{"title": "Learning Saccadic Eye Movements Using Multiscale Spatial Filters", "book": "Advances in Neural Information Processing Systems", "page_first": 893, "page_last": 900, "abstract": null, "full_text": "Learning Saccadic  Eye  Movements \n\nUsing Multiscale  Spatial Filters \n\nRajesh P.N. Rao and Dana H.  Ballard \n\nDepartment of Computer Science \n\nUniversity of Rochester \nRochester,  NY  14627 \n\n{rao)dana}~cs.rochester.edu \n\nAbstract \n\nWe  describe  a  framework  for  learning  saccadic  eye  movements  using  a \nphotometric representation of target points  in natural scenes.  The rep(cid:173)\nresentation takes the form of a high-dimensional vector comprised of the \nresponses  of spatial filters  at different  orientations and  scales.  We  first \ndemonstrate the use  of this  response  vector in  the task of locating pre(cid:173)\nviously foveated  points in a  scene and subsequently use  this property in \na  multisaccade strategy to derive  an adaptive motor map for  delivering \naccurate saccades. \n\n1 \n\nIntroduction \n\nThere has been recent interest in the use of space-variant sensors in active vision systems \nfor tasks such as visual search and object tracking [14].  Such sensors realize the simultane(cid:173)\nous need for wide field-of-view and good visual acuity.  One popular class of space-variant \nsensors  is  formed  by  log-polar  sensors  which  have  a  small  area near  the optical  axis  of \ngreatly  increased resolution  (the fovea)  coupled  with  a  peripheral region  that  witnesses \na  gradual logarithmic falloff in resolution  as  one  moves  radially  outward.  These sensors \nare  inspired  by  similar  structures  found  in  the  primate  retina  where  one  finds  both  a \nperipheral region of gradually decreasing acuity and a circularly symmetric  area  centmlis \ncharacterized by  a  greater density  of receptors  and a  disproportionate representation in \nthe optic nerve  [3].  The peripheral region,  though of low  visual  acuity,  is  more  sensitive \nto light  intensity and movement. \nThe existence of a  region optimized for  discrimination and recognition  surrounded  by  a \nregion  geared towards detection thus  allows  the image of an  object  of  interest  detected \nin  the outer region  to be placed on the more analytic  center for  closer  scrutiny.  Such  a \nstrategy  however necessitates  the  existence  of  (a)  methods  to determine  which  location \nin  the  periphery  to foveate  next,  and  (b)  fast  gaze-shifting  mechanisms  to  achieve  this \n\n\f894 \n\nRajesh P.  N.  Rao,  Dana H.  Ballard \n\nfoveation.  In  the  case  of humans,  the  \"where-to-Iook-next\"  issue  is  addressed  by  both \nbottom-up strategies such  as motion or salience clues from  the periphery as  well  as top(cid:173)\ndown strategies such as search for a particular form or color.  Gaze-shifting is accomplished \nvia  very  rapid  eye  movements  called  saccades.  Due  to  their  high  velocities,  guidance \nthrough visual feedback is  not possible and hence,  saccadic movement is  preprogrammed \nor  ballistic:  a  pattern of muscle  activation  is  calculated in  advance  that  will  direct  the \nfovea almost exactly to the desired position [3]. \n\nIn  this  paper,  we  describe  an  iconic  representation  of scene  points  that  facilitates  top(cid:173)\ndown foveal  targeting.  The  representation takes  the  form  of a  high-dimensional  vector \ncomprised of the responses of different order Gaussian derivative filters,  which are known \nto  form  the  principal  components  of natural  images  [5],  at  variety  of orientations  and \nscales.  Such a representation has been recently shown to be useful for visual tasks ranging \nfrom texture segmentation [7]  to object indexing using a sparse distributed memory [11]. \nWe  describe how  this photometric representation of scene points can be used  in locating \npreviously foveated  points  when  a  log-polar sensor  is  being  used.  