{"title": "A Rotation and Translation Invariant Discrete Saliency Network", "book": "Advances in Neural Information Processing Systems", "page_first": 1319, "page_last": 1326, "abstract": "", "full_text": "A Rotation and Translation Invariant Discrete\n\nSaliency Network\n\nLance R. Williams\n\nDept. of Computer Science\n\nUniv. of New Mexico\n\nAlbuquerque, NM 87131\n\nJohn W. Zweck\n\nDept. of CS and EE\n\nUniv. of Maryland Baltimore County\n\nBaltimore, MD 21250\n\nAbstract\n\nWe describe a neural network which enhances and completes salient\nclosed contours. Our work is different from all previous work in three\nimportant ways. First, like the input provided to V1 by LGN, the in-\nput to our computation is isotropic. That is, the input is composed of\nspots not edges. Second, our network computes a well de\ufb01ned function\nof the input based on a distribution of closed contours characterized by\na random process. Third, even though our computation is implemented\nin a discrete network, its output is invariant to continuous rotations and\ntranslations of the input pattern.\n\n1 Introduction\n\nThere is a long history of research on neural networks inspired by the structure of visual\ncortex whose functions have been described as contour completion, saliency enhancement,\norientation sharpening, or segmentation[6, 7, 8, 9, 12]. A similiar network has been pro-\nposed as a model of visual hallucinations[1]. In this paper, we describe a neural network\nwhich enhances and completes salient closed contours. Our work is different from all pre-\nvious work in three important ways. First, like the input provided to V1 by LGN, the input\nto our computation is isotropic. That is, the input is composed of spots not edges. Second,\nour network computes a well de\ufb01ned function of the input based on a distribution of closed\ncontours characterized by a random process. Third, even though our computation is imple-\nmented in a discrete network, its output is invariant to continuous rotations and translations\nof the input pattern.\n\nThere are two important properties which a computation must possess if it is to be invariant\nto rotations and translations, i.e., Euclidean invariant. First, the input, the output, and all\nintermediate representations must be Euclidean invariant. Second, all transformations of\nthese representations must also be Euclidean invariant. The models described in [6, 7, 8,\n9, 12] are not Euclidean invariant, \ufb01rst and foremost, because their input representations\nare not Euclidean invariant. That is, not all rotations and translations of the input can be\nrepresented equally well. This problem is often skirted by researchers by choosing input\npatterns which match particular choices of sampling rate and phase. For example, Li [7]\nused only six samples in orientation (including \u0002\u0001 ) and Heitger and von der Heydt[5] only\ntwelve (including \u0003\u0001 ,\n\u0005\u0001 ). Li\u2019s \ufb01rst test pattern was a dashed line of orientation,\n\u0003\u0001 , and\n\u0005\u0001 , while Heitger and von der Heydt used a Kanizsa Triangle with sides of \t\u0001 ,\n\n\u0005\u0001 and\n\n\u0006\b\u0007\n\n\u0004\n\u0004\n\f\u0006\b\u0007\nresearcher in this area has ever commented on this problem before.\n\n\u0005\u0001 orientation. There is no reason to believe that the experimental results they showed\nwould be similiar if the input patterns were rotated by as little as \n\u0001 . To our knowledge, no\n\n2 A continuum formulation of the saliency problem\n\nThe following section reviews the continuum formulation of the contour completion and\nsaliency problem as described in Williams and Thornber[11].