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Notebook[{

Cell[CellGroupData[{
Cell["The Lifting Method", "Title"],

Cell["\<\
Wavelet Workshop
June 6-9, 2007
University of St. Thomas\
\>", "Subtitle"],

Cell[CellGroupData[{

Cell["Objectives", "Section"],

Cell["\<\
The purpose of this notebook is to introduce you to the lifting method.
\
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["DiscreteWavelets", "Section"],

Cell[TextData[{
  StyleBox["You should run this cell each time you open this notebook!!",
    FontColor->RGBColor[1, 0, 0]],
  "  It loads the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " package ",
  StyleBox["DiscreteWavelets",
    FontFamily->"Courier"],
  " for use in subsequent computations."
}], "Text"],

Cell[BoxData[
    \(<< DiscreteWavelets`DiscreteWavelets`\)], "Input",
  FontColor->GrayLevel[0.500008]]
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Cell[CellGroupData[{

Cell["Help on DiscreteWavelets", "Section"],

Cell[TextData[{
  "If you ever need help with ",
  StyleBox["DiscreteWavelets",
    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["DiscreteWavelets",
    FontFamily->"Courier"],
  ".  "
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["The One-Dimensional LeGall Wavelet Transform", "Section"],

Cell["\<\
The package DiscreteWavelets has a function for computing the LeGall wavelet \
transform.  It is called LWT1D1.  Here is an example:\
\>", "Text"],

Cell[BoxData[{
    \(\(v = 
        Table[Random[Integer, {0, 10}], {16}];\)\), "\[IndentingNewLine]", 
    \(wt = LWT1D1[v]\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["A Module for the LeGall Wavelet Transform", "Section"],

Cell["", "Text"],

Cell["\<\
Our first step is to partition the input into even and odd portions:\
\>", "Text"],

Cell[BoxData[{
    \(\(z = Array[vv, {16}];\)\), "\[IndentingNewLine]", 
    \(e = Take[z, {2, Length[z], 2}]\), "\n", 
    \(o = Take[z, {1, Length[z], 2}]\)}], "Input"],

Cell["\<\
Now we need to append the o vector with the last element of o.\
\>", "Text"],

Cell[BoxData[
    \(o = Append[o, Last[o]]\)], "Input"],

Cell["Now we can easily compute the highpass portion!", "Text"],

Cell[BoxData[{
    \(\(a = Partition[o, 2, 1];\)\), "\[IndentingNewLine]", 
    \(d = e - Map[Total, a]/2\)}], "Input"],

Cell["\<\
Next we need a copy of d with the first element of d prepended to d.\
\>", "Text"],

Cell[BoxData[
    \(dd = Prepend[d, First[d]]\)], "Input"],

Cell["The last step is to compute s!", "Text"],

Cell[BoxData[{
    \(\(a = Partition[dd, 2, 1];\)\), "\[IndentingNewLine]", 
    \(\(s = Drop[o, \(-1\)] + Map[Total, a]/4;\)\)}], "Input"],

Cell["The module is", "Text"],

Cell[BoxData[
    \(\(\(sLWT1D1[v_] := 
        Module[{o, e, oo, a, d, dd, s}, \[IndentingNewLine]\t
          e = Take[v, {2, Length[z], 2}]; \n\t\ \ \ \ \ \ \ o = 
            Take[v, {1, Length[z], 2}]; \[IndentingNewLine]\t
          o = Append[o, Last[o]]; \[IndentingNewLine]\t
          a = Partition[o, 2, 1]; \[IndentingNewLine]\t
          d = e - Map[Total, a]/2; \[IndentingNewLine]\t
          dd = Prepend[d, First[d]]; \[IndentingNewLine]\t
          a = Partition[dd, 2, 1]; \[IndentingNewLine]\t
          s = Drop[o, \(-1\)] + Map[Total, a]/4; \[IndentingNewLine]\t
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    \)\)], "Input"],

Cell[BoxData[{
    \(\(v = Table[Random[Integer, {0, 100}], {16}];\)\), "\n", 
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    \(wt1 - wt\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["A Module for the Inverse LWT", "Section"],

Cell["\<\
In the cell below, write the module ILWT1D1 that computes one iteration of \
the inverse one-dimensional LWT.\
\>", "Text"],

Cell[BoxData[
    \(\(ILWT1D1[v_] := 
        Module[\({ (*\ 
              Put\ local\ variables\ here\ *) }\)\(,\)\[IndentingNewLine] (*\ 
            Put\ code\ here\ *) \[IndentingNewLine]];\)\)], "Input"],

Cell["\<\
Test your code on the cell below:
    \
\>", "Text"],

Cell[BoxData[{
    \(\(v = 
        Table[Random[Integer, {0, 100}], {16}];\)\), "\[IndentingNewLine]", 
    \(\(wt = LWT1D1[v];\)\), "\[IndentingNewLine]", 
    \(v - ILWT1D1[wt]\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Mapping Integers to Integers", "Section"],

Cell["\<\
Use the directive IntegerMap set to True to map integers to integers with the \
LeGall wavelet transform:\
\>", "Text"],

Cell[BoxData[{
    \(v = Table[Random[Integer, {0, 100}], {16}]\), "\[IndentingNewLine]", 
    \(wt = N[LWT1D1[v]]\), "\[IndentingNewLine]", 
    \(wt1 = LWT1D1[v, IntegerMap \[Rule] True]\)}], "Input"],

Cell["We can compute the inverse in the same way:", "Text"],

Cell[BoxData[
    \(newv = ILWT1D1[wt1, IntegerMap \[Rule] True]\)], "Input"]
}, Open  ]]
}, Open  ]]
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