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Notebook[{

Cell[CellGroupData[{
Cell["Haar Transform 2D", "Title"],

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

Cell["Objectives", "Subsubtitle"],

Cell["\<\
The purpose of this notebook is to show you how to code and use the \
two-dimensional Discrete Haar Wavelet Transform.\
\>", "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]\), "\
\[IndentingNewLine]", 
    \(\(flashdir = "\<E:/Images/\>";\)\), "\[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["Analyzing the Block Structure of the 2D Transform", "Section"],

Cell[TextData[{
  "Let's build the Haar matrix for an 8x8 image ",
  StyleBox["A",
    FontWeight->"Bold"],
  ".  For illustrative purposes, let's omit the ",
  Cell[BoxData[
      \(\@2\)]],
  " from the filters."
}], "Text"],

Cell[BoxData[{
    \(\(h = {1, 1}/2;\)\), "\[IndentingNewLine]", 
    \(\(g = {1, \(-1\)}/2;\)\), "\[IndentingNewLine]", 
    \(\(z = Table[0, {2}];\)\), "\[IndentingNewLine]", 
    \(\(r1 = Join[h, z];\)\), "\[IndentingNewLine]", 
    \(\(r2 = Join[g, z];\)\), "\[IndentingNewLine]", 
    \(\(tp = 
        Table[RotateRight[r1, 2*k], {k, 0, 1}];\)\), "\[IndentingNewLine]", 
    \(\(bt = 
        Table[RotateRight[r2, 2*k], {k, 0, 1}];\)\), "\[IndentingNewLine]", 
    \(\(W = Join[tp, bt];\)\), "\[IndentingNewLine]", 
    \(MatrixForm[W]\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(Clear[a, A];\)\), "\[IndentingNewLine]", 
    \(\(A = Array[a, {4, 4}];\)\), "\[IndentingNewLine]", 
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Cell[TextData[{
  "Now we compute ",
  StyleBox["WA",
    FontWeight->"Bold"],
  Cell[BoxData[
      \(W\^T\)],
    FontWeight->"Bold"]
}], "Text"],

Cell[BoxData[{
    \(\(B = W . A . Transpose[W]\  // Simplify;\)\), "\[IndentingNewLine]", 
    \(MatrixForm[B]\)}], "Input"],

Cell[TextData[{
  "Let's grab the upper left corner of ",
  StyleBox["B",
    FontWeight->"Bold"],
  ":"
}], "Text"],

Cell[BoxData[{
    \(\(blur = SubMatrix[B, {1, 1}, {2, 2}];\)\), "\[IndentingNewLine]", 
    \(MatrixForm[A]\), "\[IndentingNewLine]", 
    \(MatrixForm[blur]\)}], "Input"],

Cell[TextData[{
  "The upper right corner of ",
  StyleBox["B",
    FontWeight->"Bold"],
  ":"
}], "Text"],

Cell[BoxData[{
    \(\(vert = SubMatrix[B, {1, 3}, {2, 2}];\)\), "\[IndentingNewLine]", 
    \(MatrixForm[A]\), "\[IndentingNewLine]", 
    \(MatrixForm[vert]\)}], "Input"],

Cell[TextData[{
  "The lower left corner of ",
  StyleBox["B",
    FontWeight->"Bold"],
  ":"
}], "Text"],

Cell[BoxData[{
    \(\(horiz = SubMatrix[B, {3, 1}, {2, 2}];\)\), "\[IndentingNewLine]", 
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    \(MatrixForm[horiz]\)}], "Input"],

Cell[TextData[{
  "The lower right corner of ",
  StyleBox["B",
    FontWeight->"Bold"],
  ":"
}], "Text"],

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    \(MatrixForm[diag]\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Coding the 2D Transform", "Section"],

Cell[TextData[{
  "Let's first establish some code for applying the Haar transform to a \
vector.  We could use our module from before or simply use the 1D transform \
in ",
  StyleBox["WaveletFunctions",
    FontFamily->"Courier"],
  "."
}], "Text"],

Cell[BoxData[
    \(\(DHWT[v_] := WT1D[v, N[Haar[]], NumIterations \[Rule] 1];\)\)], "Input"],

Cell["Let's load an image to test our code:", "Text"],

Cell[BoxData[{
    \(\(A = 
        ReadImage[
          flashdir <> "\<garfield_gray.jpg\>"];\)\), "\[IndentingNewLine]", 
    \(\(ImagePlot[A];\)\)}], "Input"],

