{"title": "The g Factor: Relating Distributions on Features to Distributions on Images", "book": "Advances in Neural Information Processing Systems", "page_first": 1231, "page_last": 1238, "abstract": null, "full_text": "The 9  Factor:  Relating  Distributions on \n\nFeatures to Distributions on Images \n\nJames M.  Coughlan and  A.  L.  Yuille \nSmith-Kettlewell Eye Research Institute, \n\n2318  Fillmore Street, \n\nSan Francisco,  CA  94115,  USA. \n\nTel.  (415)  345-2146/2144.  Fax.  (415)  345-8455. \n\nEmail:  coughlan@ski.org.yuille@ski.org \n\nAbstract \n\nWe  describe  the  g-factor,  which  relates  probability  distributions \non image features  to  distributions on the  images  themselves.  The \ng-factor  depends  only  on  our choice  of features  and  lattice  quanti(cid:173)\nzation and is  independent of the training image data.  We  illustrate \nthe importance  of the g-factor by analyzing how the parameters of \nMarkov Random Field (i.e.  Gibbs or log-linear) probability models \nof images are learned from data by maximum likelihood estimation. \nIn particular, we study homogeneous MRF models which learn im(cid:173)\nage distributions in terms of clique potentials corresponding to fea(cid:173)\nture  histogram  statistics  (d.  Minimax  Entropy  Learning  (MEL) \nby  Zhu,  Wu  and  Mumford  1997  [11]) .  We  first  use  our  analysis \nof the  g-factor  to  determine  when  the  clique  potentials  decouple \nfor  different  features .  Second,  we  show  that  clique  potentials  can \nbe  computed  analytically  by  approximating  the  g-factor.  Third, \nwe  demonstrate a  connection between this  approximation and the \nGeneralized Iterative Scaling algorithm (GIS),  due to Darroch and \nRatcliff  1972  [2],  for  calculating  potentials.  This  connection  en(cid:173)\nables  us  to  use  GIS  to  improve  our  multinomial  approximation, \nusing  Bethe-Kikuchi[8]  approximations to simplify  the  GIS  proce(cid:173)\ndure.  We  support our analysis by computer simulations. \n\n1 \n\nIntroduction \n\nThere has  recently  been  a  lot  of interest  in  learning probability models  for  vision. \nThe  most  common  approach  is  to  learn  histograms  of filter  responses  or,  equiva(cid:173)\nlently,  to  learn  probability  distributions  on  features  (see  right  panel  of figure  (1)). \nSee,  for  example,  [6],  [5],  [4].  (In this  paper the features  we  are considering will  be \nextracted from the image by filters - hence we  use the terms  \"features\"  and \"filters\" \nsynonymously. ) \n\n\fAn alternative approach, however, is to learn probability distributions on the images \nthemselves.  The Minimax Entropy Learning  (MEL)  theory  [11]  uses  the maximum \nentropy  principle  to  learn  MRF  distributions  in  terms  of clique  potentials  deter(cid:173)\nmined  by  the feature  statistics  (i.e.  histograms of filter  responses).  (We  note that \nthe maximum entropy principle is equivalent to performing maximum likelihood es(cid:173)\ntimation on an MRF whose form  is  determined by the choice of feature statistics.) \nWhen applied to texture modeling it gives a way to unify the filter based approaches \n(which  are often  very effective)  with  the  MRF  distribution  approaches  (which  are \ntheoretically attractive). \n\n)  \\ \n\nFigure  1:  Distributions  on  images  vs.  distributions  on  features.  Left  and  center \npanels  show  a  natural  image  and  its  image  gradient  magnitude map,  respectively. \nRight  panel  shows  the  empirical  histogram  (i.e.  a  distribution  on  a  feature)  of \nthe  image  gradient  across  a  dataset  of natural  images.  