{"title": "A Model of Early Visual Processing", "book": "Advances in Neural Information Processing Systems", "page_first": 173, "page_last": 179, "abstract": "", "full_text": "A  Model of Early Visual Processing \n\nLaurent Itti, Jochen Braun,  Dale K.  Lee and Christof Koch \n\n{itti,  achim,  jjwen,  koch}Gklab.caltech.edu \n\nComputation &  Neural  Systems,  MSC  139-74 \n\nCalifornia Institute of Technology,  Pasadena,  CA  91125,  U.S.A. \n\nAbstract \n\nWe  propose  a  model for  early  visual  processing  in  primates.  The \nmodel consists  of a  population of linear spatial filters  which  inter(cid:173)\nact  through non-linear excitatory  and inhibitory pooling.  Statisti(cid:173)\ncal estimation theory  is  then used  to derive human psychophysical \nthresholds from the responses of the entire population of units.  The \nmodel is  able  to reproduce  human thresholds for  contrast  and ori(cid:173)\nentation discrimination tasks,  and to predict contrast thresholds in \nthe presence  of masks of varying orientation and spatial frequency. \n\n1 \n\nINTRODUCTION \n\nA remarkably wide range of human visual thresholds for  spatial patterns appears to \nbe  determined  by  the earliest  stages  of visual  processing,  namely,  orientation- and \nspatial frequency-tuned visual filters and their interactions [18,  19,  3, 22, 9].  Here we \nconsider the  possibility of quantitatively relating arbitrary spatial vision thresholds \nto  a  single  computational  model.  The  success  of such  a  unified  account  should \nreveal the extent  to which human spatial vision indeed reflects  one particular stage \nof processing.  Another  motivation for  this  work  is  the controversy  over  the neural \ncircuits  that  generate  orientation  and  spatial  frequency  tuning  in  striate  cortical \nneurons  (13,  8,  2].  We  think  it  is  likely  that  behaviorally  defined  visual  filters \nand  their  interactions reveal  at  least  some of the  characteristics  of the  underlying \nneural circuitry.  Two specific  problems  are  addressed:  (i)  what  is  the  minimal set \nof model  components  necessary  to  account  for  human  spatial  vision,  (ii)  is  there \na  general  decision  strategy  which  relates  model responses  to behavioral thresholds \nand which obviates case-by-case assumptions about the decision strategy in different \nbehavioral situations.  To investigate these  questions,  we  propose  a  computational \nmodel articulated  around three  main stages:  first,  a  population of bandpass linear \nfilters extracts visual features from a stimulus; second, linear filters interact through \nnon-linear  excitatory  and  inhibitory  pooling;  third,  a  noise  model  and  decision \nstrategy  are  assumed in order to relate  the model's output to psychophysical data. \n\n\f174 \n\nL  Itti, 1.  Braun, D.  K.  Lee and C.  Koch \n\n2  MODEL \nWe  assume  spatial  visual  filters  tuned  for  a  variety  of  orientations  e E  e  and \nspatial periods  A E A.  The filters  have  overlapping receptive  fields  in  visual space. \nQuadrature filter  pairs,  p{~(r and F{~d, are  used  to compute a  phase-independent \nlinear  energy  response,  E>.,6,  to  a  visual stimulus S.  A small constant  background \nactivity,  f,  is  added to the  linear energy  responses: \n\nE>.  6  =  . /(peven * S)2  + (podd * S)2  + f \n\n, \n\n\\I \n\n>' ,6 \n\n>.,6 \n\nFilters have separable  Gaussian tuning curves  in orientation and spatial frequency. \nTheir corresponding shape in visual space is close  to that of Gabor filters,  although \nnot  separable  along spatial dimensions. \n\n2.1  Pooling:  self excitation and  divisive inhibition \n\nA model based on linear filters  alone would not correctly  account for  the non-linear \nresponse characteristics to stimulus contrast which have been  observed psychophys(cid:173)\nically  [19].  