{"title": "Orientation, Scale, and Discontinuity as Emergent Properties of Illusory Contour Shape", "book": "Advances in Neural Information Processing Systems", "page_first": 831, "page_last": 837, "abstract": null, "full_text": "Orientation,  Scale,  and Discontinuity as \nEmergent  Properties of Illusory  Contour \n\nShape \n\nKarvel  K.  Thornber \nNEC  Research  Institute \n\n4  Independence Way \nPrinceton, NJ 08540 \n\nLance  R.  Williams \n\nDept.  of Computer Science \nUniversity of New  Mexico \nAlbuquerque,  NM  87131 \n\nAbstract \n\nA  recent  neural  model  of  illusory  contour  formation  is  based  on \na  distribution  of natural  shapes  traced  by  particles  moving  with \nconstant speed in directions given by Brownian motions.  The input \nto that model consists of pairs of position and direction constraints \nand the  output  consists  of the  distribution  of contours joining all \nsuch  pairs.  In general,  these contours will  not  be closed  and  their \ndistribution  will  not  be  scale-invariant.  In  this  paper,  we  show \nhow  to  compute  a  scale-invariant  distribution  of  closed  contours \ngiven  position  constraints  alone  and  use  this  result  to  explain  a \nwell  known illusory contour effect. \n\n1 \n\nINTRODUCTION \n\nIt has been proposed by Mumford[3] that the distribution of illusory contour shapes \ncan  be  modeled  by  particles  travelling  with constant  speed  in  directions  given  by \nBrownian motions.  More recently,  Williams  and Jacobs[7,  8]  introduced the notion \nof a  stochastic  completion field,  the distribution of particle trajectories joining pairs \nof position  and  direction  constraints,  and  showed  how  it  could  be  computed  in  a \nlocal  parallel  network.  They  argued  that  the  mode,  magnitude  and  variance  of \nthe  completion  field  are  related  to the  observed  shape,  salience,  and  sharpness  of \nillusory  contours. \n\nUnfortunately,  the Williams  and  Jacobs  model,  as  described,  has  some  shortcom(cid:173)\nings.  Recent  psychophysics  suggests  that  contour  salience  is  greatly  enhanced  by \nclosure[2].  Yet,  in  general,  the  distribution computed by the Williams  and Jacobs \nmodel  does  not  consist  of closed  contours.  Nor  i:;  it  scale-invariant-doubling the \ndistances between the constraints does not produce a comparable completion field of \n\n\f832 \n\nK.  K.  Thornber and L.  R.  Williams \n\ndouble the size without a  corresponding doubling of the particle's speeds.  However, \nthe  Williams  and  Jacobs  model  contains  no  intrinsic  mechanism  for  speed  selec(cid:173)\ntion.  The speeds  (like  the directions)  must  be specified  a priori.  In this  paper, we \nshow how to compute a scale-invariant distribution of closed contours given position \nconstraints alone. \n\n2  TECHNICAL DETAILS \n\n2.1  SHAPE DISTRIBUTION \n\nConsistent  with  our  earlier  work[5,  6],  in  this  paper  we  do  not  use  the  same  dis(cid:173)\ntribution described by Mumford[3]  but instead assume a  distribution of completion \nshapes  consisting  of  straight-line  base-trajectories  modified  by  random  impulses \ndrawn from  a  mixture of two limiting distributions.  The first  distribution  consists \nof weak  but frequently  acting impulses  (we  call  this  the  Gaussian-limit).  The dis(cid:173)\ntribution of these weak impulses has zero mean and variance equal to (7~.  The weak \nimpulses  act  at  Poisson  times  with  rate  R g .  