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                     CreatedBy='Mathematica 5.2'

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Notebook[{

Cell[CellGroupData[{
Cell["Daubechies Filters", "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 learn about Daubechies family of orthogonal \
filters.\
\>", "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[
    \(<< WaveletFunctions`WaveletFunctions`\)], "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["Solving for the Daubechies 4-Tap Filter", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "We start by defining the unknown filter.  We will use the ",
  StyleBox["Array",
    FontWeight->"Bold"],
  " command from ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ".  ",
  StyleBox["Array",
    FontWeight->"Bold"],
  " needs a variable name, the length of the array and the starting \
subscript.  Note that I've cleared the variable name before defining the \
array."
}], "Subsubtitle"],

Cell[BoxData[{
    \(\(Clear[h];\)\), "\[IndentingNewLine]", 
    \(\(n = 4;\)\), "\[IndentingNewLine]", 
    \(v = Array[h, n, 0]\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Next, let's define the Fourier series associated with h. ", \
"Subsubtitle"],

Cell[BoxData[{
    \(\(H[w_] := 
        v . Table[E^\((I*k*w)\), {k, 0, n - 1}];\)\), "\[IndentingNewLine]", 
    \(Print["\<H[w] = \>", H[w]]\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Now we form the orthogonality conditions", "Subsubtitle"],

Cell[BoxData[
    \(orth = {Total[v^2] \[Equal] 1, 
        h[0]*h[2] + h[1]*h[3] \[Equal] 0}\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["and then the lowpass/derivative conditions. ", "Subsubtitle"],

Cell[BoxData[
    \(\(\(lp = {H[0] \[Equal] Sqrt[2], 
        H[Pi] \[Equal] 0, \(H'\)[Pi] \[Equal] 0}\)\(\[IndentingNewLine]\)
    \)\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "Finally, we form the system and solve it.  ",
  StyleBox["Solve",
    FontWeight->"Bold"],
  " needs a list of equations and a list of variables."
}], "Subsubtitle"],

Cell[BoxData[{
    \(\(eqs = Join[orth, lp];\)\), "\[IndentingNewLine]", 
    \(TableForm[eqs]\), "\[IndentingNewLine]", 
    \(\(s = Solve[eqs, v];\)\), "\[IndentingNewLine]", 
    \(TableForm[s]\), "\[IndentingNewLine]", 
    \(\(Print["\<There are \>", 
        Length[s], "\< two solutions to this system.\>"];\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "Here is some ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " code to strip out the arrows from the solutions.  The ",
  StyleBox["/.",
    FontWeight->"Bold"],
  " simply replaces all the h[0], h[1], h[2], h[3] with their values"
}], "Subsubtitle"],

Cell[BoxData[{
    \(d4 = v /. s\), "\[IndentingNewLine]", 
    \(\(Print["\<The first solution is \>", 
        d4[\([1]\)], "\<.\>"];\)\), "\[IndentingNewLine]", 
    \(\(Print["\<The second solution is \>", 
        d4[\([2]\)], "\<.\>"];\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Let's look at the modulus of the Fourier series associated with the first \
solution.\
\>", "Subsubtitle"],

Cell[BoxData[{
    \(\(v = d4[\([1]\)];\)\), "\[IndentingNewLine]", 
    \(\(Plot[Abs[H[w]], {w, 0, Pi}, PlotStyle \[Rule] Red, 
        Ticks \[Rule] {{Pi/4, Pi/2, 3*Pi/4, Pi}, {Sqrt[2]}}];\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Compare this graph and that of the modulus of the Fourier transform \
associated with the Haar transform (plotted in blue):\
\>", "Subsubtitle"],

Cell[BoxData[
    \(\(Plot[{Abs[H[w]], Sqrt[2]*Cos[w/2]}, {w, 0, Pi}, 
        PlotStyle \[Rule] {Red, Blue}, 
        Ticks \[Rule] {{Pi/4, Pi/2, 3*Pi/4, Pi}, {Sqrt[2]}}];\)\)], "Input"],

Cell["The Daubechies 4-Tap Filter in WaveletFunctions", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  StyleBox["WaveletFunctions",
    FontColor->RGBColor[0.501961, 0, 0]],
  " contains a routine that will generate the Daubechies 4-tap filter.\n"
}], "Subsubtitle"],