Several  models  have  consequently  introduced  a  non-linear  transducer \nstage  following  each  linear  unit  [19].  A  more  appealing  possibility  is  to  assume  a \nnon-linear  pooling stage  [6,  21,  3,  22].  In this study,  we  propose  a  pooling strategy \ninspired  by  Heeger's  model for  gain  control  in  cat  area VI  [5,  6].  The  pooled  re(cid:173)\nsponse  R>.,6  of a  unit tuned for  (A, 0)  is  computed from  the linear energy  responses \nof the entire  population: \nR>. \n\nE'Y \n\n(1) \n\n>',6 \n\n-\n\n,6  - So  + L>'I,61 W>.,6(N, OI)E~/,61 \n\n+ 1] \n\nwhere  the  sum is  taken  over  the  entire  population  and  W>.,6  is  a  two-dimensional \nGaussian  weighting  function  centered  around  (A,O),  and  1]  a  background  activity. \nThe  numerator in  Eq.  1  represents  a  non-linear self-excitation  term.  The denomi(cid:173)\nnator represents  a  divisive  inhibitory  term which depends  not  only  on  the  activity \nof the unit  (A,O)  of interest,  but also on the  responses  of other units.  We shall  see \nin  Section  3  that,  in  contrast  to  Heeger's  model for  electrophysiological  data  in \nwhich  all units contribute equally to the pool,  it is  necessary  to assume that only a \nsubpopulation of units  with tuning close  to (A, 0)  contribute to the pool in order to \naccount for  psychophysical data.  Also,  we  assume, > 15  to obtain a  power  law  for \nhigh  contrasts  [7],  as  opposed  to Heeger's  physiological model in  which, = 15  = 2 \nto  account for  neuronal  response  saturation at high contrasts. \n\nSeveral  interesting  properties  result  from  this  pooling  model.  First,  a  sigmoidal \ntransducer function - in  agreement with contrast discrimination psychophysics - is \nnaturally obtained through pooling and thus need  not be introduced post-hoc.  The \ntransducer slope for high contrasts is determined by ,-15, the location of its inflexion \npoint by 5, and the slope at this point by the absolute value of, (and 15).  Second, the \ntuning curves  of the  pooled units for  orientation and spatial period do not depend \nof stimulus contrast,  in  agreement  with  physiological  and  psychophysical evidence \n[14].  In comparison, a model which  assumes a non-linear transducer  but no pooling \nexhibits  sharper  tuning  curves  for  lower  contrasts.  Full  contrast  independence  of \nthe  tuning is  achieved  only  when  all  units  participate in  the  inhibitory pool;  when \nonly sub-populations participate in  the pool,  some contrast dependence  remains. \n\n2.2  Noise model:  Poisson lX \n\nIt is  necessary  to assume the presence  of noise  in the  system in order to be  able  to \nderive  psychophysical  performance from  the  responses  of the  population of pooled \n\n\fA Model of Early Visual Processing \n\n175 \n\nunits.  The deterministic response  of each  unit  then represents  the  mean of a  ran(cid:173)\ndomly distributed  \"neuronal\" response which varies from trial to trial in a simulated \npsychophysical  experiment . \n\nExisting  models  usually  assume  constant  noise  variance  in  order  to  simplify  the \nsubsequent  decision  stage  [18].  Using  the  decision  strategy  presented  below,  it  is \nhowever  possible  to  derive  psychophysical  performance  with  a  noise  model  whose \nvariance  increases  with  mean  activity,  in  agreement  with  electrophysiology  [16]. \nIn  what  follows,  Poissoncx  noise  will  be  assumed  and  approximated by  a  Gaussian \nrandom variable with  variance = mean cx  (0'  is  a  constant close  to unity). \n\n2.3  Decision strategy \n\nWe  use  tools from  statistical estimation theory  to compute the system's behavioral \nresponse  based  on  the  responses  of the  population  of pooled  units.  