Computational VisionU. Minn. Psy 5036Daniel KerstenLecture 19: Motion Illusions & Bayesian modelsInitialize‡Spell check offIn[766]:=Off@General::spell1D;In[767]:=SetOptions@ArrayPlot, ColorFunction Ø "GrayTones", DataReversed Ø True,Frame Ø False, AspectRatio Ø Automatic, Mesh Ø False,PixelConstrained Ø True, ImageSize Ø SmallD;SetOptions@ListPlot, ImageSize Ø SmallD;SetOptions@Plot, ImageSize Ø SmallD;SetOptions@DensityPlot, ImageSize Ø Small, ColorFunction Ø GrayLevelD;nbinfo = NotebookInformation@EvaluationNotebook@DD;dir =H"FileName" ê. nbinfo ê. FrontEnd`FileName@d_List, nam_, ___D ßToFileName@dDL;OutlineLast time• Early motion measurement--types of models•Functional goals of motion measurements• Optic flowCost function (or energy) descent modelA posteriori and a priori constraintsGradient descent algorithmsComputer vs. human vision and optic flow-- area vs. contourToday‡Local measurements, neural systems & orientation in space-timeRepresenting motion, Orientation in space-time Fourier representation and samplingOptic flow, the gradient constraint, aperture problemNeural systems solutions to the problem of motion measurement.Space-time oriented receptive fields‡Motion phenomena & illusionsNeither the area-based nor the contour-based algorithms we've seen can account for the range of human motion phenom-ena or psychophysical data that we now have. Look at human motion perception‡Global integrationSketch a Bayesian formulation--the integrating uncertain local measurements with the right priors can be used to model a variety of human motion results. Orientation in space-time: Relating neuron responses to gradient-based motion models2 19.MotionHumanPerception.nbOrientation in space-time: Relating neuron responses to gradient-based motion modelsIn this section, we'll see how viewing motion measurement as detecting orientation in space-time is related to neurophysio-logical theories of neural motion selectivity.Demo: area-based vs. contour-based modelsLast time we asked: Are the representation, constraints, and algorithm a good model of human motion perception? The answer seems to be "no". The representation of the input is probably wrong. Human observers often give more weight to contour movement than to intensity flow. Human perception of the sequence illustrated below differs from "area-based" models of optic flow such as the above Horn and Schunck algorithm. The two curves below would give a maxi-mum correlation at zero--hence zero predicted velocity. Human observers see the contour move from left to right--because the contours are stronger features than the gray-levels. However we will see in Adelson's missing fundamental illusion that the story is not as simple as a mere "tracking of edges" --and we will return to spatial frequency channels to account for the human visual system's motion measurements. At the end of this lecture, we'll review a Bayesian model that integrates local motion information according to reliability, providing a theory that may explain a diverse set of motion illusions.size = 120;Clear[y];low = 0.2; hi = .75;y[x_] := hi /; x<1y[x_] := .5 Exp[-(x-1)^2]+.1 /; x >= 1ylist = Table[y[i],{i,0,3,3/255.}];width = Dimensions[ylist][[1]];picture1 = Table[ylist,{i,1,width/2}];picture2 = .9 - Transpose[Reverse[Transpose[picture1]]];19.MotionHumanPerception.nb 3g1 = ListPlot[picture1[[size/2]],PlotStyle->{Hue[.3]}];g2 = ListPlot[picture2[[size/2]],PlotStyle->{Hue[.6]}];Show[g1,g2]501001502002500.20.40.60.8ga1=ArrayPlot[picture1,Frame->False,Mesh->False, PlotRange->{0,1}, AspectRatio->Automatic];ga2=ArrayPlot[picture2,Frame->False,Mesh->False, PlotRange->{0,1}, AspectRatio->Automatic];ListAnimate@8Show@ga1D, Show@ga2D, Show@ga1D<, 2, Paneled Ø False,AppearanceElements Ø None, AnimationRunning Ø FalseDThere is a clear sense of motion of the edge, even though the signal inferred from an intensity, region-based integration of optic flow would produce little or no optic flow in that direction.Adelson's missing fundamental motion illusionWe first make a square-wave grating.In[779]:=realsquare[x_,y_,phase_] := Sign[Sin[x + phase]];And make a four-frame movie in which the grating gets progressively shifted LEFT in steps of Pi/2. That is we shift the grating left in 90 degree steps.4 19.MotionHumanPerception.nbAnd make a four-frame movie in which the grating gets progressively shifted LEFT in steps of Pi/2. That is we shift the grating left in 90 degree steps.In[780]:=Table[DensityPlot[realsquare[x,y,i Pi/2],{x,0,14},{y,0,1}, Frame->False,Mesh->False,PlotPoints -> 60, Axes->None, PlotRange->{-4,4},ImageSize->Tiny],{i,1,4,1}]Out[780]=: , ,, >In[783]:=Plot@realsquare@x, .5, Pi ê2D, 8x, 0, 14<DOut[783]=2468101214- 1.0- 0.50.51.019.MotionHumanPerception.nb 5In[784]:=gsq = Table@ArrayPlot@Table@realsquare@x, y, i Pi ê2D, 8y, 0, 14, .1<,8x, 0, 14, .1<D, PlotRange Ø 8-8, 8<, Frame -> False,ColorFunction -> "GrayTones",Mesh -> False, Axes -> NoneD, 8i, 1, 4, 1<D; ListAnimate@gsq, AnimationRunning Ø FalseDOut[784]=A square wave can be decomposed into its Fourier components as: realsquare(x) = (4/p)*{sin(x) + 1/3 sin(3x) + 1/5 sin(5x) + 1/7 sin(7x) + ...}Now subtract out the fundamental frequency from the square wave...leaving (4/p)*{1/3 sin(3x) + 1/5 sin(5x) + 1/7 sin(7x) + ...}In[785]:=realmissingfundamental[x_,y_,phase_] := realsquare[x,y,phase] - (4.0 / Pi) Sin[x + phase];And make another four-frame movie in which the missing fundamental grating gets progressively shifted LEFT in steps of Pi/2. That is we shift the grating left in 90 degree steps.It is well-known that a low contrast square wave with a missing fundamental appears similar to the square wave (with the fundamental). (There is a pitch analogy in audition.) One reason is that we are more sensitive to sharp than gradual changes in intensity. If you look at the luminance profile with the missing fundamental, you would probably guess that the perceived motion for this sequence would appear to move to the left, as before. But it doesn't. Surprisingly, the missing fundamental wave appears to move to the right!6 19.MotionHumanPerception.nbIn[847]:=gsqr = Table@ArrayPlot@Table@realmissingfundamental@x, y, i Pi ê2D, 8y, 0, 14, .1<,8x, 0, 14, .1<D, PlotRange Ø 8-8, 8<, Frame -> False,ColorFunction -> "GrayTones",Mesh -> False, Axes -> NoneD, 8i, 1, 4, 1<D; ListAnimate@gsqr, AnimationRunning Ø FalseDOut[847]=Play the above movie. It typically appears to be moving to the right. You can