This  property is  then \nused  in  a  simple  learning  strategy  that  makes  use  of  multiple  corrective  saccades  to \nadaptively  form  a  retinotopic motor  map similar  in  spirit  to the  one  known  to exist  in \nthe deep layers of the primate superior colliculus [13].  Our approach differs from previous \nstrategies for  learning motor maps (for  instance,  [12])  in that we use the visual modality \nto actively supply the necessary reinforcement signal required during the motor learning \nstep (Section 3.2) . \n\n2  The Multiscale Spatial Filter Representation \n\nIn the active vision framework, vision is  seen as subserving a larger context of the encom(cid:173)\npassing  behaviors  that  the agent  is  engaged in.  For  these  behaviors,  it is  often  possible \nto  use  temporary,  iconic  descriptions  of  the  scene  which  are  only  relatively  insensitive \nto  variations  in  the  view.  Iconic  scene  descriptions  can  be  obtained,  for  instance,  by \nemploying  a  bank of linear  spatial filters  at a  variety of orientations  and  scales.  In  our \napproach,  we  use  derivative of Gaussian filters  since these are known to form  the domi(cid:173)\nnant eigenvectors of natural images  [5]  and can thus be expected to yield  reliable results \nwhen used  as  basis functions  for  indexingl . \n\nThe exact  number of Gaussian derivative  basis functions  used  is  motivated  by  the need \nto make the representations invariant to rotations in the image  plane  (see  [11]  for  more \ndetails).  This  invariance  can  be  achieved  by  exploiting  the  property  of  steerability  [4] \nwhich  allows filter  responses at arbitrary orientations to be synthesized from  a finite  set \nof basis filters.  In  particular,  our implementation  uses  a  minimal  basis  set  of two first(cid:173)\norder directional derivatives at 0\u00b0  and 90\u00b0,  three second-order derivatives at 0\u00b0, 60\u00b0  and \n120\u00b0,  and four third-order derivatives oriented at 0\u00b0,  45\u00b0,  90\u00b0,  and 135\u00b0. \n\nThe response of an image patch J centered at (xo, Yo)  to a  particular basis filter  G~j can \nbe obtained by convolving the image patch with the filter  : \n\nIf  9 \n\nri,j(XO, Yo)  = (G/ * I)(xo, Yo)  =  G/ (XO  - x, Yo  - y)J(x, y)dx dy \n\ng . \n\n(1) \n\nlIn  addition,  these  filters  are  endorsed by recent  physiological  studies  [15]  which  show  that \nderivative-of-Gaussians  provide the best fit  to primate cortical receptive field profiles among the \ndifferent functions suggested in the literature. \n\n\fLearning Saccadic  Eye  Movements  Using  Multiscale  Spatial Filters \n\n895 \n\nThe  iconic  representation  for  the  local  image  patch  centered  at  (xo, Yo)  is  formed  by \ncombining into a single high-dimensional vector the responses from  the nine  basis filters, \neach  (in our current implementation)  at five  different scales: \n\nr(xo, Yo)  = (ri,j,s) , \n\ni  =  1,2, 3;j = 1, . .. , i + 1; S  = Smin , . .. , Smax \n\n(2) \n\nwhere i  denotes  the order of the filter,  j  denotes  the number  of filters  per  order,  and  S \ndenotes the number of different  scales. \n\nThe use of multiple scales increases the perspicuity of the representation and allows inter(cid:173)\npolation strategies for scale invariance (see  [9]  for more details).  The entire representation \ncan be computed using only nine convolutions done at frame-rate within a pipeline image \nprocessor with nine  constant size 8 x 8 kernels on a five-level  octave-separated low-pass(cid:173)\nfiltered  pyramid of the input image. \n\nThe 45-dimensional  vector  representation  described  above  shares  some of the favorable \nmatching properties that accrue to high-dimensional  vectors  (d.  [6]).  In particular,  the \ndistribution  of  distances  between  points  in  the  45-dimensional  space  of  these  vectors \napproximates a normal distribution; most of the points in the space lie at approximately \nthe mean distance and are thus relatively uncorrelated to a given point  [11].  