{"title": "Learning to Estimate Scenes from Images", "book": "Advances in Neural Information Processing Systems", "page_first": 775, "page_last": 781, "abstract": null, "full_text": "Learning to estimate scenes from  images \n\nWilliam T.  Freeman and  Egon C.  Pasztor \nMERL, Mitsubishi  Electric Research  Laboratory \n\n201  Broadway;  Cambridge,  MA  02139 \n\nfreeman@merl.com,  pasztor@merl.com \n\nAbstract \n\nWe  seek  the  scene  interpretation  that  best  explains  image  data. \nFor example,  we  may want to infer the projected velocities  (scene) \nwhich  best  explain  two  consecutive  image  frames  (image).  From \nsynthetic data , we  model the relationship between image and scene \npatches, and between a scene patch and neighboring scene patches. \nGiven' a  new  image,  we  propagate likelihoods in  a  Markov network \n(ignoring  the  effect  of loops)  to  infer  the  underlying  scene.  This \nyields  an  efficient  method  to form  low-level  scene  interpretations. \nWe  demonstrate the technique for  motion analysis  and estimating \nhigh  resolution images from  low-resolution ones. \n\n1 \n\nIntroduction \n\nThere  has  been  recent  interest  in  studying  the  statistical  properties  of the  visual \nworld.  Olshausen  and  Field  [23J  and  Bell  and  Sejnowski  [2J  have  derived  VI-like \nreceptive fields  from  ensembles of images;  Simon celli  and  Schwartz  [30J  account for \ncontrast  normalization  effects  by  redundancy  reduction.  Li  and  Atick  [1 J  explain \nretinal  color  coding  by  information  processing arguments.  Various research groups \nhave developed  realistic texture synthesis methods by studying the response statis(cid:173)\ntics  of VI-like  multi-scale,  oriented  receptive  fields  [12,  7,  33,  29J.  These  methods \nhelp  us  understand  the early  stages of image  representation and  processing in  the \nbrain. \n\nUnfortunately, they don't  address how  a  visual system might  interpret images, i.e., \nestimate the underlying scene.  In  this  work,  we  study the  statistical  properties of \na  labelled  visual  world ,  images  together  with  scenes,  in  order  to  infer  scenes  from \nimages.  The  image  data  might  be  single  or  multiple  frames;  the  scene  quantities \n\n\f776 \n\nW T.  Freeman and E.  C.  Pasztor \n\nto  be  estimated  could  be  projected  object  velocities,  surface  shapes,  reflectance \npatterns, or colors. \n\nWe  ask:  can  a  visual  system  correctly  interpret  a  visual  scene  if  it  models  (1) \nthe  probability  that  any  local  scene  patch  generated  the local  image,  and  (2)  the \nprobability that any local scene is  the neighbor to any other?  The first  probabilities \nallow  making  scene  estimates  from  local  image  data,  and  the  second  allow  these \nlocal  estimates to propagate.  This leads  to a  Bayesian method for  low  level  vision \nproblems, constrained by Markov assumptions.  We describe this method, and show \nit working for  two low-level vision  problems. \n\n2  Markov networks for  scene  estimation \n\nFirst,  we  synthetically generate images and their underlying scene representations, \nusing  computer  graphics.  The  synthetic  world  should  typify  the  visual  world  in \nwhich  the algorithm will  operate. \n\nFor example, for  the motion estimation problem of Sect. 3, our training images were \nirregularly  shaped  blobs,  which  could  occlude  each  other,  moving  in  randomized \ndirections at speeds  up  to 2 pixels  per frame .  The contrast values of the blobs  and \nthe  background  were  randomized.  The  image  data  were  the  concatenated  image \nintensities from  two successive frames  of an image sequence.  The scene  data were \nthe velocities  of the visible objects at each pixel  in  the two frames. \n\nSecond,  we  place  the  image  and  scene  data in  a  Markov  network  [24].  We  break \nthe images and scenes into localized patches where image patches connect with un(cid:173)\nderlying scene patches; scene patches also  connect  with  neighboring scene patches. \nThe neighbor relationship can  be with regard to position, scale, orientation, etc. \n\nFor  the  motion  problem,  we  represented  both  the  images  and  the  velocities  in  4-\nlevel  Gaussian  pyramids  [6],  to  efficiently  communicate  across  space.  