Assignment 4
Computational Vision
U. Minn. Psy 5036
Daniel Kersten

Off[General::spell1];

Themes

Modeling image variation: Scene-based vs. image-based

Geometric vs. photometric variation

Different generative models make different predictions for recognition inference

Humans easily recognize familiar 3D objects from unfamiliar views. Humans can also recognize a familiar object under an unfamiliar illumination. How is this done? There are many possible models. Thinking about how to model image variation--i.e. the generative process, suggests some basic hypotheses to test, even if we don't have a precise model of recognition.  
One can model viewpoint and illumination change  using "true" 3D scene-based models of image formation. Perhaps the visual system in effect "knows" how the 3D structure of the object interacts with viewpoint and illumination, and uses this knowledge to allow for image variations. 
Alternatively, image variation can be modeled using 2D image-based knowledge. One way this could be done is if the visual system has 2D knowledge of a few instances of images it has seen before, and then interpolates between them to allow for variation (Ullman & Basri, 1991).
In the first two exercises, you are going to see how to model viewpoint change using 3D (scene-based) and 2D (image-based) manipulations. This second (2D) method is just one of many ways to model 2D variations ( e.g. if one has 2D "affine" operations for scale or rotation, or shear that are workable approximations to model 3D depth or 3D rotations over small domains). In these exercises, you will be modeling "geometrical" variation.
In the second two exercises, you will model illumination change using 3D (scene-based) and 2D (image-based) manipulations. In these second two exercises, you will be modeling "photometric" variation.

3D Scene-based modeling of viewpoint change

Generate an n-segmented 3D wire object--"bent paper clip"

s = Table[{Random[]-0.5,Random[]-0.5,Random[]-0.5},{6}];

Connect the points with lines

lines=Partition[s,2,1];

View the object under perspective (e.g. Viewpoint at {-2,-2,-2}) allowing rotations about Viewpoint axis

Manipulate[Graphics3D[{Thick,Rotate[Line[lines],theta,{-1,-1,-1}]},ViewPoint->{-2,-2,-2},PlotRange->{{-1,1},{-1,1},{-1,1}},AspectRatio->1,Axes->True,AxesLabel->{"x","y","z"},ImageSize->Medium,PreserveImageOptions->True],{theta,0,Pi}]

Make an orthographic projection of paper clip by setting the z-coordinate to 0, and using ListPlot[ ]

orthographic[x_]:= Drop[#,-1]&/@x

ortholines=Partition[orthographic[s],2,1];

ListPlot[ortholines,Joined->True,PlotRange->{{-1,1},{-1,1}}]

Define a rotation matrix to rotate each {x, y, z} vertex of the paper clip by θ about a vector {1, 1, 1}.

Dot[RotationMatrix[Pi/16,{1,1,1}],#]&/@s

Rotate the paper clip by θ = Pi/8, and then make an orthographic projection again using ListPlot[ ]

ListPlot[orthographic[Dot[RotationMatrix[Pi/16,{1,1,1}],#]&/@s],Joined->True,PlotRange->{{-1,1},{-1,1}}]

Now make a series of plots from  θ = Pi/8 to 3*Pi/4 in steps of Pi/16. (i.e. iterate {θ,Pi/4,3*Pi/4,Pi/16}).

gorthorotateclips = Table[ListPlot[orthographic[Dot[RotationMatrix[θ,{1,1,1}],#]&/@s],AspectRatio->1,Joined->True,PlotRange->{{-1,1},{-1,1}}],{θ,Pi/8,3*Pi/4,Pi/16}];

2D Image-based modeling of viewpoint change

Make a list of the first view of the above animation, i.e.  θ = Pi/8. Call it firstview.

firstview = orthographic[Dot[RotationMatrix[Pi/8, {1, 1, 1}], #] & /@ s];

Make a list of the last view of the above animation, i.e.  θ = 3*Pi/4. Call it lastview.

lastview = orthographic[Dot[RotationMatrix[3*Pi/4, {1, 1, 1}], #] & /@ s];

Define a function of t that interpolates between the 2D vertices of firstview (t=0) and lastview (t=1)

interview[t_]:= t*lastview + (1-t)*firstview;

Make an animation using ListPlot[ ], for t=0 to 1, in steps of 0.1 (i.e. iterate {t, 0, 1, .1}).

gorthorotateclipsinterv = Table[ListPlot[interview[t],Joined->True, AspectRatio->1,PlotRange->{{-1,1},{-1,1}}],{t,0,1,.1}];

Compare "true" 3D rotations with 2D image-based interpolated approximations

Manipulate[GraphicsRow[{gorthorotateclips[[i]],gorthorotateclipsinterv[[i]]}],{i,1,Dimensions[gorthorotateclips][[1]],1}]

3D scene-based modeling of illumination change

In this section, we will make use of Mathematica's in-built 3D rendering model as used in ListPlot3D.

