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Notebook[{

Cell[CellGroupData[{
Cell["Denoising", "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 introduce you to some basic \
wavelet-based denoising methods.\n\nThe notebook also contains a \
module-writing lab that makes up part of ",
  StyleBox["Computer Session Three",
    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[{
    \(<< WaveletFunctions`WaveletFunctions`\), "\[IndentingNewLine]", 
    \(<< LinearAlgebra`MatrixManipulation`\), "\[IndentingNewLine]", 
    \(<< Graphics`Graphics`\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(<< Statistics`ContinuousDistributions`\), "\n", 
    \(<< Statistics`DescriptiveStatistics`\n\), "\[IndentingNewLine]", 
    \(\(Needs["\<Miscellaneous`Audio`\>"];\)\[IndentingNewLine]\n\
\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(flashdir = "\<F:/WaveletWorkshop06/Images/\>";\)\), "\
\[IndentingNewLine]", 
    \(\(sounddir = "\<F:/WaveletWorkshop06/Sounds/\>";\)\), "\
\[IndentingNewLine]", 
    \(\(Print["\<The flash drive directory is \>", 
        flashdir, "\<.\>"];\)\[IndentingNewLine]\), "\[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["The Shrinkage Function", "Section"],

Cell["\<\
Here is the shrinkage function defined and plotted for \[Lambda]=2.\
\>", "Text"],

Cell[BoxData[{
    \(\(shrink[v_, lambda_] := 
        Which[v > lambda, v - lambda, v < \(-lambda\), v + lambda, True, 
          0];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(Plot[shrink[v, 2], {v, \(-5\), 5}, 
        Ticks \[Rule] {{\(-4\), \(-3\), \(-2\), \(-1\), 1, 2, 3, 
              4}, {\(-3\), \(-2\), \(-1\), 1, 2, 3}}, 
        PlotStyle \[Rule] Blue];\)\)}], "Input"],

Cell["Let's try it on an example", "Text"],

Cell[BoxData[{
    \(v = {\(-3.2\), \(-2.5\), 1, 1, 1.1, 5, 3.2, 
        1.9, \(-4\), \(-6\)}\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(lambda = 2;\)\), "\[IndentingNewLine]", 
    \(Map[shrink[#, lambda] &, v]\[IndentingNewLine]\[IndentingNewLine] (*\ 
      another\ way\ to\ do\ it\ *) \[IndentingNewLine]\), \
"\[IndentingNewLine]", 
    \(\(shrink2[v_] := shrink[v, lambda];\)\), "\[IndentingNewLine]", 
    \(Map[shrink2, v]\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["A Sample Signal ", "Section"],

Cell["\<\
For our example, we will use a sample signal suggested by Donoho.  He calls \
this function the Heavisine function.
    \
\>", "Text"],

Cell[BoxData[
    \(\(heavisine[t_] := 
        4*Sin[4*Pi*t] - Sign[t -  .3] - Sign[ .72 - t];\)\)], "Input"],

Cell[TextData[{
  "Let's sample this function to create our true signal ",
  StyleBox["v",
    FontWeight->"Bold"],
  " and then ",
  StyleBox["ListPlot",
    FontWeight->"Bold"],
  " it."
}], "Text"],

Cell[BoxData[{
    \(\(n = 2048;\)\), "\n", 
    \(\(v = 
        Table[heavisine[k/n], {k, 0, n - 1}];\)\), "\[IndentingNewLine]", 
    \(\(ListPlot[v, PlotStyle \[Rule] Maroon];\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Creating Noise", "Section"],

Cell["Now we need some Gaussian white noise.", "Text"],

Cell[BoxData[{
    \(\(nd = NormalDistribution[0. , 1. ];\)\), "\n", 
    \(\(SeedRandom[];\)\), "\n", 
    \(\(noise = Table[Random[nd], {k, 1, n}];\)\), "\[IndentingNewLine]", 
    \(\(xv = Table[k/n, {k, 0, n - 1}];\)\n\), "\[IndentingNewLine]", 
    \(\(plt = 
        ListPlot[Transpose[{xv, v}], 
          Ticks \[Rule] {{0,  .2,  .4,  .6,  .8, 1}, {\(-6\), \(-4\), \(-2\), 
                0, 2, 4, 6}}];\)\)}], "Input"],