This  feature  distribution \ncan  be  used  to  create  a  MRF  distribution  over images[10].  This  paper introduces \nthe g-factor  to examine  connections  between  the  distribution over images  and the \ndistribution over features. \n\nAs  we  describe  in  this  paper  (see  figure  (1)),  distributions  on  images  and  on  fea(cid:173)\ntures can be related  by  a  g-factor  (such factors  arise in  statistical physics,  see  [3]) . \nUnderstanding the g-factor allows us to  approximate  it in  a form  that helps  explain \nwhy  the  clique  potentials  learned  by  MEL  take  the  form  that  they  do  as  functions \nof the feature  statistics.  Moreover,  the  MEL  clique  potentials for  different features \noften seem to  be  decoupled  and the g-factor can  explain  why,  and when,  this  occurs. \n(I.e.  the two clique  potentials corresponding to two features  A  and B  are identical \nwhether we  learn them jointly or independently). \n\nThe g-factor  is  determined only  by the form  of the features  chosen  and  the  spatial \nlattice  and  quantization  of the  image  gray-levels.  It is  completely  independent  of \nthe  training  image  data.  It  should  be  stressed  that  the  choice  of image  lattice, \ngray-level quantization and histogram quantization can make a big difference to the \ng-factor and hence to the probability distributions which are the output of MEL. \n\nIn Section (2),  we  briefly review Minimax Entropy Learning.  Section (3)  introduces \nthe  g-factor  and  determines  conditions  for  when  clique  potentials  are  decoupled. \nIn  Section  (4)  we  describe  a  simple  approximation  which  enables  us  to  learn  the \nclique  potentials  analytically,  and  in  Section  (5)  we  discuss  connections  between \nthis  approximation and the Generalized Iterative Scaling  (GIS)  algorithm. \n\n2  Minimax Entropy Learning \n\nSuppose  we  have training image  data which  we  assume  has  been  generated by  an \n(unknown)  probability  distribution  PT(X)  where x represents  an  image.  Minimax \nEntropy Learning (MEL)  [11]  approximates PT(X)  by selecting the distribution with \n\n\f-\n\n-\n\n-\n\nmaximum entropy constrained by observed feature statistics i(X) =  ;fobs.  This gives \n>:. \u00a2( \u00a3) \nP(xIA)  =  e Z [>:] \n,where A is  a  parameter chosen  such that  Lx P(xIA)\u00a2>(X)  =  'l/Jobs\u00b7 \nOr equivalently,  so  that  <910;{[>:]  =  ;fobs. \nWe  will  treat  the special  case  where  the  statistics i are  the  histogram of a  shift(cid:173)\ninvariant filter {fi(X)  : i  =  1, ... , N} , where N  is the total number of pixels in the im(cid:173)\nage.  So  'l/Ja  =  \u00a2>a(x)  =  -tv  L~l ba,' i(X)  where a =  1, ... , Q indicates  the  (quantized) \nfilter response values.  The potentials become A\u00b7\u00a2>(X)  =  -tv  La=l Li=l A(a)ba,fi(X) = \n-tv  L~l A(fi(X)).  Hence P(xl,X)  becomes a MRF distribution with clique potentials \ngiven by A(fi (x)).  This determines a Markov random field with the clique structure \ngiven by the filters  {fd. \nMEL  also  has  a  feature  selection  stage  based  on  Minimum  Entropy  to  determine \nwhich features  to use  in the  Maximum Entropy Principle.  The features  are  evalu(cid:173)\nated by  computing the entropy  - Lx P(xl,X) log P(xl,X)  for  each  choice of features \n(with small entropies being preferred).  