As  a result, \nthe multiscale filter  bank tends to generate almost unique location-indexed signatures of \nimage regions  which  can tolerate considerable noise  before they are confused  with other \nimage regions. \n\n2.1  Localization \n\nDenote the response vector from  an image point as fi and that from a previously foveated \nmodel point as Tm.  Then one metric for  describing the similarity between the two points \nis simply the square of the Euclidean distance (or the sum-of-squared-differences) between \ntheir response  vectors  dim  = llfi - r.n 112.  The algorithm  for  locating  model  points  in  a \nnew scene can then be described  as follows : \n\n1.  For  the response  vector  representing  a  model  point  m,  create a  distance  image \n\nI m  defined  by \n\n(3) \nwhere  t3  is  a  suitably chosen  constant  (this  makes  the best match the brightest \npoint in  Im). \n\nIm(x,y) = min [Imax  - t3dim , 0] \n\n2.  Find the best match point  (Xb~, Yb~) in the image using the relation \n\n(4) \n\nFigure  1  shows  the  use  of the  localization  algorithm  for  targeting the  optical  axis  of a \nuniform-resolution sensor in  an example scene. \n\n2.2  Extension to Space-Variant  Sensing \n\nThe localization  algorithm  as  presented  above  will  obviously  fail  for  sensors  exhibiting \nnonuniform resolution characteristics.  However,  the multiscale structure of the response \nvectors  can  be  effectively  exploited  to  obtain  a  modified  localization  algorithm.  Since \ndecreasing radial resolution results in an effective reduction in scale  (in addition to some \n\n\f896 \n\nRajesh P.  N.  Rao,  Dana H.  Ballard \n\n(a) \n\n(b) \n\n(c) \n\n(d) \n\nFigure  1:  Using  response  vectors  to  saccade  to  previously  foveated  positions.  (a)  Initial  gaze \npoint. \n(b)  New  gaze  point;  (c)  To  get  back  to  the  original  point,  the  \"distance  image\"  is \ncomputed:  the brightest spot represents the point whose response vector is  closest to that of the \noriginal  gaze  point;  (d)  Location of best  match is  marked and an oculomotor  command at that \nlocation  can  be executed to foveate  that point. \n\nother minor distortions)  of previously foveated  regions as  they move towards the periph(cid:173)\nery,  the filter  responses  previously occuring at larger scales  now  occur  at  smaller  scales. \nResponses  usually  vary  smoothly between scales;  it is  thus  possible to establish a  corre(cid:173)\nspondence  between  the  two  response  vectors  of the  same  point  on  an object  imaged  at \ndifferent  scales  by  using a simple  interpolate-and-eompare  scale matching strategy.  That \nis,  in  addition  to  comparing  an  image  response  vector  and  a  model  response  vector  di(cid:173)\nrectly as  outlined in  the previous section, scale interpolated versions of the image vector \nare also compared with the original model response vector.  In the simplest case,  interpo(cid:173)\nlation amounts to shifting image response vectors by one scale and thus, responses from  a \nnew  image are compared with original model responses  at second,  third,  .. , scales,  then \nwith  model  responses  at third,  fourth,  ... scales,  and  so  on  upto some  threshold  scale. \nThis is  illustrated in Figure 2 for  two discrete movements of a simulated log-polar sensor. \n\n3  The M ultisaccade Learning Strategy \nSince  the  high  speed  of  saccades  precludes  visual  guidance,  advance  knowledge  of the \nprecise  motor  command  to  be  sent  to  the extraocular  muscles  for  fixation  of a  desired \nretinal  location is  required.  Results  from  neurophysiological and psychophysical  studies \nsuggest that in  humans,  this  knowledge is  acquired via learning:  infants  show  a  gradual \nincrease in saccadic accuracy during their first year [1,  2]  and adults can adapt to changes \n(caused for example by weakening of eye-muscles) in the interrelation between visual input \nand the saccades needed for centering.  