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Notebook[{

Cell[CellGroupData[{
Cell["Haar Edge Detection", "Title"],

Cell["\<\
Wavelet Workshop
June 7-10, 2006
University of St. Thomas\
\>", "Subtitle"],

Cell["Objectives", "Subsubtitle"],

Cell[TextData[{
  "The purpose of this notebook is to show you how to perform naive edge \
detection using the two-dimensional Discrete Haar Wavelet Transform.\n\nThe \
notebook also contains a color image compression lab that makes up part of ",
  StyleBox["Computer Session One",
    FontWeight->"Bold"],
  "."
}], "Text"],

Cell[CellGroupData[{

Cell["WaveletFunctions", "Subsubtitle",
  InitializationCell->True],

Cell[TextData[{
  "This cell initializes every time you open the notebook.  It loads the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " package ",
  StyleBox["WaveletFunctions",
    FontFamily->"Courier"],
  " for use in subsequent computations."
}], "Text",
  InitializationCell->True],

Cell[BoxData[{
    \(<< WaveletFunctions`WaveletFunctions`\), "\n", 
    \(<< LinearAlgebra`MatrixManipulation`\), "\[IndentingNewLine]", 
    \(\(\(<< Graphics`Graphics`\)\(\[IndentingNewLine]\)
    \)\), "\[IndentingNewLine]", 
    \(\(flashdir = "\<E:/WaveletWorkshop06/Images/\>";\)\), "\
\[IndentingNewLine]", 
    \(\(Print["\<The flash drive directory is \>", 
        flashdir, "\<.\>"];\)\), "\[IndentingNewLine]", 
    \(\(imgurl = \
"\<http://cam.mathlab.stthomas.edu/pvf/Math316/TestImages/\>";\)\), "\
\[IndentingNewLine]", 
    \(\(Print["\<The URL for images is \>", imgurl, "\<.\>"];\)\)}], "Input",
  InitializationCell->True]
}, Open  ]],

Cell[CellGroupData[{

Cell["Help on WaveletFunctions", "Subsubtitle"],

Cell[TextData[{
  "If you ever need help with ",
  StyleBox["WaveletFunctions",
    FontFamily->"Courier"],
  ", go to ",
  StyleBox["Help",
    FontSlant->"Italic"],
  ", then ",
  StyleBox["Help Browser",
    FontSlant->"Italic"],
  ", and click on ",
  StyleBox["AddOns & Links",
    FontSlant->"Italic"],
  ".  If you scroll down you will find ",
  StyleBox["WaveletFunctions",
    FontFamily->"Courier"],
  ".  "
}], "Text"],

Cell[CellGroupData[{

Cell["Load an Image and Compute the Haar Wavelet Transform", "Section"],

Cell["\<\
We begin by loading the snoopy_gray.jpg image and computing it's Haar Wavelet \
Transform.  \
\>", "Text"],

Cell[BoxData[{
    \(\(\(A = 
        ReadImage[flashdir <> "\<snoopy_gray.jpg\>", PrintInfo \[Rule] True, 
          PowersOfTwo \[Rule] 3];\)\(\[IndentingNewLine]\)
    \)\), "\[IndentingNewLine]", 
    \(\(ImagePlot[A];\)\), "\[IndentingNewLine]", 
    \(\({rows, cols} = Dimensions[A];\)\)}], "Input"],

Cell[BoxData[{
    \(\(its = 2;\)\), "\[IndentingNewLine]", 
    \(\(wt = 
        WT2D[A, N[Haar[]], 
          NumIterations \[Rule] its];\)\), "\[IndentingNewLine]", 
    \(\(WaveletDensityPlot[wt, NumIterations \[Rule] its, 
        DivideLines \[Rule] True, 
        DivideLinesColor \[Rule] Coral];\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Converting the Blur Portion to the Zero Matrix", "Section"],

Cell[TextData[{
  "If you are familiar with ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ", then you probably have used the ",
  StyleBox["SubMatrix",
    FontWeight->"Bold"],
  ", ",
  StyleBox["BlockMatrix",
    FontWeight->"Bold"],
  ", or ",
  StyleBox["ZeroMatrix",
    FontWeight->"Bold"],
  " commands.    These commands will be helpful since you want to convert the \
upper left block of the wavelet transform to a matrix consisting of 0's.\n\n\
Rather than use those commands, there are a couple of ",
  StyleBox["UNDOCUMENTED ",
    FontWeight->"Bold"],
  "commands in WaveletFunctions that will assist you with splitting out \
various portions of the wavelet transform.\n\nThe command ",
  StyleBox["WaveletMatrixToList",
    FontWeight->"Bold"],
  " takes a wavelet transform and the ",
  StyleBox["NumIterations",
    FontWeight->"Bold"],
  " directive and returns a list that stores various parts of the wavelet \
transform.  It's best to explain via example:\n\nSuppose wt is the wavelet \
transform of some image using 2 iterations and you issue the command"
}], "Text"],