\n\n2.1 Shape distribution\n\nMumford[3] observed that the probability distribution of object boundary shapes could be\nmodeled by a Fokker-Planck equation of the following form:\n\n(1)\n\n\u0004+*\n\n\u001d\u001f\u001e\n\n\u0002\"!\n\n\u0010)(\n\n\u0001\u0003\u0002\n\u0001\u0005\u0012\u0013\b\u0014\u000e\u0016\u0015\u0018\u0017\u0019\u0010\n\n\u0001\u0003\u0002\n\u0001\u0005\u0004\u0007\u0006\t\b\u000b\n\r\f\u000f\u000e\u0011\u0010\n\nis the probability that a particle is located at position,\n\n\u0001\u0003\u0002\n\u0001\u0005\u001a\u001c\u001b\n#,\u0012\"'\u0016\u001a-* , and\nwhere \u0002$#&%\n\u0012$'\nis moving in direction, \u0010 , at time, \u0004 . This partial differential equation can be viewed as\na set of independent advection equations in \u0012 and \u001a\nin the \u0010 dimension by the diffusion equation (the third term). The advection equations\ntranslate probability mass in direction, \u0010 , with unit speed, while the diffusion term models\nthe Brownian motion in direction, with diffusion parameter, \u001d\n\u001e . Finally, the effect of the fourth term is\nthe left or right by an amount proportional to \u001d\nthat particles decay over time, with a half-life given by the decay constant, \n\n. The combined effect of\nthese three terms is that particles tend to travel in straight lines, but over time they drift to\n\n(the \ufb01rst and second terms) coupled\n\n.\n\n2.2 The propagators\n\n\u0010)(\n\n\u0010)(\n\nGL#,\u0004+*\n\n\u0004@24%\n\n'16\n\n, and direction,\n\nis the Dirac delta function. The\n\nobject. 1 The short-time propagator:\n\nGreen\u2019s function is used to de\ufb01ne two propagators. The long-time propagator:\n\n'76\n\u000410324%\n#&%\n\u0012/'\n\u000498:* , gives the probability that a particle observed at\nThe Green\u2019s function,.\n\u0010)(\n8 , will later be observed at position, %\n, and direction,6\n, at time,\u0004\nposition, %\n0 . It is the solution,\u0002$#&%\n\u0012/'\n\u0010 , at time,\u0004\n* , of the Fokker-Planck initial value problem with\n\u0010)(\n6>* where <\ninitial value, \u0002$#&%\n#&%\n\u000498;*\n\u0012\"'\n\u0010\u0007\b\n\u0006=<\n\u0010)(\n2@%\n'76>*\n#&%\n\u0012/'\n\u0004/GH#I\u0004+*\n#&%\n\u0012/'\n8FE\n\u0006BADC\ngives the probability that #&%\n* and #&%\n'16\u0005* are distinct edges from the boundary of a single\n\u0012/'\nGL#,\u0004+*NM\n\u0004KJ\n'16\u0005*\n2@%\n\u0006BA\ngives the probability that #&%\n\u0012$'\n* and #:%\n'16>* are from the boundary of a single object but are\nreally the same edge. In both of these propagators, GH#O!P*\nandQ\nGL#,\u0004+*\n\u0017W\\I]\nThe cut-off function is characterized by three parameters, _\nspeci\ufb01es where the cut-off is and ]\nikj\nthem, and thate/fOg,h\n\n#&%\n\u0012/'\nis a cut-off function with GL#\n\b`_\"a&bca\n, ]\n\n\u001eWV\nspeci\ufb01es how hard hard it is. The parameter, d\n\nm@n\u0014e/fpo9m for particles travelling at unit speed.\n\n1We assume that the probability that two edges are the same depends only on the distance between\n\n. The parameter, _\n\nscale of the edge detection process.\n\n, and d\n\n,\n, is the\n\nX\u001cY[Z7Y\n\n\u0004@24%\n\n'76\n\n#:%\n\u0012$'\n\n:\n\n\u0015\u0018RTSIU\n\n(2)\n\n(3)\n\n(4)\n\n\u0007\n\u0001\n\u001e\n\u0002\n\u0001\n\u0010\n\u001e\n\b\n\u0006\n \n%\n\u0012\n\u0006\n5\n(\n5\n\u0012\n\u0004\n0\n\u0012\n\b\n%\n5\n*\n<\n#\n?\n8\n\u0010\n5\n.\n5\n(\n\n*\n\u0010\n5\n?\n0\n\u0010\n5\nC\n8\nE\n\u0006\n\b\n.\n5\n(\n\n*\n\u0010\n5\n\n*\n\u0006\n\nC\n\u0006\n\u0006\n\u0006\n0\n\u0006\n\u001b\n\u001e\nV\nS\n^\n!\nh\nl\ng\n\f2.3 Eigenfunctions\n\nThe integral linear operator, \n#&%\n\u0012/'\n\nthat two edges belong to the same object; 2) the probability that the two edges are distinct;\nand 3) the probability that the two edges exist. It is de\ufb01ned as follows:\n\n#+!