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Notebook[{
Cell[TextData[{
  "Computational Vision\nU. Minn. Psy 5036\nDaniel Kersten\nLecture 17: Shape \
from shading cont'd\n",
  StyleBox[ButtonBox["http://vision.psych.umn.edu/www/kersten-lab/courses/\
Psy5036/SyllabusF2000.html)",
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SyllabusF2000.html"], None},
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Cell[CellGroupData[{

Cell["Initialize", "Subsection"],

Cell[BoxData[{
    \(\(Off[General::spell1];\)\), "\[IndentingNewLine]", 
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    \(<< Graphics`PlotField3D`\), "\[IndentingNewLine]", 
    \(<< Graphics`PlotField`\)}], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Outline", "Section"],

Cell[CellGroupData[{

Cell["Last time", "Subsection"],

Cell["\<\
Inference of shape from shading--set in context of other cues to \
shape, \
\>", "Subsubsection"],

Cell["Perception of shape from shading", "Subsubsection"],

Cell[CellGroupData[{

Cell["\<\
Introduce models for the inverse problem \"shape from shading\": \
\
\>", "Subsubsection"],

Cell[TextData[{
  "Inverting the generative model: L(pn,qn) =",
  StyleBox["n.e",
    FontWeight->"Bold"],
  "= ",
  Cell[BoxData[
      \(TraditionalForm\`{pn, qn, \(-1\)} . {pe, qe, \(-1\)}\/\@\(\((pn\^2 + \
qn\^2 + 1)\) \((pe\^2 + qe\^2 + 1)\)\)\)]],
  "\nClassic Ikeuchi & Horn solution:\n\tAssume light source direction is \
known. Constant reflectance. Known surface normals at the smooth occluding \
boundary.\n\tFind p's and q's such that ",
  Cell[BoxData[
      \(TraditionalForm\`E\_D\)]],
  "+",
  Cell[BoxData[
      \(TraditionalForm\`E\_S\)]],
  "is small.\n\t\tGenerative model -> data  cost function ",
  Cell[BoxData[
      \(TraditionalForm\`E\_D\)]],
  "(p(x,y), q(x,y))\n\t\tPrior assumption of surface smoothness \n\t\t\t(i.e. \
spatial second derivatives of p(x,y) and q(x,y) should be small) -> \
data-independent cost function ",
  Cell[BoxData[
      \(TraditionalForm\`\(E\_S\)(\((\[Del]\^2 p)\)\^2, \ \[Del]\^2\( \(q\)\(\
\ \)\)\^2)\)]],
  ". \n\tEquivalent to Bayes. "
}], "Text"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Today", "Subsection"],

Cell["Perceptual ambiguities continued", "Subsubsection"],

Cell["\<\
Formal ambiguities seen by analyzing the generative model, the \
bas-relief transform\
\>", "Subsubsection"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["\<\
Solutions to the shape from shading problem for the linear \
case\
\>", "Section"],

Cell[CellGroupData[{

Cell["Learning scene parameters from images", "Subsection"],

Cell[TextData[{
  "Last time I mentioned that if one has available a set of images {",
  Cell[BoxData[
      \(TraditionalForm\`L\_i\)]],
  "}, together with a set of unit surface normals for each image {",
  Cell[BoxData[
      \(TraditionalForm\`p\_i\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`q\_i\)]],
  "}(e.g. derived from a set of range images), then a linear model suggests \
that one could simply do a linear regression to find a map: L -> {",
  Cell[BoxData[
      \(TraditionalForm\`p\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`q\)]],
  "}, that might be a workable approximation to shape from shading from an  \
image L that was not in the training class {",
  Cell[BoxData[
      \(TraditionalForm\`L\_i\)]],
  "} . \nThis was done by Knill and Kersten (1990). They used a neural \
network-like technique (Widrow-Hoff learning rule for synaptic modification) \
to do regression learning \"on-line\", rather than in  \"batch mode\". This \
method was general and empirical in the sense, that one can imagine applying \
it to other scene-from-image problems (Kersten et al., 1988). How well  it \
works depends on the linear approximation. Non-linear neural networks and \
learning techniques such as back-propagation can be used to find solutions \
for non-linear formulations (Lehky & Sejnowski, 1988)."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Pentland's solution to the inverse problem", "Subsection"],

Cell[CellGroupData[{

Cell["Recall from last time ", "Subsubsection"],

Cell[TextData[{
  "Last time we derived a linear approximation to the lambertian shading \
model, L= ",
  StyleBox["n.e. ",
    FontWeight->"Bold"],
  "The image luminance is given by",
  " \nL(pn,qn) =",
  StyleBox["n.e",
    FontWeight->"Bold"],
  "= ",
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      \(TraditionalForm\`{pn, qn, \(-1\)} . {pe, qe, \(-1\)}\/\@\(\((pn\^2 + \
qn\^2 + 1)\) \((pe\^2 + qe\^2 + 1)\)\)\)]],
  ", \nwhere n={pn,qn,-1}/",
  Cell[BoxData[
      \(TraditionalForm\`\@\((pn\^2 + qn\^2 + 1)\)\)]],
  " and e={pe,qe,-1}/",
  Cell[BoxData[
      \(TraditionalForm\`\@\(\(pe\^2 + qe\^2 + 1\)\()\)\)\)]],
  " are the unit surface normal vector and illumination vector respectively. \
\nUsing a Taylor series expansion about {pn,qn}={0,0}, we were able to derive \
a linear approximation to the shading equation."
}], "Text"],

Cell["\<\
Lmodel[pn_,qn_,pe_,qe_]:= ({pn,qn,-1}/Sqrt[pn^2+qn^2 + \
1]).({pe,qe,-1}/Sqrt[pe^2+qe^2 + 1]);
Series[Lmodel[pn,qn,pe,qe], {pn, 0, 1}, {qn, 0, 1}]\
\>", "Input"],

