{"title": "Convergence Rates of Algorithms for Visual Search: Detecting Visual Contours", "book": "Advances in Neural Information Processing Systems", "page_first": 641, "page_last": 647, "abstract": null, "full_text": "Convergence Rates of Algorithms for \n\nVisual  Search:  Detecting Visual  Contours \n\nA.L.  Yuille \n\nSmith-Kettlewell Inst . \n\nSan Francisco,  CA 94115 \n\nJames M.  Coughlan \nSmith-Kettlewell Inst. \n\nSan Francisco, CA 94115 \n\nAbstract \n\nThis  paper  formulates  the  problem  of  visual  search  as  Bayesian \ninference  and  defines  a  Bayesian  ensemble  of  problem  instances . \nIn  particular,  we  address  the  problem  of  the  detection  of  visual \ncontours  in  noise/clutter  by  optimizing  a  global  criterion  which \ncombines  local  intensity  and  geometry  information.  We  analyze \nthe  convergence  rates  of  A * search  algorithms  using  results  from \ninformation theory to bound  the  probability of rare  events  within \nthe Bayesian ensemble.  This analysis determines characteristics of \nthe  domain ,  which  we  call  order  parameters,  that  determine  the \nconvergence  rates.  In  particular,  we  present  a  specific  admissible \nA * algorithm with pruning which converges, with high probability, \nwith  expected  time  O(N)  in  the  size  of  the  problem. \nIn  addi(cid:173)\ntion,  we  briefly  summarize  extensions  of this  work  which  address \nfundamental  limits  of target  contour  detectability  (Le.  algorithm \nindependent results)  and the use  of non-admissible heuristics. \n\n1 \n\nIntroduction \n\nMany  problems  in  vision,  such  as  the  detection of edges  and  object  boundaries  in \nnoise/clutter,  see  figure  (1),  require  the  use  of search  algorithms .  Though  many \nalgorithms have  been proposed, see Yuille and  Coughlan  (1997)  for  a  review, none \nof them are clearly optimal and it is difficult to judge their relative effectiveness.  One \napproach has been to compare the results of algorithms on a representative dataset \nof  images.  This  is  clearly  highly  desirable  though  determining  a  representative \ndataset is  often rather subjective. \nIn this paper we are specifically interested in the convergence rates of A * algorithms \n(Pearl  1984).  It can  be  shown  (Yuille  and  Coughlan  1997)  that  many  algorithms \nproposed  to  detect  visual  contours  are  special  cases  of  A * .  We  would  like  to \nunderstand what characteristics of the problem domain  determine the convergence \n\n\f642 \n\nA.  L.  Yuille and J.  M  Coughlan \n\nFigure  1:  The  difficulty  of  detecting  the  target  path  in  clutter  depends,  by  our \ntheory  (Yuille  and  Coughlan  1998),  on the order  parameter  K.  The larger  K  the \nless  computation  required.  Left,  an  easy  detection  task  with  K  =  3.1.  Middle,  a \nhard detection  task K  =  1.6.  Right,  an impossible task with  K  =  -0.7. \n\nrates. \n\nWe formulate the problem of detecting object curves in images to be one of statistical \nestimation.  This  assumes  statistical  knowledge  of the  images  and  the  curves,  see \nsection  (2).  Such statistical knowledge  has  often  been  used  in  computer  vision  for \ndetermining optimization criteria to be minimized.  We  want to go one step further \nand  use  this  statistical  knowledge  to determine  good search  strategies  by  defining \na  Bayesian  ensemble of problem instances.  For this ensemble, we  can prove certain \ncurve and  boundary  detection  algorithms,  with  high  probability,  achieve  expected \ntime  convergence  in  time  linear  with  the  size  of the  problem.  Our  analysis  helps \ndetermine important characteristics of the problem, which we  call order parameters, \nwhich  quantify the difficulty of the problem. \n\nThe  next  section  (2)  of this  paper  describes  the  basic  statistical  assumptions  we \nmake about the domain and describes the mathematical tools used in the remaining \nsections.  