Similar tools \nhave been used  by Seung and Sompolinsky [12]  under the simplifying assumption of \npurely  Poisson noise  and for  the  particular task of orientation discrimination in the \nlimit of an  infinite  population  of oriented  units.  Here,  we  extend  this  framework \nto  the  more  general  case  in  which  any  stimulus  attribute  may differ  between  the \ntwo stimulus presentations to  be discriminated by  the model.  Let's assume that we \nwant to estimate psychophysical performance at discriminating between two stimuli \nwhich  differ  by  the  value  of  a  stimulus  parameter  ((e.g .  contrast,  orientation, \nspatial period). \n\nThe  central  assumption  of our  decision  strategy  is  that  the  brain  implements  an \nunbiased  efficient statistic T(R; (), which  is  an estimator of the  parameter (  based \non  the  population  response  R  = {R).,I/; A E  A, ()  E  0}.  The  efficient  statistic  is \nthe  one  which,  among  all  possible  estimators  of (,  has  the  property  of minimum \nvariance in the estimated value of ( .  Although  we  are  not suggesting any  putative \nneuronal  correlate  for  T,  it  is  important  to  note  that  the  assumption  of efficient \nstatistic does  not  require  T  to be  prohibitively complex;  for  instance,  a  maximum \nlikelihood  estimator  proposed  in  the  decision  stage  of several  existing  models  is \nasymptotically (with respect  to the number of observations)  a  efficient  statistic. \n\nBecause  T  is  efficient,  it  achieves  the  Cramer-Rao bound  [1].  Consequently,  when \nthe number  of observations  (i.e.  simulated psychophysical  trials)  is  large, \n\nE[T] =  ( \n\nand \n\nvar[T]  =  1/3(() \n\nwhere  E[.]  is  the  mean  over  all  observations,  var[.]  the  variance,  and  3(()  is  the \nFisher information.  The Fisher information can be computed using the noise model \nassumption  and  tuning  properties  of the  pooled  units:  for  a  random  variable  X \nwith  probability density  f(x; (), it is  given  by  [1]: \n\nJ(() =  E [:c In/(X;()r \n\nFor  our  Poissoncx  noise  model  and  assuming  that  different  pooled  units  are  inde(cid:173)\npendent  [15],  this  translates into: \n\nOne unit R). ,I/: \nAll independent units: \n\nThe Fisher  information computed for  each  pooled  unit and three  types  of stimulus \nparameters  (  is  shown  in  Figure  1.  This  figure  demonstrates  the  importance  of \nusing  information from  all  units  in  the  population rather  than from only  one  unit \noptimally tuned for  the stimulus:  although the unit carrying  the most information \nabout contrast is the one optimally tuned to the stimulus pattern, more information \n\n\f176 \n\nL. lui, 1  Braun, D.  K.  Lee and C.  Koch \n\nabout orientation or spatial frequency is carried by units which are tuned to flanking \norientations  and spatial periods  and  whose  tuning curves  have  maximum slope  for \nthe  stimulus  rather  than  maximum  absolute  sensitivity.  In  our  implementation, \nthe derivatives of pooled responses  used  in the expression of Fisher information are \ncomputed numerically. \n\norientation \n\nspatial frequency \n\nFigure  1:  Fisher  information  computed  for  contrast,  orientation  and  spatial  frequency. \nEach  node  in  the  tridimensional  meshes  represents  the  Fisher  information  for  the  corre(cid:173)\nsponding pooled  unit  (A, B)  in  a  model  with 30 orientations  and  4 scales.  Arrows indicate \nthe  unit  (A, B)  optimally  tuned  to  the stimulus.  