Managing Uncertainty in Cue Combination
Authors: Zhiyong Yang, Richard S. Zemel
Published in: Advances in Neural Information Processing Systems 12  (NIPS 1999) (Main Conference Track), 1999

Abstract: We develop a hierarchical generative model to study cue combi(cid:173) nation.  The model maps a global shape parameter to local cue(cid:173) specific  parameters,  which in tum generate  an intensity image.  Inferring shape from images is achieved by inverting this model.  Inference produces a probability distribution at each level; using  distributions rather than a single value of underlying variables at  each stage preserves information about the validity  of each local  cue for the given image.  This allows the model, unlike standard  combination models, to adaptively weight each cue based on gen(cid:173) eral  cue  reliability  and  specific  image context.  We  describe  the  results  of a cue combination psychophysics experiment we con(cid:173) ducted that allows a direct comparison with the model. The model  provides a good fit to our data and a natural account for some in(cid:173) teresting aspects of cue combination. 
Understanding cue  combination is  a  fundamental  step in developing  computa(cid:173) tional models of visual perception, because many aspects of perception naturally  involve multiple cues, such as binocular stereo, motion, texture, and shading. It is  often formulated as a problem of inferring or estimating some relevant parameter,  e.g., depth, shape, position, by combining estimates from individual cues.  An important finding  of psychophysical studies of cue combination is that cues  vary in the degree to which they are used in different visual environments. Weights  assigned to estimates derived from a particular cue seem to reflect  its estimated  reliability  in  the  current  scene  and  viewing  conditions.  For  example,  motion  and stereo are  weighted approximately equally at near distances, but motion is  weighted more at far distances,  presumably due to distance limits on binocular  disparity.3  Experiments have also found these weightings sensitive to image ma(cid:173) nipulations; if a cue is weakened, such as by adding noise, then the uncontami(cid:173) nated cue is utilized more in making depth judgments.9 A recent study2 has shown  that observers can adjust the weighting they assign to a cue based on its relative  utility for a particular task. From these and other experiments, we can identify two  types of information that determine relative cue weightings:  (1) cue reliability:  its  relative utility in the context of the task and general viewing conditions; and (2)  region informativeness:  cue information available locally in a given image.  A central question in computational models of cue combination then concerns how  these forms of uncertainty can be combined. We propose a hierarchical generative 
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Z.  Yang and R.  S.  Zemel 
model. Generative models have a rich history in cue combination, as thel underlie  models of Bayesian perception that have been developed in this area. lO ,  The nov(cid:173) elty in the generative model proposed here lies in its hierarchical nature and use  of distributions throughout, which allows for both context-dependent and image(cid:173) specific uncertainty to be combined in a principled manner.  Our aims in this paper are dual: to develop a combination model that incorporates  cue reliability and region informativeness (estimated across and within images),  and to use this model to account for data and provide predictions for psychophys(cid:173) ical experiments. Another motivation for the approach here stems from our recent  probabilistic framework,11  which posits that every step of processing entails  the  representation of an entire probability distribution, rather than just a single value  of the relevant underlying variable(s).  Here we use separate local probability dis(cid:173) tributions for each cue estimated directly from an image. Combination then entails  transforming representations and integrating distributions across both space and  cues, taking across- and within-image uncertainty into account. 
1  IMAGE GENERATION 
In this paper we study the case of combining shading and texture. Standard shape(cid:173) from-shading models exclude texture, l, 8  while standard shape-from-texture mod(cid:173) els  exclude  shading.7  Experimental  results  and  computational arguments  have  supported a strong interaction between these cues}O but no model accounting for  this interaction has yet been worked out.  The shape used in our experiments is a simple surface: 
Z = B(l - x2 ), Ixl  <= 1, Iyl  <= 1 
(1) 
where Z is the height from the xy plane.  B is the only shape parameter.  Our image formation model is a hierarchical generative model (see Figure 1). The  top layer contains the global parameter B.  The second layer contains local shad(cid:173) ing and texture parameters S, T  = {Sj, 11}, where i  indexes image regions.  The  generation of local cues from a global parameter is intended to allow local uncer(cid:173) tainties to be introduced separately into the cues.  This models specific conditions  in realistic images, such as shading uncertainty due to shadows or specularities,  and texture uncertainty when prior assumptions such as isotropicity are violated.4  Here we introduce uncertainty by adding independent local noise to the underly(cid:173) ing shape parameter; this manipulation is less realistic but easier to control.