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Notebook[{

Cell[CellGroupData[{
Cell["Haar Transform 1D", "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 \
one-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", 
    \(\(\(Needs["\<Miscellaneous`Audio`\>"];\)\(\n\)
    \)\), "\[IndentingNewLine]", 
    \(\(flashdir = "\<E:/Images/\>";\)\), "\[IndentingNewLine]", 
    \(\(Print["\<The flash drive directory is \>", 
        flashdir, "\<.\>"];\)\), "\[IndentingNewLine]", 
    \(\(sounddir = "\<F:/Sounds/\>";\)\), "\[IndentingNewLine]", 
    \(\(Print["\<The flash drive directory for sounds is \>", 
        sounddir, "\<.\>"];\)\), "\[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["Applying the DHWT to a Vector", "Section"],

Cell[TextData[{
  StyleBox["WaveletFunctions",
    FontFamily->"Courier"],
  " comes with a command called ",
  StyleBox["WT1D",
    FontWeight->"Bold"],
  ".  This is the 1D wavelet transform command.  It needs a vector, a filter, \
and a directive that gives the number of iterations (more on that later - for \
now, we'll set it to 1).  We can also produce the Haar filter with the \
command ",
  StyleBox["Haar[ ]",
    FontWeight->"Bold"],
  ".",
  "\n\nHere are some examples.  Remember, any vector you give to ",
  StyleBox["WT1D",
    FontWeight->"Bold"],
  " must have ",
  StyleBox["even length",
    FontColor->RGBColor[1, 0, 0]],
  "."
}], "Text"],

Cell[BoxData[{
    \(Haar[]\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(v = {1, 1, 1, 1, 1, 1, 1, 1};\)\), "\[IndentingNewLine]", 
    \(wt = 
      WT1D[v, Haar[], 
        NumIterations \[Rule] 
          1]\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(w = Table[\((\(-1\))\)^k, {k, 1, 8}]\), "\[IndentingNewLine]", 
    \(wt = WT1D[w, Haar[], NumIterations \[Rule] 1]\)}], "Input"],

Cell["\<\
Here is a fun example to work for students.  First, sample and plot a common \
function from Calculus.\
\>", "Text"],

Cell[BoxData[{
    \(\(f[t_] := Cos[2*Pi*t];\)\), "\[IndentingNewLine]", 
    \(\(v = Table[f[k/200], {k, 0, 199}];\)\), "\[IndentingNewLine]", 
    \(\(ListPlot[v, PlotStyle \[Rule] Red];\)\)}], "Input"],

Cell["\<\
Now compute its Haar transform.  You can use the command WaveletVectorPlot \
from WaveletFunctions to plot the transform.  (For more options on \
WaveletVectorPlot, see the Help Browser or type ?WaveletVectorPlot and/or \
Options[WaveletVectorPlot].)\
\>", "Text"],

Cell[BoxData[{
    \(\(wt = 
        WT1D[v, Haar[], 
          NumIterations \[Rule] 
            1];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(WaveletVectorPlot[wt, NumIterations \[Rule] 1, Axes \[Rule] True, 
        DivideLines \[Rule] True, UseColors \[Rule] True];\)\)}], "Input"],

Cell["\<\
The above graph isn't real illuminating until you plot the individual \
portions of the output:\
\>", "Text"],

Cell[BoxData[{
    \(\(WaveletVectorPlot[wt, NumIterations \[Rule] 1, Axes \[Rule] True, 
        DivideLines \[Rule] True, UseColors \[Rule] True, 
        Region \[Rule] 
          LowPass];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(WaveletVectorPlot[wt, NumIterations \[Rule] 1, Axes \[Rule] True, 
        DivideLines \[Rule] True, UseColors \[Rule] True, 
        Region \[Rule] HighPass];\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Applying the DHWT to a Digital Audio File", "Section"],

Cell[TextData[{
  "In the initialization cell, I added a command to load ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "'s functions for processing audio.  We can read audio (unfortunately, not \
as easily from the web) from the flash drive."
}], "Text"],

Cell[BoxData[
    \(\(data = 
        ReadSoundFile[sounddir <> "\<hellocmp.wav\>", 
          PrintHeader \[Rule] True];\)\)], "Input"],

