Computational VisionU. Minn. Psy 5036Daniel KerstenLecture 19: Motion Illusions & Bayesian modelsInitialize‡Spell check offIn[360]:=Off@General::spell1D;<< VectorFieldPlots`In[362]:=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‡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‡Local measurements & neural systemsRepresenting 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‡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. Human motion perception2 19.MotionHumanPerception.nbHuman motion perceptionDemo: 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.In[367]:=size = 120;Clear[y];low = 0.2; hi = .75;y[x_] := hi /; x<1y[x_] := .5 Exp[-(x-1)^2]+.1 /; x >= 1In[372]:=ylist = Table[y[i],{i,0,3,3/255.}];width = Dimensions[ylist][[1]];In[374]:=picture1 = Table[ylist,{i,1,width/2}];picture2 = .9 - Transpose[Reverse[Transpose[picture1]]];In[376]:=g1 = ListPlot[picture1[[size/2]],PlotStyle->{Hue[.3]}];g2 = ListPlot[picture2[[size/2]],PlotStyle->{Hue[.6]}];Show[g1,g2]Out[378]=501001502002500.20.40.60.819.MotionHumanPerception.nb 3In[379]:=ga1=ArrayPlot[picture1,Frame->False,Mesh->False, PlotRange->{0,1}, AspectRatio->Automatic]ga2=ArrayPlot[picture2,Frame->False,Mesh->False, PlotRange->{0,1}, AspectRatio->Automatic]Out[379]=Out[380]=There 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.Aperture effectsIn[381]:=niter = 8; width = 32; theta1 = Pi/4.; contrast1 = 0.5; freq1 = 4.; period1 = 1/freq1; stepx1 = Cos[theta1]*(period1/niter); stepy1 = Sin[theta1]*(period1/niter);grating[x_,y_,freq_,theta_] := Cos[(2. Pi freq)*(Cos[theta]*x + Sin[theta]*y)];4 19.MotionHumanPerception.nb‡Circular apertureIn[386]:=Animate[DensityPlot[If[(x-0.5)^2+(y-0.5)^2<0.3^2,grating[x+i*stepx1,y+i*stepy1,freq1,theta1],0],{x,0,1},{y,0,1},Mesh->False,Frame->None,PlotRange->{-2,2},PlotPoints->width],{i,1,niter}]Out[386]=i19.MotionHumanPerception.nb 5‡Square apertureIn[387]:=Animate[DensityPlot[grating[x+i*stepx1,y+i*stepy1,freq1,theta1],{x,0,1},{y,0,1},Mesh->False,Frame->None,PlotRange->{-2,2},PlotPoints->width],{i,1,niter}]Out[387]=iWhat do you see at the vertical boundaries? The horizontal boundaries?‡Rectangular horizontal apertureIn[388]:=Animate[DensityPlot[grating[x+i*stepx1,y+i*stepy1,freq1,theta1],{x,0,1},{y,0,.25},Mesh->False,Frame->None,PlotRange->{-2,2},PlotPoints->width, AspectRatio->Automatic],{i,1,niter}]Out[388]=i6 19.MotionHumanPerception.nb‡Rectangular vertical apertureIn[389]:=Animate[DensityPlot[grating[x+i*stepx1,y+i*stepy1,freq1,theta1],{x,0,.25},{y,0,1},Mesh->False,Frame->None,PlotRange->{-2,2},PlotPoints->width, AspectRatio->Automatic],{i,1,niter}]Out[389]=iProject idea: Try the above with stereo-defined aperturesAdelson's missing fundamental motion illusionWe first make a square-wave grating.In[390]:=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.19.MotionHumanPerception.nb 7And 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[391]:=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[391]=: , , , >In[392]:=Plot@realsquare@x, .5, Pi ê 2D, 8x, 0, 14<DOut[392]=2468101214-1.0-0.50.51.08 19.MotionHumanPerception.nbIn[393]:=gsq = Table@ArrayPlot@Table@realsquare@x, y, i Pi ê 2D, 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@gsqDOut[393]=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[394]:=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