In  section  (3)  we  specify  our  search algorithm and  establish  converEence \nrates.  We  conclude by placing this work in a larger context and summarizing recent \nextensions. \n\n2  Statistical Background \n\nOur approach assumes that both the intensity properties and the geometrical shapes \nof the target path (i.e.  the edge contour) can be determined statistically.  This path \ncan be considered .to be a set of elementary path segments joined together.  We first \nconsider  the intensity properties along the edge and then the geometric properties. \nThe set of all  possible paths can be represented by a  tree structure, see  figure  (2). \n\nThe  image  properties  at  segments  lying  on  the  path  are  assumed  to  differ,  in  a \nstatistical sense, from those off the path.  More precisely,  we  can design a  filter  \u00a2(.) \nwith output  {Yx  =  \u00a2(I(x))}  for  a  segment at point x  so  that: \n\nP(Yx)  = Pon(Yx),  if \"XII  lies  on the true path \nP(Yx)  =  Poff(Yx),  if \"X'I  lies off the true path. \n\n(1) \n\nFor example,  we  can think of the {Yx}  as being values of the edge strength at point \nx  and  Pon, Poll  being  the probability  distributions of the  response  of \u00a2(.)  on  and \noff an  edge.  The set  of  possible  values  of the  random  variable  Yx  is  the  alphabet \nwith  alphabet  size M  (Le.  Yx  can take any of M  possible values).  See  (Geman and \nJedynak  1996)  for  examples  of distributions  for  Pon, Pol I  used  in  computer  vision \napplications. \n\nWe  now  consider  the  geometry  of the  target  contour.  We  require  the  path  to  be \nmade up of connected segments  Xl, X2, ... , x N.  There will be a  Markov probability \ndistribution Pg(Xi+I!Xi)  which specifies  prior probabilistic knowledge of the target. \n\n\fConvergence Rates of Algorithmsfor Visual Search: Detecting Visual Contours \n\n643 \n\nIt is  convenient, in terms of the graph search algorithms we will use, to consider that \neach  point  x  has a  set of Q neighbours.  Following terminology from  graph theory, \nwe  refer  to Q as  the  branching  factor.  We  will  assume  that  the  distribution  Pg \ndepends only on the relative positions of XHI  and Xi.  In other words, Pg(XHllxi)  = \nPLlg(XHl  - Xi).  An  important  special  case  is  when  the  probability  distribution \nis  uniform  for  all  branches  (Le.  PLlg(Ax)  = U(Ax)  =  I/Q,  VAx).  The  joint \ndistribution P(X, Y)  of the road geometry X  and filter responses Y  determines the \nBayesian  Ensemble. \nBy standard Bayesian analysis, the optimal path X*  =  {xi, ... , XN}  maximizes the \nsum of the log posterior: \n\n(2) \n\nwhere the sum i  is  taken over all  points on the target.  U(Xi+l  - Xi)  is  the  uniform \ndistribution and its  presence merely changes the log  posterior E(X)  by a  constant \nvalue.  It is  included to make the form of the intensity and geometric terms similar, \nwhich simplifies our later analysis. \nWe  will refer to E(X) as the  reward of the path X  which is the sum of the  intensity \nrewards log  Pon (Y(~jl)  and the geometric  rewards log PL:>.g (Xi+l -Xi) \nU(Xi+l -Xi) \n\nPoll (Y(~i\u00bb \n\nIt  is  important  to  emphasize  that  our  results  can  be  extended  to  higher-order \nMarkov  chain  models  (provided  they  are  shift-invariant).  We  can,  for  example, \ndefine  the  x  variable  to represent  spatial orientation  and position  of a  small  edge \nsegment.  This  will  allow  our  theory  to  apply  to  models,  such  as  snakes,  used  in \nrecent successful  vision applications  (Geman and Jedynak 1996).  (It is  straightfor(cid:173)\nward to transform the standard energy function formulation of snakes into a Markov \nchain  by  discretizing and replacing the  derivatives  by  differences.  The smoothness \nconstraints, such  as  membranes and thin  plate terms,  will  transform into first  and \nsecond  order Markov  chain connections  respectively).  