Cell["\<\
Note that the data size is even so we can apply the DHWT.  We can now plot \
data just like any other vector.\
\>", "Text"],

Cell[BoxData[
    \(\(ListPlot[data, PlotStyle \[Rule] Brown, 
        PlotRange \[Rule] All];\)\)], "Input"],

Cell["\<\
Once we get the right sampling rate (11025 in this case), we can also play \
the vector:\
\>", "Text"],

Cell[BoxData[
    \(\(ListPlay[data, SampleRate \[Rule] 11025];\)\)], "Input"],

Cell[TextData[{
  "We can compute its DHWT and plot it as well.  Note that I've \"wrapped\" \
the ",
  StyleBox["Haar[ ] ",
    FontWeight->"Bold"],
  "in a ",
  StyleBox["N[ ]",
    FontWeight->"Bold"],
  ".  This converts the filter to a numerical representation and makes the \
computations faster."
}], "Text"],

Cell[BoxData[{
    \(\(wt = 
        WT1D[data, N[Haar[]], 
          NumIterations \[Rule] 1];\)\), "\[IndentingNewLine]", 
    \(\(WaveletVectorPlot[wt, NumIterations \[Rule] 1, Axes \[Rule] True, 
        DivideLines \[Rule] True, UseColors \[Rule] True, 
        PlotRange \[Rule] All];\)\)}], "Input"],

Cell[TextData[{
  StyleBox["WaveletFunctions",
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  " also comes with a routine to let you \"listen\" to the transform.  For \
more options on ",
  StyleBox["WaveletVectorPlay",
    FontWeight->"Bold"],
  ", see the ",
  StyleBox["Help Browser",
    FontSlant->"Italic"],
  "."
}], "Text"],

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        SampleRate \[Rule] 11025, Delay \[Rule] 2, 
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Cell[CellGroupData[{

Cell["Coding the One-Dimensional DHWT", "Section"],

Cell["\<\
Let's write some code for the inverse DHWT.  Let's follow the example in \
Lecture 4.  First we create a generic vector:\
\>", "Text"],

Cell[BoxData[{
    \(\(Clear[v];\)\), "\[IndentingNewLine]", 
    \(w = Array[v, 8, 1]\)}], "Input"],

Cell[TextData[{
  "Now let's build the matrix ",
  StyleBox["V",
    FontWeight->"Bold"],
  ".  How do we do this?"
}], "Text"],

Cell[BoxData[{
    \(\(\(\(V\)\(=\)\) (*\ Put\ code\ here\ *) ;\)\), "\[IndentingNewLine]", 
    \(MatrixForm[V]\)}], "Input"],

Cell[TextData[{
  "We create the filters ",
  StyleBox["h",
    FontWeight->"Bold"],
  " and ",
  StyleBox["g",
    FontWeight->"Bold"],
  " and dot both with ",
  StyleBox["V",
    FontWeight->"Bold"],
  ":"
}], "Text"],

Cell[BoxData[{
    \(\(h = Sqrt[2]*{1, 1}/2;\)\), "\[IndentingNewLine]", 
    \(\(g = Sqrt[2]*{1, \(-1\)}/2;\)\), "\[IndentingNewLine]", 
    \(a = V . h\), "\[IndentingNewLine]", 
    \(b = V . g\)}], "Input"],

Cell["\<\
How to intertwine them?  We could make a matrix out of the two, transpose it, \
and then Flatten the result:\
\>", "Text"],

Cell[BoxData[{
    \(\(A = {b, a};\)\), "\[IndentingNewLine]", 
    \(MatrixForm[A]\), "\[IndentingNewLine]", 
    \(MatrixForm[Transpose[A]]\), "\[IndentingNewLine]", 
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Cell[TextData[{
  "Here is a module for computing the inverse ",
  StyleBox["IDHWT",
    FontWeight->"Bold"],
  ":"
}], "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["Let's try our module on an example.", "Text"],

Cell[BoxData[{
    \(v = 
      Table[Random[Integer, {0, 100}], {k, 1, 16}]\), "\[IndentingNewLine]", 
    \(\(wt = 
        WT1D[v, Haar[], 
          NumIterations \[Rule] 
            1];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(iwt = IDHWT[wt]\  // Simplify\), "\[IndentingNewLine]", 
    \(\)}], "Input"]
}, Open  ]]
}, Open  ]]
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