The total  Fisher information  in  the pop(cid:173)\nulation  is  the  sum of the information  for  all  units. \n\nUsing  the  estimate  of  (  and  its  variance  from  the  Fisher  information,  it  is  pos(cid:173)\nsible  to  derive  psychophysical  performance  for  a  discrimination  task  between  two \nstimuli with parameters (1  ~ (2  using standard ideal observer signal discrimination \ntechniques  [4] .  For such  discrimination, we  use  the  Central  Limit Theorem  (in  the \nlimit of large  number  of trials)  to model  the  noisy  responses  of the  system  as  two \nGaussians  with  means (1  and  (2,  and  variances  lTi  = 1/:1((d  and  lTi  = 1/:1((2) \nrespectively.  A decision  criterion  D  is  chosen  to minimize the overall probability of \nerror;  since  in  our  case  lT1  =f. \nlT2  in  general,  we  derive  a  slightly more  complicated \nexpression for performance  P  at a Yes/No  (one  alternative forced  choice)  task than \nwhat  is  commonly used  with  models assuming constant  noise  [18]: \n\n(2 lTi - (llT~ -\n\nD  = \n\nlT1lT2J((1  - (2)2 +  2(lTr  -\n\n2 \nlT1  -\n\n2 \nlT2 \n\nlTi) log(lT!/lT2) \n\nP= ~+~erf((2-D) +  ~erf(D-(l) \n\nlT2..J2 \n\n4 \n\n2 \n\n4 \n\nlT1..J2 \n\nwhere  erf is  the Normal error function.  The expression  for  D  extends by continuity \nto  D  =  ((2  - (1)/2 when  lT1  =  lT2 .  This  decision  strategy  provides  a  unified,  task(cid:173)\nindependent framework for the computation of psychophysical performance from the \ndeterministic responses  of the pooled units.  This strategy can easily be extended to \nallow the model to perform discrimination tasks with respect  to additional stimulus \nparameters,  under exactly  the  same theoretical  assumptions. \n\n3  RESULTS \n\n3.1  Model  calibration \n\nThe parameters of the model were  automatically adjusted to fit  human psychophys(cid:173)\nical  thresholds measured in our laboratory [17]  for  contrast and orientation discrim(cid:173)\nination  tasks  (Figure  2).  The  model  used  in  this  experiment  consisted  of  60 \norientations evenly distributed between  0 and 180deg.  One spatial scale  at 4 cycles \nper degree  (cpd)  was sufficient to account for  the data.  A multidimensional simplex \nmethod  with  simulated  annealing  overhead  was  used  to  determine  the  best  fit  of \nthe  model  to  the  data  [10].  The  free  parameters  adjusted  during  the  automatic \n\n\fA Model of Early VlSUal Processing \n\n177 \n\nfits  were:  the  noise  level  a,  the  pooling exponents  'Y  and &,  the  inhibitory pooling \nconstant 5,  and  the  background firing  rates,  E  and rJ. \n\nThe  error  function  minimized  by  the  fitting  algorithm  was  a  weighted  average  of \nthree  constraints:  1)  least-square  error  with  the  contrast  discrimination  data  in \nFigure 2.a;  2)  least-square error  with  the orientation discrimination data in  Fig(cid:173)\nure  2.h;  3)  because  the  data was  sparse  in  the  \"dip-shaped\"  region  of the  curve \nin  Figure  2.a,  and  unreliable  due  to  the  limited  contrast  resolution  of  the  dis(cid:173)\nplay used for  the psychophysics,  we  added  an additional constraint favoring  a  more \npronounced  \"dip\", as has been observed  by  several other  groups  [11,  19,  22] . \n\nData fits used for model calibration: \n\n. -_____  ---..:a:::..,  a \n\niii \n\n~u  ~ 0 -\n\n~ \n\n~ ~ 10-2 \nc:Ch \nQ)~ E :5  10-3 L..-__ _ ___  --...J \no \n0 \n10 \nc: .-\n\nmask contrast \n\n\u00b72 \n10 \n\nQ) \n\n0.2 \n\n0.4 \nstimulus contrast \n\nQ) \nCh \n\nTransducer function: \n\n-~50.----________ c~ \nc: o a. \n~ \nu \nQ) \n(5 \n8. \n\nstimulus contrast \n\n0.5 \n\nOrientation tuning: \nd \n\nI ; \n\nQ) \nCh \n\nC o a. \n~ 0.5 \n\n~ \nQ) \n> \n~  O~~----~----=-~ \n100 \n~  -100 \nstimulUS orientation  (deg) \n\n0 \n\nFigure  2:  The  model  (solid  lines)  was  calibrated  using  data from  two  psychophysical \nexperiments:  (a)  discrimination  between a pedestal contrast  (a.a)  and the same pedestal \nplus an increment contrast (a.