computation should vary continuously under continuoustransformation of its input. We believe that an understanding of how thehuman visual system achieves this invariance will lead to significantimprovements in the performance of machine vision systems.Conversely, by studying the fundamental information processing problemscommon to both human and machine vision we can lead achieve to a deeperunderstanding of the structure and function of the human visualcortex.

There is a long history of research on neural networks inspired by thestructure of visual cortex whose functions have been described ascontour completion, saliency enhancement, orientation sharpening, orsegmentation, e.g., August (2000), Li (1998), Yen and Finkel(1996), Guy and Medioni (1996), Iverson (1993), Heitger and von derHeydt (1993), Parent and Zucker (1989), Sashua and Ullman (1988), andGrossberg and Mingolla (1985). In this paper, we describe a neuralnetwork which that enhances and completes salient closed contours inimages. Our work is different from all previous work in threeimportant ways. First, like the input provided to primary visualcortex (V1) by the lateral geniculate nucleus (LGN), the input to ourcomputation is isotropic. That is, the input is composed of spots, notedges.\footnote{Like our model, Guy and Medioni’s (1996)'s can also be applied to input patterns consisting of spots.} Second, our networkcomputes a well- defined function of the input based on a distributionof closed contours characterized by a random process. Third, eventhough our computation is implemented in a discrete network, itsoutput is invariant to continuous rotations and translations of theinput image.\footnote{Guy and Medioni (1996) represent a distribution on ${\mathbf R}^2 \times S^1$ as an $N \times N$ array of $2\times 2$ scatter matrices. Ignoring the issue of the biological plausibility of such an encoding scheme, we observe that although the representation of $S^1$ is clearly continuous, the rectangular grid which is used to sample ${\mathbf R}^2$ is not. It follows that their computation is not Euclidean invariant.}

There are two important properties which a computation must possess ifit is to be invariant to rotations and translations, i.e.,Euclidean invariant. First, the input, the output, and allintermediate representations must be Euclidean invariant. Second, alltransformations of these representations must also be Euclideaninvariant. Previous models are not Euclidean invariant, first andforemost, because their input representations are not Euclideaninvariant. That is, not all rotations and translations of the inputcan be represented equally well. This problem is often skirted bychoosing input patterns which that match particular choices of samplingrate and phase. For example, Li (1998) used only six samples inorientation (including $0^\circ$) and Heitger and von der Heydt (1993)only twelve (including $0^\circ$, $60^\circ$ and $120^\circ$). Li'sfirst test pattern was a dashed line of orientation, $0^\circ$, whileHeitger and von der Heydt (1993) used a Kanizsa Triangle with sides of$0^\circ$, $60^\circ$, and $120^\circ$ orientation.\footnote{Nor are we blameless in this regard. Williams and Jacobs (1997) used 36 discrete orientations including $0^\circ$, $60^\circ$, and $120^\circ$, and also show results on a conveniently oriented Kanizsa Triangle.} There is no reason to believe that theexperimental results shown in these papers would be similar if theinput images were rotated by as little as $5^\circ$. To our knowledge,

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until now no one has ever commented on this problem before.\section{A continuum formulation of the saliency problem}The following section reviews the continuum formulation of the contourcompletion and saliency problem as described in Williams andThornber (2001).\subsection{Shape distribution}Mumford (1994) observed that the probability distribution ofobject boundary shapes could be modeled by a {\it Fokker-Planck}equation of the following form,\begin{eqnarray}\frac{\partial p}{\partial t} = -\cos\theta\frac{\partialp}{\partial x} -\sin\theta\frac{\partial p}{\partial y} +\frac{\sigma^2}{2}\frac{\partial^2 p}{\partial \theta^2}-\frac{1}{\tau}p,\end{eqnarray}\noindent \noindent where $p(\vec{x},\theta\,;\,t)$ is the probabilitythat a particle is located at position, $\vec{x}=(x,y)$, and is movingin direction, $\theta$, at time, $t$. This partial differentialequation can be viewed as a set of independent {\it advection}equations in $x$ and $y$ (the first and second terms) coupled in the$\theta$ dimension by the {\it diffusion} equation (the third term).The advection equations translate probability mass in direction,$\theta$, with unit speed, while the diffusion term implements aBrownian motion in direction, with {\it diffusion parameter},$\sigma$. The combined effect of these three terms is that particlestend to travel in straight lines, but over time they drift to the leftor right by an amount proportional to $\sigma^2$. Finally, the effectof the fourth term is that particles decay over time, with a half-lifegiven by the decay constant, $\tau$.\subsection{The propagators}The Green's function,$G(\vec{x},\theta\,;\,t_1\,|\,\vec{u},\phi\,;\,t_0)$, gives theprobability that a particle observed at position, $\vec{u}$, anddirection, $\phi$, at time, $t_0$, will later be observed at position,$\vec{x}$, and direction, $\theta$, at time, $t_1$. It is thesolution, $p(\vec{x},\theta\,;\,t_1)$, of the Fokker-Planck initialvalue problem with initial value, $p(\vec{x},\theta\,;\,t_0) =\delta(\vec{x}-\vec{u})\delta(\theta-\phi)$ where $\delta$ is theDirac delta function. Unlike Williams and Thornber (2001), who assumedthat the location of input edges could be specified with arbitraryprecision, we explicitly model the limited spatial acuity of the edge-detection process. We consider two edges to be indistinguishable ifthey are separated by a distance smaller than some scale, $\Delta$.The Green's function is used to define two {\it propagators}. Thelong-time propagator:\begin{equation}P_0(\vec{x}, \theta\;|\;\vec{u}, \phi) ={\int_0^\infty dt\; \chi(t)\; G(\vec{x},\theta\,;\,t\,|\,\vec{u},\phi\,;\,0)}\end{equation}\noindent gives the probability that $(\vec{x},\theta)$ and $(\vec{u},\phi)$are edges that are distinguishable edges from the boundary of a singleobject. The short-time propagator:\begin{equation}P_1(\vec{x}, \theta\;|\;\vec{u}, \phi) ={\int_0^\infty dt\; \left[1-\chi(t)\right]\;G(\vec{x},\theta\,;\,t\,|\,\vec{u},\phi\,;\,0)}\end{equation}