Range data defines surface list

Range data to define rface list -- BIG 64x64 file. 
Just initialize this cell--don't bother opening cell (from: http://sampl.eng.ohio-state.edu/)
If you want to read in a different file, use the Import[ ]

rface = 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size=Dimensions[rface][[1]]; hsize=size/2;

(Range data from: http://sampl.eng.ohio-state.edu/)
Intensity in a ArrayPlot is proportional to the z (depth) from the camera

GraphicsRow[{ArrayPlot[rface, Mesh -> False, DataReversed->True,ImageSize -> Small, ColorFunction->"GrayTones"],ListPlot3D[rface,PlotRange->{0,255}]}]

Define ShadedListPlot3D[ ]

ShadedListPlot3D[list_,ref_,illum_] := ListPlot3D[rface,PlotRange->{0,255},Mesh->False,
  Lighting -> {{"Directional", GrayLevel[1], {{0,0,0},illum}}},ViewPoint->{0,0,-10},
    AspectRatio->1,Ticks->False,Axes->False,Boxed->False,PlotRangePadding -> None,Background->Black,ImageSize->Tiny];

theta = N[Pi/4];
s={Cos[theta],1,Sin[theta]};
ref = 0.9;
gillum=Table[ShadedListPlot3D[rface,ref,{Cos[theta],1,Sin[theta]}],{theta,Pi/4,3*Pi/4,Pi/8}];

2D image-based modeling of illumination change

Initialize left and right illuminated faces (image data in cells is hidden, i.e. cell property is not "open")

firstframe

firstframe=Graphics[{
Raster[{
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{{0, 0, 0}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0, 0, 0}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0, 0, 0}, {0.0118, 0.0118, 0.0118}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0.325, 0.325, 0.325}, {0.0314, 0.0314, 0.0314}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0, 0, 0}, {0, 0, 0}, {0, 0, 0}, {0.00392, 0.00392, 0.00392}, {0.00784, 0.00784, 0.00784}, {0.00392, 0.00392, 0.00392}, {0.00392, 0.00392, 0.00392}, {0, 0, 0}}}, {{62, 4082}, {162, 4182}}]}, ImageSize->{100, 101}];

Extract graylevel images

leftillumface =Map[0.3333#[[1]]+0.3333#[[2]]+0.333#[[3]]&,N[firstframe[[1,1,1]]],{2}];
rightillumface =Map[0.3333#[[1]]+0.3333#[[2]]+0.333#[[3]]&,N[secondframe[[1,1,1]]],{2}];

Define interpolation function

interillumface[t_] := t*rightillumface + (1-t)*leftillumface;

gilluminterp=Table[ArrayPlot[interillumface[t],Mesh->False,DataReversed->True,ImageSize -> Tiny, ColorFunction->"GrayTones",PlotRange->Max[leftillumface]],{t,0,1,.25}];

Compare "true" 3D illumination with linearly image-based interpolated illumination

Manipulate[GraphicsRow[{gillum[[i]],gilluminterp[[i]]}],{i,1,Dimensions[gilluminterp][[1]],1}]

QUESTIONS

You will be graded on the following

1. Evaluate and try to understand each step in this notebook.

Then type your answers to the following questions:

2a. Explain in words the differences between how the images are generated using 3D vs. 2D models of viewpoint change. Describe one way the 2D approximation seems to depart from the "ground truth" represented by the 3D views.

2b. Suppose an experimentalist tells you that his human subjects could recognize an object equally well from all views (i.e. from any point on the 3D viewing sphere), even though they had initially been trained on only two views. As a modeler, would you be inclined to favor a 3D scene-based or a 2D image-based theory? Why?

3a.  Explain in words the differences between how the images are generated using 3D vs. 2D models of illumination change.

3b. Suppose an experimentalist tells you that after training subjects on two different but specific illuminations (e.g. point light source on the left and then one on the right), her human observers recognize the object given "in-between" illuminations equally well, but not others (e.g. ones generated with light sources from above or below).
Would this support the 2D image-based model of illumination change described above or not? Why?

4. There are two major theoretical/computational problems with the image-based model for dealing with view-point change in non-wire objects: a) self-occlusion; b) identifying geometrical feature points. 

Explain why these are minor problems for the wire-objects above, but are major problems for a typical "everyday" object.

5. Explain how the problem of identifying a particular rigid 3D object differs from recognizing the category of an object.

References

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Tarr, M. J., Kersten, D., & Bulthoff, H. H. (1998). Why the visual recognition system might encode the effects of illumination. Vision Res, 38(15-16), 2259-2275.
Tarr, M. J., & Bülthoff, H. H. (1995). Is human object recognition better described by  geon-structural-descriptions or by multiple-views? Journal of Experimental Psychology: Human Perception and Performance, 21(6), 1494-1505.
Ullman, S., & Basri, R. (1991). Recognition by linear combinations of models. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(10), 992-1006.
Ullman, S. (1996). High-level Vision: Object Recognition and Visual Cognition. Cambridge, Massachusetts: MIT Press.
Vogels, R., & Biederman, I. (2002). Effects of illumination intensity and direction on object coding in macaque inferior temporal cortex. Cereb Cortex, 12(7), 756-766.