Cell["If you want, you can actually \"play\" the noise:", "Text"],

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

Cell["\<\
We create the noisy vector by picking a noise level and simply adding v to a \
scaled version of the noise.\
\>", "Text"],

Cell[BoxData[{
    \(\(sigma =  .5;\)\), "\[IndentingNewLine]", 
    \(\(y = v + sigma*noise;\)\n\), "\[IndentingNewLine]", 
    \(\(plt = 
        ListPlot[Transpose[{xv, y}], 
          Ticks \[Rule] {{0,  .2,  .4,  .6,  .8, 1}, {\(-6\), \(-4\), \(-2\), 
                0, 2, 4, 6}}, PlotStyle \[Rule] Brown];\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Estimating the Noise Level", "Section"],

Cell[TextData[{
  "To estimate the noise level, we use the built-in ",
  StyleBox["MedianDeviation",
    FontWeight->"Bold"],
  " function in the statistics package.  We have to first compute the wavelet \
transform.  We'll use the Daubechies 4 filter for this example and compute 2 \
iterations."
}], "Text"],

Cell[BoxData[{
    \(\(h = N[Daub[4]];\)\), "\[IndentingNewLine]", 
    \(\(Print["\<We will use the Daubechies 4-term filter: \>", 
        h, "\<.\>"];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(its = 5;\)\), "\[IndentingNewLine]", 
    \(\(wt = 
        WT1D[y, h, NumIterations \[Rule] its];\)\), "\[IndentingNewLine]", 
    \(\(WaveletVectorPlot[wt, NumIterations \[Rule] its, Frame \[Rule] True, 
        DivideLines \[Rule] True, UseColors \[Rule] True, 
        PlotRange \[Rule] All];\)\)}], "Input"],

Cell[TextData[{
  "We need to extract the first highpass portion.  There is a command in ",
  StyleBox["WaveletFunctions",
    FontFamily->"Courier"],
  " to do this:"
}], "Text"],

Cell[BoxData[{
    \(\(hp1 = 
        ExtractVectorPart[wt, NumIterations \[Rule] its, Iteration \[Rule] 1, 
          Region \[Rule] 
            HighPass];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(ListPlot[hp1, PlotStyle \[Rule] Olive];\)\)}], "Input"],

Cell["Now we estimate the noise:", "Text"],

Cell[BoxData[{
    \(\(sigmahat = 
        MedianDeviation[
            hp1]/ .6745;\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(Print["\<Our estimate of the noise is \>", 
        sigmahat, "\<.\>"];\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["VISUShrink", "Section"],

Cell["\<\
To perform VISUShrink, we compute the universal tolerance and use it in the \
shrink functions on the highpass portions.  

We use the UNDOCUMENTED function WaveletVectorToList to easily extract all \
the highpass portions.\
\>", "Text"],

Cell[BoxData[{
    \(\(wtlist = 
        WaveletVectorToList[wt, 
          NumIterations \[Rule] its];\)\), "\[IndentingNewLine]", 
    \(\(lowpass = First[wtlist];\)\), "\[IndentingNewLine]", 
    \(\(highpass = 
        Flatten[Drop[wtlist, 
            1]];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(lambda = 
        sigmahat*Sqrt[2*Log[Length[highpass]]];\)\), "\[IndentingNewLine]", 
    \(\(Print["\<The value of the universal threshold is \>", 
        lambda, "\<.\>"];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(shrink2[v_] := 
        shrink[v, lambda];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(newhighpass = 
        Map[shrink2, 
          highpass];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(newwt = Join[lowpass, newhighpass];\)\), "\[IndentingNewLine]", 
    \(\(WaveletVectorPlot[newwt, NumIterations \[Rule] its, 
        DivideLines \[Rule] True, Frame \[Rule] True, UseColors \[Rule] True, 
        PlotRange \[Rule] All];\)\)}], "Input"],