The  second  distribution  consists  of \nstrong but  infrequently acting impulses  (we  call  this  the  Poisson-limit).  Here,  the \nmagnitude of the random  impulses  is  Gaussian  distributed  with  zero  mean.  How(cid:173)\never,  the  variance  is  equal  to  (72  (where  (7~  \u00bb \n(7;).  The  strong  impulses  act  at \nPoisson times with rate Rp  < < kg.  Particles decay with half-life equal to a  param(cid:173)\neter T.  The effect is  that particles tend to travel in smooth, short paths punctuated \nby occasional orientation discontinuities.  See  [5,  6]. \n\n2.2  EIGENSOURCES \n\nLet i  and j  be position and velocity constraints,  (xi,id and  (xj,Xj).  Then P(jl i) \nis  the conditional  probability that a  particle beginning at i  will  reach  j.  Note that \nthese transition probabilities are not symmetric, i.e., P(j 1 i)  1=  P(i 1 j).  However, by \ntime-reversal symmetry,  P(j 1 i) = P(I 1 J)  where I = (Xi, -Xi)  and J = (Xj, -Xj). \nGiven  only the matrix of transition  probabilities,  P, we  would  like  to compute the \nrelative number of closed contours satisfying a given position and velocity constraint. \nWe  begin  by  noting that,  due  to  their  randomness,  only  increasingly  smaller  and \nsmaller fractions of contours are likely  to satisfy increasing numbers of constraints. \nSuppose we  let  s~l)  contours start at Xi  with  Xi.  Then \n\n(2)  _  ~ P( 'I .)  (1) \n- ui  J  '/,  Si \nSj \n\nis  the relative number  of contours  through  Xj  with  Xj,  i.e.,  which  satisfy  two con(cid:173)\nstraints.  In general, \n\n- ui \nNow  suppose we  compute the eigenvector, \n\nSj \n\n(n+1)  _  ~ P( 'I .)  (n) \n\nJ \n\n'/,  Si \n\nwith largest, real positive eigenvalue, and take s~1)  =  Si.  Then clearly si n+1)  =  AnSi. \nThis implies that as the number of constraints satisfied increases by one, the number \nof contours remaining in  the sample of interest  decreases by A.  However, the ratios \nof the  Si  remain  invariant.  Letting  n  pass  to  infinity,  we  see  that  the  Si  are  just \nthe  relative number of contours  through  i.  To summarize,  having started with  all \npossible contours,  we  are now  left  with  only  those  bridging  pairs  of constraints at \nall  past-times.  By solving  AS  =  Ps for  s  we  know  their relative numbers.  We  refer \nto the components of s  as the  eigensources of the stochastic completion field. \n\n\fEmergent Properties of Illusory Contour Shape \n\n833 \n\n2.3  STOCHASTIC  COMPLETION FIELDS \n\nNote that the eigensources alone do not represent a  distribution of closed contours. \nIn fact, the majority of contours contributing to s will not satisfy a single additional \nconstraint.  However, the following recurrence equation gives the number of contours \nwhich  begin at constraint i  and end  at constraint j  and  satisfy  n  - 1 intermediate \nconstraints \n\np(n+1) (j I i)  =  Lk P(j I k)p(n) (k I i) \n\nwhere  p( 1) (j I i)  =  P(j I i).  Given  the above recurrence equation, we  can  define  an \nexpression  for  the  relative  number  of contours  of any length  which  begin  and  end \nat constraint i: \n\nCi  =  limn -+ oo  p(n)(i I i)/ Lj p(n)(j I)) \n\nUsing  a  result  from  the  theory  of positive  matrices[l},  it  is  possible  to  show  that \nthe above expression is  simply \n\nCi  =  Si 8d Lj Sj8j \n\nwhere  sand s are  the  right  and  left  eigenvectors  of P  with  largest  positive  real \neigenvalue,  i.e.,  AS  =  Ps and  AS  =  pTs.  