\n* , combines three sources of information: 1) the probability\n'76>*\n24%\n\u0006\u0002\u0001\n\u0012>* , gives the probability that an edge exists at\n#&%\nwhere the input bias function, \u0001\n* and \u0007\n#+!\ndescribed in Williams and Thornber[11], the right and left eigenfunctions, \u0006\nof \n\n#&%\n\u0012>*\u0004\u0003\n#O!P* with largest positive real eigenvalue, \b\n\u0005\r\f\u000f\u000e\n\f\u000f\u000e\n\n(6)\n(7)\nis invariant under a transformation which reverses the order and direction of\n\n, play a central role in the computation of\n\n#:%\n'16>*\n24%\n\u0012\"'\n\n'76>*\n'16\n#&%\n\n(5)\n. As\n\n#&%\n\u0012/'\n#&%\n\u0012$'\n\n8\u0003#&%\n\u0012/'\n\nsaliency:\n\n#O!P* ,\n\n'76>*\n\n\b\t\u0006\n\b\u0010\u0007\n\n2@%\n\n*\u0004\u0003\n\n24%\n\n#&%\n\nBecause \n\nits arguments:\n\n#+!\n\n#&%\n\u0012$'\n'16>*\n#:%\n24%\n\u0012\"'\n\nAWATA\u000b\n\nAWATA\n'76>*\n24%\n#:%\n\u0012$'\n\n#&%\n\u0012/'\n\n'76\n#&%\n#:%\n\u0012$'\n\n#&%\n\nterms:\n\n(8)\n\n(9)\n\n(10)\n\n\u001b\u0013\u0011\n\n#&%\n\n'76>*\n\n'16>*\n\n#&%\n#:%\n\n#&%\n'76>*\n\n8\u0003#&%\n\f\u000f\u000e\n'76>*\n\nwhere\u0002\t\u0015\n\n2.4 Stochastic completion \ufb01eld\n\nthe right and left eigenfunctions are related as follows:\n\n\u001b\u0012\u0011\n\u001b\u0012\u0011\n'16\u0005* , equals the probability that a\nThe magnitude of the stochastic completion \ufb01eld, \u0014\n'76>* . It is the sum of three\nclosed contour satisfying a subset of the constraints exists at #&%\n\u0002\u001f0\n'16\u0005*\n'76>*\n#&%\n\u0002\u001f0\n'76>*\n\u0012\"'\n#&%\n#&%\n\u0012$'\n2@%\n'16\n\u0012\"'\n#&%\n\u0012>*\n\n'76>*\n#&%\n'76>*\nAWATA\n\u0002\u0016\u0015\nis a source \ufb01eld, and \u0007\n'16>*\n#:%\n\f\u0017\u000e\n#:%\n'16>*\n\f\u0017\u000e\n'16\u0005* ,\nThe purpose of writing \u0014\nof closed contours at scales smaller than d which would otherwise dominate the com-\n#O!P* with largest positive real eigenvalue.\nproblem is computing the eigenfunction, \u0006\n#O!P* , to the function,\npower method involves repeated application of the linear operator, \n#O!P* , followed by normalization:\n'76>*\n#&%\nATATA\n#&%\n'76>*\n\f\u000f\u000e\nAWATA\n\u0015\u0019\u0018\ngets very large, \u0006\u0004\u001b\n\n\u0012$'\n#&%\nATATA\n'16\u0005*\r!\n2@%\nATATA\nin this way is to remove the contribution,\u0002\n'16\u0005*\n#&%\n\n'76>*\n* converges to the eigenfunction of \n\nlargest positive real eigenvalue. We observe that the above computation can be considered\na continuous state, discrete time, recurrent neural network.\n\nTo accomplish this, we can use the well known power method (see [4]). In this case, the\n\npletion \ufb01eld. Given the above expression for the completion \ufb01eld, it is clear that the key\n\n#&%\n\u0012\"'\nIn the limit, as \u001a\n\n'76>*\n#&%\n\u0012$'\n\n* , with\n\n#O!P* , of \n\n#&%\n#&%\n\u0012/'\n\nis a sink \ufb01eld:\n\n#&%\n\u0012>*\n#:%\n\u0012\"'\n\n\f\u0017\u000e\n#&%\n\u0012/'\n\n(11)\n\n(12)\n\n'76>*\n\n8\u0003#&%\n\n#&%\n\n(13)\n\n#+!\n\n\f\u000f\u000e\n0\u001d\u001c\n\n#&%\n\n\u0015\u0019\u0018\n\n#&%\n\u0012$'\n\nAWA\n\n#&%\n\n3 A discrete implementation of the continuum formulation\n\nThe continuous functions comprising the state of the computation are represented as\nweighted sums of a \ufb01nite set of shiftable-twistable basis functions. The weights form the\ncoef\ufb01cient vectors for the functions. The computation we describe is biologically plausible\nin the sense that all transformations of state are effected by linear transformations (or other\nvector parallel operations) on the coef\ufb01cient vectors.\n\n\n\u0010\n5\n\u0005\n?\n\u0010\n5\n\u0001\n5\n\u0005\n%\n\u0012\n\u0006\n\u0010\n*\n\u0006\n\u0003\nE\n%\n5\nE\n6\n\n\u0010\n5\n\u0006\n5\n\u0006\n\u0010\n*\n\u0006\n\n\u0005\n\u0003\nE\n%\n5\nE\n6\n\u0007\n\u0006\n5\n\n5\n\u0010\n*\n!