Recent  work  by  Zhu  (1998) \nshows that Markov chain models of this type can be learnt using Minimax Entropy \nLearning theory from  a  representative set of examples.  Indeed  Zhu goes  further by \ndemonstrating that other Gestalt grouping laws can be expressed in this framework \nand learnt from  representative data. \nMost Bayesian vision theories have stopped at this point.  The statistics of the prob(cid:173)\nlem domain are used only  to determine the optimization criterion to be minimized \nand are not exploited to analyze the complexity of algorithms for  performing the op(cid:173)\ntimization.  In this paper, we go a stage further.  We use the statistics ofthe problem \ndomain to define  a  Bayesian ensemble and hence  to  determine  the  effectiveness  of \nalgorithms for  optimizing criteria such as  (2).  To do this requires the use of Sanov's \ntheorem  for  calculating the  probability  of rare  events  (Cover  and  Thomas  1991). \nFor  the  road  tracking  problem  this  can  be  re-expressed  as  the  following  theorem, \nderived in  (Yuille  and Coughlan 1998): \nTheorem  1.  The  probabilities  that  the  spatially  averaged  log-likelihoods  on,  and \noff,  the  true  curve  are  above,  or below,  threshold T  are  bounded  above  as  follows: \n\nPr{.!. t {log  Pon(y(Xi\u00bb)  }on  < T} :s;  (n + I)M2-nD(PTlfPon) \n\nn  i=l \n\nPoff (Y(Xi\u00bb) \n\nPr{.!. t{lOg Pon(Y(Xi\u00bb)  }off > T}:S;  (n+ I)M2-nD(PTIIPOI/) , \n\nn  i=l \n\nPOff(Y(Xi\u00bb) \n\n(3) \n\n(4) \n\n\f644 \n\nA.  L.  Yuille and J.  M.  Coughlan \n\nwhere  the  subscripts  on  and  off  mean  that  the  data  is  generated  by  Pon, Po\", \nPT(y)  =  p;;;>'(T) (y)P;;p jZ(T)  where  a ::;  \"\\(T)  ::;  1  is  a  scalar  which  depends \non  the  threshold T  and Z(T)  is a normalization factor.  The  value  of \"\\(T)  is  deter(cid:173)\nmined by  the  constraint 2: y  PT (y) log ;'\u00b0In}(~)  =  T. \nIn  the  next  section,  we  will  use  Theorem  1  to  determine  a  criterion  for  pruning \nthe search based on comparing the intensity reward to a  threshold T  (pruning will \nalso be done  using the  geometric reward).  The choice  of T  involves  a  trade-off.  If \nT  is  large  (Le.  close  to D(PonllPoff))  then  we  will  rapidly  reject  false  paths  but \nwe  might  also  prune  out  the  target  (true)  path.  Conversely,  if T  is  small  (close \nto  -D(PoffllPon))  then  it  is  unlikely  we  will  prune  out  the  target  path  but  we \nmay waste a  lot of time exploring false  paths.  In this paper we  choose T  large and \nwrite  the  fall-off  factors  (Le.  the  exponents  in  the  bounds  of equations  (3,4))  as \nD(PTllPon)  =  tl (T),  D(PTilPoff) =  D(PonilPoff) - t2(T)  where  tl (T), t2(T)  are \npositive  and  (tl(T),t2(T))  t-+  (0,0)  as T  t-+  D(PonilPoff ).  We  perform  a  similar \nanalysis for  the geometric rewards by substituting P 6.g , U for  Pon , Pol I' We  choose \na  threshold T satisfying  -D(UIIP6.g)  < T < D(P6.gllU).  The results of Theorem \n1 apply with the obvious substitutions.  In particular, the alphabet factor  becomes \nQ (the  branching factor).  Once again,  in this  paper,  we  choose T to be large  and \nobtain fall-off factors  D(Pt'IIP6.g)  = El (T),  D(Pt'IIU) = D(P6.gllU)  - E2(T). \n\n3  Tree  Search:  A *,  heuristics, and block pruning \n\nWe  now  consider  a  specific  example,  motivated  by  Geman  and  Jedynak  (1996), \nof searching  for  a  path  through  a  search  tree.  In  Geman  and  Jedynak  the  path \ncorresponds  to a  road in  an aerial  image  and  they  assume  that  they  are  given  an \ninitial point and direction on the target path.  They have a  branching factor Q =  3 \nand, in their first  version, the prior probability of branching is  considered to be the \nuniform  distribution  (later they consider more sophisticated priors).  They assume \nthat no  path segments overlap  which  means that the search space is  a  tree of size \nQN  where  N  is  the  size  of the  problem  (Le.  the  longest length).  