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  "Now we want to apply ",
  StyleBox["DHWT",
    FontWeight->"Bold"],
  " to each column of ",
  StyleBox["A",
    FontWeight->"Bold"],
  ".    The quirky thing about ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " is that it is a row-oriented language.  Thus we need to apply our ",
  StyleBox["DHWT",
    FontWeight->"Bold"],
  " to the transpose of ",
  StyleBox["A",
    FontWeight->"Bold"],
  " and then transpose the result:"
}], "Text"],

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    \(\(B = Transpose[Map[DHWT, Transpose[A]]];\)\), "\[IndentingNewLine]", 
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Cell[TextData[{
  "The last step is to apply the ",
  StyleBox["DHWT",
    FontWeight->"Bold"],
  " to each of the columns of the transpose of ",
  StyleBox["B",
    FontWeight->"Bold"],
  ".  But this is actually an easier step, since transposing the transpose \
gets us back to the original!"
}], "Text"],

Cell[BoxData[{
    \(\(wt = Map[DHWT, B];\)\), "\[IndentingNewLine]", 
    \(\(WaveletDensityPlot[wt, NumIterations \[Rule] 1, 
        DivideLines \[Rule] True];\)\)}], "Input"],

Cell["Here is a working module:", "Text"],

Cell[BoxData[
    \(\(DHWT2D[a_] := 
        Module[{b, wt}, \[IndentingNewLine]\t
          b = Transpose[Map[DHWT, Transpose[a]]]; \[IndentingNewLine]\t
          wt = Map[DHWT, b]; \[IndentingNewLine]\t
          Return[wt];\[IndentingNewLine]];\)\)], "Input"],

Cell[TextData[{
  "You can use this module or the one in ",
  StyleBox["WaveletFunctions",
    FontFamily->"Courier"],
  ":"
}], "Text"],

Cell[BoxData[{
    \(\(wt = DHWT2D[A];\)\), "\[IndentingNewLine]", 
    \(\(WaveletDensityPlot[wt, NumIterations \[Rule] 1, 
        DivideLines \[Rule] 
          True];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(wt = 
        WT2D[A, N[Haar[]], 
          NumIterations \[Rule] 1];\)\), "\[IndentingNewLine]", 
    \(\(WaveletDensityPlot[wt, NumIterations \[Rule] 1, 
        DivideLines \[Rule] True];\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Features of WaveletDensityPlot", "Section"],

Cell[TextData[{
  "One of the nice options with ",
  StyleBox["WaveletDensityPlot ",
    FontWeight->"Bold"],
  "is you can pick out various parts of the transform.  For example,"
}], "Text"],

Cell[BoxData[{
    \(\(WaveletDensityPlot[wt, NumIterations \[Rule] 1, 
        Region \[Rule] Vertical];\)\), "\[IndentingNewLine]", 
    \(Dimensions[wt]\)}], "Input"],

Cell[BoxData[
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        Region \[Rule] Vertical, 
        ImageSize \[Rule] Dimensions[wt]];\)\)], "Input"],

Cell[BoxData[
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        ImageSize \[Rule] Dimensions[wt]];\)\)], "Input"],

Cell[BoxData[
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        Region \[Rule] Horizontal, 
        ImageSize \[Rule] Dimensions[wt]];\)\)], "Input"],

Cell[BoxData[
    \(\(WaveletDensityPlot[wt, NumIterations \[Rule] 1, 
        Region \[Rule] Diagonal, 
        ImageSize \[Rule] Dimensions[wt]];\)\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Coding the Inverse 2D Haar Wavelet Transform", "Section"],

Cell[TextData[{
  "Can you produce a module ",
  StyleBox["IHWT2D",
    FontWeight->"Bold"],
  " that computes the inverse 2D Haar Wavelet Transform? \n\nIt will look \
much like ",
  StyleBox["HWTD2D",
    FontWeight->"Bold"],
  ", but you'll need the module from the last notebook.  I've copied it in \
the cell below:"
}], "Text"],

Cell[BoxData[
    \(\(IDHWT[v_] := 
        Module[{V, a, b, h, g, ans}, \[IndentingNewLine]\t
          h = Sqrt[2]*{1, 1}/2; \[IndentingNewLine]\t
          g = Sqrt[2]*{1, \(-1\)}/2; \[IndentingNewLine]\t
          V = Transpose[Partition[v, Length[v]/2]]; \[IndentingNewLine]\t
          a = V . h; \[IndentingNewLine]\tb = V . g; \[IndentingNewLine]\t
          ans = Flatten[Transpose[{b, a}]]; \[IndentingNewLine]\t
          Return[ans];\[IndentingNewLine]];\)\)], "Input"],