Each  scene \npatch then additionally connects with  the patches at neighboring resolution levels. \nFigure  2  shows  the  multiresolution  representation  (at  one  time  frame)  for  images \nand scenes. 1 \n\nThird, we propagate probabilities.  Weiss showed the advantage of belief propagation \nover regularization methods for severall-d problems [31];  we apply related methods \nto  our  2-d  problems.  Let  the  ith  and  jth  image  and  scene  patches  be  Yi  and \nXj,  respectively.  For  the  MAP  estimate  [3]  of  the  scene  data,2  we  want  to  find \nargmaxxl ,X2 , ... ,XNP(Xl,X2,'\"  ,xNIYl,Y2, .. . ,YM), where Nand M  are the number \nof scene  and  image  patches.  Because  the joint probability  is  simpler  to  compute, \nwe  find,  equivalently,  argmaxx1,X2, ... ,XNP(Xl , X2,\u00b7  . . , XN, Yl , Y2, \u00b7 .. , YM) . \n\nThe  conditional  independence  assumptions of the Markov network let us factorize \nthe desired joint probability into quantities involving only local measurements and \ncalculations  [24,  32].  Consider  the  two-patch  system  of Fig.  1.  We  can  factorize \nP(Xl , X2,Yl,Y2)  in  three steps:  (1)  P(XI,X2 ,Yl,Y2)  =  P(X2 ,Yl,Y2Ixt}P(Xl)  (by  el(cid:173)\nementary  probability);  (2)  P(X2,Yl,Y2Ixl)  =  P(ydXl)P(X2 ,Y2Ixl)  (by  conditional \n\nITo maintain the desired conditional independence relationships, we  appended the im(cid:173)\n\nage data to the scenes.  This provided the scene elements with image contrast information , \nwhich  they would otherwise lack. \n\n2Related arguments follow  for  the MMSE or other estimators. \n\n\fLearning to Estimate Scenes from Images \n\n777 \n\nindependence);  (3)  P(X2,Y2IxI)  =  P(x2Ixt}P(Y2Ix2)  (by  elementary  probability \nand the Markov assumption).  To estimate just Xl  at node 1,  the argmaxx2  becomes \nmax X 2 ,  and  then  slides  over  constants,  giving  terms involving only  local  computa(cid:173)\ntions  at  each node: \n\nargmaxX1 max x2 P(xI, X2, YI, Y2)  =  argmaxx1 [P(XI )P(Yllxl)maxX2 [P(x2Ixt}P(Y2I x 2)]]. \n\n(1) \n\nThis  factorization  generalizes  to  any  network  structure  without  loops.  We  use  a \ndifferent factorization  at each scene node:  we  turn the initial joint probability into \na conditional by factoring out that node's prior, P(Xj) , then proceeding analogously \nto the example above.  The resulting factorized computations give local propagation \nrules,  similar  to  those  of  [24,  32] :  Each  node,  j,  receives  a  message  from  each \nneighbor,  k ,  which  is  an  accumulated  likelihood  function,  Lkj  = P(Yk ... Yzlxj), \nwhere  Yk  . .. Yz  are  all  image  nodes  that  lie  at  or  beyond  scene  node  k,  relative  to \nscene node j.  At each iteration, more image nodes Y enter that likelihood function. \nAfter each iteration, the MAP estimate at node j  is argmaxXj P(x j  )P(Yj Ix j) Ilk L kj , \nwhere  k  runs over all  scene node neighbors of node j .  We  calculate Lkj  from: \n\nL kj  =  maxxkP(xklxj)P(Yklxk) II \u00a3lk, \n\nl#j \n\n(2) \n\nwhere  Llk  is  Llk  from  the  previous  iteration.  The  initial  \u00a3lk'S  are  1.  Using  the \n\nFigure  1:  Markov network  nodes used in  example. \n\nfactorization  rules described  above, one can verify  that the local  computations will \ncompute argmaxx1 ,X2 , . .. , XN P(XI' X2,  ... ,xNIYI, Y2,  ... ,YM), as desired.  To learn the \nnetwork parameters,  we  measure P(Xj),  P(Yjlxj),  and  P(xklxj) , directly from  the \nsynthetic training data. \n\nIf the network contains loops,  the above factorization  does  not hold .  Both learning \nand  inference  then  require  more  computationally  intensive  methods  [15].  Alterna(cid:173)\ntively,  one  can  use  multi-resolution  quad-tree  networks  [20],  for  which  the  factor(cid:173)\nization rules  apply,  to propagate information spatially.  However, this  gives  results \nwith  artifacts  along  quad-tree boundaries,  statistical  boundaries  in  the  model  not \npresent  in  the  real  problem.  We  found  good  results  by  including  the loop-causing \nconnections between adjacent nodes at the same tree level  but applying the factor(cid:173)\nized  propagation rules,  anyway.  