An adaptive mechanism for  automatically learning \nthe  transfer  function  from  retinal  image  space into motor  space is  also  desirable  in  the \ncontext of active vision systems  since  an autonomous  calibration of the saccadic system \nwould  (a)  avoid  the need  for  manual  calibration,  which  can  sometimes  be  complicated, \nand (b)  provide resilience amidst changing circumstances caused by, for  instance, changes \nin the camera lens  mechanisms or degradation of the motor apparatus. \n\n3.1  Motor Maps \n\nIn primates,  the  superior  eollieulus  (SC),  a  multilayered neuron complex  located  in  the \nupper regions of the brain stem, is  known to playa crucial role in the saccade generation \n[13].  The  upper  layers  of the  SC  contain  a  retinotopie  sensory  map  with  inputs  from \n\n\fLearning Saccadic  Eye  Movements  Using  MuLtiscaLe  SpatiaL  Filters \n\n897 \n\n(a) \n\n(b) \n\n(c) \n\nScale I \n\nScale 2 \n\nScale 3 \n\nScale 4 \n\nScale 5 \n\n(a)  III' I \" \n\n(b) \n\n(c) \n\n111'1\"\"  I I \"  III, ' II \" ,li'I\"\"  111'1 \nseNe 4 \n\ns1I3 \n\nScale 2 \n\nScille  I \n\nScale 5 \n\n111 ' 1 \"'1 1\" 111 \" 11,, 1 \" 1' ,11 ,,1 ' 11 1,1 \n\nScale I \n\nScale 2 \n\nScaje 3 \n\nScale 4 \n\nScale 5 \n\n11\"1 11 ,  II '  ,11'''1, ,111,, 111 11 1 ,1'11'1 \n\nI \n\n(d) \n\nFigure 2:  Using response vectors with a  log-polar sensor,  (a) through (c) represent a sequence of \nimages (in Cartesian coordinates) obtained by movement of a simulated log-polar sensor from  an \noriginal  point  (marked by  '+') in  the foveal  region  (indicated by  a  circle)  towards  the right.  (d) \ndepicts the process of interpolating  (in  this case,  shifting)  and matching response vectors of the \nsame point as  it  moves  towards  the  periphery  of the sensor  (Positive  responses  are  represented \nby  proportional  upward  bars  and  negative  ones  by  proportional  downward  bars  with  the  nine \nsmallest scale responses  at  the beginning and the nine largest ones at  the end). \n\nthe retina while  the deeper  layers  contain a  motor map  approximately  aligned  with  the \nsensory  map.  The  motor  map  can  be  visualized  as  a  topologically-organized  network \nof neurons  which  reacts  to a  local  activation  caused  by  an input  signal  with  a  vectorial \noutput quantity that can be transcoded into a saccadic motor command. \nThe alignment of the sensory and motor maps suggests the following  convenient strategy \nfor  foveation:  an excitation in the sensory layer  (signaling  a foveal  target)  is  transferred \nto the underlying neurons  in  the motor layer  which  deliver  the required saccade.  In our \nframework, the excitation in  the sensory layer before a goal-directed saccade corresponds \nto the brightest spot (most likely match) in the distance image (Figure 1 (c)  for example), \nThe formation  of sensory  map  can  be  achieved  using  Kohonen's  well-known  stochastic \nlearning algorithm  by  using  a  Gaussian input density function  as  described in  [12].  Our \nprimary  interest  lies  not  in  the  formation  of  the  sensory  map  but  in  the  development \nof  a  learning  algorithm  that  assigns  appropriate  motor  vectors  to  each  location  in  the \ncorresponding retinotopically-organized motor map.  In particular, our algorithm employs \na  visual reinforcement signal obtained using iconic scene representations to determine the \nerror vector during the learning step. \n\n3.2  Learning the Motor Map \n\nOur multisaccade learning strategy is inspired by the following observations in [2] : During \nthe first  few  weeks  after birth, infants  appear to fixate randomly.  At  about  3 months of \nage,  infants  are  able  to  fixate  stimuli  albeit  with  a  number  of  corrective  saccades  of \nrelatively  large  dispersion.  There  is  however  a  gradual  decrease  in  both  the  dispersion \n\n\f898 \n\nRajesh P.  N.  Rao,  Dana H.  Ballard \n\nand  the  number  of saccades  required  for  foveation  in  subsequent  months  (Figure  3  (a) \ndepicts  a  sample set  of fixations).  After  the first  year,  saccades  are generally  accurate, \nrequiring at most one corrective saccade2 \u2022 \n\nThe  learning  method  begins  by  assigning  random  values  to  the  motor  vectors  at  each \nlocation.  