{"title": "A Cortically-Plausible Inverse Problem Solving Method Applied to Recognizing Static and Kinematic 3D Objects", "book": "Advances in Neural Information Processing Systems", "page_first": 59, "page_last": 66, "abstract": null, "full_text": "A  Cortically-Plausible Inverse  Problem \nSolving  Method Applied  to  Recognizing \n\nStatic  and  Kinematic  3D  Objects \n\nDavid W.  Arathorn \n\nCenter for  Computational Biology, \n\nMontana State University \n\nBozeman, MT  59717 \n\ndwa@cns . montana . edu \n\nGeneral Intelligence Corporation \n\ndwa@giclab . com \n\nAbstract \n\nRecent neurophysiological evidence suggests the ability to  interpret \nbiological  motion  is  facilitated  by  a  neuronal  \"mirror  system\" \nwhich  maps  visual  inputs  to  the  pre-motor  cortex.  If the  common \narchitecture  and  circuitry  of  the  cortices  is  taken  to  imply  a \ncommon  computation  across  multiple  perceptual  and  cognitive \nmodalities, this visual-motor interaction  might be  expected to  have \na unified computational basis.  Two  essential tasks underlying such \nvisual-motor  cooperation  are  shown  here  to  be  simply  expressed \nand  directly  solved  as  transformation-discovery  inverse  problems: \n(a)  discriminating  and  determining  the  pose  of a  primed  3D  object \nin  a  real-world  scene,  and  (b)  interpreting  the  3D  configuration  of \nan  articulated kinematic object in  an  image.  The recently developed \nmap-seeking  method  provides \ntractable, \ncortically-plausible  solution  to  these  and  a  variety  of other  inverse \nproblems  which  can be  posed  as  the  discovery  of a  composition of \ntransformations  between  two  patterns.  The  method  relies  on  an \nordering  property  of superpositions  and  on  decomposition  of the \ntransformation  spaces  inherent  in  the  generating  processes  of the \nproblem. \n\na  mathematically \n\n1  Introduction \nA  variety  of  \"brain  tasks\"  can  be  tersely  posed  as  transformation-discovery \nproblems.  Vision is  replete  with such problems, as  is  limb  control.  The problem of \nrecognizing  the  2D  projection  of  a  known  3D  object  is  an  inverse  problem  of \nfinding  both  the  visual  and  pose  transformations  relating  the  image  and  the  3D \nmodel  of the  object.  When  the  object  in  the  image  may  be  one  of many  known \nobjects  another  step  is  added  to  the  inverse  problem,  because  there  are  multiple \n\n\fcandidates each of which must be mapped to  the  input image  with possibly different \ntransformations.  When  the  known  object  is  not  rigid,  the  determination  of \narticulations  and/or  morphings  is  added  to  the  inverse  problem.  This  includes  the \ngeneral  problem of recognition of biological  articulation  and  motion,  a task recently \nattributed to  a neuronal  mirror-system  linking visual  and  motor cortical  areas  [1]. \n\nThough  the  aggregate  transformation  space  implicit  in  such  problems  is  vast,  a \nrecently  developed  method  for  exploring  vast  transformation  spaces  has  allowed \nsome  significant progress with  a  simple unified approach.  The map-seeking method \n[2,4]  is  a  general  purpose  mathematical  procedure  for  finding  the  decomposition  of \nthe  aggregate  transformation  between  two  patterns,  even  when  that  aggregate \ntransformation  space  is  vast  and  there  is  no  prior information is  available  to  restrict \nthe  search  space.  The  problem  of  concurrently  searching  a  large  collection  of \nmemories can be treated as  a subset of the  transformation problem and consequently \nthe  same  method  can  be  applied  to  find  the  best  transformation  between  an  input \nimage  and  a  collection  of memories  (numbering  at  least  thousands  in  practice  to \ndate) during a single convergence.  