Cell[BoxData[{
    RowBox[{"d4", "=", 
      StyleBox[\(Daub[4]\),
        FontColor->RGBColor[0.501961, 0, 
        0]]}], "\[IndentingNewLine]", \(Print["\<Here is a numerical version \
of WaveletFunctions d4 filter: \>", 
        N[d4]];\), "\[IndentingNewLine]", \(Print["\<Here is a numerical \
version of the filter we derived: \>", N[v]];\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "Note that the solution (v)  we obtained is a reflection of the solution \
given by ",
  StyleBox["Daub[4]",
    FontColor->RGBColor[0.501961, 0, 0]],
  "."
}], "Subsubtitle"],

Cell[CellGroupData[{

Cell["The Corresponding Highpass Filter", "Section"],

Cell[BoxData[{
    \(\(g = Reverse[d4]*{1, \(-1\), 1, \(-1\)};\)\), "\[IndentingNewLine]", 
    \(\(Print["\<The corresponding highpass filter for the Daubechies 4-tap \
filter is :\>", g];\)\)}], "Input"]
}, Open  ]],

Cell["The Transform Matrix", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell["We can verify orthogonality for size M x M matrices:", "Subsubtitle"],

Cell[BoxData[{
    StyleBox[\(M = 8;\),
      FontColor->RGBColor[0, 0, 
      1]], "\[IndentingNewLine]", \(r1 = 
        Join[d4, 
          Table[0, {M - Length[d4]}]];\), "\[IndentingNewLine]", \(tp = 
        Table[RotateRight[r1, 2*k], {k, 0, 
            M/2 - 1}];\), "\[IndentingNewLine]", \(r2 = 
        Join[g, Table[0, {M - Length[d4]}]];\), "\[IndentingNewLine]", \(bt = 
        Table[RotateRight[r2, 2*k], {k, 0, 
            M/2 - 1}];\), "\[IndentingNewLine]", \(W = 
        N[Join[tp, 
            bt]];\), "\[IndentingNewLine]", \(Print["\<The transform matrix \
for size \>", M, "\< is \>", 
        MatrixForm[
          W], "\<.\>"];\), "\[IndentingNewLine]", \(Print["\<The inverse \
transform matrix for size \>", M, "\< is \>", 
        MatrixForm[Chop[Inverse[W]]], "\<.\>"];\)}], "Input"],

Cell["Using the Daubechies 4-Tap Filter", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
We can easily compute wavelet transforms using the Daubechies 4-tap filter.  \
We use the command WT1D for the transform and IWT1D for the inverse.  We need \
to provide these routines with the input vector, the filter, and the number \
of iterations.  I've defined the filter numerically to speed up computation \
time.\
\>", "Subsubtitle"],

Cell[BoxData[{
    RowBox[{
      RowBox[{"d4", "=", 
        RowBox[{"N", "[", 
          StyleBox[\(Daub[4]\),
            FontColor->RGBColor[0.501961, 0, 0]], "]"}]}], 
      ";"}], "\[IndentingNewLine]", 
    StyleBox[\(x = Table[k, {k, 0, 31}];\),
      FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", 
    RowBox[{
      RowBox[{"wt", "=", 
        StyleBox[\(WT1D[x, d4, NumIterations \[Rule] 1]\),
          FontColor->RGBColor[0.501961, 0, 0]]}], 
      StyleBox[";",
        FontColor->RGBColor[0.501961, 0, 0]]}], "\[IndentingNewLine]", 
    StyleBox[\(WaveletVectorPlot[wt, NumIterations \[Rule] 1, 
          Axes \[Rule] True, DivideLines \[Rule] True, 
          UseColors \[Rule] True];\),
      FontColor->RGBColor[0.501961, 0, 0]]}], "Input"],

Cell["D4 versus Haar", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Let's look at the approximation and differences portion of the Daubechies \
transform: \
\>", "Subsubtitle"],

Cell[BoxData[{
    StyleBox[\(WaveletVectorPlot[wt, NumIterations \[Rule] 1, 
          Axes \[Rule] True, DivideLines \[Rule] True, 
          UseColors \[Rule] True, Region \[Rule] LowPass];\),
      FontColor->RGBColor[0.501961, 0, 0]], "\[IndentingNewLine]", 
    RowBox[{
      StyleBox[\(WaveletVectorPlot[wt, NumIterations \[Rule] 1, 
            Axes \[Rule] True, DivideLines \[Rule] True, 
            UseColors \[Rule] True, Region \[Rule] HighPass];\),
        FontColor->RGBColor[0.501961, 0, 0]], 
      "\[IndentingNewLine]"}], "\[IndentingNewLine]", }], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
You can see the effects of the wrapping rows in the transform in the \
approximation portion of the transform.    For the linear data, note that the \
difference portion of the transform is essentially 0.     