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\noindent gives the probability that $(\vec{x},\theta)$ and $(\vec{u},\phi)$are from the boundary of a single object, but are indistinguishable. Inboth of these propagators, $\chi(.)$ is a cut-off function with$\chi(0) = 0$ and $\lim_{t\to\infty} \chi(t)= 1$:\begin{equation}\chi(t) \,\,=\,\, \tfrac 12 \left[ 1 + \tfrac 2\pi\;{\rm atan}\left(\mu\left[\tfrac{t}{\Delta} - \alpha \right]\right) \right].\end{equation}The cut-off function is characterized by three parameters, -- $\alpha$,$\mu$, and $\Delta$. The parameter, $\Delta$, is the scale of the edge-detection process. Relative to this scale, the parameter, $\alpha$,specifies the location of the cut-off, and $\mu$ specifies itshardness.\subsection{Eigenfunctions}We use the long-time propagator,, $P_0$,, to define an integral linearoperator, $Q(.)$, which combines three sources of information: (1) theprobability that two edges belong to the same object,; (2) theprobability that the two edges are distinguishable,; and (3) theprobability that the two edges exist. It is defined asfollows:\begin{equation}Q(\vec{x},\theta\,|\,\vec{u},\phi) = b(\vec{x})^\frac{1}{2}P_0(\vec{x}, \theta\;|\;\vec{u}, \phi) b(\vec{u})^\frac{1}{2}\end{equation}\noindent where the {\it input bias function}, $b(\vec{x})$, givesthe probability that an edge exists at $\vec{x}$. Notethat, as defined, $b(\vec{x})$ depends only on the position,$\vec{x}$, and not the direction, $\theta$, of the edge. This reflectsthe fact that the input to the computation is isotropic.\footnote{The model could easily be extended to include non-isotropic input by making $b(.)$ a function of both $\vec{x}$ and $\theta$. This would allow us to model any orientation bias present in feedforward connections from LGN to V1.}

As described in Williams and Thornber (2001), the right and lefteigenfunctions, $s(.)$ and $\bar{s}(.)$, of $Q(.)$ with the largestpositive real eigenvalue, $\lambda$, play a central role in thecomputation of saliency:\begin{eqnarray}\!\!\!\!\lambda s(\vec{x},\theta) & = &{\int\int\int_{{\mathbf R}^2 \times S^1} d\vec{u} d\phi\;Q(\vec{x},\theta\,|\,\vec{u},\phi)s(\vec{u},\phi)}\\[3mm]\!\!\!\!\lambda \bar{s}(\vec{x},\theta) & = &{\int\int\int_{{\mathbf R}^2 \times S^1} d\vec{u} d\phi\;\bar{s}(\vec{u},\phi)Q(\vec{u},\phi\,|\,\vec{x},\theta)}.\end{eqnarray}Because $Q(.)$ is invariant under the transformation\begin{equation}Q(\vec{x},\theta\,|\,\vec{u},\phi) = Q(\vec{u},\phi +\pi\,|\,\vec{x},\theta+\pi),\end{equation}\noindent which reverses the order and direction of its arguments,the right and left eigenfunctions are related by\begin{equation}\bar{s}(\vec{x},\theta) = s(\vec{x},\theta+\pi).