In  the  last several  years the  map-seeking method \nhas  been  applied to  a variety of practical  problems,  most of them related to  vision,  a \nfew  related to  kinematics,  and  some  which  do  not correspond to  usual  categories of \n\"brain  functions.\"  The  generality  of the  method  is  due  to  the  fact  that  only  the \nmappings  are  specialized  to  the  task.  The  mathematics  of  the  search,  whether \nexpressed  in  an  algorithm  or  in  a  neuronal  or  electronic  circuit,  do  not  change. \nFrom an evolutionary biological point of view this  is a  satisfying characteristic for  a \nmodel  of  cortical  function  because  only  the  connectivity  which  implements  the \nmappings must be  varied to  specialize a cortex to  a task.  All  the  rest - organization \nand dynamics - would remain the same across cortical areas. \n\nf  =  = \n\nt' / '\" \n(; y \n\nL1 \n\n~  f' \n\ne/ '\" \n\nL2 \n\nE \n\n~  f 2 =  = \n\n~ \n\nL3 \n\nt\n\nI \n\nE \n\n, \nq \n\nt \nb' \n\n.. ' \n\n~ \n\nt\"' \n\n~ V \n\n~ \n\nb\\ \n)  ~  I \n\n, \n\n. ~ 2 \n)'1 \n\n~ \n\nt'2 \n\n~ V \n\n~ \n\nb3 \nE  t \n\n.~ \n\nq3 \n\n93 \n\nw, \n\nW2 \n\nw,~ \n\nc:::!:::J wt \n\n~ b3 \n\n} ~ t \n\n1  q3 \n1 \n\nw, \n\nW2 \n\nw,~ \n\nFigure  1.  Data flow  in  map-seeking circuit \n\n\fCortical neuroanatomy offers  emphatic  hints about the  characteristics of its  solution \nin the  vast neuronal resources allocated to  creating reciprocal top-down and bottom(cid:173)\nup  pathways.  More  specifically,  recent  evidence  suggests  this  reciprocal  pathway \narchitecture  appears  to  be  organized  with  reciprocal,  co-centered  fan  outs  in  the \nopposing  directions  [3],  quite  possibly  implementing  inverse  mappings.  The  data \nflow  of map-seeking  computations, seen  in  Figure  I, is  architecturally compatibility \nwith  these  features  of cortical  organization.  Though  not  within  the  scope  of this \ndiscussion,  it  has  been  demonstrated  [4]  that  the  mathematical  expression  of the \nmap-seeking  method, \nisomorphic \nimplementation in  neuronal  circuitry  with reasonably realistic  dendritic  architecture \nand dynamics (e.g.  compatible with [5]  ) and oscillatory dynamics. \n\nin  equations  6-9  below,  has  an \n\nseen \n\n2  The basis for tractable transformation-discovery \nThe  related  problems  of  recognition/interpretation  of  2D  images  of  static  and \narticulated  kinematic  3D  objects  illustrate  how  cleanly  significant  vision  problems \nmay be  posed and  solved as  transformation-discovery inverse ?roblems.  The visual \nand  pose  (in  the  sense  of orientation)  transformations,  tVIS ua  and  fo se,  between  a \ngiven  3D  model  ml  and  the  extent  of an  input  image  containing  a  2D  projection \nP(OI)  of an object 01  mappable to  ml  can be expressed \nff sllal  E  T Visuol , trse  E  T pose \n\neq.  I \n\nIf we  now  consider  that  the  model  ml  may  be  constructed  by  the  one-to-many \nmapping  of a  base  vector  or  feature  e,  and  that  arbitrarily  other  models  mj  may  be \nsimilarly  constructed  by  different  mappings,  then \nf ormation \ncorresponding  to  the  correct  \"memory\"  converts  the  memory  database  search \nproblem  into  another \ntransformation-discovery  problem  with  one  more  composed \ntransformation I \n\nthe  transformation \n\np( 0  )  =  r :isual  0  { pose  0  ( formation (e) \n\nI \n\nJ \n\nk \n\n1111 \n\nt~~rmatiol1 E  T formatioll \nt fo rmatioll (e) =  m \n\n\"\" \n\nI  m l E \n\nM \n\neq.  2 \n\nFinally,  if we  allow  a  morphable  object to  be  \"constructed\"  by  a  generative  model, \nwhose  various  configurations or articulations  may  be generated by a  composition of \ntransformations  f ell erative  of some  root  or  seed  feature  e,  the  problem  of explicitly \nrecognizing  the  particular  configuration  of  morph  becomes  a  transformation(cid:173)\ndiscovery problem of the form \n\np( C ( 0) ) = t,/\",al  0  tfse  0  Wile/alive ( e) \n\nt lenerative E  T generative \n\neq.  3 \n\nThese unifying  formulations  are  only useful,  however,  if there  is  a  tractable  method \nof solving  for  the  various  transformations.  