The response  vector for  the current fixation  point is  first  stored and a  random \nsaccade is  executed  to  a  different  point.  The goal  then  is  to refixate  the  original  point \nwith  the  help  of the  localization  algorithm  and a  limited  number of multiple  corrective \nsaccades.  The  algorithm  keeps  track  of the  motor  vector  with  minimum  error  during \neach  run and updates  the  motor  vectors  for  the neighborhood  around  the  original  unit \nwhenever  an  improvement  is  observed.  The  current  run  ends  when  either  the  original \npoint was  successfully foveated or the limit  MAX for  the maximum number of allowable \ncorrective saccades was exceeded.  A  more detailed  outline of the algorithm is  as follows: \n\n1.  Initialize  the  motor  map  by  assigning  random  values  (within  an  appropriate \nrange)  to  the  saccadic  motor  vectors  at  each  location.  Align  the  optical  axis \nof the sensor so that a  suitable salient point falls  on the fovea.  Initialize the run \nnumber to t := O. \n\n2.  Store in memory the filter  response vector of the point p  currently in the center \n\nof the foveal  region.  Let t  := t + 1. \n\n3.  Execute a  random saccade to move the fovea to a  different  location in the scene. \n4.  Use  the localization  algorithm described  in  Section  2.2  and  the  stored response \nvector to find the location [ of the previously foveated point in the current retinal \nimage.  Execute a  saccade using the motor vector St  stored in this location in the \nmotor map. \n\n5.  If the currently foveated  region  contains  the original point p,  return  to  2  (SI  is \n\naccurate);  otherwise, \n(a)  Initialize the number of corrective saccades N  := 0 and let s:= St. \n(b)  Determine the new location /' of p in the new image as in (4)  and let emin  be \nthe error vector, i.e.  the vector from  the foveal  center to /',  computed from \nthe output of the localization algorithm. \n\n(c)  Execute a saccade using the motor vector Stl  stored at [' and let ebe the error \nvector  (computed  from  the  output  of the  localization  algorithm)  from  the \nfoveal  center to the new location [II of point p found  as in 4.  Let N  :=  N + 1 \nand let s:= s+ SI'  . \n\n(d)  If lie'll  <  lliminll,  then  let  emin  := e and  update the  motor  vectors  for  the \nunits  k  given  by  the  neighborhood  function  N(l, t)  according  to  the  well(cid:173)\nknown Kohonen rule: \n\nwhere 'Y(t)  is  an appropriate gain function  (0 < 'Y(t)  < 1). \n\n(e)  If the  currently  foveated  region  contains  the  original  point  p,  return  to  2; \notherwise, if N  < MAX, then determine the new location [' of p in the new \nimage  as  in  (4)  and  go  to  5(c)  (i.e.  execute  the  next  saccade);  otherwise, \nreturn to  2. \n\n(5) \n\n2Large saccades in adults are usually hypometric i.e.  they undershoot,  necessitating a slightly \nslower  corrective  saccade.  There  is  currently  no  universally  accepted  explanation  for  the  need \nfor  such  a  two-step strategy. \n\n\fLearning Saccadic  Eye  Movements  Using  Multiscale  Spatial  Filters \n\n899 \n\n! \" 1 \nI \nI .. \u2022 \n\nf.! \n\n+ IMX-IO \n\n...... , \n\n'  ..... 1 \n\n(a) \n\nN!.IBfIll __ \n\n(b) \n\n\",~;:;!l-;;:;;;--l-;;;;\n\nll1I)\"'---;;:ll1Ol;;;--;;\"=-lI) ---;\",=-,C:;;,.,:---;;:; ... ;;-;:;!,,\", \n\nN_la rillelllltN \n\n(c) \n\nFigure 3:  (a)  Successive saccades  executed  by  a  3-month old  (left)  and a  5-month old  (right) \ninfant when presented with a single illuminated stimulus (Adapted from  [2]) .  (b) Graph showing \n%  of saccades  that  end  directly  in  the  fovea  plotted  against  the  number  of  iterations  of  the \nlearning  algorithm  for  different  values  of  MAX.  (c)  An  enlarged  portion  of the  same  graph \nshowing points when convergence was  achieved. \n\nThe  algorithm  continues  typically  until  convergence  or  the  completion  of  a  maximum \nnumber of runs.  