\end{equation}\subsection{Stochastic completion field}The fact that the distribution of closed contours, or the {\it stochastic completion field}, can be factored into a {\it source field} derivedfrom $s(.)$ and a {\it sink field} derived from $\bar{s}(.)$ isdiscussed in detail in Williams and Thornber (2001). The stochasticcompletion field, $c(\vec{u},\phi)$, gives the probability that aclosed contour containing any subset of the edges exists at$(\vec{u},\phi)$. It is the sum of three terms:\begin{eqnarray}&& c(\vec{u},\phi) =\\&& \frac{p_0(\vec{u},\phi)\bar{p}_0(\vec{u},\phi) +p_0(\vec{u},\phi)\bar{p}_1(\vec{u},\phi) +p_1(\vec{u},\phi)\bar{p}_0(\vec{u},\phi)}{\lambda \int\int\int_{{\mathbf R}^2 \times S^1} d\vec{x}d\theta\;s(\vec{x},\theta)\bar{s}(\vec{x},\theta)}\nonumber\\\nonumber\end{eqnarray}\noindent where $p_m(\vec{u},\phi)$ is a source field, and$\bar{p}_m(\vec{u},\phi)$ is a sink field:\begin{eqnarray*}p_m(\vec{u},\phi) & = &{\int\int\int_{{\mathbf R}^2 \times S^1} d\vec{x} d\theta\;P_m(\vec{u},\phi\;|\;\vec{x},\theta)b(\vec{x})^\frac{1}{2}s(\vec{x},\theta)}\\[3mm]\bar{p}_m(\vec{u},\phi) & = &{\int\int\int_{{\mathbf R}^2 \times S^1} d\vec{x} d\theta\;\bar{s}(\vec{x},\theta) b(\vec{x})^\frac{1}{2}P_m(\vec{x},\theta\;|\;\vec{u},\phi)}.\end{eqnarray*}\noindent The purpose of writing $c(\vec{u},\phi)$ in this way is to remove thecontribution, $p_1(\vec{u},\phi)\bar{p}_1(\vec{u},\phi)$, of closedcontours at scales smaller than $\Delta$ which that would otherwisedominate the completion field. Given the above expression for thecompletion field, it is clear that the key problem is computing theeigenfunction, $s(.)$, of $Q(.)$ with the largest positive realeigenvalue. To accomplish this, we can use the well- known powermethod. See Golub and Van Loan (1996). In this case, the power methodinvolves repeated application of the linear operator, $Q(.)$, to thefunction, $s(.)$, followed by normalization:\begin{eqnarray}&& s^{(m+1)}(\vec{x},\theta) = \\&& \frac{\int\int\int_{{\mathbf R}^2 \times S^1} d\vec{u}d\phi\; Q(\vec{x},\theta|\vec{u},\phi) s^{(m)}(\vec{u},\phi)}{\int\int\int_{{\mathbf R}^2 \times S^1} \int\int\int_{{\mathbf R}^2 \times S^1}d\vec{x}d\theta d\vec{u}d\phi\;Q(\vec{x},\theta|\vec{u},\phi) s^{(m)}(\vec{u},\phi)} \nonumber\end{eqnarray}\noindent In the limit, as $m$ gets very large,$s^{(m+1)}(\vec{x},\theta)$ converges to the eigenfunction of $Q(.)$,with the largest positive real eigenvalue.\footnote{It is only in the limit of large $m$, i.e., after the power- method iteration has converged, that $c(\vec{x},\theta)$ represents a distribution of {\bf closed} contours through location $(\vec{x},\theta)$. Prior to convergence, $s^{(m)}(\vec{x},\theta)\bar{s}^{(m)}(\vec{x},\theta)$ can be considered to be a distribution of non-closed contours of length $2m+1$ edges centered at location $(\vec{x},\theta)$.} Weobserve that the above computation can be considered a continuous

-state, discrete -time, recurrent neural network.\section{A neural implementation}The fixed -point of the neural network we describe is the eigenfunctionwith the largest positive real eigenvalue of the linear integral operator,$Q(.)$, formed by composing the input- independent linear operator,$P_0(.)$, and the input- dependent linear operator, $b(.)$. Thedynamics of the neural network are derived from the update equationfor the standard power method for computing eigenvectors. It is usefulto draw an analogy between our neural network for contour completionand the models for the emergence of orientation selectivity in primaryvisual cortexV1 described by Somers et al. (1995) and Ben-Yishaiet al. (1995). See Fig. ~\ref{fig:recurrent10}. First, we canidentify $s(.)$ with simple cells in V1 and the input- bias function,$b(.)$, which modulates $s(.)$ in the numerator of the updateequation, with the feedforward excitatory connections from the LGN.Second, we can identify $P_0(.)$ with the intra-cortical excitatoryconnections, which which Somers et al. hypothesize are primarilyresponsible for the emergence of orientation selectivity in V1. As inthe model of Somers et al., these connections are highlyspecific and mainly target mainly cells of similar orientation preference.Third, we identify the denominator of the update equation with theorientation- non-specific intra-cortical inhibitory connections, whichwhichSomers et al. hypothesize keep the level of activity withinbounds. Finally, we identify $c(.)$, the stochastic completion field,with the population of cells in V2 described by von der Heydt {et al.} (1984).\begin{figure}[t]%f1\psfig{figure=recurrent10.ps,width=\columnwidth,clip=}%{0.45}%{7.0,3.25}{0.6,0.4}%{A% model of orientation selectivity in V1.}\caption{Thin solid lines indicate feedforward connections from LGN, which provide a weak orientation bias, i.e., $b(.)$, to simple cells in V1, i.e., $s(.)$. Solid lines with arrows indicate orientation- specific intra-cortical excitatory connections, i.e., $P_0(.)$. Dashed lines with arrows indicate orientation -non-specific intra-cortical inhibitory connections. Thick solid lines indicate feedforward connections between V1 and V2, i.e., $c(.)$}\label{fig:recurrent10}\end{figure}\section{A discrete implementation\new line of the continuum formulation}Following Zweck and Williams (2000), the continuous functionscomprising the state of the computation are represented as weightedsums of a finite set of {\it shiftable-twistable} basis functions. Theweights form the coefficient vectors for the functions. Thecomputation we describe is biologically plausible in the sense thatall transformations of state are effected by linear transformations(or other vector parallel operations) on the coefficient vectors.\subsection{Shiftable-twistable bases}The input and output of the above computation are functions defined onthe continuous space, $\mathbf R^2 \times S^1$, of positions in theplane, ${\mathbf R}^2$, and directions in the circle, $S^1$. For suchcomputations, the important symmetry is determined by thosetransformations, $T_{\vec x_0, \theta_0}$, of $\mathbf R^2\times S^1$,which that perform a shift in $\mathbf R^2$ by $\vec x_0$, followed by atwist in $\mathbf R^2\times S^1$ through an angle, $\theta_0$. A {\it