{3);  (b) discrimination between two orientations near vertical \n(b.a and b.{3).  After calibration,  the transducer function  of each pooled unit  (c)  correctly \nexhibits  an  accelerating  non-linearity  near  threshold  (contrast  ~ 1%)  and  compressive \nnon-linearity  for  high  contrasts  (Weber's  law).  We  can  see  in  (d)  that  pooling  among \nunits with similar tuning properties sharpens  their tuning curves.  Model  parameters were: \na  ~ 0.75,,),  ~ 4,\u00ab5  ~ 3.5,E  ~ 1%, '1  ~ 1.7Hz,S  such  that  transducer  inflexion  point  is \nat 4x  detection  threshold  contrast,  orientation  tuning  FWHM=68deg  (full  width  at half \nmaximum),  orientation  pooling  FWHM=40deg. \n\nTwo  remaining parameters are  the  orientation  tuning  width,  (7'8,  of the  filters  and \nthe  width,  (7'We,  of the  pool.  It  was  not  possible  from  the  data in  Figure 2  alone \nto  unambiguously  determine  these  parameters.  However,  for  any  given  (7'8,  (7'W8 \nis  uniquely  determined  by  the  following  two  qualitative constraints:  first,  a  small \npool  size  is  not  desirable  because  it  yields  contrast-dependent  orientation  tuning; \nit  however  appears from  the data in  Figure  2.h that  this  tuning should  not  vary \nmuch over a  wide  range of contrasts.  The second constraint is qualitatively derived \nfrom  Figure 3.a:  for  large  pool sizes,  the model predicted  significant  interference \nbetween  mask  and  test  patterns  even  for  large orientation differences.  Such  inter-\n\n\f178 \n\nL  Itti,  1.  Braun, D.  K.  Lee and C.  Koch \n\nference  was  not  observed  in  the  data for  orientation differences  larger than  45deg. \nIt consequently seems that a  partial inhibitory pool, composed only of a fraction  of \nthe  population of oriented filters  with tuning similar to the central excitatory  unit, \naccounts  best  for  the  psychophysical  data.  Finally,  (76  was  fixed  so  as  to  yield  a \ncorrect  qualitative curve  shape for  Figure 3.a. \n\n3.2  Predictions \n\nWe  used  complex  stimuli  from  masking  experiments  to  test  the  predictive  value \nof the  model  (Figure  3).  Although  it  was  necessary  to  use  some of the  qualita(cid:173)\ntive  properties  of the  data seen  in  Figure  3.a to calibrate  the  model  as  detailed \nabove,  the calibrated model correctly  produced  a quantitative fit  of this data.  The \ncalibrated model also correctly  predicted  the complex data of Figure 3.h. \n\nc::10 \n0 \n~ \n> \nQ)  5 \nCD \n\"C \n(5 \n.r; \n(/J \nQ) \n~ \n.r; \n...... \n\n0 \nmask orientation (deg)  mask \n\n90  no \n\n30 \n\n60 \n\na \n\na \n\n~ \n\nb \n\nc:: \n0  6 \n~ \n> \n4 \nQ) \nQ) \n\"C  2 \n(5 \n.r; \n(/J \n8 \nQ) \n~ \n.r;  mask spatial freq.  (cpd) \n...... \n\n2 \n\n4 \n\nFigure  3:  Prediction  of psychophysical  contrast  thresholds  in  the presence  of an oblique \nmask.  The  mask  was  a  50%-contrast  stochastic  oriented  pattern  (a).  and  the  superim(cid:173)\nposed  test pattern was a  sixth-derivative  of Gaussian  bar (j3).  In  (a),  threshold  elevation \n(i.e.  ratio  of threshold  in the  presence  of mask  to threshold  in  the  absence  of mask)  was \nmeasured for  varying  mask orientation,  for  mask  and  test patterns at  4 cycles  per degree \n(cpd).  In  (b), orientation difference  between test and mask was fixed  to 15deg, and thresh(cid:173)\nold  elevation  was  measured  as  a function  of mask spatial  frequency.  