Cell["\<\
We apply the inverse wavelet transform to obtain our estimate of v:\
\>", "Text"],

Cell[BoxData[{
    \(\(vhat = 
        IWT1D[newwt, h, 
          NumIterations \[Rule] its];\)\), "\[IndentingNewLine]", 
    \(\(ListPlot[vhat, PlotStyle \[Rule] Brown];\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["SUREShrink", "Section"],

Cell["\<\
To apply SUREShrink, we need the following function to find \[Lambda]\
\>", "Text"],

Cell[BoxData[
    \(\(DonohoSure[v_] := 
        Module[{n, a, b, c, k, s, r, idx}, n = Length[v]; 
          a = Sort[v^2]; \[IndentingNewLine]b\  = 
            Drop[FoldList[Plus, 0, a], 1]; \[IndentingNewLine]c\  = 
            Reverse[Table[k, {k, 0, n - 1}]]; \n\ \ \ \ \ \ \ s\  = 
            b + c*a; \n\ \ \ \ \ \ \ r\  = 
            n - 2*Table[k, {k, 1, n}] + 
              s; \[IndentingNewLine]idx = \(Flatten[
                Position[r, Min[r]]]\)[\([1]\)]; \[IndentingNewLine]Return[
            Sqrt[a[\([idx]\)]]];\n];\)\)], "Input"],

Cell[TextData[{
  "Remember we want to compute a different \[Lambda] for each highpass \
portion.  Let's first redefine the highpass portion from wtlist without ",
  StyleBox["Flatten",
    FontWeight->"Bold"],
  ".  Now highpass is a list of ",
  StyleBox["its",
    FontWeight->"Bold"],
  " elements."
}], "Text"],

Cell[BoxData[
    \(\(highpass = Drop[wtlist, 1];\)\)], "Input"],

Cell[TextData[{
  "Remember, for Stein's theorem to work, the variance must be one.    So we \
need to divide all the highpass portions by ",
  StyleBox["sigmahat",
    FontWeight->"Bold"],
  " when computing the tolerances.  We will do all the computations with a ",
  StyleBox["Map",
    FontWeight->"Bold"],
  " command:"
}], "Text"],

Cell[BoxData[{
    \(\(lambdaS = 
        Map[DonohoSure, highpass/sigmahat];\)\), "\[IndentingNewLine]", 
    \(\(Table[
        Print["\<The SURE threshold for highpass portion \>", 
          its - k + 1, "\< is \>", lambdaS[\([k]\)], "\<.\n\>"], {k, 1, 
          its}];\)\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Apply the SUREShrink Threshold", "Section"],

Cell["\<\
We will again use a Table to apply the tolerances to each of the highpass \
portions:\
\>", "Text"],

Cell[BoxData[
    \(\(newhighpass = 
        Table[Map[shrink[#, lambdaS[\([k]\)]] &, 
            highpass[\([k]\)]/sigmahat], {k, 1, its}];\)\)], "Input"],

Cell["Now we put the wavelet transform back together and plot it:", "Text"],

Cell[BoxData[{
    \(\(newwt = 
        Join[lowpass, 
          Flatten[newhighpass]];\)\[IndentingNewLine]\[IndentingNewLine]\), "\
\n", 
    \(\(WaveletVectorPlot[wt, Frame \[Rule] True, PlotRange \[Rule] All, 
        DivideLines \[Rule] True, UseColors \[Rule] True, 
        NumIterations \[Rule] its];\)\n\), "\[IndentingNewLine]", 
    \(\(WaveletVectorPlot[newwt, Frame \[Rule] True, PlotRange \[Rule] All, 
        DivideLines \[Rule] True, UseColors \[Rule] True, 
        NumIterations \[Rule] its];\)\)}], "Input"],

Cell["Finally, we invert to obtained the denoised signal.", "Text"],

Cell[BoxData[{
    \(\(vhat = 
        IWT1D[newwt, h, 
          NumIterations \[Rule] its];\)\), "\[IndentingNewLine]", 
    \(\(ListPlot[vhat, PlotStyle \[Rule] Brown];\)\)}], "Input"]
}, Open  ]]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