twist} through an angle, $\theta_0$, consists of two parts: (1) arotation, $R_{\theta_0}$, of $\mathbf R^2$ and (2) a translation in$S^1$, both by $\theta_0$. The symmetry, $T_{\vec x_0, \theta_0}$,which is called a {\it shift-twist transformation}, is given by theformula,\begin{equation}T_{(\vec x_0, \theta_0)}(\vec x, \theta) \,\,=\,\, (R_{\theta_0}(\vec x -\vec x_0)\, , \, \theta - \theta_0).\end{equation}\noindent The relationship between continuous shift-twist transformations andcomputations in primary visual cortex has been hypothesized byWilliams and Jacobs (1997) and by Bressloff et al. (2001). Thecontinuous shift-twist group characterizes the symmetry of the Green'sfunction of the Fokker-Planck equation described by Mumford (1994):\begin{eqnarray}\nonumber & G(\vec{x},\theta\,;\,t_1 | \,\vec{u},\phi\,;\,t_0) = &\\& G(R_{\phi}(\vec{x}-\vec{u}),\theta-\phi\,;\,t_1-t_0 | \,0,0\,;\,0). &\end{eqnarray}The {\bf continuous} shift-twist symmetry of the Green's function canbe understood by considering the analogous {\bf discrete} symmetry ofhighest order. An idealized model of V1 might be based on a set ofhypercolumns arranged in a hex lattice and with six discreteorientation preferences (see Fig. ~\ref{fig:hex}). By virtue of theshift-twist symmetry of the interconnection matrix, the long- range,orientation- specific, intra-cortical excitatory connections labeled$A$ and $B$ have equal weight. However, because this lattice is notinvariant under continuous shift-twist transformations, the output ofsuch a model (unlike ours) would not be Euclidean invariant.\begin{figure}[]%f2\psfig{figure=hex2.ps,width=5.5cm,clip=}%{0.5} % puts figure at top of%page% {7.0, 3.4}{1.25,0.2} %width, height, offset from top right%corner%{The highest- order discrete shift-twist group.}\caption{The {\it continuous} shift-twist symmetry of the Green'sfunction canmaybe understood by considering the analogous {\it discrete} symmetry ofhighest order}\label{fig:hex}\end{figure}A visual computation, $C$, on $\mathbf R^2 \times S^1$ is called {\it shift-twist invariant} if, for {\bf all} $(\vec x_0, \theta_0)\in\mathbf R^2\times S^1$, a shift-twist of the input by $(\vec x_0,\theta_0)$ produces an identical shift-twist of the output. Thisproperty can be depicted in the following commutative diagram,:\[\begin{array}{ccc}b(\vec{x},\theta) & \stackrel{C}{\rightarrow} & c(\vec{x},\theta)\\\downarrow T_{\vec x_0, \theta_0} & & \downarrow T_{\vec x_0, \theta_0}\\b(R_{\theta_0}(\vec{x}-\vec{x}_0),\theta-\theta_0)& \stackrel{C}{\rightarrow} &c(R_{\theta_0}(\vec{x}-\vec{x}_0),\theta-\theta_0),\\\end{array}\]\noindent where $b(.)$ is the input, $c(.)$, is the output,$\stackrel{C}{\rightarrow}$ is the computation, and $\stackrel{T_{\vec

x_0, \theta_0}}{\rightarrow}$ is the shift-twist transformation.Correspondingly, we define a {\it shiftable-twistable basis}\footnote{We use this terminology even though the basis functions need not be linearly independent.} of functions on${\mathbf R}^2 \times S^1$ to be a set of functions on ${\mathbf R}^2\times S^1$ with the property that whenever a function, $f(\vec x,\theta)$, is in their span, then so is $f(T_{\vec x_0, \theta_0}(\vecx, \theta))$, is as well for every choice of $(\vec x_0, \theta_0)$ in ${\mathbf R}^2 \times S^1$. As such, the notion of a shiftable-twistablebasis on $\mathbf R^2\times S^1$ generalizes that of atheshiftable-steerable basis on $\mathbf R^2$ introduced by Freeman andAdelson (1991) and Simoncelli et al. (1992).

Shiftable-twistable bases can be constructed as follows. Let$\Psi(\vec x, \theta)$ be a function on $\mathbf R^2 \times S^1$ whichthatis periodic (with period $X$) in both spatial variables, $\vec x$. Inanalogy with the definition of a shiftable-steerable function on$\mathbf R^2$, we say that $\Psi$ is {\it shiftable-twistable} on$\mathbf R^2\times S^1$ if there are integers, $K$ and $M$, and {\it interpolation functions}, $a_{\vec k, m}(\vec x_0, \theta_0)$, suchthat for each $(\vec x_0, \theta_0)\in \mathbf R^2\times S^1$, theshift-twist of $\Psi$ by $(\vec x_0,\theta_0)$ is a linear combinationof a finite number of basic shift-twists of $\Psi$ by amounts $(\veck\Delta,m\Delta_\theta)$, i.e., if\begin{equation}\Psi( T_{\vec x_0, \theta_0}(\vec x, \theta))={\sum_{\vec k, m} a_{\vec k, m}(\vec x_0, \theta_0)\Psi (T_{\vec k\Delta, m\Delta_\theta}(\vec x, \theta))}.\label{e:shift-twist}\end{equation}Here $\Delta = X/K$ is the {\it basic shift amount} and $\Delta_\theta= 2\pi/M$ is the {\it basic twist amount}. The sum inEq.~(\ref{e:shift-twist}) is taken over all pairs of integers,$\vec k = (k_x,k_y)$, in the range, $0 \leq k_x,k_y < K$, and allintegers, $m$, in the range, $0\leq m < M$. As in Simoncelli {et al.} (1993) the interpolation functions, $a_{\vec k, m}$, are sincfunctions.