\n*\n\n\u0010\n5\n\u0006\n\n5\n\u0010\n*\n\u0007\n\u0006\n\u0010\n*\n\u0006\n\u0006\n\u0010\n*\n!\n5\n5\n\u0014\n5\n\u0006\n\u0002\n5\n\u0007\n\u0002\n8\n5\n\u001b\n\u0002\n5\n\u0007\n5\n\u001b\n5\n\u0007\n\u0002\n8\n5\n\b\n\n\u0005\n\u0003\nE\n%\n\u0012\nE\n\u0010\n\u0006\n\u0010\n*\n\u0007\n\u0006\n\u0010\n*\n5\n5\n\u0002\n\u0015\n5\n\u0006\n\n\u0005\n\u0003\nE\n%\n\u0012\nE\n\u0010\n?\n\u0015\n5\n\u0010\n*\n\u0001\n\u0003\n\u0005\n\u0006\n\u0010\n*\n\u0007\n\u0002\n\u0015\n5\n\u0006\n\n\u0005\n\u0003\nE\n%\n\u0012\nE\n\u0010\n\u0007\n\u0006\n\u0010\n*\n\u0001\n\u0003\n\u0005\n?\n\u0015\n\u0010\n5\n5\n0\n5\n\u0007\n\u0002\n0\n5\n\u0006\n\u0006\n0\n\u0010\n*\n\u0006\n\n\u0005\n\u0003\nE\n%\n5\nE\n6\n\n\u0010\n2\n%\n5\n\u0006\n\u0015\n5\nA\n\n\u0005\n\u0003\n\n\u0005\n\u0003\nE\n%\n\u0012\nE\n\u0010\nE\n%\n5\nE\n6\n\n\u0010\n2\n%\n5\n\u0006\n\u0015\n5\n!\n\u0010\n\f3.1 Shiftable-twistable bases\n\ntransformation, is given by the formula,\n\nThe input and output of the above computation are functions de\ufb01ned on the continuous\n\n0 . For such\n0 , of positions in the plane, \n\u001e , and directions in the circle, \u0003\nspace, \ncomputations, the important symmetry is determined by those transformations, \u0005\u0007\u0006\n\r\u000e\t , of\n\b\n\t\f\u000b\n\u001e by %\n8 , followed by a twist in \n\u001e\u0002\u0001\u000f\u0003\n\u001e\u0002\u0001\u000f\u0003\n8 , consists of two parts: (1) a rotation, \u0010\nangle, \u0010\n\t , of \n8 . The symmetry, \u0005\nand (2) a translation in \u0003\n\t , which is called a shift-twist\n8:*>'\n8:*\n\n\u001e\u0002\u0001\u0004\u0003\n0 , which perform a shift in \n8 . A twist through an angle, \u0010\n0 , both by \u0010\n\u0012$'\n#&%\n\u001e\u0012\u0001\u0004\u0003\n0 , a shift-twist of the input by #&%\n\n\u000e\t\nis called shift-twist invariant if, for all #&%\n8:*\u0014\u0013\n\u0012\u00058\u000f'\n* produces an identical shift-twist of the output.\n\nThis property can be depicted in the following commutative diagram:\n\nA visual computation, \u0011\n\u001e\u0015\u0001\u0016\u0003\n\n, on \n\nthrough an\n\n\u0010k\b\n\n(14)\n\n#&%\n\n\u001b\t\n*c'\n\n#&%\nis the input, \u0014\nto be a set of functions on \n\n\u0012\"'\n#&%\n\b\n\t\f\u000b\n\u0005\u001a\u0006\n* , is the output,\n\u0001\u0014\u0003\n#&%\n\u0012/'\n\n\u0010k\b\n\n#+!\n\n\u0012/'\n#&%\n\b\n\t\u001c\u000b\n\u0005\u001a\u0006\n\n\u001b\t\n*c'\n\n\u0010k\b\n\n#&%\n\n#&%\n\n\t\u001c!\n\n\"#\t\n\n\u001e [2, 10].\n\n) in both spatial variables,\n\n. In analogy with the\nis shiftable-twistable on\n\nif there are integers, *\n\u001e\u0002\u0001\u000f\u0003\n*+*\n#&%\n\u0012/'\n\nis the shift-\ntwist transformation. Correspondingly, we de\ufb01ne a shiftable-twistable basis2 of functions\n\nis the computation, and \u001d\u001f\u001e\n#O!P*\nwhere \u0001\n0 with the property that whenever a function,\n\u0001\u0012\u0003\non \n0 .\n*+* , for every choice of #&%\n#&%\n\u0012$'\n* , is in their span, then so is $\n\u001e\u001a\u0001\u0014\u0003\nin \n0 generalizes that of a shiftable-\n\u001e%\u0001&\u0003\nAs such, the notion of a shiftable-twistable basis on \nsteerable basis on \n* be a function on \n#&%\n\u0012$'\n\u001e\u001a\u0001\nShiftable-twistable bases can be constructed as follows. Let '\n0 which is periodic (with period (\n\u001e , we say that '\nde\ufb01nition of a shiftable-steerable function on \n* , such\n\u001e\u0014\u0001)\u0003\nand +\n0 , the shift-twist of '\nby #:%\nthat for each #:%\n*/\u0013\nof a \ufb01nite number of basic shift-twists of ' by amounts #\n\u001a\u000bd\n8;*\n8\u0003'\n#&%\n\b2\t\f\u000b\n\u000643\n\u00056\u0006\n\u00051\u0006\nis the basic