Others have obtained  good  results using the  same \napproach  for  inference  [8,  21,  32];  Weiss  provides  theoretical  arguments  why  this \nworks for  certain cases  [32]. \n\n3  Discrete Probability Representation  (motion example) \n\nWe  applied  the training method  and propagation rules  to motion estimation, using \na  vector  code  representation  [11]  for  both  images  and  scenes.  We  wrote  a  tree(cid:173)\nstructured  vector  quantizer,  to code  4  by  4  pixel  by  2  frame  blocks  of image  data \n\n\f778 \n\nW.  T.  Freeman and E.  C.  Pasztor \n\nfor  each  pyramid  level  into  one  of 300  codes  for  each  level.  We  also  coded  scene \npatches into one of 300 codes. \n\nDuring training, we presented approximately 200,000 examples of irregularly shaped \nmoving  blobs,  some  overlapping,  of  a  contrast  with  the  background  randomized \nto  one  of  4  values.  Using  co-occurance  histograms,  we  measured  the  statistical \nrelationships  that embody our algorithm:  P(x) , P(ylx), and P(xnlx), for  scene  Xn \nneighboring scene x. \n\nFigure 2 shows an input test image, (a) before and (b)  after vector quantization.  The \ntrue underlying scene,  the desired  output,  is  shown  (c)  before and  (d)  after vector \nquantization.  Figure 3 shows six iterations of the algorithm  (Eq. 2)  as it converges \nto  a  good  estimate for  the  underlying scene  velocities.  The local  probabilities  we \nlearned  (P(x),  P(ylx), and P(xnlx))  lead  to figure/ground segmentation, aperture \nproblem constraint propagation, and filling-in  (see caption). \n\nFigure  2:  (a)  First  of two  frames  of image  data  (in  gaussian  pyramid),  and  (b) \nvector quantized.  (c)  The optical flow  scene information , and (d)  vector quantized. \nLarge arrow  added to show  small vectors'  orientation. \n\n4  Density Representation  (super-resolution example) \n\nFor super-resolution, the input  \"image\"  is the high-frequency components (sharpest \ndetails)  of a  sub-sampled image.  The  \"scene\"  to be estimated is  the high-frequency \ncomponents of the full-resolution  image,  Fig. 4. \n\nWe  improved  our method for  this second  problem.  A faithful  image representation \nrequires  so  many vector  codes  that it  becomes  infeasible  to measure the prior and \nco-occurance statistics  (note unfaithful  fit  of Fig. 2) .  On the other hand,  a  discrete \nrepresentation allows fast  propagation.  We  developed a  hybrid method that allows \nboth good fitting  and fast  propagation. \n\nWe  describe  the  image  and  scene  patches  as  vectors  in  a  continuous  space,  and \nfirst  modelled  the  probability  densities,  P(x) ,  P(y, x),  and  P(xn, x),  as  gaussian \nmixtures [4] .  (We reduced the dimensionality some by principal components analysis \n[4]).  We  then  evaluated  the prior  and  conditional distributions  of Eq.  2  only  at  a \ndiscrete  set  of scene  values,  different  for  each  node.  (This  sample-based  approach \nrelates to  [14,  7]).  The scenes  were  a  sampling of those scenes which  render to the \nimage  at  that  node.  This  focusses  the  computation  to  the  locally  feasible  scene \ninterpretations.  P(xkIXj)  in  Eq.  2  becomes  the  ratios  of  the  gaussian  mixtures \nP(Xk , Xj)  and P(Xj), evaluated at the scene samples at nodes k  and j, respectively. \nP(Yklxk)  is P(Yk ,Xk)/P(Xk)  evaluated  at the scene samples of node k. \n\nTo  select  the  scene  samples,  we  could  condition  the  mixture  P(y , x)  on  the  Y ob(cid:173)\nserved  at  each  node,  and  sample  x's  from  the  resulting  mixture  of gaussians.  We \nobtained  somewhat  better  results  by  using  the  scenes  from  the  training set  whose \n\n\fLearning to Estimate Scenes from Images \n\n779 \n\nT -\n\n1 \nI \nk.... \n,~\"'....  . \n'<6. \n... .J 4 \n.. ..,. \n~  ...... ~ \nit::;) \n\n1 \nI \n(a) \n1 \nl. ............. :ol. \n~ ....................... .. \n! ...... \n.... .... \"1\u00b7 \nr-:;; \nt'~ \nI \nI \nI\n\u00b7 ... \n1 \n\n.. ... \n... .. \n\nI \nI \n\n2 \n\nI \n: \nt..;i' ....... ~ ....  ,\"', \n\nI; \u2022 \n\n~ .. \n\n3 \n\n1 \n\n..  I;. \n\nI ,. ..... ;i),. ;1, , ~  ,; ~. \n\nt \nI \n\n~A~ \nA~ \n,. ,,;~ : \n\n............. ;:::: \n:!::::::~:::: \n~;;;;~ ... ~\",,~~~~~~ \n\n! \nI \n\n#::~;:.. \n\n\" \n~~ \n\n... \n\ni \"~~A'''1~''!