Because  of the  time-reversal  symmetry \nof P, the right and left  eigenvectors are related by a  permutation which exchanges \nopposite directions,  i.e. ,  8i  =  St. \nFinally,  given  sand s,  it  is  possible  to  compute  the  relative  number  of  closed \ncontours through  an  arbitrary  position  and  velocity  in  the  plane,  i.e.,  to  compute \nthe stochastic  completion  field.  If\", =  (x, x)  is  an  arbitrary position  and  velocity \nin  the plane,  then \n\nC(\",)  =  >.s~s Li P(\", I i)Si . Lj P(j I \",)8j \n\ngives  the relative probability that a  closed  contour will  pass  through \",.  Note,  that \nthis  is  a  natural  generalization  of the  Williams  and  Jacobs[7]  factorization  of  the \ncompletion field  into the product of source and sink fields. \n\n2.4  SCALE-INVARIANCE \n\nUnder the restriction that particles  have constant speed, the transition  probability \nmatrix,  P, becomes  block-diagonal.  Each block  corresponds  to a  different  possible \nspeed, 'Y- Since the components of any given eigenvector will  be confined to a  single \nblock, we  can  consider P  to be a  function  of, and solve: \n\nA(r) s(r)  =  P(r)s(r) \n\nLet Amax (r) be the largest positive real eigenvalue of P(r) and let ,max be the speed \nwhere  Amax (r)  is  maximized.  Then  Smax (rma x),  i.e.,  the  eigenvector  of P (rmax) \nassociated with  Amax (rma x), is  the limiting distribution over  all  spatial scales. \n\n3  EXPERIMENTS \n\n3.1  EIGHT  POINT CIRCLE \n\nGiven  eight  points  spaced  uniformly  around the  perimeter  of a  circle  of diameter, \nd =  16,  we  would like to find  the distribution of directions  through each  point  and \nthe corresponding completion field  (Figure  1 (left)).  Neither the order of traversal , \ndirections,  i.e.,  xdlxil,  or  speed,  i.e. ,  ,  =  IXil.  are  specified  a  priori. \nIn  all  of \nour  experiments,  we  sample  direction  at  5\u00b0  intervals.  Consequently,  there  are  72 \ndiscrete directions  and 576  position-direction pairs,  i.e., P(r)  is  of size 576 x  576. 1 \nlThe parameters defining the distribution of completion shapes are T  = Rga~ = 0.0005 \n\nand  'T  = 9.5.  For  simplicity,  we  assume the pure Gaussian-limit case described in  [6] . \n\n\f834 \n\n\u2022 \n\u2022 \n\u2022 \n\n\u2022 \n\n\u2022 \n\na \n\n\u2022 \n\u2022 \n\u2022 \n\n/ \nI \n\n\" I \n\"  / \n\nc \n\nK.  K.  Thornber and L.  R.  Williams \n\nEight  Point  Circle  (two  sizes) \n\nI \ni \n\n,..,,, \no \nw~ \n,0 \nI \n. \no \n\n.-\nw \n\n\u2022 a \n\nIl. \n\n'0 \nxo \nog~~~~~~~~~~~~~~~ \n:l~ \no \n\n30 \n\n,0 \n\n0 \n\n\" \n\n'\" \n\n20 \n\nb \n\nd \n\nFigure 1:  Left:  (a)  The eight  position  constraints.  Neither  the order  of traversal,  direc(cid:173)\ntions,  or speed  are specified  a priori.  (b)  The eigenvector,  Smax (,max)  represents  the lim(cid:173)\niting distribution  over  all  spatial scales.  (c)  The product of smaxC!max)  and smaxC!max). \nOrientations  tangent  to  the circle  dominate the  distribution  of closed  contours.  (d)  The \nstochastic completion field,  C, due to smaxC!max).  Right:  Plot of magnitude of maximum \npositive real  eigenvalue,  >'max,  vs.  logl.l (1/,) for  eight  point  circle  with  d = 16.0  (solid) \nand d =  32.0  (dashed). \n\n==  \n\n~ \n~ \n\n~ ==::J \n\n11 \nU \n\nFigure  2:  Observers  report  that  as  the  width  of  the  arms  increases,  the  shape  of  the \nillusory  contour  changes from  a circle  to a square[4]. \n\nFirst, we evaluated Amax b) over the velocity interval [1.1- 1 , 1.1-3oJ using standard \nnumerical routines and plotted the magnitude of the largest, real positive eigenvalue, \nAmax  vs.  logl.l(l/,).  