shift amount and d\n, in the range, >=\n, and all integers, \u001a\n#&%\n\u0012\u001f*\n(H74d\n#&%\n\u0012$'\n\nHere d\n* , in the range,\n>=\nThe Gaussian-Fourier basis is the product of a shiftable-steerable basis of Gaussians in %\nand a Fourier series basis in \u0010 . For the experiments in this paper, the standard deviation of\n\u0005FE\nthe Gaussian basis function, @\n^&A<BDC\n#&%\n\u0012\u0005* as a periodic function of period, (\nregard @\n, so that @\n(87\nshift amount, d\nfunctions, '\n\n, and interpolation functions, ,-\u0006\n* , i.e., if\n*+*\r!\n#&%\n\u0012\"'\n0<;\n\n\u0005 , equals the basic shift amount, d\n* and its derivatives are essentially zero. For each frequency, G\n\n. We\n, which is chosen to be much larger than\n, and\nis an integer), we de\ufb01ne the Gaussian-Fourier basis\n\nThe sum in equation (15) is taken over all pairs of integers,\n\nis the basic twist amount.\n\n(87\n(where *\n\nis a linear combination\n\nZweck and Williams[13] showed that the Gaussian-Fourier basis is shiftable-twistable.\n\n(879*\n\n;\u0012?\n\n(16)\n\nIL\n\nAKJ\n\n(15)\n\n7:+\n\n.\n\n, by\n\n\u000e\t\n\n,5\u0006\n\n#&%\n\n2We use this terminology even though the basis functions need not be linearly independent.\n\n\n\u0012\n0\n\n\u001e\n\u0006\n\b\n\t\n\u000b\n\n\u0005\n\u001b\n\u0006\n\b\n\t\n\u000b\n\n\t\n\u001c\n\u0010\n*\n\u0006\n#\n\u0010\n\u0012\n\b\n%\n\u0012\n\u0010\n!\n0\n\u0010\n\n\u0012\n8\n'\n\u0010\n8\n\u0001\n\u0010\n*\n\u0017\n\u0018\n\u0014\n\u0010\n*\n\u0019\n\u0019\n\u0001\n#\n\u0010\n\n\t\n\u0012\n\b\n%\n\u0012\n8\n\u0010\n8\n*\n\u0017\n\u0018\n\u0014\n#\n\u0010\n\n\t\n\u0012\n\b\n%\n\u0012\n8\n\u0010\n8\n*\n\u0017\n\u0018\n \n\u0018\n\u001e\n0\n\u001e\n$\n\u0010\n#\n\u0005\n\u0006\n\b\n\t\n\u000b\n\n\t\n\u0010\n\u0012\n8\n'\n\u0010\n8\n*\n\u0010\n\u0003\n%\n\u0012\n\n0\n.\n\u000b\n\u0015\n\u0012\n8\n'\n\u0010\n8\n\u0012\n8\n'\n\u0010\n8\n\n\u0012\n8\n'\n\u0010\n8\n*\n%\n0\nd\n'\n\n'\n#\n\u0010\n\u0006\n.\n\u000b\n\u0015\n.\n\u000b\n\u0015\n\u0012\n\u0010\n'\n#\n.\n^\n\u000b\n\u0015\n^\n\"\n\u0010\n\u0006\n\n\u0006\n\u0007\n\u0011\n%\n0\n\u0006\n#\n0\n\b\n'\n0\n\b\n'\n0\n*\n\u001a\n?\n+\n\u0012\n\u0006\n0\n\u0006\n\b\nC\n\u001e\n^\nd\n#\n\u0007\n'\n\u0007\n\u0006\n\u0006\n.\n\u000b\nI\n'\n\u0006\n.\n\u000b\nI\n\u0010\n*\n\u0006\n@\n\u0012\n\b\n%\n0\nd\n*\n!\n\f3.2 Power method update formula\n\nSuppose that \u0006\n\n* can be represented in the Gaussian-Fourier basis as\n\n#&%\n\u0012/'\n\n*c!\n\n#&%\n\u0012$'\n#&%\n\u0012/'\n\u001c , with components, \u0006\n\u0015\u0019\u0018\nAWA\n\n#:%\n\u0012$'\n\nThe vector, \u0001\nIn the next two sections, we demonstrate how the following integral linear transform:\n\n, will be called the coef\ufb01cient vector of \u0006\n'16>*\n\n'16>*\n\n#:%\n\u0012$'\n\n2@%\n\n#:%\n\n#&%\n\n\f\u000f\u000e\n\n(i.e., the basic step in the power method) can be implemented as a discrete linear transform\nin a Gaussian-Fourier shiftable-twistable basis:\n\u0002\u0004\u0003\n\n\u0015\u0019\u0018\n\n(19)\n\n0\u001d\u001c\n\n3.3 The propagation operator P\nIn practice, we do not explicitly represent the matrix, \u0002\n. Instead we compute the necessary\nmatrix-vector product using the advection-diffusion-decay operator in the Gaussian-Fourier\nshiftable-twistable basis, \u0005\u0007\u0006\t\b\n\u0015\u0019\u0018\n\n, described in detail in Zweck and Williams[13]:\n\n(20)\n\n\u0002\u0004\u0003\n\n\u001c and where:\n\nwhere \f\n\n\u0006\u000e\n\n\u0004+*\nIn the shiftable-twistable basis, the advection operator, \u0005\n\n\u0005\u0007\u0006\t\b\n\nGH#\u0010\u000f\n\nwith the following kernel:\n\nI\u0015\u0014\n\n(17)\n\n#&%\n\u0012/'\n\n* .