That  is  what  the  map-seeking  method \nthe  discovery  of  a  composition  of \nprovides. \ntransformations  between  two  patterns.  In  general  the  transformations  express  the \ngenerating process  of the  problem.  Define  correspondence  c  between vectors  rand \nw  through a  composition of L  transformations  tJ, ,t]2  , .. \u00b7,tfL  where t~t  E  ti ,t~,\u00b7 \u00b7\u00b7,t;'t \n\nthe  problem \n\nAbstractly \n\nis \n\n1  This  illustrates  that  forming  a  superposItion  of memories  is  equivalent  to  forming \nsuperpositions  of transformations.  The  first  is  a  more  practical  realization,  as  seen  in \nFigure  1.  Though  not  demonstrated  in  this  paper,  the  multi-memory  architecture  has \nproved robust with  1000 or more  memory patterns from real-world datasets. \n\n\fc( j) = (~I tj i  (  r) , w) \n\neq.  4 \n\nwhere the composition operator is defined \n\nL\n\no  t l  ( r) = \n\n. \n\n;=0. 1  ;. \n\n(I = 1\u00b7\u00b7\u00b7L \n\n1=0 \n\n( L  o ( L- I  ... o (1  (r) \n\nJ 1 \n\n)L \n\nJL-I \n\nr \n\nLet  C  be  an  L  dimensional  matrix  of values  of c(j)  whose  dimensions  are  n, .. . nL. \nThe problem, then is to  find \n\nx =  argmax c(j) \n\neq.  5 \n\nThe  indices  x  specify  the  sequence  of  transformations  that  best  correspondence \nbetween  vectors  rand w.  The  problem  is  that C  is  too  large  a space to  search  for  x \nInstead,  a  continuous  embedding  of C  permits  a  search \nby  conventional  means. \nwith  resources  proportional  to  the  sum  of sizes  of the  dimensions  of C  instead  of \ntheir product. \nC is  embedded in a superposition dot product space Q defined \n\neq.  6 \n\nnm  is  number of t  in  layer m,  g;:,  E  [0,1] , \n\nwhere  G = [g;:\" ]  m = 1\u00b7\u00b7\u00b7L,x\",  = 1\u00b7\u00b7 \u00b7nm \nt: I  is adjoint of tf . \nIn  Q  space,  the  solution  to  eq.  5  lies  along  a  single  axis  in  the  set  of  axes \nis,  gIll  =< 0,.\u00b7 \u00b7'U'm\" \u00b7 \u00b7,0>  U'm > 0  which \nrepresented  each  row  of  G. \ncorresponds  to  the  best  fitting  transformation  tx  ,  where  Xm  is  the  mth  index  in  x  in \neq.  5.  This  state  is  reached  from  an  initial  ~'~ate  G = [1]  by  a  process  termed \nsuperposition  culling in which the components of grad Q are used to  compute a path \nin steps Llg  , \n\nThat \n\neq.  7 \n\neq.  8 \n\nThe  functionfpreserves  the  maximal  component and reduces the  others:  in  neuronal \nterms, lateral inhibition . The resulting path  along the  surface Q can be  thought of as \na  \"high  traverse\"  in  contrast  to  the  gradient  ascent  or  descent usual  in  optimization \nmethods .  The price  for  moving the  problem into  superposition dot product space  is \nthat collusions  of components of the  superpositions can  result  in  better  matches  for \nincorrect mappings  than  for  the  mappings  of the  correct solution.  If this  occurs  it  is \nalmost always  a  temporary state  early in  the  convergence.  This  is  a  consequence of \nthe  ordering  property  of  superpositions  (OPS)  [2,4],  which,  as  applied  here, \nthree \ndescribes \n\nsurface  Q.  For  example, \n\nthe  characteristics  of \n\nthe \n\nlet \n\n\fsuperpositions  r = :t U;  ,  S = :t V j \n\nand  s' = :t Vk be  formed  from  three  sets  of sparse \n\ni= 1 \n\nj = l \n\nk = l \n\nvectors  u; ER,  Vj ES  and  VkES'  where  R n S=0  and  R n S'=vq \u2022  Then  the \nfollowing relationship expresses the OPS: \n\ndefine Pco\" ec!  = p( r \u2022 s'  > r. s),  P'\"co,,'eCl  = p( r. s' :::;  r. s) \nthen  Pcorrecl  > R ncorrect  or  R orrecf  > 0.5 \nand as n, m --+ 1  Pco\"'ect  --+ l.0 \n\nApplied  to  eq.  8,  this  means  that  for  superposItIOns  composed  of vectors  which \nsatisfy the  distribution  properties of sparse,  decorrelating encodings 2  (a  biologically \nplausible  assumption  [6]),  the  probability  of the  maximum  components  of grad  Q \nmoving the  solution in the  correct direction is  always  greater than 0.5  and  increases \ntoward  1.0  as  the  G  becomes  sparser.  In  other  words,  the  probability  of  the \noccurrence  of  collusion  decreases  with  the  decrease  in  numbers  of  contributing \ncomponents in  the  superposition(s), and/or the  decrease in  their gating coefficients. \n\n3  The map-seeking method and  application \n\nA  map-seeking  circuit  (MSC)  is  composed  of several  transformation  or  mapping \nlayers  between  the  input  at  one  end  and  a  memory  layer  at  the  other,  as  seen  in \nFigure  l.  The  compositional  structure  is  evident  in  the  simplicity  of the  equations \n(eqs.  9-12  below)  which  define  a  circuit of any  dimension.  In  a  multi-layer  circuit \nof L  layers plus memory with n{ mappings in layer I the forward path signal for layer \nm  is computed \n\n11m \n\nf m =  Lg;\"  t;' (rm-l) \n\n) = 1 \n\nfor  m =  1. .. L \n\neq.  9 \n\nThe \n\nsignal \n\nfor \n\nlayer \n\nm \n\nis \n\ncomputed \n\nform=1. .. L \n\nor  !gZ\" Wk  or  W \n\nfor  m = L+ I \n\nk=1 \n\nThe mapping coefficients g  are  updated by the recurrence \n\ngi\"  := K( gi\", ti\" (f m- I  ) .  b \",+I) for m = 1. .. L ,i = 1. .. n, \ng/+I := K( g/+I , f' \u2022 W k ) for k = l... nw (optional) \n\neq.  10 \n\neq.  11 \n\nwhere match operator u \u2022  v = q,  q is  a  scalar measure of goodness-of-match between \nu  and  v,  and  may be  non-linear.  When.  is  a  dot product,  the  second  argument of K \nis  the  same  as  oQlg  in  eq.  7.  The  competition  function  K  is  a  realization  of lateral \ninhibition  function/in  eq.  8.  It may  optionally be  applied  to  the  memory  layer,  as \nseen in eq.  11. \n\n2  A restricted  case of the  superposition ordering property using non-sparse  representation \nis exploited by HRR distributed memory.  See [7]  for an  analysis which  is  also applicable \nhere. \n\n\fK(g;, q;) = max [0,  g; - k,  -(1- m:~ q J J \n\neq.  12 \n\nThresholds  are  normally  applied  to  q  and  g,  below  which  they  are  set  to  zero  to \nspeed convergence.  In  above,  f  is  the  input  signal,  tT , (Ill  are  the  /h  forward  and \nbackward  mappings  for  the  m  th  layer,  Wk  is  the  kth  memory  pattern,  z( )  is  a  non(cid:173)\nlinearity  applied  to  the  response  of  each  memory. \ngill  is  the  set  of  mapping \ncoefficients  gT for  the  m  th  layer,  each of which  is  associated with  mapping  tT and \nis  modified over time by the competition function  K( ). \n\nRecognizing 2D  projections of  3D objects under real operating conditions \n\n(a)  3D  memory model \n\n(b )source image \n\n(c)  input image - blurred \n\n2 00 , - - - - - - - - - ,   200 . - - - - - - - - ,   ,.-------, 0. \n\n150 \n\n100 \n\n50 \n\n150 \n\n100 \n\n50 \n\no. \n\no. \n\nos \n\no ':: \no \n\n50 \n\n1 00  15 0  200 \n\n0 \n\n0 \n\n50 \n\n100  150  2 00 \n\n0 ':---::cc----:-o-:---:-:-----:,-:' \n\n100  150  2 00  oL..-o-.-o-.--os-....lo 0 \n\n0 \n\n50 \n\n(d)  iter  1 \n\n(e)  iter 3 \n\n(f)iterI2 \n\n(g)  final model pose \n\nFigure  2.  Recognizing  target among  distractor vehicles.  (a)  M60  3D  memory  model ; \n(b)  source  image,  Fort  Carson  Data  Set;  (c)  Gaussian  blurred  input  image;  (d-f) \nisolation  of  target  in  layer  0,  iterations  1,  3,  12;  (g)  pose  determination  in  final \niteration,  layer  4  backward  - presented  left-right  mirrored  to  reflect  mirroring \ndetermined in  layer 3.  