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                SeriesData[ qn, 0, {}, 0, 2, 1]]}],
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                  Power[ qe, 2]], 
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                    Power[ pe, 2], 
                    Power[ qe, 2]], 
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              Power[ pe, 2], 
              Power[ qe, 2]], 
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                Power[ pe, 2], 
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                Power[ pe, 2], 
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Cell["\<\
Lapprxom[x_,y_] := (1+qe*qn[x,y] + pe*pn[x,y])/Sqrt[1+pe^2 + qe^2];\
\
\>", "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["\<\
Using the fourier transform to estimate surface depth z, from image \
intensity L\
\>", "Subsubsection"],

Cell[TextData[{
  "A standard result from fourier transform theory is, given g(x) and its \
fourier transform Fg(fx), then it is easy to write down the fourier transform \
of the derivative of g(x). It is just the fourier transform of g(x) times 2\
\[Pi]ifx. So, FourierTransform[",
  Cell[BoxData[
      \(TraditionalForm\`dg\/dx\)]],
  "] = 2\[Pi]ifx Fg(fx).(cf. Gaskill).\nLet z(x,y) be any differentiable \
function.  Let Fz[fx,fy] be the fourier transform of z, which can be written \
in terms of the (complex) amplitude and phases as: \na[fx,fy] ",
  Cell[BoxData[
      \(TraditionalForm\`e\^\(i\[Phi](fx, fy)\)\)]],
  ". Using the above result, we have:"
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Cell[TextData[{
  "Fp[fx,fy] = 2\[Pi] i fx a(fx,fy)  ",
  Cell[BoxData[
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  " and Fq[fx,fy] = 2\[Pi] i fy a(fx,fy)  ",
  Cell[BoxData[
      \(TraditionalForm\`e\^\(i(\[Phi](fx, fy))\)\)]],
  ". "
}], "Text"],

Cell["\<\
Let's simplify L = (1+qe*qn[x,y] + pe*pn[x,y])/Sqrt[1+pe^2 + qe^2] \
by ignoring the constant term 1.\
\>", "Text"],

Cell[TextData[{
  "Let's express the direction of the light source in terms of tilt and \
slant. Then \[Tau]=",
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  ", and \[Sigma] = ",
  Cell[BoxData[
      \(TraditionalForm\`\(tan\^\(-1\)\)(\@\(q\_e\^2 + p\_e\^2\))\)]],
  ". \nSo we can write L = cos\[Sigma] + p sin\[Sigma] + q sin\[Tau] sin \
\[Sigma].  Let the fourier transform of the image function L be: ",
  Cell[BoxData[
      \(TraditionalForm\`a\_L\)]],
  "[fx,fy] ",
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sin\[Tau])\)\)\)],
    FontSize->18]
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Cell["\<\
The solution requires known point light source, constant \
reflectance. Further the linear approximation is good for small p's and q's \
--i.e. for shallow bas-relief like shapes.
The inverse fourier transform provides an estimate of the surface depth z. \
Note that if ignore the constant term in image luminance, we can replace z-> \
z + k1x + k2 y + k3, we obtain the same solution. 
Shortly, we will look at recent results that show how general shape ambiguity \
is when the light source is unknown.\
\>", "Text"]
}, Closed]]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Shading & the bas-relief transform", "Section"],

Cell[CellGroupData[{

Cell["Generalizing the simple lambertian generative model", "Subsection"],

Cell[TextData[{
  "The lambertian model thus far is pretty limiting. It doesn't take into \
account ",
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  " or cast shadows or multiple light sources. An attached shadow occurs \
whenever the surface normal is at an angle greater than 90 degrees relative \
to the light source vector. The cosine of an oblique angle is negative, but \
we can't have negative light. If there is no ambient light, then an attached \
shadow region is black. So a better model would be:\nL = Max[",
  StyleBox["n.e",
    FontWeight->"Bold"],
  ",0]. We may have more than one light source, and since light is nice and \
linear, we can just add up all the contributions:\nL = ",
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      FormBox[
        RowBox[{\(\[Sum]\+i\), 
          RowBox[{"Max", "[", 
            RowBox[{
              RowBox[{"n", ".", 
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}], "Text"],

Cell["\<\
A cast shadow boundary occurs on a receiving surface whenever \
another surface (or part of one) is between the receiving surface and the \
light source.\
\>", "Text"],

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Cell[CellGroupData[{

Cell["The bas-relief effect", "Subsection"],

Cell["\<\
Recently, Belhumeur, P. N., Kriegman, D. J., & Yuille (1997) showed \
that there is a simple transformation of a surface (i.e. adding a slanted and \
tilted plane to a surface z(x,y) and a compression in depth) that induces \
(together with an albedo and illumination adjustment) an well-defined \
equivalence class of surfaces for a given image. For each image of a \
lambertian surface z(x,y), there is an identical image of a bas-relief \
produced by a transformed light source. Further this holds for both shaded \
and shadowed regions. And for the classical bas-relief in sculpture (no added \
slant, just compression), the image is insensitive to small motions and \
illumination changes.\
\>", "Text"],

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Cell[CellGroupData[{

Cell["First surface", "Subsection"],

Cell[CellGroupData[{

Cell["Define First surface  bump", "Subsubsection"],

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              1/\((1 + 
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.584 .584 .584 r
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.584 .584 .584 r
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.588 .588 .588 r
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.587 .587 .587 r
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.587 .587 .587 r
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.588 .588 .588 r
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.589 .589 .589 r
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.59 .59 .59 r
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.591 .591 .591 r
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