The  size  of the \nproblem  requires  an algorithm that converges in O(N) time and they demonstrate \nan algorithm which empirically performs at this speed.  But no proof of convergence \nrates  are  given  in  their  paper.  It  can  be shown,  see  (Yuille  and  Coughlan  1997), \nthat the Geman and Jedynak algorithm is  a  close  approximation to A * which  uses \npruning.  (Observe that  Geman and  Jedynak's tree representation is  a  simplifying \nassumption of the  Bayesian model  which  assumes  that once  a  path diverges  from \nthe true path it can never recover,  although we  stress that the  algorithm is  able to \nrecover from  false  starts - for  more  details see  Coughlan and Yuille  1998). \nWe  consider  an  algorithm  which  uses  an  admissible  A * heuristic  and  a  pruning \nmechanism.  The idea is  to examine the paths  chosen by the A * heuristic.  As  the \nlength of the candidate path reaches an integer multiple of No  we  prune it based on \nits intensity reward and its geometric reward evaluated on the previous No  segments, \nwhich  we  call  a  segment  block.  The reasoning  is  that  few  false  paths  will  survive \nthis  pruning for  long but the target path will  survive  with high probability. \n\nWe prune on the intensity by eliminating all paths whose intensity reward, averaged \nover the last No  segments, is  below a threshold T  (recall that -D(PoffllPon) < T  < \nD(PonllPoff)  and  we  will  usually  select  T  to take  values  close  to  D(PonllPoff)). \nIn  addition,  we  prune  on  the  geometry  by  eliminating  all  paths  whose  geometric \nrewards,  averaged  over  the  last  No  segments,  are  below T (where  -D(UIIP6.g)  < \nT < D(P6.gllU)  with T typically  being  close  to  D(P6.gllU)).  More  precisely,  we \n\n\fConvergence Rates of AlgOrithms for Visual Search : Detecting Visual Contours \n\ndiscard  a  path provided  (for  any integer  z  ~ 0): \n\n1  (z~o I \n- ~  og \nNo  i=zNo+l \n\nPon(Yi) \n\nPoff(yd \n\n< T,  or  No  L  log  U(Llx.)  < T. \n\nPLlg(Llxi) \n\nt \n\n1  (z+l)No \n\ni=zNo+l \n\n645 \n\n(5) \n\nThere  are  two  important  issues  to  address:  (i)  With  what  probability  will  the \nalgorithm  converge?,  (ii)  How  long  will  we  expect  it  take  to converge?  The  next \ntwo subsections put bounds on these issues. \n\n3.1  Probability of Convergence \n\nBecause of the pruning, there is  a  chance that there will  be no paths which survive \npruning.  To  put  a  bound  on  this  we  calculate  the  probability  that  the  target \n(true)  path survives  the  pruning.  This  gives  a  lower  bound  on  the  probability  of \nconvergence  (because  there  could  be  false  paths  which  survive  even  if the  target \npath is  mistakenly pruned out). \n\nThe  pruning rules  removes  path segments for  which  the intensity reward r I  or the \ngeometric  reward r 9  fails  the pruning test.  The probability of failure  by  removing \na  block segment of the true path, with rewards r~, r~, is  Pr(r~ < T  or  r~ < T)  ::; \nPr(r~ < T) + Pr(r~ < T)  ::;  (No  + 1)M2- NoE1 (T)  + (No  + 1)Q2-NoilCT),  where we \nhave used Theorem 1 to put bounds on the probabilities.  The probability of pruning \nout  any  No  segments  of the  true  path  can  therefore  be  made  arbitrarily  small  by \nchoosing No, T, T so as to make  Notl  and  NOtl  large. \nIt  should  be  emphasized  that  the  algorithm  will  not  necessarily  converge  to  the \nexact target path.  The admissible nature of the heuristic means that the algorithm \nwill  converge  to the  path with highest  reward  which  has survived  the  pruning.  It \nis  highly  probable  that  this  path  is  close  to  the  target  path.  Our  recent  results \n(Coughlan  and  Yuille  1998,  Yuille  and  Coughlan  1998)  enable  us  to quantify  this \nclaim. \n\n3.2  Bounding the  Number of False  Paths \n\nSuppose we  face  a  Q-nary tree.  We  can order the false  paths by the stage at which \nthey  diverge from  the  target  (true)  path,  see  figure  (2).  