The  function  reaches  its  maximum  value  at '\"'(max::::::  1.1- 2\u00b0. \nConsequently, the eigenvector, Smax (1.1 - 2\u00b0) represents the limiting distribution over \nall  spatial scales  (Figure  1  (right)). \n\nNext,  we  scaled  the  test  Figure  by  a  factor  of  two,  i.e.,  d'  =  32.0  and  plotted \nA~axb) over the same interval  (Figure  1  (right)).  We observe that A~ax(1.1-x+7) \n::::::  Amax (1.1- X ),  i.e.,  when  plotted  using  a  logarithmic  x-axis,  the  functions  are \nidentical except for a translation.  It follows  that '\"'(~ax  ::::::  logl.1  7 x '\"'(max::::::  2.0 x '\"'(max' \nThis  confirms  the  scale-invariance of the  system-doubling the  size  of the  Figure \nresults in  a  doubling of the selected speed. \n\n3.2  KOFFKA  CROSS \n\nThe Koffka Cross stimulus (Figure 2)  has two basic degrees of freedom which we call \ndiameter  (i.e. ,  d)  and arm width  (i.e.,  w)  (Figure  3  (a)).  We  are interested in how \n\n\fEmergent Properties of Illusory Contour Shape \n\n(a) \n\n(e) \n\n835 \n\n(d) \n\nr---~  .--......, \n\no \nU \n\nd \n\n(b) \n\n(-0 5w . O.5d) \n\n( O.5w ,O.Sd ) \n\n(--O.Sd , 05w) \n\n(--OSd,  --05w) \n\n(--05w.  -O.5d) \n\n(O.5w. --OSd ) \n\n(OSd , 05w) \n\nn \n( 0 Sd ,-O.5w )  u \n\nr - - - - -\n\n- - - - - - ,  \n\nFigure 3:  (a) Koffka Cross showing diameter, d,  and width , w.  (b) Orientation and \nposition constraints in  terms of d and w.  The normal orientation at each endpoint \nis  indicated  by  the  solid  lines  while  the  dashed  lines  represent  plus  or  minus  one \nstandard deviation  (i.e. ,  12.8\u00b0)  of the  Gaussian  weighting  function.  (c)  Typically \nperceived  as  square.  (d)  Typically  perceived  as  circle.  The  positions  of  the  line \nendpoints is  the same. \n\nthe  stochastic  completion  field  changes  as  these  parameters are varied.  Observers \nreport  that  as  the  width  of the  arms  increases,  the  shape  of the  illusory  contour \nchanges  from  a  circle  to  a  square[4].  The  endpoints  of  the  lines  comprising  the \nKoftka  Cross  can  be  used  to  define  a  set  of  position  and  orientation  constraints \n(Figure  3  (b)).  The position constraints are specified in terms of the parameters, d \nand w.  The orientation constraints take the form  of a  Gaussian weighting function \nwhich  assigns  higher  probabilities  to contours  passing through the endpoints  with \norientations normal to the lines. 2  The prior probabilities assigned to each position(cid:173)\ndirection  pair by the Gaussian weighting function form  a  diagonal matrix, D: \n\nwhere  P(r)  is  the  transition  probability  matrix  for  the  random  process  at  scale \n\"  A(r)  is  an  eigenvalue  of Q(,),  and  s(r)  is  the  corresponding  eigenvector.  Let \nAmax(r)  be  the  largest  positive real  eigenvalue of Q(r)  and  let  ,max  be  the  scale \nwhere  Amax(r)  is  maximized.  Then  smax(rmax),  i.e.,  the  eigenvector  of Q(rmax) \nassociated with  Amax (rma x),  is  the limiting distribution over  all  spatial scales. \n\nFirst, we used a Koffka Cross where d =  2.0 and w  =  0.5 and evaluated Amax (r) over \nthe velocity interval  [8.0  x  1.1- 1 , 8.0 x  1.1-8\u00b0]  using standard numerical routines. 3 \nThe function  reaches its  maximum  value at ,max::::;  8.0  X  1.1-62  (Figure  4  (left)). \nObserve  that  the  completion  field  due  to  the  eigenvector,  smax(8.0  x  1.1-62 ),  is \ndominated  by contours  of a  predominantly  circular  shape  (Figure  4  (right)).  