A  filter  pursuit procedure was  described to \ndetermine which filters/features should be considered (our approximations work for \nthis also). \n\n~  ~ \n\nQ \n\nN \n\n3  The g-Factor \n\nThis  section  defines  the  g-factor  and  starts  investigating  its  properties  in  subsec(cid:173)\ntion  (3.1). \nIn  particular,  when,  and  why,  do  clique  potentials  decouple?  More \nprecisely,  when  do  the  potentials for  filters  A  and  B  learned simultaneously  differ \nfrom  the potentials for  the two filters  when they are learned independently? \n\nWe  address these issues  by introducing the g-factor g(;f)  and the associated distri(cid:173)\nbution Po (;f): \n\n(1) \n\nx  space  -----+  iii  space GG \n\ng(ijiJ = number of images x \n\nwith histogram  iii \n\nFigure  2:  The g-factor g(;f)  counts  the number of images  x  that have statistics ;f. \nNote  that  the  g-factor  depends  only  on  the  choice  of filters  and  is  independent  of \nthe training image data. \n\nHere L  is the number of grayscale levels of each pixel, so that LN is the total number \nof possible images.  The g-factor is  essentially  a  combinational factor  which  counts \nthe  number of ways  that  one  can obtain statistics  ;f, see  figure  (2).  Equivalently, \nPo  is  the default  distribution on ;f if the images  are generated by  white  noise  (i.e. \ncompletely random images). \n\n\fWe can use the g-factor to compute the induced distribution P(~I'x) on the statistics \ndetermined by MEL: \n\n~ ~ \n\nA \n\nP(1/1 I'\\)  = \n\nL \n\n-\nX\n\n~ ~ \n\n6;;: 2(-)P(xl'\\) = \n'j','j'  x \n\ng( ~)eX.,j; \n\n~, Z[,\\]  = \n\nZ[,\\] \n\n~ \n\nL \n\n,j; \n\n~  X,j; \n. \n\ng(1/1)e\u00b7\n\n(2) \n\nA \n\n~ \n\n~ ~  ~ \n\nObserve that both P(~I'x) and log Z[,X]  are sufficient for  computing the parameters \nX.  The ,X  can be found by solving either of the following two  (equivalent)  equations: \nL:,j; P(1/1 I,\\) 1/1  = 1/1obs,  or \n= 1/1obs,  which shows that knowledge  of the g-factor \nand eX. ,j;  are  all  that  is  required  to  do  MEL. \nObserve  from  equation  (2)  that  we  have  P(~I'x  =  0)  =  Po(~) . \nsetting ,X  =  0 corresponds to a  uniform  distribution on the images x. \n\nIn  other  words, \n\n8 10  zrXl \n\n;X \n\n~ \n\n3.1  Decoupling Filters \n\nWe  now  derive an important property of the minimax entropy approach.  As  men(cid:173)\ntioned  earlier,  it  often  seems  that  the  potentials  for  filters  A  and  B  decouple. \nIn  other  words,  if one  applies  MEL  to  two  filters  A, B  simultaneously  bv  letting \n...... \n....A \n1/1  = (1/1 \nto \nthe equations: \n\n...... B...... \n...... B \n,1/1  ), '\\ = (,\\  ,'\\  ),  and 1/1obs  = (1/1 obs ' 1/1 obs)'  then the solutIOns'\\  , '\\ \n\n....A  -B \n\n:..tA \n\n...... B \n\n\"'\"\"'A \n\n...... \n\n. \n\nLP(xl,XA , ,XB)(iA(x) , iB(x))  =  (~:bs'~!s)' \nx \n\n(3) \n\nare the same (approximately) as the solutions to the equations L:x p(xl,XA )iA(x) = \n~!s and  L:x P(xl,XB)iB(x)  =  ~!s, see figure  (3)  for  an example. \n\nFigure  3:  Evidence for  decoupling  of features.  The left  and right  panels  show  the \nclique potentials learned for the features a I ax and a I ay respectively.  The solid lines \ngive  the potentials when they  are  learned individually.  The dashed lines  show the \npotentials when they are learned simultaneously.  Figure courtesy of Prof.  Xiuwen \nLiu, Florida State University. \n\nWe  now  show  how  this  decoupling  property arises  naturally if the g-factor for  the \ntwo  filters  factorizes.  