Let's plot the highpass portion of the Haar transform:\
\>", "Subsubtitle"],

Cell[BoxData[{
    RowBox[{
      RowBox[{"wtHaar", "=", 
        StyleBox[\(WT1D[x, N[Haar[]], NumIterations \[Rule] 1]\),
          FontColor->RGBColor[0.501961, 0, 0]]}], 
      StyleBox[";",
        FontColor->RGBColor[0.501961, 0, 0]]}], "\[IndentingNewLine]", 
    StyleBox[\(WaveletVectorPlot[wtHaar, NumIterations \[Rule] 1, 
          Axes \[Rule] True, DivideLines \[Rule] True, 
          UseColors \[Rule] True, Region \[Rule] HighPass];\),
      FontColor->RGBColor[0.501961, 0, 0]]}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "Note the Haar differences are constant.  If you study more about the \
classical development of orthogonal wavelets, you will learn that the ",
  StyleBox["wavelet functions",
    FontSlant->"Italic"],
  " associated with the D4 filter can exactly reproduce (in an L^2 sense) \
linear polynomials while the ",
  StyleBox["wavelet functions",
    FontSlant->"Italic"],
  " associated with the Haar filter can exactly reproduce only constant \
functions."
}], "Subsubtitle"],

Cell["Problem - Find the Daubechies 6-Term Filter", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
In the cells below, I have recopied the syntax we used to set up and solve \
for the Daubechies 4-tap filter.  Using pencil and paper, can you write down \
orthogonality/lowpass conditions for the 6-tap filter ( use a matrix of size \
say 10 x 10 and remember you only need the conditions for the lowpass filter \
h).  Then modify the code below to find all 6-tap filters that work.  (Hint:  \
There will be three orthogonality conditions, 2 lowpass conditions, and the \
derivative condition H'(\[Pi]) = 0.  You will need to add another \
condition....)\
\>", "Subsubtitle"],

Cell["Define the unknown filter variables", "Subsubtitle"],

Cell[BoxData[{
    \(\(Clear[h];\)\), "\[IndentingNewLine]", 
    \(\(n = 4;\)\), "\[IndentingNewLine]", 
    \(v = Array[h, n, 0]\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Set up the Fourier series.", "Subsubtitle"],

Cell[BoxData[{
    \(\(H[w_] := 
        v . Table[E^\((I*k*w)\), {k, 0, n - 1}];\)\), "\[IndentingNewLine]", 
    \(Print["\<H[w] = \>", H[w]]\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["The orthogonality conditions:", "Subsubtitle"],

Cell[BoxData[
    \(orth = {Total[v^2] \[Equal] 1, 
        h[0]*h[2] + h[1]*h[3] \[Equal] 0}\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["The lowpass/derivative conditions:", "Subsubtitle"],

Cell[BoxData[
    \(lp = {H[0] \[Equal] Sqrt[2], 
        H[Pi] \[Equal] 0, \(H'\)[Pi] \[Equal] 0}\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "Set up and solve the system (I have replaced ",
  StyleBox["Solve",
    FontWeight->"Bold"],
  " with ",
  StyleBox["NSolve",
    FontWeight->"Bold"],
  ")"
}], "Subsubtitle"],

Cell[BoxData[{
    \(\(eqs = Join[orth, lp];\)\), "\[IndentingNewLine]", 
    \(TableForm[eqs]\), "\[IndentingNewLine]", 
    \(\(s = NSolve[eqs, v];\)\), "\[IndentingNewLine]", 
    \(TableForm[s]\), "\[IndentingNewLine]", 
    \(\(Print["\<There are \>", 
        Length[s], "\< two solutions to this system.\>"];\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Print the solutions", "Subsubtitle"],