Cell[BoxData[
    \(wtlist = WaveletMatrixToList[wt, NumIterations \[Rule] 2]\)], "Text"],

Cell[TextData[{
  "\nThen the output is a list with three elements.  The first element of the \
list is the lowpass portion of the transform.  The second element of the list \
is a list of three matrices - the vertical, horizontal, and diagonal portions \
of the second iteration of the transform.  The third element of the list is \
also a list of three matrices - the vertical, horizontal, and diagonal \
portions of the first iteration of the transform.\n\nThe command ",
  StyleBox["WaveletListToMatrix",
    FontWeight->"Bold"],
  " converts a list of transform portions back to a matrix.  \n\nThe reason \
we use this is we can convert to a list, drop the first element and replace \
it with a zero matrix, and then convert back.\n"
}], "Text"],

Cell[BoxData[{
    \(\(wtlist = 
        WaveletMatrixToList[wt, 
          NumIterations \[Rule] its];\)\), "\[IndentingNewLine]", 
    \(Length[wtlist]\)}], "Input"],

Cell[TextData[{
  "Here are some plots to help you understand the components of ",
  StyleBox["wtlist",
    FontWeight->"Bold"],
  "."
}], "Text"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(\(ImagePlot[wtlist[\([1]\)], 
        LinearScaling \[Rule] True];\)\[IndentingNewLine]\[IndentingNewLine]
    \(ImagePlot[wtlist[\([2, 1]\)], 
        LinearScaling \[Rule] True];\)\[IndentingNewLine]\[IndentingNewLine]
    \(ImagePlot[wtlist[\([3, 2]\)], 
        LinearScaling \[Rule] True];\)\)\)\)], "Input"],

Cell[TextData[{
  "We drop the first element of ",
  StyleBox["wtlist",
    FontWeight->"Bold"],
  " and replace it with a zero matrix of appropriate size:"
}], "Text"],

Cell[BoxData[{
    \(\(z = ZeroMatrix[rows/2^its, cols/2^its];\)\), "\[IndentingNewLine]", 
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    \(\(\(wtlist = Prepend[tmp, z];\)\(\[IndentingNewLine]\)
    \)\), "\[IndentingNewLine]", 
    \(\(newwt = 
        WaveletListToMatrix[wtlist, 
          NumIterations \[Rule] its];\)\), "\[IndentingNewLine]", 
    \(\(WaveletDensityPlot[newwt, NumIterations \[Rule] its, 
        DivideLines \[Rule] True, \ 
        DivideLinesColor \[Rule] Coral];\)\)}], "Input"],

Cell["\<\
Now we simply apply the inverse Haar wavelet transform to obtain the edges:\
\>", "Text"],

Cell[BoxData[{
    \(\(edges = 
        IWT2D[newwt, N[Haar[]], 
          NumIterations \[Rule] its];\)\), "\[IndentingNewLine]", 
    \(\(ImagePlot[edges, LinearScaling \[Rule] True];\)\)}], "Input"],

Cell["\<\
If you want a better idea of where the nonzero pixels are, you can try \
something radical like what's done in the cell below.\
\>", "Text"],

Cell[BoxData[{
    \(\(bright[t_] := 
        Which[t \[Equal] 0, 0, True, 255];\)\), "\[IndentingNewLine]", 
    \(\(s = Flatten[edges];\)\), "\[IndentingNewLine]", 
    \(\(newedges = Map[bright, s];\)\), "\[IndentingNewLine]", 
    \(\(newedges = Partition[newedges, cols];\)\), "\[IndentingNewLine]", 
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Cell["\<\
Of course, we don't want to keep ALL the nonzero values in the highpass \
portions - that's where the next exercise comes in!\
\>", "Text"]
}, Open  ]]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Computer Session One", "Title"],

Cell["\<\
There is one task in this computer session.  You will employ the thresholding \
scheme of Gonzalez, et. al., to (hopefully) improve the edge detection \
algorithm.\
\>", "Text"],

Cell[CellGroupData[{

Cell["Task One", "Subtitle"],

Cell[TextData[{
  "In this task, you will employ the thresholding scheme of Gonzalez, et. al, \
to hopefully improve the edge detection process.\n\nTo get started, simply \
choose ",
  StyleBox["S",
    FontWeight->"Bold"],
  " to be the ",
  StyleBox["absolute value",
    FontWeight->"Bold"],
  " of modified wavelet transform matrix.  The blur is already 0, so there's \
nothing to convert there.\n\nTake \[Alpha]=1 - that's the lowest level in \
change in pixel values."
}], "Text"],

Cell[BoxData[{
    \(\(S = Abs[newwt];\)\), "\[IndentingNewLine]", 
    \(\(WaveletDensityPlot[S, NumIterations \[Rule] its, 
        DivideLines \[Rule] True, 
        DivideLinesColor \[Rule] Coral];\)\)}], "Input"],