This  factorization, of course,  is  a  property only  of the  form \nof the statistics  and  is  completely  independent  of whether  the statistics  of the  two \nfilters  are  dependent for  the training  data. \n\nProperty  I:  Suppose  we  have  two  sufficient  statistics  iA(x), iB (x)  which  are  in(cid:173)\ndependent  on  the  lattice  in  the  sense  that  g(~A,~B )  =  gA (~A )gB(~B) , \nlogZ[,XA,,XB]  = \n\nthen \nlogZA[,XA] + logZB[,XB]  and  p(~A,~B )  =  pA(~A)pB(~B ). \n\n\f. \n\nThis  implies  that  the  parameters  XA,  XB  can  be  solved  from  the  independent \n81ogZA[XA]  _ \n-A \n'ljJobs' \n-\nequatwns \nL.,j;B  pB(;fB );fB  =  ;f~s ' \n\n-B \n'ljJobs  or  L.,j;A  P  ('ljJ)'ljJ \n\nA  A  -A  -A \n\n8 1ogZ B [XB ] \n\n-A \n'ljJobs' \n\n8XA \n\n8XB \n\n_ \n-\n\n= \n\nMoreover,  the  resulting  distribution  PUC)  can  be  obtained  by  multiplying the  distri(cid:173)\nbutions  (l/Z A)eXA .,j;A(x)  and (l/ZB) eXB.,j;B(x)  together. \n\nThe point here is  that the potential terms for  the two statistics ;fA,;fB  decouple if \nthe phase factor  g(;fA,;fB)  can be factorized.  We conjecture  that  this  is  effectively \nthe case for many linear filters  used in vision processing.  For example, it is plausible \nthat  the  g-factor  for  features  0/ ox  and  0/ oy  factorizes  - and  figure  (3)  shows \nthat  their  clique  potentials  do  decouple  (approximately).  Clearly,  if factorization \nbetween filters  occurs then it gives  great simplification to the system. \n\n4  Approximating the g-factor for  a  Single  Histogram \n\nWe  now  consider  the  case  where the statistic is  a  single  histogram.  Our aim  is  to \nunderstand  why  features  whose  histograms  are  of stereotypical  shape  give  rise  to \npotentials  of the  form  given  by  figure  (3).  Our  results,  of course,  can  be  directly \nextended  to  multiple  histograms  if the  filters  decouple,  see  subsection  (3.1).  We \nfirst  describe the approximation and then discuss  its  relevance for  filter  pursuit. \nWe  rescale the X variables by N  so  that we  have: \n\neNX.\u00a2(x) \n\nA \n\n_ \n\n_  eNX.,j; \n\nP(X'I-\\)  = \n\nZ[X] \n\n,  P('ljJ I-\\)  =  g('ljJ)  Z[X]  , \n\n(4) \n\nWe  now  consider  the approximation that the filter  responses  {Ii}  are  independent \nof each  other  when  the  images  are  uniformly  distributed.  This  is  the  multinomial \napproximation.  (We  attempted a  related approximation [1]  which was  less  success(cid:173)\nful.)  It  implies  that  we  can  express  the  phase  factor  as  being  proportional  to  a \nmultinomial distribution: \n(nt:) \n9  <P  = \n\n(N'ljJd!. .. (N'ljJQ)!Ol \n\n(N'ljJd!. .. (N'ljJQ)!o \n\nn (nt:)  _ \n-\n\n... 0Q \n\n'  TO  <p \n\nN1/JQ \n\nN1/JQ \n\nLN \n\n\"'OQ \n\nN1/Jl \n\nN! \n\nN! \n\nN1/Jl \n\n(5) \nwhere  L.~=1 'ljJa  =  1 (by  definition)  and the  {oa} are the means of the components \nNa }  with  respect  to  the  distribution  Po (;f).  As  we  will  describe  later,  the  {oa} \nwill  be determined  by the filters  {fi}.  See  Coughlan and Yuille,  in preparation, for \ndetails  of how  to compute the  {oa}. \n\nThis approximation enables us  to calculate MEL analytically. \n\nTheorem  With  the multinomial  approximation  the  log  partition function  is: \n\nlog Z[X]  =  N  log L + N  log{~= e \" a+1og aa } , \n\nQ \n\n(6) \n\na=l \n\nand  the  \"potentials\"  P a}  can  be  solved  in  terms  of the  observed  data  {'ljJobs ,a}  to \nbe: \n\na  =  1, .. . ,Q. \n\n(7) \n\n-\n\n\\ \nAa - og--, \n\n'ljJobs,a \n\nI \n\nOa \n\n\fFigure  4:  Top  row:  the  multinomial  approximation.  