M-60 model courtesy Colorado  State University. \n\nReal  world  problems  of  the  form  expressed  in  eq.  1  often  present  objects  at \ndistances  or  in  conditions  which  so  limit  the  resolution  that  there  are  no  alignable \nfeatures  other  than  the  shape  of the  object  itself, which  is  sufficiently  blurred  as  to \nprevent  generating  reliable  edges  in  a  feed-forward  manner  (e.g.  Fig.  2c). \nIn  the \nmap-seeking  approach,  however,  the  top-down  (in  biological  parlance)  inverse(cid:173)\nmappings  of  the  3D  model  are  used  to  create  a  set  of  edge  hypotheses  on  the \nbackward  path  out of layer  1  into  layer  O. \nIn  layer  0  these  hypotheses  are  used  to \ngate  the  input image.  As  convergence  proceeds,  the edge hypotheses are  reduced to \na single edge hypothesis that best fits  the  grayscale  input image.  Figure 2 shows this \nprocess  applied  to  one  of a  set  of deliberately blurred  images  from  the  Fort Carson \nImagery  Data  Set.  The  MSC  used  four  layers  of visual  transformations:  14,400 \ntranslational,  31  rotational,  41  scaling,  481  3D  projection.  The  MSC  had  no \ndifficulty  distinguishing  the  location  and  orientation of the  tank,  despite  distractors \n\n\fand  background  clutter:  in  all  tests  in  the  dataset  target  was  correctly  located.  In \neffect,  once  primed  with  a  top-down  expectation,  attentional  behavior  IS  an \nemergent property of application of the map-seeking method to  vision [8]. \n\nAdapting generative models by transformation \n\n\"The  direct-matching  hypothesis  of the  interpretation  of biological  motion]  holds \nthat  we  understand  actions  when  we  map  the  visual  representation  of the  observed \naction  onto  our  motor  representation  of  the  same  action.\"  [1]  This  mapping, \nattributed  to  a  neuronal  mirror-system  for  which  there  is  gathering  neurobiological \nevidence  (as  reviewed  in  [1]),  requires  a  mechanism  for  projecting  between  the \nvisual  space  and  the  constrained  skeletal  joint  parameter  (kinematic)  space  to \ndisambiguate the 2D projection of body structure. [4]  Though this  problem has  been \nsolved  to  various  degrees  by  other  computational  methods,  a  review  of which  is \nbeyond  the  scope  of this  discussion,  to  the  author's  knowledge  none  of these  have \nbiological  plausibility.  The  present purpose  is  to  show  how  simply the  problem can \nbe expressed by the generative model  interpretation  problem  introduced in  eq.  3  and \nsolve by map-seeking  circuits.  An  idealized  example  is  the problem of interpreting \nthe  shape  of a  featureless  \"snake\"  articulated  into  any  configuration,  as  appears  in \nFig.  3. \n\n(e) \n\n(d) \n\nFigure 3. Projection between visual and kinematic spaces with two  map-seeking \ncircuits.  (a)  input view, (b) top view,  (c) projection of 3D  occluding contours, \n(d ,e)  projections of relationship of occluding contours to  generating spine. \n\nThe  solution  to  this  problem  involves  two  coupled  map-seeking  circuits.  The \nkinematic  circuit  layers  model  the  multiple  degrees  of freedom  (here  two  angles, \nvariable  length  and  optionally variable  radius  from  spine  to  surface)  of each  of the \nconnected  spine  segments.  The  other circuit determines  the  visual  transformations, \nas  seen  in  the  earlier  example.  The  surface  of the  articulated  cylinder  is  mapped \nfrom  an  axial  spine.  The  points  where  that  surface  is  tangent  to  the  viewpoint \nvectors  define  the  occluding  contours  which,  projected  in  2D,  become  the  object \nsilhouette.  The problem is to  find  the articulations,  segment lengths  (and optionally \nsegment diameter)  which  account  for  the  occluding contour matching the  silhouette \nin  the  input image.  In the  MSC  solution,  the  initial  state  all possible articulations of \nthe  snake  spine  are  superposed,  and  all  the  occluding  contours  from  a  range  of \nviewing  angles  are  projected  into  2D.  