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

Cell["\<\
There are two tasks in this computer session.  You will re-visit image \
compression using Daubechies filters and you will write modules to perform \
the 1D Daubechies transform and its inverse.\
\>", "Text"],

Cell[CellGroupData[{

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

Cell[TextData[{
  "Repeat the demos above using the Daubechies 6-term filter and the Coiflet \
6-term filter.  Remember the Coiflet 6-term filter generates an orthogonal \
matrix but its Fourier series satisfies H(0) = ",
  Cell[BoxData[
      \(\@2\)]],
  ", H'(0) = 0, H(\[Pi]) = H'(\[Pi]) = 0.  Which filter did a better job \
denoising?  Which method?  Try varying the number of iterations.\n\nHere is \
some more on the Coiflet 6-term filter:\n"
}], "Text"],

Cell[BoxData[{
    \(\(h = Coif[1];\)\), "\[IndentingNewLine]", 
    \(\(Print["\<The Coiflet 6-term filter is \>", 
        h, "\<.  It is not causal - the first index is -2\>"];\)\
\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(H[w_] := 
        h . Table[
            E^\((I*k*w)\), {k, \(-2\), 3}];\)\), "\[IndentingNewLine]", 
    \(\(Plot[Abs[H[w]], {w, 0, Pi}, 
        Ticks \[Rule] {{0, Pi/2, Pi}, {Sqrt[2]/2, 
              Sqrt[2]}}];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(H[0]\  // Simplify\), "\[IndentingNewLine]", 
    \(\(H'\)[0]\  // Simplify\), "\[IndentingNewLine]", 
    \(H[Pi]\  // Simplify\), "\[IndentingNewLine]", 
    \(\(H'\)[Pi]\  // Simplify\)}], "Input"],

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

Cell[CellGroupData[{

Cell["Task Two", "Subtitle"],

Cell["\<\
Let's try denoising on an audio file.  The cell below reads, chops, and plays \
an audio file for you:\
\>", "Text"],

Cell[BoxData[{
    \(\(data = 
        ReadSoundFile[sounddir <> "\<wargames.wav\>", 
          PrintHeader \[Rule] True];\)\), "\[IndentingNewLine]", 
    \(\(v = 
        ChopVector[data, PowersOfTwo \[Rule] 6];\)\), "\[IndentingNewLine]", 
    \(\(n = Length[v];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(Print["\<The length of data is now \>", 
        n, "\< and we can perform as many as 6 iterations of the wavelet \
transform.\>"];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(ListPlay[v, SampleRate \[Rule] 16000];\)\)}], "Input"],

Cell["\<\
In the cell below, add some noise to the audio file - I picked sigma=1000 - \
feel free to change it.\
\>", "Text"],

Cell[BoxData[{
    \(\(nd = NormalDistribution[0. , 1. ];\)\), "\n", 
    \(\(SeedRandom[];\)\), "\n", 
    \(\(noise = 
        Table[Random[nd], {k, 1, 
            n}];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(sigma = 1000;\)\), "\[IndentingNewLine]", 
    \(\(y = v + sigma*noise;\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(ListPlay[y, SampleRate \[Rule] 16000];\)\)}], "Input"],

Cell["", "Text"],

Cell["\<\
In the cell below, use either VISUShrink or SUREShrink to denoise the \
audio:\
\>", "Text"],

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

Cell[CellGroupData[{

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

Cell[TextData[{
  "Go to the Math 316 web site and under Media Files, copy the \
LaurelHardy.wav file to your flashdrive Sounds directory.  This is an OLD \
recording and is very noisy.  Thus we don't need to add any noise!  See how \
good a job you can do denoising it!\n\nSuggestion:  The file is long - after \
reading it in, use the ",
  StyleBox["Take ",
    FontWeight->"Bold"],
  "command to grab the first quarter of the file.  Just work with that."
}], "Text"]
}, Open  ]]
}, Open  ]]
}, Open  ]]
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End of Mathematica Notebook file.
*******************************************************************)