Bottom row:  full  implemen(cid:173)\ntation of MEL  (see  text).  (Left  panels)  the potentials,  (center  panels)  synthesized \nimages,  and  (right  panels)  the  difference  between  the  observed histogram  (dashed \nline)  and  the  histogram  of the  synthesized  images  (bold  line).  Filters  were  d/dx \nand d/dy. \n\nWe  note  that  there  is  an  ambiguity  Aa  r-+  Aa  + K  where  K  is  an  arbitrary  number \n(recall  that L~=l 'IjJ(a)  =  1).  We  fix  this  ambiguity  by  setting X =  0  if a. =  \"Jobs. \nProof.  Direct  calculation. \n\nOur  simulation  results  show  that  this  simple  approximation  gives  the  typical  po(cid:173)\ntential  forms  generated  by  Markov  Chain  Monte  Carlo  (MCMC)  algorithms  for \nMinimax Entropy Learning.  Compare the multinomial approximation results with \nthose  obtained from  a  full  implementation  of MEL  by  the  algorithm  used  in  [11], \nsee figure  (4). \n\nFilter  pursuit  is  required  to  determine  which  filters  carry  most  information. \nMEL  [11]  prefers  filters  (statistics)  which  give  rise  to  low  entropy  distribu(cid:173)\ntions  (this  is  the  \"Min\"  part  of  Minimax).  The  entropy  is  given  by  H(P)  = \n- L x P(xIX) log P(xIX)  =  log Z[X]  - L~=l Aa'IjJa \u00b7  For  the  multinomial  approxi(cid:173)\nmation  this  can  be  computed  to  be  N log L  - N L~=l 'ljJa  log ~.  This  gives  an \nintuitive interpretation of feature pursuit:  we  should prefer filters  whose statistical \nresponse  to  the  image  training data is  as  large  as  possible  from  their  responses to \nuniformly distributed images.  This is  measured by the Kullback-Leibler divergence \nL~=l 'ljJa  log ~. Recall that if the multinomial approximation is  used for  multiple \nfilters  then we  should simply add together the entropies of different filters. \n\n\f5  Connections to  Generalized Iterative  Scaling \n\nIn  this  section  we  demonstrate  a  connection  between  the  multinomial  approxima(cid:173)\ntion  and  Generalized  Iterative  Scaling  (GIS)[2].  GIS  is  an  iterative  procedure  for \ncalculating clique potentials that is  guaranteed to converge to the maximum likeli(cid:173)\nhood values of the potentials given the desired empirical filter  marginals  (e.g.  filter \nhistograms).  We  show  that  estimating the  potentials  by  the  multinomial  approx(cid:173)\nimation  is  equivalent  to  the  estimate obtained  after  performing  the  first  iteration \nof GIS.  We  also outline an efficient  procedure that allows us  to continue additional \nGIS  iterations to improve upon the multinomial approximation. \n\nThe  GIS  procedure  calculates  a  sequence  of  distributions  on  the  entire  image \n(and  is  guaranteed  to  converge  to  the  correct  maximum  likelihood  distribu-\ntion),  with  an  update  rule  given  by  p(t+1)(x)  ex  P(O)(x)Il~=l{ :F; } <pa(x),  where \n'lfJit )  =<  <Pa(X)  >P(t)(x)  is  the  expected  histogram  for  the  distribution  at  time  t. \nThis  implies  that  the  corresponding  clique  potential  update  equation  is  given  by: \n>.it +1)  =  >.it )  + log 'lfJ~bs - log 'lfJit ). \nIf we  initialize  GIS  so  that  the  initial  distribution  is  the  uniform  distribution, \np(O) (x)  =  L -N,  then  the  distribution  after  one  iteration  is  p(1) (x)  ex \ni.e. \ne2::a <Pa(X) log(1j;~bs /aa) .  