Cell[TextData[{
  "Here are some commands that might help you get started.  For our purposes, \
we can ",
  StyleBox["Flatten",
    FontWeight->"Bold"],
  " ",
  StyleBox["S",
    FontWeight->"Bold"],
  " into a vector."
}], "Text"],

Cell[BoxData[{
    \(\(S = Flatten[S];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(tau1 = \((Min[S] + Max[S])\)/2;\)\), "\[IndentingNewLine]", 
    \(\(S1 = Select[S, # < tau1 &];\)\), "\[IndentingNewLine]", 
    \(\(S2 = 
        Select[S, # \[GreaterEqual] tau1 &];\)\), "\[IndentingNewLine]", 
    \(tau2 = \((Mean[S1] + Mean[S2])\)/2\), "\[IndentingNewLine]", 
    \(Abs[tau2 - tau1]\), "\[IndentingNewLine]", 
    \(\)}], "Input"],

Cell[TextData[{
  "You will want to iterate the above process.  You can use a ",
  StyleBox["While",
    FontWeight->"Bold"],
  " loop or a ",
  StyleBox["Table",
    FontWeight->"Bold"],
  " or ask me if you're not sure.\n\nOnce you have the iterative scheme for \
selecting tau working, can you turn it into a module?  Put your module in the \
cell below:"
}], "Text"],

Cell[BoxData[
    \(\(FindTau[v_] := 
        Module[\({ (*\ 
              Local\ variables\ replace\ this\ \(\(comment\)\(.\)\)\ \
*) }\)\(,\)\[IndentingNewLine]\t\t\t (*\ 
            Mathematica\ commands\ go\ here\ *) \[IndentingNewLine]];\)\)], \
"Input",
  FontColor->RGBColor[0, 0, 1]],

Cell[TextData[{
  "\nHere is some ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " code for applying your final ",
  StyleBox["tau",
    FontWeight->"Bold"],
  " to ",
  StyleBox["S",
    FontWeight->"Bold"],
  " and then ",
  StyleBox["Partition",
    FontWeight->"Bold"],
  "ing back into a matrix.  I picked ",
  StyleBox["tau=40",
    FontWeight->"Bold"],
  " to illustrate.  You will want to use the tau returned to you by ",
  StyleBox["FindTau",
    FontWeight->"Bold"],
  "."
}], "Text"],

Cell[BoxData[{
    \(\(tau = 
        40. ;\)\[IndentingNewLine]\[IndentingNewLine] (*\ \(or\ use\ your\ \
\(module : \[IndentingNewLine]\[IndentingNewLine]tau\) = 
          FindTau[S];\)\[IndentingNewLine]\[IndentingNewLine]*) \
\[IndentingNewLine]\[IndentingNewLine] (*\ 
      Below\ is\ some\ simple\ code\ for\ the\ threshold\ \(\(function\)\(.\)\
\)*) \), "\[IndentingNewLine]", 
    \(\(thresh[v_, tau_] := 
        Which[v > tau, v, True, 
          0];\)\ \ \[IndentingNewLine]\[IndentingNewLine] (*\ 
      tmp\ is\ the\ result\ of\ applying\ the\ threshold\ function\ \((with\ \
whatever\ tau\ you\ decided\ to\ use)\)\ to\ the\ list\ \(\(S\)\(.\)\)\
\[IndentingNewLine]*) \[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(tmp = 
        Map[thresh[#, tau] &, 
          S];\)\[IndentingNewLine]\[IndentingNewLine] (*\[IndentingNewLine]We\
\ need\ to\ ultimately\ make\ tmp\ into\ a\ matrix, \ 
      so\ we\ simply\ partition\ it\ into\ lists\ that\ are\ of\ length\ \
\(\(cols\)\(.\)\)\[IndentingNewLine]*) \[IndentingNewLine]\), "\
\[IndentingNewLine]", 
    \(\(threshWT = 
        Partition[tmp, cols];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(WaveletDensityPlot[threshWT, NumIterations \[Rule] its, 
        DivideLines \[Rule] True, 
        DivideLinesColor \[Rule] Coral];\)\)}], "Input"],

Cell["\<\
In the cell below, you can apply the inverse transform to threshWT to obtain \
the edges.\
\>", "Text"],

Cell[BoxData[
    \( (*\ Put\ Mathematica\ code\ here\ *) \)], "Input",
  FontColor->RGBColor[0, 0, 1]],

Cell[CellGroupData[{

Cell["Extra Credit Question", "Section"],

Cell["\<\
Now that you have a module for FindTau, let's put it to use!  Can you do \
thresholding with FindTau on each individual highpass component of the \
wavelet transform?  If so, put the code below and then determine if \
thresholding individual highpass channels improved the detected edges.\
\>", "Text"],

Cell[BoxData[
    \( (*\ Put\ Mathematica\ code\ here\ *) \)], "Input",
  FontColor->RGBColor[0, 0, 1]]
}, Open  ]]
}, Open  ]]
}, Open  ]]
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