\"'''-!:-:'' I \nt \nI \nI \nI \n\nI \n\nI \nI \nI \n\nI \nI \nI \n\nI \n\nFigure 3:  The most probable scene code for Fig. 2b at first 6 iterations of Bayesian \nbelief propagation.  (a)  Note initial  motion  estimates occur only  at edges.  Due to \nthe  \"aperture  problem\",  initial  estimates  do  not  agree.  (b)  Filling-in  of motion \nestimate occurs.  Cues for figure/ground determination may include edge curvature, \nand information  from  lower  resolution  levels.  Both  are  included  implicitly  in  the \nlearned  probabilities.  (c)  Figure/ground  still  undetermined  in  this  region  of low \nedge curvature.  (d)  Velocities have filled-in,  but do not yet all  agree.  (e)  Velocities \nhave  filled-in ,  and  agree  with  each  other  and  with  the correct  velocity  direction, \nshown  in Fig.  2. \n\nimages  most  closely  matched  the  image  observed  at that  node  (thus  avoiding one \ngaussian mixture modeling step). \n\nUsing 40 scene samples per node, setting up the P(xklxj) matrix for  each link took \nseveral  minutes for  96x96  pixel  images.  The scene  (high resolution)  patch size was \n3x3; the image (low resolution)  patch size was  7x7.  We  didn't feel  long-range scene \npropagation  was  critical  here,  so  we  used  a  flat,  not  a  pyramid,  node  structure. \nOnce  the  matrices  were  computed,  the  iterations  of  Eq.  2  were  completed  within \nseconds. \n\nFigure 4 shows  the results.  The training images  were random' shaded  and painted \nblobs  such  as  the  test  image  shown.  After  5  iterations,  the synthesized  maximum \nlikelihood estimate of the high  resolution  image is  visually close to the actual  high \nfrequency  image  (top  row).  (Including  P(x)  gave  too  flat  results,  we  suspect  due \nto errors modeling  that  highly  peaked  distribution).  The dominant  structures are \nall in approximately the correct position.  This may enable high  quality zooming of \nlow-resolution images, attempted with  limited success  by  others  [28,  25]. \n\n5  Discussion \n\nIn  related  applications of Markov random fields  to vision,  researchers typically  use \nrelatively simple,  heuristically derived expressions (rather than learned) for the like(cid:173)\nlihood  function  P(ylx)  or for  the spatial  relationships  in  the prior term  on  scenes \n\n\f780 \n\nW.  T.  Freeman and E.  C. Pasztor \n\nsub-sampled \n\nimage \n\nzoomed high freqs. \n\nof sub-sampled image \n\n(algorithm input) \n\nfull-detail \n\nimage \n\nhigh freqs. of \n\nfull-detail image \n(desired output) \n\n. ,-\n/ , \n\nf \n\nI \n\n. ' r \n/ \n\n/ \n\n'f' , \n\u2022  J \nI \"\" .. \n/ \nf  . \n} \nI \n, \n1'( \n\u2022 \nI \n\n, \nt \n, \n\\: \n'. \niteration 5 \n(output) \n\nw/o \n\nwith \n\ncomputed output \n\niteration 0 \n\niteration 1 \n\nFigure 4:  Superresolution example.  Top row:  Input and desired output (contrast \nnormalized,  only those orientations around vertical).  Bottom row:  algorithm  out(cid:173)\nput and comparison of image with and without estimated high vertical frequencies. \n\n[10,  26,  9,  17,  5,  20,  19,  27].  Some  researchers  have  applied  related  learning  ap(cid:173)\nproaches  to  low-level  vision  problems,  but  restricted  themselves  to  linear  models \n[18,  13].  For  other learning or constraint propagation approaches in motion analy(cid:173)\nsis,  see  [20,  22,  16]. \n\nIn  summary,  we  have developed  a  principled  and practical  learning based  method \nfor  low-level  vision problems.  Markov assumptions lead to factorizing the posterior \nprobability.  The parameters of our Markov random field  are probabilities specified \nby the training data.  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Technical Report \n1616,  AI Lab  Memo,  MIT , Cambridge,  MA  02139,  1998. \nS.  C.  Zhu  and  D.  Mumford.  Prior  learning  and  Gibbs  reaction-diffusion. \nPattern  Analysis  and  Ma chine  Int elligence,  19(11),  1997. \n\nIEEE \n\n[27] \n\n[28] \n\n[25] \n\n[26] \n\n[21] \n\n[22] \n\n[23] \n\n[24] \n\n[32] \n\n[33] \n\n[29] \n\n[30] \n\n[17] \n\n[20] \n\n[31] \n\n\f", "award": [], "sourceid": 1629, "authors": [{"given_name": "William", "family_name": "Freeman", "institution": null}, {"given_name": "Egon", "family_name": "Pasztor", "institution": null}]}