The  latter  superposition  serves  as  the \nbackward  input  to  the  visual  space  map-seeking  circuit.  Since  the  snake  surfaceis \ndetermined  by  all  of  the  layers  of  the  kinematic  circuit,  these  are  projected  in \n\n\fparallel  to  form  the  backward  (biologically  top-down)  2D  input  to  the  visual \ntransformation-discovery  circuit.  A  matching operation between the  contributors  to \nthe 2D  occluding contour superposition and the forward  transformations of the  input \nimage  modulates  the  gain  of each  mapping  in  the  kinematic  circuit  via a/n \nin  eqs. \n13,  14  (modified  from  eq.  11). \nIn  eqs.  13 ,  14  K  indicates  kinematic  circuit,  V \nindicates visual circuit. \n\nK \ngin := camp  gt', at' \u00b7 tin \n\n( K  VKK ( K )  K ) \n\n\u2022  b m+1 \n\nf m- l \n\nK \nfor m =  l. .. L,i =  l. .. n! \n\nK \n\nVK \na!\"  =  f L\n\n( V ) \n\n.  t 3D....::,.2D  0  t mrjace  0  t: III  bm+ ' \n\nK (  K  ) \n\neq.  13 \n\neq.  14 \n\nThe  process  converges  concurrently in  both  circuits  to  a  solution,  as  seen  in  Figure \n3.  The  match  of the  occluding  contours  and  the  input  image,  Figure  3a,  is  seen  in \nFigure 3b,c,  with  its  three dimensional  structure  is  clarified in  Figure 3d.  Figure 3e \nshows  a  view  of  the  3D  structure  as  determined  directly  from  the  mapping \nparameters defining the snake \"spine\" after convergence. \n\n4  Conclusion \nThe  investigations  reported  here  expand  the  envelope  of vision-related  problems \namenable  to  a  pure  transformation-discovery  approach  implemented  by  the  map(cid:173)\nseeking  method.  The  recognition  of static  3D  models,  as  seen  in  Figure  2,  and \nother  problems  [9]  solved  by  MSC  have  been  well  tested  with  real-world  input. \nNumerous  variants  of  Figure  3  have  demonstrated  the  applicability  of  MSC  to \nrecognizing generative models of high dimensionality,  and the principle has recently \nbeen  applied  successfully to  real-world domains.  Consequently,  the  research  to  date \ndoes  suggest  that  a  single  cortical  computational  mechanism  could  span  a \nsignificant range of the brain's visual and kinematic computing. \nReferences \n[IJ  G.  Rizzo lati,  L.  Fogassi,  V.  Gallese,  Neurophysiological  mechanisms  underlying  the \nunderstanding and imitation of action, Nature Reviews Neuroscience, 2,  2001 , 661-670 \n\n[2J  D.  Arathorn,  Map-Seeking:  Recognition  Under  Transformation  Using  A  Superposition \nOrdering Property. Electronics Letters 37(3), 2001  pp164-165 \n[3J  A.  Angelucci,  B.  Levitt,  E.  Walton,  J.M.  Hupe,  J.  Bullier, J. Lund, Circuits for  Local and \nGlobal  Signal  Integration  in  Primary Visual  Cortex,  Journal  of Neuroscience,  22(19)  , 2002 \npp  8633-8646 \n[4J  D.  Arathorn,  Map-Seeking  Circuits  in  Visual  Cognition,  Palo  Alto,  Stanford  Univ  Press, \n2002 \n[5J  A.  Polsky,  B.  Mel, J. Schiller,  Computational  Subunits  in  Thin  Dendrites  of Pyramidal \nCells, Nature Neuroscience 7(6), 2004 pp  621-627 \n[6J  B.A.  OIshausen,  D.J.  Field,  Emergence  of Simple-Cell  Receptive  Field  Properties  by \nLearning a Sparse Code for Natural  Images, Nature,  381,  1996 pp607-609 \n[7J  T.  Plate,  Holographic  Reduced Representation,  CSLI  publications,  Stanford,  California, \n2003 \n\n[8J  D.  Arathorn,  Memory-driven  visual  attention:  an  emergent  behavior  of  map-seeking \ncircuits,  in  Neurobiology  of Attention,  Eds  Itti  L,  Rees  G,  Tsotsos  J,  Academic/Elsevier, \n2005 \n\n[9J  C.  Vogel,  D.  Arathorn,  A.  Parker,  and  A.  Roorda,  \"Retinal  motion  tracking  in  adaptive \noptics  scanning  laser ophthalmoscopy\" , Proceedings  of OSA  Conference  on  Signal Recovery \nand Synthesis, Charlotte NC, June 2005. \n\n\f", "award": [], "sourceid": 2936, "authors": [{"given_name": "David", "family_name": "Arathorn", "institution": null}]}