Solid  lines  represent \nmodel  predictions,  and  dashed lines  represent  unity  threshold  elevation. \n\n4  DISCUSSION  AND  CONCLUSION \n\nWe have developed a model of early visual processing in humans which accounts for \na  wide  range  of measured  spatial  vision  thresholds  and  which  predicts  behavioral \nthresholds  for  a  potentially unlimited number  of spatial discriminations.  In  addi(cid:173)\ntion to orientation- and spatial-frequency-tuned units, we  have found it necessary  to \nassume  two  types  of interactions  between  such  units:  (i)  non-linear self-excitation \nof each  unit  and  (ii)  divisive  normalization  of each  unit  response  relative  to  the \nresponses  of similarly tuned  units.  All  model  parameters  are  constrained  by  psy(cid:173)\nchophysical data and an  automatic fitting  procedure  consistently  converged  to  the \nsame parameter set regardless  of the initial position in parameter space. \n\nOur two main contributions are the small number of model components and the un i(cid:173)\n.fied,  task-independent decision strategy.  Rather than making different assumptions \nabout  the  decision  strategy in different  behavioral  tasks,  we  combine the  informa(cid:173)\ntion  contained  in  the  responses  of all  model  units  in  a  manner that  is  optimal for \nany  behavioral  task.  We  suggest  that  human observers  adopt  a  similarly optimal \ndecision  procedure  as they  become familiar with a  particular task  (\" task set\").  Al(cid:173)\nthough  here  we  apply  this decision  strategy  only  to  the discrimination of stimulus \ncontrast,  orientation,  and  spatial  frequency,  it  can  readily  be  generalized  to  arbi(cid:173)\ntrary discriminations such  as, for  example, the discrimination of vernier  targets. \n\n\fA Model of Early Vzsual Processing \n\n179 \n\nSo  far  we  have  considered  only  situations  in  which  the  same  decision  strategy  is \noptimal for  every  stimulus presentation.  We  are  now  studying situations in  which \nthe  optimal  decision  strategy  varies  unpredictably  from  trial  to  trial  (\" decision \nuncertainty\").  For example, situations in which  the observer  attempts to detect  an \nincrease  in either  the  spatial frequency  or  the  contrast of stimulus.  In  this  way,  we \nhope  to learn  the extent  to  which  our  model reflects  the decision  strategy  adopted \nby  human observers  in  an  even  wider  range  of situations.  We  have  also  assumed \nthat  the  model's  units  were  independent,  which  is  not  strictly  true  in  biological \nsystems  (although  the  main source  of correlation  between  neurons  is  the  overlap \nbetween  their  respective  tuning curves,  which  is  accounted  for  in  the  model).  The \nmathematical developments  necessary  to  account  for  fixed  or  variable  covariance \nbetween  units are  currently  under  study. \n\nIn  contrast  to  other  models  of early  visual  processing  [5,  6],  we  find  that  the  psy(cid:173)\nchophysical data is  consistent  only  with interactions  between  similarly tuned  units \n(e.g., \"near-orientation inhibition\")' not with interactions between  units of very dif(cid:173)\nferent  tuning  (e.g.,  \"cross-orientation  inhibition\") .  Although  such  partial  pooling \ndoes not render tuning functions completely contrast-independent, an additional de(cid:173)\ngree  of contrast-independence  could be  provided by  pooling across  different  spatial \nlocations.  This issue  is  currently  under  investigation. \n\nIn  conclusion,  we  have  developed  a  model  based  on  self-excitation  of each  unit, \ndivisive  normalization  [5,  6]  between  similarly  tuned  units,  and  an  ideal  observer \ndecision strategy.  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