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Notebook[{

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

Cell["\<\
Patrick J. Van Fleet
Minicourse #5, Joint Mathematics Meetings
New Orleans, LA\
\>", "Subsubtitle"],

Cell["Objective", "Section"],

Cell[CellGroupData[{

Cell["\<\
In this notebook, we will explore the two-dimensional discrete Haar wavelet \
transform.\
\>", "Subsubtitle"],

Cell["Conventions", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "This notebook uses the package ",
  StyleBox["WaveletFunctions",
    FontColor->RGBColor[0.501961, 0, 0]],
  " (written by Patrick Van Fleet).  All commands from the WaveletFunctions \
library will be denoted in ",
  StyleBox["maroon",
    FontColor->RGBColor[0.501961, 0, 0]],
  ".  Help is available for almost every command in ",
  StyleBox["WaveletFunctions",
    FontColor->RGBColor[0.501961, 0, 0]],
  ".  Click on Help and then Help Browser (or Shift F1).  Next click the \
Add-ons & Links tab.  Scroll down until you find ",
  StyleBox["WaveletFunctions",
    FontColor->RGBColor[0.501961, 0, 0]],
  ".  Like all ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " help screens, the help is \"live\" - you can either execute the commands \
in the help to see the effects of the command or cut and paste them into your \
own notebook.\n\nValues that can be changed by the user are defined using ",
  StyleBox["blue",
    FontColor->RGBColor[0, 0, 1]],
  ".\n\nComments are useful within cells of code.  Any code enclosed by (*  \
*) is a comment and ignored by the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " kernel."
}], "Subsubtitle"],

Cell[CellGroupData[{

Cell["Load WaveletFunctions", "Section"],

Cell[BoxData[
    \(\(\(<< WaveletFunctions`WaveletFunctions`\)\(\[IndentingNewLine]\)
    \)\)], "Input"]
}, Open  ]],

Cell["Spelling Off", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
This turns off the annoying error messages you get when the spelling of two \
variable names is close.\
\>", "Subsubtitle"],

Cell[BoxData[{
    \(\(Off[General::spell];\)\), "\[IndentingNewLine]", 
    \(\(Off[General::spell1];\)\)}], "Input"],

Cell["Two-Dimensional Discrete Haar Transform Modules", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "\nWe will use the 2D modules from ",
  StyleBox["WaveletFunctions",
    FontColor->RGBColor[0.501961, 0, 0]],
  " to compute Haar Transforms of matrices.  The modules are ",
  StyleBox["WT2D",
    FontColor->RGBColor[0.501961, 0, 0]],
  " and ",
  StyleBox["IWT2D",
    FontColor->RGBColor[0.501961, 0, 0]],
  ".  Both modules require three arguments.  The first is the input matrix.  \
The second is {1/2, 1/2}, {1, 1}, for ",
  StyleBox["WT2D",
    FontColor->RGBColor[0.501961, 0, 0]],
  ", ",
  StyleBox["IWT2D",
    FontColor->RGBColor[0.501961, 0, 0]],
  ", respectively.  The last argument is the number of iterations."
}], "Subsubtitle"],

Cell["Locating Images", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
If you have access to the internet, then use the imgdir variable that is \
listed for the internet.  Else, use the imgdir for images already stored on \
your computer.  If you have your own image, create a url/folder location as \
the imgdir variable.\
\>", "Subsubtitle"],

Cell[BoxData[
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        This\ variable\ is\ the\ location\ of\ images\ on\ the\ \
\(\(web\)\(.\)\)\ *) \), "\[IndentingNewLine]", 
      RowBox[{
        StyleBox[\(imgdir = \
"\<http://cam.mathlab.stthomas.edu/pvf/Math316/TestImages/\>";\),
          FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", 
        "\[IndentingNewLine]", \( (*\ 
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                FontColor->RGBColor[0, 0, 1]]}], 
            StyleBox[";",
              FontColor->RGBColor[0, 0, 1]]}], " ", "*)"}], 
        "\[IndentingNewLine]", 
        "\[IndentingNewLine]", \( (*\ 
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\(\(below\)\(.\)\)\ *) \), "\[IndentingNewLine]", 
        RowBox[{"(*", " ", 
          StyleBox[\(\(\(imgdir\)\(\ \)\(=\)\)\ ;\),
            FontColor->RGBColor[0, 0, 1]], " ", "*)"}], 
        "\[IndentingNewLine]"}]}]], "Input"],