\n\n(18)\n\n(21)\n(22)\n\n(23)\n\n(24)\n\n, used\n, is a\n\n(25)\n\n, is a discrete convolution:\n\n\u0004+*\n*\t\u0018\u001a\u0019\u001c\u001b\"#\n\n:*\n\nwhere the ,\n. are sinc functions. Let !\nin the shiftable-twistable basis, and let d\n\ndiagonal matrix:\n\n\u0004+*\n\nE$#\n\n\f\u000f\u000e-\u0010\n\nM\u0017\u0016\n\u000e\u0016\u0015\u0018\u0017)\u0010\nbe the number of Fourier series frequencies, G\n. The diffusion-decay operator, \b\n\n\b\u001e\u001d \u001f\n\n7\"!\n\n\u0005 .\n\n\u0006\u0015%\n\nwhere \b\n3.4 The bias operator \u0003\n#:%\n\u0012>* , by the in-\nIn the continuum, the bias operator effects a multiplication of the function, \u0006\n#:%\n\u0012\u0005* . Our aim is to identify an equivalent linear operator in the shiftable-\nput bias function, \u0001\n#:%\n\u0012>* . Their\ntwistable basis. Suppose that both \u0006 and \u0001 are represented in a Gaussian basis, @\n#&%\n\u0012>*\n#&%\n\u0012\u001f*\n#&%\n\u0012>*\n(26)\n@\u0015\u0006\n\u0012 , is a Gaussian of smaller vari-\n#:%\n#&%\n\u0012>*\n\u0012>*\nis a linear\n. . Because \u0001\n\n#&%\n\u0012\u001f*\nNow, the product of two Gaussian basis functions, @\n. and @\nance which cannot be represented in the Gaussian basis, @\u0015\u0006\n\n#&%\n\u0012>*'&\n\nproduct is:\n\n#&%\n\u0012>*\n\n@\u0015\u0006\n\n\u0006\u001f\u0006\n\n\u0006\u001f\u0006\n\n\u001b\n\u0015\n\u001c\n\u0010\n\u0006\n\u001b\n\u0015\n\u001c\n\u0010\n*\n\u0006\n3\n\u0006\n.\n\u000b\nI\n\u0006\n\u001b\n\u0015\n\u001c\n\u0006\n.\n\u000b\nI\n'\n\u0006\n.\n\u000b\nI\n\u0010\n\u001b\n\u0015\n\u001b\n\u0015\n\u001c\n\u0006\n.\n\u000b\nI\n\u001b\n\u0015\n\u001c\n\u0010\n\u0006\n\u001b\n0\n\u001c\n\u0010\n*\n\u0006\nA\n\u0005\n\u0003\nE\n%\n5\nE\n6\n?\n8\n\u0010\n5\n\u0001\n5\n*\n\u0006\n\u001b\n\u0015\n\u001c\n5\n\u0001\n\u001b\n\u0006\n\u0001\n\u001b\n\u0015\n\u001c\n!\n\u0001\n\u001b\n0\n\u001c\n\u0006\n\u0001\n\u001b\n\u0015\n\u001c\n\nQ\n\u0015\nR\n\u000b\nU\nC\n\f\n\u001b\n\u0015\n\u000b\n\u000b\n\u001c\n\u001b\n\u0015\n\u000b\n8\n\u001c\n\u001b\n\u0015\n\u000b\n8\n\u001c\n\u0006\n\u0003\n\u0001\n\u001b\n\u0015\n\f\n\u001b\n\u0015\n\u000b\n\u000b\n\u0018\n0\n\u001c\n\u0006\n\f\n\u001b\n\u000b\n\u001c\n\u001b\nd\n\n\u001b\n\u0015\n\u000b\n\u000b\n\u0018\n0\n\u001c\n\n\u001b\n\u0015\n\u000b\n\u000b\n\u0018\n0\n\u001c\n\u0006\n#\n*\n\n\u001b\n\u0015\n\u000b\n\u000b\n\u001c\n!\n\u0011\n\u001b\n\u0015\n\u000b\n\u000b\n\u0018\n\u0003\n\u0005\n\u001c\n\u0006\n\u0012\n\u000b\n\u0013\n\u0006\n3\n\u0006\n.\n\u000b\n,\n\u0006\n\u0012\nB\n\u0006\n.\n\u000b\n\u0013\nB\nI\n#\nd\n\u0011\n\u001b\n\u0015\n\u000b\n\u000b\n\u001c\n\u0006\n.\n\u000b\nI\n\u0014\n,\n\u0006\n.\n\u000b\n\u0013\n#\nd\n\u0006\n0\n\u001e\nX\nA\n\u001e\nX\n8\n,\n\u0006\n.\n#\nd\n\u0004\nJ\n\n'\n\u0010\n*\nE\n\u0010\n\u0006\n\u0010\n\u0006\n\u0007\n\u0011\n\u0011\n\u001b\n\u0015\n\u000b\n\u000b\n\u0018\n0\n\u001c\n\u0006\n.\n\u000b\nI\n\u0006\nA\nB\n^\nS\n#\n\b\nA\nB\nJ\nI\n^\n\n\u001b\n#\n\u0006\n\b\n\u0007\n\b\n*\n\u001b\n\b\nA\nJ\nI\n^\n\u0011\n\u001b\n\u0015\n\u000b\n\u000b\n\u0018\n\u0003\n\u0005\n\u001c\n\u0006\n.\n\u000b\nI\n\u0005\n\u001e\n^\nS\n\u001b\n^\n\n\u001c\n\u0006\n.\n\u0001\n\u0006\n\u0006\n3\n\u0006\n.\n.\n@\n\u0006\n.\n3\n\u0006\n\u0012\n\u0001\n\u0006\n\u0012\n\u0012\n\u0006\n3\n\u0006\n.\n3\n\u0006\n\u0012\n.\n\u0001\n\u0006\n\u0012\n@\n\u0006\n.\n\u0012\n!\n\u0006\n\u0006\n\u0006\n\f#:%\n\n\u001d$\b\n\n#&%\n\n7\f\u000b\n\n\u001d@\u001b\n\nB\u0004\u0003\n\n#:%\n\u0012\u0005*\n\n#:%\n\u0012>*\n\n#&%\n\u0012\u001f*\u0002\u0001\n\n, such that:\n\nin the Gaussian basis either. However, we observe that the convolution of \u0001\na Gaussian,\nGaussian basis. It follows that there exists a matrix, \u0003\n\ncombination of the products of pairs of Gaussian basis functions, it cannot be represented\n#:%\n\u0012>* and\n\u0005 , can be represented in the\n#:%\n\u0012>*\n#&%\n\u0012\u001f*c!