For  example,  at the  first \nbranch point the target path lies  on only one of the Q branches and there are Q - 1 \nfalse  branches which  generate the first  set  of false  paths  Fl'  Now  consider  all  the \nQ -1 false branches at the second target branch, these generate set F2 .  As  we follow \nalong the true path we  keep  generating these false  sets  Fi .  The set  of all  paths is \ntherefore  the  target  path  plus  the  union  of the  Fi  (i  =  1, ... , N).  To  determine \nconvergence rates we  must bound the amount of time we spend searching the Fi.  If \nthe  expected time  to search each  Fi  is  constant then searching for  the  target path \nwill  at most take constant\u00b7 N  steps. \nConsider the set Fi  of false  paths which leave the true path at stage i.  We will apply \nour analysis to block segments of Fi  which are completely off the true path.  If (i -1) \nis  an integer multiple of No  then all block segments of Fi  will satisfy this condition. \nOtherwise,  we  will  start  our  analysis  at  the  next  block  and  make  the  worse  case \nassumption  that  all  path  segments  up  till  this  next  block  will  be  searched.  Since \nthe distance to the next block is  at most No  - 1,  this gives  a  maximum number of \nQNo-l  starting blocks for  any branch of Fi .  Each Fi  also has Q - 1 branches and \nso this gives a  generous upper bound of (Q  - l)Q N o-l  starting blocks for  each  Fi . \n\n\f646 \n\nA. L.  Yuille and J.  M.  Coughlan \n\nFigure  2:  The  target  path  is  shown  as  the  heavy  line.  The  false  path  sets  are \nlabelled  as  Fl ,F2 ,  etc.  with the  numbering depending on how  soon  they  leave  the \ntarget path.  The branching factor  Q  =  3. \n\nFor  each  starting block,  we  wish  to  compute  (or  bound)  the  expected  number  of \nblocks that are explored thereafter.  This requires computing the fertility of a  block, \nthe average number of paths in the block that survive pruning.  Provided the fertility \nis smaller than one, we can then apply results from the theory of branching processes \nto determine the expected number of blocks searched in  Fi . \n\nThe fertility  q is  the number of paths that survive the geometric pruning times the \nprobability that each  survives  the  intensity pruning.  This can  be  bounded  (using \nTheorem 1)  by q :s q where: \nq =  QN0(No + I)Q2-No{D(hgIIU)-\u20ac2(T)}(No  + I)M2-No{D(PonIIPoff)-E2(T)} \n=  (No + I)Q+M 2- No {D(Pon IIPof! )-H(Pag)-E2(T)-\u20ac2(T)}, \n\n(6) \n\nIn  other  words,  the  better  the  edge  detector  and  the  more \n\nwhere we  used  the fact  that D(PLlgIIU)  =  10gQ - H(PLlg). \nObserve that the condition q < 1 can be satisfied provided D(PonllPolf )-H(PLlg) > \nO.  This condition is  intuitive, it requires that the edge detector information,  quan(cid:173)\ntified by D(PonIIPolf )' must be greater than the uncertainty in the geometry mea(cid:173)\nsured  by  H(PLlg). \npredictable the path geometry then the smaller q will  be. \nWe  now apply the theory of branching processes to determine the expected number \nof blocks  explored from  a  starting block in  Fi  'L~o qZ  =  1/(1 - q).  The number \nof branches of Fi  is  (Q  - 1), the total number of segments explored  per block  is  at \nmost QNo ,  and we  explore at most QNo-l  segments before reaching the first  block. \nThe  total  number  of Fi  is  N.  Therefore  the  total  number  of segments  wastefully \nexplored is  at most  N(Q - 1) 1~qQ2No-1. We  summarize this result in  a  theorem: \nTheorem  2.  Provided q =  (No  + I)Q+M2- NoK  <  1,  where  the  order  parameter \nK  =  D(PonllPolf)  - H(PLlg)  -\nthen  the  expected  number  of false \nsegments  explored  is  at  most N(Q - 1) 1~qQ2No-1. \nComment The requirement that q < 1 is  chiefly determined by the order parameter \nK  =  D (Pon IlPolf ) - H (P Llg)  - \u20ac2  (T) - f2 (T).  Our convergence proofrequires that \nK  > 0 and will  break down  if K  < O.  Is  this a  limitation of our proof?  