Cell["Available Images", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Here is a list of images for use with this notebook.  Surprisingly cartoons \
work really well!  Some images work better than others.  Uncomment one of the \
images.  You can also add your own image!\
\>", "Subsubtitle"],

Cell[BoxData[
    RowBox[{
      RowBox[{"(*", " ", 
        StyleBox[\(image\  = \ "\<beatles.png\>";\),
          FontColor->RGBColor[0, 0, 1]], " ", "*)"}], "\[IndentingNewLine]", 
      RowBox[{
        StyleBox[\(image = \ "\<calvin_hobbes_gray.jpg\>";\),
          FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", 
        RowBox[{"(*", " ", 
          StyleBox[\(image\  = \ "\<fingerprint.gif\>";\),
            FontColor->RGBColor[0, 0, 1]], " ", "*)"}], "\[IndentingNewLine]",
         
        RowBox[{"(*", " ", 
          StyleBox[\(image\  = \ "\<garfield_gray.jpg\>";\),
            FontColor->RGBColor[0, 0, 1]], " ", "*)"}], "\[IndentingNewLine]",
         
        RowBox[{"(*", " ", 
          StyleBox[\(image\  = \ "\<moonsurface.png\>";\),
            FontColor->RGBColor[0, 0, 1]], " ", "*)"}], "\[IndentingNewLine]",
         
        RowBox[{"(*", " ", 
          StyleBox[\(image = "\<snoopy_gray.jpg\>";\),
            FontColor->RGBColor[0, 0, 1]], " ", "*)"}], "\[IndentingNewLine]",
         
        RowBox[{"(*", " ", 
          StyleBox[\(image\  = \ "\<spockkirkmccoy.png\>";\),
            FontColor->RGBColor[0, 0, 1]], " ", "*)"}], "\[IndentingNewLine]",
         
        RowBox[{"(*", " ", 
          StyleBox[\(image\  = \ "\<stooges.gif\>";\),
            FontColor->RGBColor[0, 0, 1]], " ", "*)"}], "\[IndentingNewLine]",
         
        RowBox[{"(*", " ", 
          StyleBox[\(\(\(image\)\(\ \)\(=\)\)\ ;\),
            FontColor->RGBColor[0, 0, 1]], " ", "*)"}]}]}]], "Input"],

Cell["Loading and Plotting the Image", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "To load and plot an image, use ",
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  " and ",
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  ".  The ",
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  " function has two optional arguments.  You can print information using ",
  StyleBox["PrintInfo",
    FontColor->RGBColor[0.501961, 0, 0]],
  " regarding the image (dimensions and whether the image was color or gray \
scale).    You can also use ",
  StyleBox["PowersOfTwo",
    FontColor->RGBColor[0.501961, 0, 0]],
  " to chop rows/columns off the bottom/right to make the dimensions \
divisible by ",
  Cell[BoxData[
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  ".\n\n",
  StyleBox["Warning:",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  "  If you use your own image, make sure to use the ",
  StyleBox["PrintInfo",
    FontColor->RGBColor[0.501961, 0, 0]],
  " option.  If the image is read as a color image (even though it looks gray \
scale), then uncomment the line below ",
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    FontColor->RGBColor[0.501961, 0, 0]],
  " before processing it with ",
  StyleBox["WT2D",
    FontColor->RGBColor[0.501961, 0, 0]],
  "."
}], "Subsubtitle"],

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          StyleBox[\(ReadImage[imgdir <> image, PrintInfo \[Rule] True]\),
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        StyleBox[";",
          FontColor->RGBColor[0.501961, 0, 0]]}], 
      "\[IndentingNewLine]", \( (*\ 
        If\ your\ image\ was\ processed\ as\ a\ color\ image, \ 
        uncomment\ the\ line\ below\ to\ convert\ \(\(it\)\(.\)\)\ *) \)}], "\
\[IndentingNewLine]", 
    RowBox[{\( (*\ \(A\  = \ Total[A]/3;\)\ *) \)}]}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "The dimensions of the image below will be divisible by ",
  Cell[BoxData[
      \(2\^6\)]],
  "."
}], "Subsubtitle"],

Cell[BoxData[{
    RowBox[{
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        RowBox[{"A", "=", 
          