The Gaussian-Fourier basis is the product of a shiftable-steerablebasis of Gaussians in $\vec x$ and a Fourier series basis in $\theta$.For the experiments in this paper, the standard deviation of theGaussian basis function, $g(\vec x)=\tfrac 1\Delta e^{-\| \vec x \|^2/2\Delta^2}$, equals the basic shift amount, $\Delta$. We regard$g(\vec{x})$ as a periodic function of period, $X$, which is chosen tobe much larger than $\Delta$, so that $g(X/2,X/2)$ and its derivativesare essentially zero. For each frequency, $\omega$, and shift amount,$\Delta$ (where $K=X/\Delta$ is an integer), we define the {\it Gaussian-Fourier basis functions}, $\Psi_{\vec k,\omega}$, by\begin{equation}\Psi_{\vec k,\omega}(\vec x,\theta) =g(\vec x - \vec k \Delta)\,\,e^{i\omega \theta}.\label{e:gaussian-fourier}\end{equation}Zweck and Williams (2000) showed that the Gaussian-Fourier basis iswasshiftable-twistable.\subsection{Power method update formula}Suppose that $Q(.)$'s approximate eigenfunction, $s^{(m)}(\vec

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x,\theta)$, can be represented in the Gaussian-Fourier basis as\begin{equation}s^{(m)}(\vec x,\theta)={\sum_{\vec k, \omega}s^{(m)}_{\vec k, \omega} \Psi_{\vec k, \omega}(\vec x,\theta)}.\end{equation}The vector, ${\mathbf s}^{(m)}$, with components,$s^{(m)}_{\vec{k},\omega}$, will be called the {\it coefficient vector} of $s^{(m)}(\vec x, \theta)$. In the next two sections, wedemonstrate how the following integral linear transform:\begin{eqnarray}&& s^{(m+1)}(\vec{x},\theta)= \\&& {\int\int\int_{{\mathbf R}^2 \times S^1} d\vec{u} d\phi\;P_0(\vec{x}, \theta\;|\;\vec{u}, \phi)b(\vec{u})s^{(m)}(\vec{u},\phi)}\nonumber\end{eqnarray}\noindent (i.e., the basic step in the power method) can be implementedas a discrete linear transform in a Gaussian-Fouriershiftable-twistable basis:\begin{equation}{\mathbf s}^{(m+1)} = {\mathbf P}{\mathbf B}{\mathbf s}^{(m)}.\end{equation}\subsection{The propagation operator {\bf P}}In practice, we do not explicitly represent the matrix, ${\mathbf P}$.This is because it is quite large, and explicitly computing the abovematrix-vector product using it the matrix would be prohibitively expensive.Instead, we compute the matrix-vector product using theadvection-diffusion-decay operator in the Gaussian-Fouriershiftable-twistable basis, ${\mathbf A}\circ{\mathbf D}$, which isderived and described in detail in Zweck and Williams (2000). Thecoefficient vector, ${\mathbf s}^{(m+1)}$, is a weighted average ofsolutions, ${\mathbf q}^{(m,n)}$, of a Fokker-Planck initial- valueproblem. The average is over all positive times, $t = n \Delta t$.That is,\begin{equation}{\mathbf s}^{(m+1)} =\sum_{n=0}^\infty \chi(n \Delta t) {\mathbf q}^{(m,n)}\end{equation}\noindent where $\chi(.)$ is the cutoff function, and the initial value for{\bf q} is the product of the bias operator, {\bf B}, and thecoefficient vector, ${\mathbf s}^{(m)}$,\begin{equation}{\mathbf q}^{(m,0)}={\mathbf B}{\mathbf s}^{(m)}.\end{equation}For convenience, we introduce a variable, ${\mathbf p}^{(m,n)}$, torepresent the partial sum of ${\mathbf q}^{(m,n)}$ from $t = 0$ to $t = n\Delta t$. This leads to the following update equations:\begin{eqnarray}{\mathbf q}^{(m,n+1)} & = & \left({\mathbf A}\circ{\mathbf D}\right){\mathbf q}^{(m,n)}\\{\mathbf p}^{(m,n+1)} & = & {\mathbf p}^{(m,n)}+\chi(n \Delta t)\;{\mathbf q}^{(m,n+1)},\end{eqnarray}\noindent where ${\mathbf A}$ is the the advection operator and ${\mathbf D}$ isthe diffusion- decay operator. In the shiftable-twistable basis, theadvection operator, ${\mathbf A}$, is a discrete convolution:\begin{equation}{q_{\vec \ell,\eta}^{(m,n+\frac{1}{2})}=\sum_{\vec k, \omega}\,\,