\n#&%\n\u0012\u0005*NM\n(27)\n, is derived by \ufb01rst completing the square in the exponent of\n#\b\u0007\n*+*\u0016*\nto obtain a function, $\n#&%\n\u0012\u001f*\n#:%\n\n\u0012>*9M , where\n#&%\n#&%\n\u0012\u0005*\u0005\u0001\n#:%\n\u0012\u0005*\nThe formula for the matrix, \u0003\nthe product of two Gaussians to obtain:\n#&%\n#&%\n\u0012>* , which is a shift of the\n#:%\n\u0012\u0005* . Finally we use the shiftability formula:\n* , and d\n#:%\n\u0012\u0005*\n*c!\n\n8;*\n. are the interpolation functions, @\n\u001b$#\n\nGaussian basis function, @\nwhere ,\u0015\u0006\nshift amount, to express $\n\n8[*\n,5\u0006\n#&%\n\u0012\u001f* equals @\n\nThis product is then convolved with\n\nin the Gaussian basis. The result is:\n\n(879*\n\nis the\n\n*\u0016*c!\n\n(29)\n\n(28)\n\n(30)\n\n#&%\n\n#&%\n\n(879*\n\ntranslates (in each\nharmonic signals in\nthe orientation dimension. The standard deviation of the Gaussian was set equal to the shift\n. For illustration purposes, all functions were rendered at a resolution of\n, equaled\n\n4 Experimental results\nIn our experiments the Gaussian-Fourier basis consisted of *\nspatial dimension) of a Gaussian (of period, (\n ), and !\namount,d\n, and the decay constant, \n, equaled \n\u00074\n\u00074\n . The time step, d\n. The parameters for the cut-off function used to eliminate self-loops were _`\u0006\n\u0006; .\nIn the \ufb01rst experiment, the input bias function, \u0001\n\n\u0004 , used to solve the Fokker-Planck equation in the basis equaled\n#&%\n\u0012>* , consisted of twenty randomly posi-\n\ntioned spots and twenty spots on the boundary of an avocado. The positions of the spots\nare real valued, i.e., they do not lie on the grid of basis functions. See Fig. 1 (left). The\nstochastic completion \ufb01eld computed using 32 iterations of the power method is shown in\nFig. 1 (right).\n\n. The diffusion parameter,\u001d\n\n\u0006\u0010\u000f\n\u0006\u0011\u000b\n\n\u0006\u000e\r\u0005\u0007\n\r\u0005\u0007\n\nand\n\n\u000f\u0013\u0012\n\n\u0006\b\u0007\n\nM\u0017\u0016\n\n#&%\n\nIn the second experiment, the input bias function from the \ufb01rst experiment was rotated by\n\u0001 and translated by half the distance between the centers of adjacent basis functions,\n\u000b\u0003\n*+* . See Fig. 2 (left). The stochastic completion \ufb01eld is identical (up\n\u0010\u0015\u0014\u0017\u0016\u0017\u0018\nto rotation and translation) to the one computed in the \ufb01rst experiment. This demonstrates\nthe Euclidean invariance of the computation. See Fig. 2 (right). The estimate of the largest\n, the power method iteration is shown in Fig.\npositive real eigenvalue, \b\n5 Conclusion\n\n, as a function of \u001a\n\n3.\n\nWe described a neural network which enhances and completes salient closed contours.\nEven though the computation is implemented in a discrete network, its output is invariant\nunder continuous rotations and translations of the input pattern.\n\nReferences\n\n[1] Cowan, J.D., Neurodynamics and Brain Mechanisms, Cognition, Computation and\n\nConsciousness, Ito, M., Miyashita, Y. and Rolls, E., (Eds.), Oxford UP, 1997.\n\n\u0006\n\nJ\n\u0001\n\u0006\n\n\u0006\n0\n^\n\u0005\nX\nA\n\u0003\n\u0006\n\b\n\u0003\n\u0003\n\u0005\nE\n^\n\nJ\n\u0001\n\u0006\n\u0006\n3\n\u0006\n.\nJ\n\u0003\n\u0001\nM\n\u0006\n.\n@\n\u0006\n.\n@\n\u0012\n\b\nd\n%\n0\n*\n@\n\u0012\n\b\nd\n%\n\u0006\n*\n\u0006\n@\n\u0007\n\u0012\n\b\n^\n\u001e\n#\n%\n0\n\u001b\n%\n\u0006\n@\n#\n^\n\t\n\u001e\n#\n%\n0\n\b\n%\n\u0006\n\n@\n\u0012\n\b\n%\n\u0012\n\u0006\n3\n\u0006\n.\n.\n\u0012\n@\n\u0006\n.\n\u0006\n.\n\u0012\n\b\nd\n%\n0\n\u0006\n\n\u0006\n.\n\u000b\n\u0006\n\u0012\n\u0006\n3\n\u0006\nJ\n\u0001\n\u0006\nJ\n\u0018\n\u0019\n\b\n2\n2\n%\n%\n\u0006\n2\n2\n\u001e\n*\n,\n\u0006\n.