In  other  words,  the  distribution  after  one  iteration  is  the \nMEL  distribution  with  clique  potential  given  by  the  multinomial  approximation. \n(The result can be adapted to the case of multiple filters,  as  explained in Coughlan \nand Yuille,  in  preparation.) \n\nWe  can  iterate  GIS  to  improve  the  estimate  of the  clique  potentials  beyond  the \naccuracy of the multinomial  approximation.  The main  difficulty  lies  in  estimating \n'lfJit )  for  t  >  0  (at  t  =  0  this  expectation  is  just  the  mean  histogram with  respect \nto the uniform distribution, <l:a,  which may be calculated efficiently as  described in \nCoughlan and Yuille, in preparation).  One way to approximate these expectations is \nto apply a Bethe-Kikuchi approximation technique [8], used for estimating marginals \non  Markov  Random  Fields,  to  our  MEL  distribution.  Our  technique,  which  was \ninspired  by  the  Unified  Propagation and  Scaling  Algorithm  [7],  consists  of writing \nthe  Bethe  free  energy  [8]  for  our  2-d  image  lattice,  simplifying  it  using  the  shift \ninvariance of the lattice  (which enables the algorithm to run swiftly), and using the \nConvex-Concave  Procedure  (CCCP)  [9]  procedure  to  obtain  an  iterative  update \nequation  to  estimate  the  histogram  expectations.  The  GIS  algorithm  is  then  run \nusing these histogram expectations  (the results were  accurate and did not improve \nappreciably  by  using  the  higher-order  Kikuchi  free  energy  approximation).  See \nCoughlan and Yuille,  in preparation, for  details  of this  procedure. \n\n6  Discussion \n\nThis  paper  describes  the  g-factor,  which  depends  on  the  lattice  and  quantization \nand is independent of the training image data.  Alternatively it can be thought of as \nbeing  proportional to the  distribution  of feature  responses  when  the input  images \nare uniformly distributed. \n\nWe  showed  that  the  g-factor  can  be  used  to  relate  probability  distributions  on \nfeatures  to  distributions  on  images.  In  particular,  we  described  approximations \n\n\fwhich,  when  valid,  enable  MEL  to  be  computed  analytically.  In  addition,  we  can \ndetermine when the clique potentials for features  decouple, and evaluate how infor(cid:173)\nmative each feature  is.  Finally, we  establish a  connection between the multinomial \napproximation and GIS, and outline an efficient  procedure based on Bethe-Kikuchi \napproximations that allows us to continue additional GIS iterations to improve upon \nthe multinomial approximation. \n\nAcknowledgements \n\nWe  would like to thank Michael Jordan and Yair Weiss for  introducing us to Gener(cid:173)\nalized Iterative Scaling and related algorithms.  We  also thank Anand Rangarajan, \nXiuwen Liu, and Song Chun Zhu for helpful conversations.  Sabino Ferreira gave use(cid:173)\nful  feedback on the manuscript.  This work was supported by the National Institute \nof Health  (NEI)  with grant number R01-EY 12691-01. \n\nReferences \n\n[1]  J.M. Coughlan and A.L.  Yuille.  \"A Phase Space Approach to Minimax Entropy \nLearning and The Minutemax approximation\". In  Proceedings  NIPS '98.  1998. \n\n[2]  J.  N.  Darroch  and  D.  Ratcliff.  \"Generalized  Iterative  Scaling  for  Log-Linear \nModels\".  The  Annals  of Mathematical  Statistics.  1972.  Vol.  43,  No.5,  1470-\n1480. \n\n[3]  C.  Domb and M.S.  Green (Eds). Phase Transitions and Critical Phenom(cid:173)\n\nena. Vol.  2.  Academic Press. London.  1972. \n\n[4]  S.  M.  Konishi,  A.L.  Yuille,  J.M.  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