Cell[BoxData[{
    \(d4 = v /. s\), "\[IndentingNewLine]", 
    \(\(Print["\<The first solution is \>", 
        d4[\([1]\)], "\<.\>"];\)\), "\[IndentingNewLine]", 
    \(\(Print["\<The second solution is \>", 
        d4[\([2]\)], "\<.\>"];\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Does one of your solutions match this one?", "Subsubtitle"],

Cell[BoxData[
    RowBox[{"N", "[", 
      StyleBox[\(Daub[6]\),
        FontColor->RGBColor[0.501961, 0, 0]], "]"}]], "Input"],

Cell[CellGroupData[{

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

Cell[BoxData[
    \(\(imgdir = $BaseDirectory <> "\</Applications/Images/\>";\)\)], "Input"]
}, Open  ]]
}, 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]], 
        StyleBox[" ",
          FontColor->RGBColor[0, 0, 1]], "*)"}], "\[IndentingNewLine]", 
      RowBox[{"(*", " ", 
        StyleBox[\(image = \ "\<calvin_hobbes_gray.jpg\>";\),
          FontColor->RGBColor[0, 0, 1]], 
        StyleBox[" ",
          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"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "Here 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"],

Cell["Daubechies Filters and Images", "Section"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
We can easily process images using Daubechies filters.  The commands are \
exactly the same as those used for 2D Haar Wavelet Transformations.\
\>", "Subsubtitle"],

Cell[BoxData[{
    RowBox[{
      RowBox[{"A", "=", 
        StyleBox[\(ReadImage[imgdir <> image, PrintInfo \[Rule] True]\),
          FontColor->RGBColor[0.501961, 0, 0]]}], 
      StyleBox[";",
        FontColor->RGBColor[0.501961, 0, 0]]}], "\[IndentingNewLine]", 
    StyleBox[\(ImagePlot[A];\),
      FontColor->RGBColor[0.501961, 0, 0]]}], "Input"],

Cell[BoxData[{
    RowBox[{
      RowBox[{"wtD4", "=", 
        StyleBox[\(WT2D[A, N[Daub[4]], NumIterations \[Rule] 2]\),
          FontColor->RGBColor[0.501961, 0, 0]]}], 
      StyleBox[";",
        FontColor->RGBColor[0.501961, 0, 0]]}], "\[IndentingNewLine]", 
    RowBox[{
      RowBox[{"wtHaar", "=", 
        StyleBox[\(WT2D[A, N[Haar[]], NumIterations \[Rule] 2]\),
          FontColor->RGBColor[0.501961, 0, 0]]}], 
      StyleBox[";",
        FontColor->RGBColor[0.501961, 0, 0]]}], "\[IndentingNewLine]", 
    StyleBox[\(WaveletDensityPlot[wtD4, NumIterations \[Rule] 2, 
          DivideLines \[Rule] True];\),
      FontColor->RGBColor[0.501961, 0, 0]]}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Here is a comparison of the cumulative energies.", "Subsubtitle"],

Cell[BoxData[{
    RowBox[{
      RowBox[{"ceD4", "=", 
        StyleBox[\(CE[wtD4]\),
          FontColor->RGBColor[0.501961, 0, 0]]}], 
      StyleBox[";",
        FontColor->RGBColor[0.501961, 0, 0]]}], "\[IndentingNewLine]", 
    RowBox[{
      RowBox[{"ceHaar", "=", 
        StyleBox[\(CE[wtHaar]\),
          FontColor->RGBColor[0.501961, 0, 0]]}], 
      StyleBox[";",
        FontColor->RGBColor[0.501961, 0, 0]]}], "\[IndentingNewLine]", \(p1 = 
        ListPlot[ceD4, 
          PlotStyle \[Rule] Red];\), "\[IndentingNewLine]", \(p2 = 
        ListPlot[ceHaar, 
          PlotStyle \[Rule] Blue];\), "\[IndentingNewLine]", \(Show[{p1, 
          p2}];\)}], "Input"]
}, Open  ]]
}, Open  ]]
},
FrontEndVersion->"5.2 for Microsoft Windows",
ScreenRectangle->{{0, 1024}, {0, 695}},
WindowSize->{1015, 651},
WindowMargins->{{0, Automatic}, {Automatic, 0}}
]

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