The gain term -y(t)  and the neighborhood N(l, t)  for  any location l  are \ngradually decreased with increasing number of iterations t. \n\n4  Results and  Discussion \n\nThe  simulation  results  for  learning  a  motor  map  comprising  of 961  units  are  shown  in \nFigures 3  (b)  and (c)  which depict the variation in saccadic accuracy with the number of \niterations of the algorithm for  values of MAX (maximum number of corrective saccades) \nof  1,  5  and  10.  From  the graphs,  it  can  be seen  that starting with  an  initially  random \nassignment  of vectors,  the  algorithm  eventually  assigns  accurate  saccadic vectors  to all \nunits.  Fewer iterations  seem  to be required  if more  corrective saccades  are  allowed  but \nthen each iteration itself takes more time. \nThe localization algorithm described in Section 2.1  has been implemented on a  Datacube \nMaxVideo  200  pipeline  image  processing  system  and  takes  1-2  seconds  for  location  of \npoints.  Current work includes the integration of the multisaccade learning algorithm de(cid:173)\nscribed above  with  the Datacube implementation and further  evaluation of the learning \nalgorithm.  One  possible  drawback  of  the  proposed  algorithm  is  that  for  large  retinal \nspaces,  learning saccadic motor vectors for  every retinal location can  be time-consuming \nand in some cases, even infeasible [1].  In order to address this problem, we  have recently \nproposed  a  variation  of  the  current  learning  algorithm  which  uses  a  sparse  motor  map \nin conjunction  with  distributed  coding  of the saccadic motor vectors.  This organization \nbears some striking similarities to Kanerva's sparse distributed memory model  [6]  and is \nin concurrence with recent neurophysiological evidence [8]  supporting a distributed popu(cid:173)\nlation encoding of saccadic movements in the superior colliculus.  We  refer the interested \nreader to [10]  for  more details. \n\n\f900 \n\nAcknowledgments \n\nRajesh P.  N.  Rao,  Dana H.  Ballard \n\nWe  thank the NIPS*94 referees for  their helpful comments.  This work was supported by \nNSF research grant no. CDA-8822724, NIH/PHS research grant no.  1 R24 RR06853, and \na  grant from  the Human Science Frontiers Program. \n\nReferences \n\n[1]  Richard N.  Aslin.  Perception of visual direction in human infants.  In C.  Granlund, \neditor,  Visual  Perception  and  Cognition  in  Infancy,  pages  91-118.  Hillsdale,  NJ: \nLawrence Erlbaum Associates,  1993. \n\n[2]  Gordon  W.  Bronson.  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Rao and Dana H.  Ballard.  An active vision architecture based on iconic \nrepresentations.  Technical Report 548,  Department of Computer Scienc~, University \nof Rochester,  1995. \n\n[10]  Rajesh  P.N.  Rao and  Dana H.  Ballard.  A computational model  for  visual  learning \nof saccadic eye movements.  Technical Report 558,  Department of Computer Science, \nUniversity of Rochester,  January 1995. \n\n[11]  Rajesh  P.N.  Rao  and  Dana  H.  Ballard.  Object  indexing  using  an  iconic  sparse \n\ndistributed  memory.  Technical  Report 559,  Department of Computer Science,  Uni(cid:173)\nversity of Rochester,  January 1995. \n\n[12]  Helge Ritter, Thomas Martinetz, and Klaus Schulten.  Neural  Computation  and Self(cid:173)\n\nOrganizing  Maps:  An Introduction.  Reading,  MA:  Addison-Wesley,  1992. \n\n[13]  David L.  Sparks and Rosi Hartwich-Young.  The deep  layers of the superior collicu(cid:173)\n\nIus.  In R.H.  Wurtz  and  M.E.  Goldberg,  editors,  The  Neurobiology  of Saccadic  Eye \nMovements,  pages 213-255. Amsterdam:  Elsevier,  1989. \n\n[14]  Massimo Tistarelli and Giulio Sandini.  Dynamic aspects in active vision.  Computer \nVision,  Graphics,  and Image Processing:  Image  Understanding, 56(1):108-129, 1992. \n[15]  R.A. Young.  The Gaussian derivative theory of spatial vision:  Analysis of cortical cell \nreceptive  field  line-weighting  profiles.  General  Motors  Research  Publication  GMR-\n4920,  1985. \n\n\f", "award": [], "sourceid": 923, "authors": [{"given_name": "Rajesh", "family_name": "Rao", "institution": null}, {"given_name": "Dana", "family_name": "Ballard", "institution": null}]}