\hat a_{\vec \ell - \vec k, \eta - \omega}(\Delta t)\,\,q_{\vec k,\omega}^{(m,n)}}\label{e:basis-advection}\end{equation}\noindent with the kernel,\begin{equation}\hat a_{\vec k, \eta}(\Delta t)={\frac{1}{2\pi} \int_0^{2\pi}a_{\vec k} (\Delta t[\cos\theta,\sin\theta]^{\rm T})\, \exp(-i\eta \theta)\, d\theta}\end{equation}\noindent where the $a_{\vec k}$ are interpolation functions. Let $N$ be thenumber of Fourier- series frequencies, $\omega$, used in theshiftable-twistable basis, and let $\Delta\theta = 2\pi / N$. Thediffusion-decay operator, ${\mathbf D}$, is given by the diagonalmatrix-vector product:,\begin{eqnarray}\nonumber & q_{\vec k,\omega}^{(m,n+1)}= &\\&\!\!\!\!\! e^{-\Delta t /\tau}(\lambda e^{-i\omega \Delta\theta} + (1-2\lambda) +\lambda e^{i\omega \Delta\theta})\,\, q_{\vec k,\omega}^{(m,n+\frac{1}{2})}, &\end{eqnarray}where $\lambda = \frac {\sigma^2}2 \frac {\Delta t}{(\Delta\theta)^2}$.\subsection{The bias operator ${\mathbf B}$}In the continuum, the bias operator multiplies the function, $s(.)$,by the input- bias function, $b(.)$. Our aim is to identify anequivalent linear operator in the shiftable-twistable basis. Supposethat both $s(.)$ and $b(.)$ are represented in a Gaussian basis,$g_{\vec k}(\vec x)=g(\vec x - \vec k \Delta)$. Their product is:\begin{eqnarray}b(\vec x)s(\vec x) & = &{\sum_{\vec k} s_{\vec k}\;g_{\vec k}(\vec x)\;\cdot\;\sum_{\vec \ell} b_{\vec \ell}\;g_{\vec \ell}\;(\vec x)}\\& = & {\sum_{\vec k}\sum_{\vec \ell} s_{\vec k} b_{\vec \ell}\;g_{\vec k}(\vec x)g_{\vec \ell}\;(\vec x)}.\end{eqnarray}Now, the product of two Gaussian basis functions, $g_{\vec k}$ and$g_{\vec \ell}$, is a Gaussian of smaller variance (withhigher-frequency content) which that cannot be represented in the Gaussianbasis, $g_{\vec k}$. Because $b(\vec x)s(\vec x)$ is a linearcombination of the products of pairs of Gaussian- basis functions, itcannot be represented in the Gaussian basis, either. However, weobserve that the convolution of $b(\vec x) s(\vec x)$ and a Gaussian,$h(\vec x) * \left[b(\vec{x})\; s(\vec{x})\right]$, where\begin{equation}h(\vec{x})=\frac{1}{\Delta^2\pi} e^{-|\!|\vec x|\!|^2/\Delta^2}\end{equation}\noindent can be represented in the Gaussian basis. It follows that there exists amatrix, ${\mathbf B}$, such that\begin{equation}h(\vec{x}) * \left[b(\vec x)\;s(\vec{x})\right]={\sum_{\vec k}\left[{\mathbf B}{\mathbf s}\right]_{\vec{k}} g_{\vec k}(\vec x)}.\end{equation}

The formula for the matrix, ${\mathbf B}$, is derived by first completingthe square in the exponent of the product of two Gaussians to obtain:\begin{equation}g_{\mathbf k}(\vec x) g_{\mathbf \ell}(\vec x) =g(\sqrt 2 ( \vec x - \tfrac \Delta 2 (\vec k + \vec \ell\;)))g(\tfrac {\Delta}{\sqrt 2} (\vec k -\vec\ell\;)).\end{equation}This product is then convolved with $h$ to obtain a function, $f(\vecx)$, which is a shift of the Gaussian basis function, $g(\vec x)$.Finally, we use the shiftability formula,\begin{equation}{g(\vec x - \vec x_0)=\sum_{\vec k} a_{\vec k} (\vec x_0) g_{\vec k} (\vec x)},\end{equation}\noindent where $a_{\vec k}$ are is the interpolation functions,to express $f(\vec x)$ in the Gaussian basis. The result is\begin{equation}{B_{\vec k,\vec\ell}=\sum_{\vec i}b_{\;\vec{i}}\exp(-|\!| \vec i - \vec \ell\; |\!|^2/4)a_{\vec k}(\Delta (\vec i + \vec \ell\;)/2)}.\label{biasmatrix1}\end{equation}Although this formula is very general, using it to compute thematrix-vector product would be prohibitively expensive due to thenon-sparseness of the matrix, ${\mathbf B}$. To overcome this problem wehave developed an alternative representation for the bias operator inthe Gaussian-Fourier basis. For this representation, we assume thatthe input- bias function is of the form,\begin{equation}{b(\vec x)=\sum_{j=1}^N g(\vec x - \vec x_j)},\label{specialinputbias}\end{equation}\noindent where the centers, $\vec x_j$, of the Gaussian spotsare assumed to be far apart from each other relative to their widths.In this case, the matrix ${\mathbf B}$ representing the bias operatorcan be expressed as a sum of $N$ outer products of vectorsrepresenting values of the Gaussian- basis functions $g_{\vec \ell}$and values of the sinc interpolation functions $a_{\vec k}$ at thelocations $\vec x_j$ of the Gaussian spots:\begin{equation}{B_{\vec k, \vec \ell}=\sum_{j=1}^N g_{\vec \ell}\,(\vec x_j)\;a_{\vec k} (\vec x_j).}\end{equation}\noindent Consequently, the action of the bias operator on a coefficient vectoris given by:\begin{equation}{\left[\mathbf{B}{\mathbf s}\right]_{\vec k}=\sum_{j=1}^N \left[\sum_{\vec \ell} s_{\vec \ell}\;g_{\vec \ell}\;(\vec x_j) \right]\;\cdot\;a_{\vec k}(\vec x_j)}.\end{equation}\noindent In this form, the bias operator is relatively inexpensive to compute.For this reason, it was used in the experimental results describedbelow.However, in the continuum, this representation of the biasoperator does not agree with the representation given in Eq.~\ref{biasmatrix1}. To understand how the second representation is