We \nthen uniformly scaled the Koffka  Cross  Figure by a  factor  of two,  i.e.,  d'  =  4.0 and \n\n20bserve that Figure  3 (c)  is  perceived as a square while Figure  3 (d) is  perceived as  a \ncircle.  Yet the positions  of the line endpoints is  the same.  It follows  that the orientations \nof the lines  affect  the percept.  We have chosen  to model  this dependence through the use \nof a  Gaussian  weighting function  which favors  contours  passing through  the endpoints of \nthe lines  in  the normal  direction.  It is  possible to motivate this  based  on  the statistics of \nnatural scenes.  The distribution of relative  orientations  at contour  crossings  is  maximum \nat  90\u00b0  and drops to nearly zero at  0 0  and 180 0 \u2022 \n\n3The  parameters  defining  the  distribution  of completion  shapes  were:  T  =  RgO'~  = \n0.0005, T  = 9.5, \u20acp  = O'~/T = 100.0  and Rp  = 1.0  X  10- 8 .  As  an  anti-aliasing measure, the \ntransition probabilities, P(j I i) , were averaged over initial conditions modeled as Gaussians \nof variance 0';  = 0';  = 0.00024  and O'J  = 0.0019.  See  [6]. \n\n\f836 \n\nK.  K.  Thornber and L.  R.  Williams \n\nKoffke  Crosses  (TWO  sizes) \n\n_ -_=0>== \n\n... \n'\"'~ \nI  a , \nLIl~ \no \n\nWo \n\n-o \n> c \nCI~ - . \n00 \n-o , \nIra ... a \na \nD.. \n\n~o \n\ni \n/ \n\n; \n\n/ \n\n~/--\n\nX 0 \nog-HTnTrnTnTrnTnTrnTnTnTrnTn~~ \n~ci \n.. 0 \n\n60 \n\na \n\n20 \n\n'0 \nx \n\nFigure  4:  Left:  Plot  of  magnitude  of  maximum  positive  real  eigenvalue,  >'max,  vs. \nlogl.l (1h)  for  Koffka  Crosses  with  d  =  2.0  and  w  =  0.5  (solid)  and  d  =  4.0  and  w  =  1.0 \n(dashed).  Right:  The completion  field  due to the eigenvector,  smax (8 .0  x 1.1-62 ) . \n\nw'  =  1.0  and  plotted  Anax (,)  over  the  same  interval  (Figur~  4  (left)) .  Observe \nthat  A~ax (8.0  X  1.1-x+  ) ::::::  Amax(8.0 x  1.1- X).  As  before,  thls confirms  the scale(cid:173)\ninvariance of the system. \n\nNext,  we  studied  how  the  relative  magnitudes  of  the  local  maxima  of  Amax (,) \nchange as  the parameter w  is  varied.  We  begin with a  Koffka  Cross where d =  2.0 \nand  w  =  0.5  and  observe  that  Amax(r)  has  two  local  maxima  (Figure  5  (left)). \nWe  refer  to  the  larger  of these  maxima  as  ,circle .  As  previously  noted,  this  max(cid:173)\nimum  is  located  at  approximately  8.0  x  1.1-62 .  The second  maximum  is  located \nat approximately 8.0 x  1.1 -32.  When  the  completion  field  due  to the  eigenvector, \nsmax(8.0  x  1.1-32 ),  is  rendered,  we  observe  that  the  distribution  is  dominated  by \ncontours  of predominantly  square shape  (Figure  5(a)).  For  this  reason ,  we  refer \nto this local  maximum as  ,square.  Now  consider a  Koffka  Cross  where the widths \nof the  arms  are  doubled  but  the  diameter  remains  the  same,  i.e.,  d'  =  2.0  and \nw'  =  1.0.  We  observe  that  A~ax (r)  still  has  two  local  maxima,  one  at  approxi(cid:173)\nmately 8.0  x  1.1-63  and a  second at approximately 8.0 x  l.1-29  (Figure  5  (left)). \nWhen we render the completion fields  due to the eigenvectors, s~ax(8.0x 1.1-63 )  and \ns~ax(8.0 x  1.1- 29 ),  we  find  that the completion fields  have the same general  char(cid:173)\nacter  as  before-the contours  associated  with  the smaller spatial  scale  (i.e.,  lower \nspeed)  are approximately circular and those associated with the larger spatial scale \n(Le.,  higher speed)  are approximately square  (Figure  5  (d)  and  (c)).  