Or does  it \ncorrespond to a  fundamental difficulty  in solving this tracking problem? \n\n\u20ac2(T) \n\n- \u20ac2(T), \n\nIn  more  recent  work  (Yuille  and  Coughlan  1998)  we  extend  the  concept  of order \nparameters and show that they characterize the difficulty  of visual  search problem \nindependently  of the  algorithm.  In  other  words,  as  K  1----7  0  the  problem  becomes \nimpossible  to  solve  by  any  algorithm.  There  will  be  too  many  false  paths  which \nhave  better rewards than the target path.  As  K  1----7  0 there is  a  phase transition in \nthe ease of solving the problem. \n\n\fConvergence Rates of Algorithmsfor Visual Search:  Detecting Visual Contours \n\n647 \n\n4  Conclusion \n\nOur analysis shows it is  possible to detect certain types of image contours in linear \nexpected  time  (with  given  starting points).  We  have  shown  how  the  convergence \nrates depend on  order parameters which  characterize the problem domain.  In  par(cid:173)\nticular,  the  entropy  of  the  geometric  prior  and  the  Kullback-Leibler  distance  be(cid:173)\ntween Pon  and Pof f  allow  us  to quantify intuitions about the  power of geometrical \nassumptions and edge detectors to solve these tasks. \n\nOur more recent work (Yuille  and Coughlan 1998) has extended this work by show(cid:173)\ning  that  the  order  parameters  can  be  used  to  specify  the  intrinsic  (algorithm  in(cid:173)\ndependent)  difficulty of the search  problem and that  phase transitions occur  when \nthese  order  parameters  take  critical  values.  In  addition,  we  have  proved  conver(cid:173)\ngence  rates for  A * algorithms which  use  inadmissible heuristics or  combinations of \nheuristics and pruning (Coughlan and Yuille  1998). \n\nAs  shown  in  (Yuille  and  Coughlan  1997)  many of the search  algorithms  proposed \nto  solve  vision  search  problems,  such  as  (Geman  and  Jedynak  1996),  are  special \ncases  of A * (or  close  approximations).  We  therefore  hope  that  the  results  of this \npaper  will  throw  light  on the success  of the  algorithms  and  may  suggest  practical \nimprovements and speed ups. \n\nAcknow ledgements \n\nWe  want to acknowledge funding from  NSF with award number IRI-9700446, from \nthe  Center  for  Imaging  Sciences  funded  by  ARO  DAAH049510494,  and  from  an \nASOSRF  contract  49620-98-1-0197  to  ALY.  We  would  like  to  thank  L.  Xu,  D. \nSnow,  S.  Konishi,  D.  Geiger,  J.  Malik,  and D.  Forsyth for  helpful  discussions. \n\nReferences \n[1]  J .M.  Coughlan and A.L.  Yuille.  \"Bayesian A * Tree Search with Expected O(N) \nConvergence  Rates  for  Road  Tracking.\"  Submitted  to  Artificial  Intelligence. \n1998. \n\n[2]  T.M.  Cover  and  J.A.  Thomas.  Elements  of Information  Theory.  Wiley \n\nInterscience Press.  New  York.  1991. \n\n[3]  D.  Geman.  and  B.  Jedynak.  \"An  active  testing  model  for  tracking  roads  in \nsatellite images\".  IEEE  Trans.  Patt.  Anal.  and  Machine  Intel.  Vol.  18.  No.1, \npp 1-14.  January.  1996. \n\n[4]  J.  Pearl.  Heuristics. Addison-Wesley.  1984. \n[5]  A.L. Yuille and J. Coughlan. \" Twenty Questions, Focus of Attention, and A *\" . \nIn  Energy  Minimization Methods  in  Computer  Vision  and  Pattern \nRecognition. Ed. M.  Pellilo and E. Hancock.  Springer-Verlag.  (Lecture Notes \nin  Computer Science  1223).  1997. \n\n[6]  A.L.  Yuille  and  J .M~ Coughlan.  \"Visual Search:  Fundamental Bounds,  Order \nParameters, Phase Transitions, and Convergence Rates.\" Submitted to Pattern \nAnalysis  and Machine  Intelligence.  1998. \n\n[7]  S.C.  Zhu.  \"Embedding  Gestalt  Laws  in  Markov  Random  Fields\".  Submitted \nto IEEE Computer Society  Workshop  on  Perceptual Organization in  Computer \nVision. \n\n\f", "award": [], "sourceid": 1513, "authors": [{"given_name": "Alan", "family_name": "Yuille", "institution": null}, {"given_name": "James", "family_name": "Coughlan", "institution": null}]}