related to the first, we now describe how it was derived. Assume thatthe input- bias function is simply $b(\vec x)=g(\vec x-\vec x_0)$ andthat $s(\vec x) = \sum_{\vec \ell} s_{\vec \ell}\; g_{\vec \ell}\;(\vec x)$. Then\begin{eqnarray}\!\!\!\!\!{\sum_{\vec{\ell}}\left[{\mathbf B}{\mathbf s}\right]_{\vec{\ell}}g_{\vec{\ell}}\,(\vec{x})} & = & {b(\vec{x})s(\vec{x})\sum_{\vec \ell} s_{\vec \ell}\;g(\vec x - \vec x_0) g_{\vec \ell}\,(\vec x)}\\& \simeq & {\left[\sum_{\vec \ell}s_{\vec \ell} \,\,g_{\vec \ell}\; (\vec x_0)\right]\cdot g(\vec x - \vec x_0)},\end{eqnarray}\noindent where the final equality holds exactly in the limit that$g(\vec x - \vec x_0) \to \delta(\vec x - \vec x_0)$, where $\delta$denotes the Dirac delta function. The second representation of the biasoperator in the basis then follows by expressing $g(\vec x - \vecx_0)$ in the Gaussian basis using the shiftability formula.\section{Experimental results}In our experiments the Gaussian-Fourier basis consisted of $K=192$translates (in each spatial dimension) of a Gaussian (of period,$X=70.0$), and $N=92$ harmonic signals in the orientation dimension.The standard deviation of the Gaussian was set equal to the shiftamount, $\Delta=X/K$. For illustration purposes, all functions wererendered at a resolution of $256 \times 256$. The diffusion parameter,$\sigma$, equaled $0.1473$, and the decay constant, $\tau$, equaled$12.5$. The time step, $\Delta t$, used to solve the Fokker-Planckequation in the basis equaled $\Delta/2$. The parameters for thecut-off function used to eliminate self-loops were $\alpha=4$ and$\mu=15$.

In the first experiment, the input- bias function, $b(\vec{x})$,consists of eight Gaussian spots spaced equally on a circle. Thepositions are real valued, i.e., they do not lie on the grid ofbasis functions. The stochastic completion field, $\int_{S^1}c(\vec{u},\phi)\;d\phi$, after five iterations of the power method, isshown in Fig. ~\ref{fig:circle}. Although the local orientations ofthe circle at the locations of the spots were not specified in theinput- bias function, they are an emergent property of the completionfield.

In the second experiment, the input- bias function was created byadding twenty randomly positioned Gaussian spots to twenty spots onthe boundary of an avocado. See Fig. ~\ref{fig:avoca1} (left). Thestochastic completion field computed using 32 iterations of the powermethod is shown in Fig. ~\ref{fig:avoca1} (right).

\begin{figure}[t]%f3\psfig{figure=circlen.ps,width=8.7cm,clip=}\caption{Left: The input- bias function, $b(\vec{x})$, consists of eightGaussian spots spaced equally on a circle.The positions are real valued, i.e., they do not lie on the gridof basis functions.Right: The stochastic completion field,$\int_{S^1} c(\vec{u},\phi)\;d\phi$, computed using$192\times192\times92$ basis functions}\label{fig:circle}

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\end{figure}

\begin{figure}[t]%f4\psfig{figure=avoca1n.ps,width=\columnwidth,clip=}\caption{Left: The input- bias function was created by adding twenty20randomly positioned Gaussian spots to twenty 20 spots on the boundary of anavocado. The positions are real valued, i.e., they do not lie onthe grid of basis functions. Right: The stochastic completion field,$\int_{S^1} c(\vec{u},\phi)\;d\phi$, computed using$192\times192\times92$ basis functions}\label{fig:avoca1}\end{figure}

In the last experiment, the input- bias function from the firstexperiment was rotated by $45^\circ$ and translated by half thedistance between the centers of adjacent basis functions,$b(R_{45^\circ}(\vec{x}-[\tfrac{\Delta}{2},\tfrac{\Delta}{2}]^{\rm T}))$. See Fig. ~\ref{fig:avoca2} (left). The stochastic completionfield is identical (up to rotation and translation) to the onecomputed in the first experiment. See Fig. ~\ref{fig:avoca2} (right).Finally, we plot the estimate of the largest positive real eigenvalue,$\lambda$, as a function of $m$, the power- method iteration for boththe rotated and non-rotated input patterns. See Fig.~\ref{fig:avoca_ev}. The final value of $\lambda$ (and allintermediate values) are is identical in both cases. It follows that the$Q(.)$ operator has the same eigenvalues for the rotated andnon-rotated inputs, i.e., the $Q(.)$'s are related by asimilarity transform. This vividly demonstrates the Euclideaninvariance of the computation.\begin{figure}[]%f5\psfig{figure=avoca2n.ps,width=\columnwidth,clip=}%{0.8} % puts figure\caption{Left: The input -bias function from Fig. ~\ref{fig:avoca1},rotated by $45^\circ$ and translated by half the distance betweenthe centers of adjacent basis functions,$b(R_{45^\circ}(\vec{x}-[\tfrac{\Delta}{2},\tfrac{\Delta}{2}]^{\rm T}))$.Right: The stochastic completion field, is identical(up to rotation and translation) to the one shownin Fig.~\ref{fig:avoca1}.This demonstrates the Euclidean invariance of the computation}\label{fig:avoca2}\end{figure}

\begin{figure}[]%f6\psfig{figure=avoca3.ps,width=\columnwidth,clip=}%{0.35} % puts\caption{The estimate of the largest positive real eigenvalue,$\lambda$, as a function of $m$, the power- method iteration for therotated (line 2) and non-rotated (line 1) input patterns.The final value of $\lambda$ (and all intermediate values) areisidentical in both cases,; hence line 1 and line 2overlap completely}\label{fig:avoca_ev}\end{figure}

\section{Discussion}In this section we discuss the relationship of our model to the knownstructure and function of areas V1 and V2 of the visual cortex.\subsection{Isotropy vs. anisotropy}

Given the physical anisotropy of our environment (a consequence of thegravitational field), the observation of a corresponding anisotropy inthe statistics of natural scenes is not surprising. However, it doesnot follow that this environmental anisotropy is necessarily manifestin the computations performed by V1, or in its structure, especiallysince it is not manifest in the circularly symmetric shapes of retinaland LGN receptive fields--areas which that immediately precede V1 in thevisual pathway.