Accordingly, \nwe  refer to the locations of the respective local maxima as '~ircle and ,~quare '  How(cid:173)\never,  what  is  most  interesting is  that  the relative magnitudes of the local  maxima \nhave reversed.  Whereas we  previously observed that Amax(,circle) >  Amax(rsquare), \nwe now observe that A~ax(r~quare) > A~ax(r~ircle)'  Therefore, the completion field \ndue  to  the  eigenvector,  s~ax(r~quare )  [not  s~ax(r~ircle)!l  represents  the  limiting \ndistribution over all spatial scales.  This is  consistent with the transition from  circle \nto square reported  by human observers when the widths  of the arms of the Koffka \nCross  are increased. \n\n\fEmergent Properties of Illusory Contour Shape \n\n837 \n\nKoffke  Crosses  (two \n\nwidths) \n\n-\n\n--, ~:...-\n\nb \n\na \n\nc \n\nd \n\no \n> \n( \nGl o \nOI~ \n\n.- . wo \n-o \u2022 (l:o \u2022 o \n\n~o \n\no \nQ. \n\n20 \n\n40 \nX \n\n60 \n\n80 \n\na \n\nc \n\nb \n\nd \n\nFigure 5:  Plot  of magnitude of maximum positive real  eigenvalue,  Ama x,  vs.  log 1.1 (1/\"'() \nfor  Koffka  Crosses  with  d  =  2.0  and  w  =  0.5  (solid)  and d  =  2.0  and  w  =  1.0  (dashed) . \nStochastic completion fields  for  Koffka Cross  due to  (a)  Smax (\"'(.quar e )  is  a local  optimum \nfor  w  =  0.5  (b)  Sma x (\"'(ci rcl e )  is  the global  optimum for  w  =  0.5  (c)  s~ax(\"'(~quar e )  is  the \nglobal  optimum  for  w  =  1.0  (d)  s~a x (\"'(~quar e )  is  a  local  optimum  for  w  =  1.0.  These \nresults  are  consistent  with  the  circle-to-square  transition  perceived  by  human  subjects \nwhen  the width of the arms of the Koffka  Cross  are increased. \n\n4  CONCLUSION \n\nWe  have  improved  upon  a  previous  model  of illusory  contour formation  by  show(cid:173)\ning how  to compute a  scale-invariant distribution of closed contours given position \nconstraints  alone.  We  also  used  our  model  to  explain  a  previously  unexplained \nperceptual effect. \n\nReferences \n\n[1]  Horn,  R.A .,  and  C.R.  Johnson,  Matrix  Analysis,  Cambridge  Univ.  Press ,  p.  500, \n\n1985. \n\n[2]  Kovacs,  I.  and  B.  Julesz,  A  Closed  Curve  is  Much  More  than  an  Incomplete  One: \nEffect  of Closure  in  Figure-Ground Segmentation,  Pmc.  Natl.  Acad.  Sci.  USA,  90, \npp.  7495-7497,  1993. \n\n[3]  Mumford, D.,  Elastica  and  Computer Vision,  Algebraic  Geometry  and  Its  Applica(cid:173)\n\ntions,  Chandrajit  Bajaj  (ed.) , Springer-Verlag,  New York,  1994. \n\n[4)  Sambin,  M.,  Angular  Margins  without  Gradients,  Italian  Journal  of Psychology  1, \n\npp.  355-361,  1974. \n\n[5]  Thornber,  KK  and  L.R.  Williams,  Analytic  Solution  of  Stochastic  Completion \n\nFields,  Biological  Cybernetics  75, pp.  141-151,  1996. \n\n[6]  Thornber , KK and L.R.  Williams,  Characterizing the Distribution of Completion \nShapes  with  Corners  Using  a  Mixture  of  Random  Processes,  Intl.  Workshop  on \nEnergy  Minimization  Methods  in  Computer  Vision,  Venice, Italy, 1997. \n\n[7]  Williams,  L.R.  and D.W . Jacobs,  Stochastic Completion Fields:  A Neural Model  of \nIllusory  Contour  Shape and Salience,  Neural  Computation 9(4) , pp.  837-858,  1997. \n[8)  Williams,  L.R.  and D.W.  Jacobs,  Local  Parallel  Computation  of  Stochastic  Com(cid:173)\n\npletion  Fields,  Neural  Computation  9(4),  pp. 859-881 ,  1997. \n\n\f", "award": [], "sourceid": 1528, "authors": [{"given_name": "Karvel", "family_name": "Thornber", "institution": null}, {"given_name": "Lance", "family_name": "Williams", "institution": null}]}