Cell[TextData[{
  "Put your code for the ",
  StyleBox["IDHWT2D",
    FontWeight->"Bold"],
  " module in the cell below.  Once you have it working, run the following \
cell to see the results."
}], "Text"],

Cell[BoxData[
    RowBox[{
      RowBox[{\(IDHWT2D[a_]\), ":=", 
        RowBox[{"Module", "[", 
          RowBox[{
            RowBox[{"{", " ", 
              StyleBox[\( (*Put\ local\ variables\ here*) \),
                FontColor->RGBColor[0, 0, 1]], " ", "}"}], ","}], 
          "\[IndentingNewLine]", 
          StyleBox[\( (*Put\ your\ code\ here*) \),
            FontColor->RGBColor[0, 0, 1]], 
          StyleBox["\[IndentingNewLine]",
            FontColor->RGBColor[0, 0, 1]], "]"}]}], ";"}]], "Input"],

Cell[BoxData[{
    \(\(orig = IDHWT2D[wt];\)\), "\[IndentingNewLine]", 
    \(\(ImagePlot[orig];\)\)}], "Input"],

Cell[TextData[{
  "The routine in ",
  StyleBox["WaveletFunctions",
    FontFamily->"Courier"],
  " is"
}], "Text"],

Cell[BoxData[{
    \(\(orig = 
        IWT2D[wt, N[Haar[]], 
          NumIterations \[Rule] 1];\)\), "\[IndentingNewLine]", 
    \(\(ImagePlot[orig];\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Iterating the Transform", "Section"],

Cell[TextData[{
  "It is really easy to iterate the wavelet transform.  Just set the \
NumIterations directive to the number of iterations you want.\n    \n    An \
important note though:\n    \n    The dimensions of the data must be \
divisible by ",
  Cell[BoxData[
      \(2\^p\)]],
  " if you wish to perform p iterations.  One way to ensure this is to use \
the ",
  StyleBox["PowersOfTwo",
    FontWeight->"Bold"],
  " directive when importing an image:"
}], "Text"],

Cell[BoxData[
    \(\(A = 
        ReadImage[flashdir <> "\<calvin_hobbes_gray.jpg\>", 
          PowersOfTwo \[Rule] 4, PrintInfo \[Rule] True];\)\)], "Input"],

Cell[TextData[{
  "Now we can take as many as four iterations of the wavelet transform.  Feel \
free to change the value of ",
  StyleBox["its",
    FontWeight->"Bold"],
  "."
}], "Text"],

Cell[BoxData[{
    \(\(its = 3;\)\), "\[IndentingNewLine]", 
    \(\(wt = 
        WT2D[A, N[Haar[]], 
          NumIterations \[Rule] its];\)\), "\[IndentingNewLine]", 
    \(\(WaveletDensityPlot[wt, NumIterations \[Rule] its, 
        DivideLines \[Rule] True];\)\)}], "Input"],

Cell["You can extract different parts of the transform.", "Text"],

Cell[BoxData[
    \(\(WaveletDensityPlot[wt, NumIterations \[Rule] its, 
        Iteration \[Rule] 1, Region \[Rule] Vertical, 
        ImageSize \[Rule] Dimensions[wt]];\)\)], "Input"],

Cell[BoxData[
    \(\(\( (*\ 
      Assuming\ its\ is\ 2\ or\ bigger\ \
*) \)\(\[IndentingNewLine]\)\(WaveletDensityPlot[wt, 
        NumIterations \[Rule] its, Iteration \[Rule] 2, 
        DivideLines \[Rule] True, 
        ImageSize \[Rule] Dimensions[wt]];\)\)\)], "Input"],

Cell["Here is a fun one to see how images are often reconstructed", "Text"],

Cell[BoxData[{
    \(\(WT = 
        Table[WT2D[A, N[Haar[]], NumIterations \[Rule] k], {k, 1, 
            4}];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(plts = Table[{}, {k, 1, 5}];\)\), "\[IndentingNewLine]", 
    \(\(plts[\([1]\)] = 
        ImagePlot[A, 
          ImageSize \[Rule] 
            Dimensions[A]];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(plts = 
        Table[WaveletDensityPlot[WT[\([k - 1]\)], 
            NumIterations \[Rule] k - 1, Region \[Rule] Blur, 
            ImageSize \[Rule] Dimensions[A]], {k, 2, 
            5}];\)\[IndentingNewLine]\[IndentingNewLine] (*\ 
      Double\ click\ any\ picture\ below\ to\ animate\ *) \), "\
\[IndentingNewLine]", 
    \(\)}], "Input"]
}, Open  ]]
}, Open  ]]
}, Open  ]]
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