\n#\nd\n#\n%\n%\n\u0006\n*\n7\n\u0007\n\u0006\n\n!\n\u0006\n\u0006\n\u0004\n\u0001\n\u0004\n!\n!\nd\n7\n\u0007\n\u000b\n]\n\u0006\n\u0001\n#\n\u0012\n\b\nJ\n^\n\u001e\n'\n^\n\u001e\n\fFigure 1: Left: The input bias function, \u0001\n\nadded to twenty spots on the boundary of an avocado. The positions are real valued, i.e.,\nthey do not lie on the grid of basis functions. Right: The stochastic completion \ufb01eld,\n\n#:%\n\u0012\u0005* . Twenty randomly positioned spots were\n\n, computed using\n\nbasis functions.\n\n#&%\n\n'76>*\n\n\u0003\u0007\n\n\u0003\u0007\n\n\u0003\u0007\n\nFigure 2: Left: The input bias function from Fig.\n\n\u0001 and translated by\n\u0016/*+* .\n#&%\nRight: The stochastic completion \ufb01eld, is identical (up to rotation and translation) to the\none shown in Fig. 1. This demonstrates the Euclidean invariance of the computation.\n\nhalf the distance between the centers of adjacent basis functions, \u0001\n\n1, rotated by\n\n\u000b\u0003\n\nA\n\u000e\n\u0003\n\u0014\n5\nE\n6\n\u0006\n\u0001\n\u0006\n\u0001\n#\n\u0010\n\u0014\n\u0016\n\u0018\n\u0012\n\b\nJ\n^\n\u001e\n'\n^\n\u001e\nM\n\f0.16\n\n0.14\n\n0.12\n\n0.1\n\n0.08\n\n0.06\n\n0.04\n\n0.02\n\n0\n\n0\n\n5\n\n10\n\n15\n\n20\n\n25\n\n30\n\n35\n\nFigure 3: The estimate of the largest positive real eigenvalue, \b\n\n, the\npower method iteration. Both the \ufb01nal value and all intermediate values are identical in the\nrotated and non-rotated cases.\n\n, as a function of \u001a\n\n[2] Freeman, W., and Adelson, E., The Design and Use of Steerable Filters, IEEE Trans-\n\nactions on Pattern Analysis and Machine Intelligence 13 (9), pp.891-906, 1991.\n\n[3] Mumford, D., Elastica and Computer Vision, Algebraic Geometry and Its Applica-\n\ntions, Chandrajit Bajaj (ed.), Springer-Verlag, New York, 1994.\n\n[4] Golub, G.H. and C.F. Van Loan, Matrix Computations, Baltimore, MD, Johns Hop-\n\nkins Univ. Press, 1996.\n\n[5] Heitger, R. and von der Heydt, R., A Computational Model of Neural Contour Pro-\ncessing, Figure-ground and Illusory Contours, Proc. of 4th Intl. Conf. on Computer\nVision, Berlin, Germany, 1993.\n\n[6] Iverson, L., Toward Discrete Geometric Models for Early Vision, Ph.D. dissertation,\n\nMcGill University, 1993.\n\n[7] Li, Z., A Neural Model of Contour Integration in Primary Visual Cortex, Neural\n\nComputation 10(4), pp. 903-940, 1998.\n\n[8] Parent, P., and Zucker, S.W., Trace Inference, Curvature Consistency and Curve\nDetection, IEEE Transactions on Pattern Analysis and Machine Intelligence 11, pp.\n823-889, 1989.\n\n[9] Shashua, A. and Ullman, S., Structural Saliency: The Detection of Globally Salient\nStructures Using a Locally Connected Network, 2nd Intl. Conf. on Computer Vision,\nClearwater, FL, pp. 321-327, 1988.\n\n[10] Simoncelli, E., Freeman, W., Adelson E. and Heeger, D., Shiftable Multiscale Trans-\n\nforms, IEEE Trans. Information Theory 38(2), pp. 587-607, 1992.\n\n[11] Williams, L.R., and Thornber, K.K., Orientation, Scale, and Discontinuity as Emer-\ngent Properties of Illusory Contour Shape, Neural Computation 13(8), pp. 1683-\n1711, 2001.\n\n[12] Yen, S. and Finkel, L., Salient Contour Extraction by Temporal Binding in a\nCortically-Based Network, Neural Information Processing Systems 9, Denver, CO,\n1996.\n\n[13] Zweck, J., and Williams, L., Euclidean Group Invariant Computation of Stochastic\nCompletion Fields Using Shiftable-Twistable Functions, Proc. European Conf. on\nComputer Vision (ECCV \u201900), Dublin, Ireland, 2000.\n\n\f", "award": [], "sourceid": 1991, "authors": [{"given_name": "Lance", "family_name": "Williams", "institution": null}, {"given_name": "John", "family_name": "Zweck", "institution": null}]}