As far as the representation of orientation in V1 is concerned, theevidence from neuroscience is inconclusive. On the one hand, there isevidence for the overrepresentation of horizontal and verticalorientations in the primary visual cortex of ferrets. See Chapman {et al.} (1998). On the other hand, we are aware of no correspondingresult for primates, and many authors have described the distributionof orientation preference in area V1 of monkeys as uniform, { e.g.}, Bartfeld and Grinwald (1992), Hubel and Wiesel (1974).\subsection{Continuity vs. isotropy}A computation is continuous if and only if a continuous transformationof the input produces a continuous (not necessarily identical)transformation of the output. It follows that continuity with respectto rotation is a weaker property than isotropy. Indeed, there are well-known examples of visual computations which that are continuous with .respect .to.rotation but not isotropic, e.g., the dependence ofshape-from-shading on illumination direction. See Sun and Perona(1995). Admittedly, in the model we describe here, we have achievedthe first (weaker) property by enforcing the second (stronger)property. However, to the best of our knowledge, it is an openquestion whether contour computations in V1 are isotropic or aremerely continuous w.r.t. rotation.

In any case, it would be a mistake to conclude that, because humanvision is not completely isotropic, , that there is a lack of continuityw.r.t. rotation in contour computations in V1. Yet, apart from ourown model, the output of no existing model of contour completion has output that variescontinuously under continuous rotation of its input. This is due tothe non-shiftability-twistability of the representations of positionand orientation used in these models.\subsection{Receptive fields vs. basis functions}In neuroscience, a receptive field characterizes the response of aneuron to a stimulus which that varies in position, size, phase, and/ororientation. In contrast, a basis function is a concept from linearalgebra. We hypothesize that the functions which that comprise the statesof visual computations are represented by expansions in basesconsisting of a family of functions related by translation, dilation,phase-shift, and/or rotation. In other words, basis functions are thebuilding blocks of visual intermediate representations.

Under the assumption that the retina, LGN, and V1 can be modeled aslinear shift-invariant systems, the notion of a receptive field can begiven a precise meaning. NamelyThat is, it can be regarded as the {\it impulse response function} of the system. Under this assumption,the relationship of receptive fields to basis functions can also beformally characterized as a {\it frame} and its {\it dual}. Recently,Salinas and Abbott (2000) have argued that simple- cell receptive

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fields do not form a {\it tight frame}. If they are correct, then thecorresponding basis functions, which are only implicitly defined, donot have the same shape as the receptive fields. Consequently, evenunder the strong assumption of linearity, and even when thedimensionalities of the input and the intermediate representation arethe same, it is a mistake to equate receptive fields and basisfunctions.

With regard to V1 and V2, this is a moot point, since the assumptionof linearity is known to be unrealistic. Many researchers havedemonstrated that the response of simple cells in V1 is modulated bystimuli outside of the classical (i.e., linear) receptive field.Furthermore, the dynamics of this modulatory activity is consistentwith a computation implemented in a recurrent network, not a simplefeedforward convolution of the input from LGN. The assumption oflinearity is even less justified in V2, where von der Heydt {et al.} (1984) have identified neurons which that "respond" to illusorycontours of specific position and orientation. Since these neurons donot respond to an isolated spot, the concept of impulse responsefunction is not useful.\subsection{Relative resolution of V1 and V2}The variance of the Gaussian- basis functions required to represent thecompletion field is half of that required to represent the source andsink fields. See Zweck and Williams (2000). It follows that our modelwould be supported if the spatial resolution of V2 was were twice as highas that of V1. However, the fact that V2 receptive fields are, (onaverage), larger than those of V1, does not mean that our model iswrong. Given the non-linearity of the proposed computation, it wouldbe a mistake to equate the size of receptive fields and with the size ofthe basis functions used to represent the completion field. Indeed,given their spatial support of V2 receptivefields in V1, one would {\it expect} them V2 receptivefields to be significantly larger. A better measure of spatialresolution would be average distance (in visual angle) betweenorientation ``pinwheel'' centers.

However, it may simply be the case that the completion field is notcomputed at its theoretically maximum resolution relative to theresolutions of the source and sink fields, i.e., V2 represents alowpass, downsampled completion field. We note that V1 directlyprojects to brain regions other than V2, primarily area MT but alsoback to LGN. It is possible that the computational requirements of MT(or perhaps another computation within V2) constrains the resolutionof V1 more strongly than does contour completion.\subsection{Locality in orientation}One significant difference between our computer model and the visualcortex is the exact form of the basis functions which that span ${\mathbf R}^2\times S^1$. In a previous section, we proposed that observed long-range, intra-cortical, excitatory connections within V1 are were the neuralembodiment of the $P_0(.)$ operator. The fact that these connectionsare orientation- specific implies that the basis functions whichthatrepresent the source field are localized in $S^1$. However, ourimplementation uses harmonic signals, which have zero locality, torepresent the orientation dimension. The problem of identifying abasis of shiftable-twistable functions which that are localized inorientation, and specifying the corresponding $P_0(.)$ operator, is atopic for future research.

\section{Conclusion}We described a neural network which that enhances and completes salientclosed contours. Even though the computation is implemented in adiscrete network, its output is invariant under continuous rotationsand translations of the input.\begin{acknowledgement}L.R.W. was supported (in part) by Los Alamos National Laboratory.J.W.Z. was supported (in part) by the Albuquerque High PerformanceComputing Center. We wish to thank Jonas August and Steve Zucker fortheir insightful comments.\end{acknowledgement}

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