          StyleBox[\(ReadImage[imgdir <> image, PrintInfo \[Rule] True, 
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        StyleBox[";",
          FontColor->RGBColor[0.501961, 0, 0]]}], 
      "\[IndentingNewLine]", \( (*\ 
        If\ your\ image\ was\ processed\ as\ a\ color\ image, \ 
        uncomment\ the\ line\ below\ to\ convert\ \(\(it\)\(.\)\)\ *) \)}], "\
\[IndentingNewLine]", 
    RowBox[{\( (*\ \(A\  = \ Total[A]/3;\)\ *) \)}]}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Plotting images is really easy!", "Subsubtitle"],

Cell[BoxData[
    StyleBox[\(ImagePlot[A];\),
      FontColor->RGBColor[0.501961, 0, 0]]], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "You can add the ",
  StyleBox["LinearScaling",
    FontColor->RGBColor[0.501961, 0, 0]],
  " option.  This maps the minimum value of the image to 0 and the maximum \
value of the image to 255.  Try this on stooges.gif."
}], "Subsubtitle"],

Cell[BoxData[
    StyleBox[\(ImagePlot[A, LinearScaling \[Rule] True];\),
      FontColor->RGBColor[0.501961, 0, 0]]], "Input"],

Cell["Using WT2D", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
In the cell below, we compute and plot the two-dimensional HWT of image A.  \
{.5, .5} is used instead of {1/2, 1/2} to speed up computation.\
\>", "Subsubtitle"],

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      RowBox[{"B", " ", "=", " ", 
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        FontColor->RGBColor[0.501961, 0, 0]]}], "\[IndentingNewLine]", 
    StyleBox[\(WaveletDensityPlot[B, NumIterations \[Rule] 1, 
          DivideLines \[Rule] True];\),
      FontColor->RGBColor[0.501961, 0, 0]]}], "Input"],

Cell["Using IWT2D", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
In the cell below, we compute the inverse transform of the image.\
\>", "Subsubtitle"],

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            DivideLines \[Rule] True];\),
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      "\[IndentingNewLine]"}], "\[IndentingNewLine]", 
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      RowBox[{"origA", " ", "=", " ", 
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      StyleBox[";",
        FontColor->RGBColor[0.501961, 0, 0]]}], "\[IndentingNewLine]", 
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      FontColor->RGBColor[0.501961, 0, 0]]}], "Input"],

Cell["Iterating the Process", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "Just as with the one-dimensional HWT, we can iterate the two-dimensional \
HWT.  In this case, the ",
  StyleBox["WT2D",
    FontColor->RGBColor[0.501961, 0, 0]],
  " is applied the approximation (blur) portion of the transform.\n\n",
  StyleBox["Warning: ",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  " Make sure to know the maximum amount of iterations you can compute before \
proceeding!! ",
  StyleBox["WT2D",
    FontColor->RGBColor[0.501961, 0, 0]],
  " doesn't check (yet!) if the number of iterations divides the dimensions \
of A.\n\nHere are some maximum iteration numbers for our image list:\n\n",
  StyleBox["File",
    FontWeight->"Bold"],
  "\t\t\t\t",
  StyleBox["Max Iterations",
    FontWeight->"Bold"],
  "\nbeatles.png\t\t\t\t2\ncalvin_hobbes_gray.jpg\t\t5\nfingerprint.gif\t\t\t\
\t7\ngarfield_gray.jpg\t\t\t3\nmoonsurface.png\t\t\t3\t\nsnoopy_gray.jpg\t\t\t\
1\nspockkirkmccoy.png\t\t\t3\nstooges.gif\t\t\t\t5"
}], "Subsubtitle"],

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}, Open  ]],

Cell[CellGroupData[{

Cell["You can plot various portions of the transform.", "Subsubtitle"],

Cell[BoxData[{
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      StyleBox[\(WaveletDensityPlot[B, NumIterations \[Rule] its, 
            DivideLines \[Rule] True, Iteration \[Rule] 3, 
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Cell[CellGroupData[{

Cell["We can \"blow up\" the blur.", "Subsubtitle"],

Cell[BoxData[{\({rows, cols} = Dimensions[A];\), "\[IndentingNewLine]", 
    StyleBox[\(WaveletDensityPlot[B, NumIterations \[Rule] its, 
          DivideLines \[Rule] True, Region \[Rule] Blur, 
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(*******************************************************************
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*******************************************************************)