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Cell[CellGroupData[{
Cell["Daubechies Filters", "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 Daubechies filters and \
see how they can be used in applications.\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],

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  " package ",
  StyleBox["WaveletFunctions",
    FontFamily->"Courier"],
  " for use in subsequent computations."
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    \(\(flashdir = "\<F:/WaveletWorkshop06/Images/\>";\)\), "\
\[IndentingNewLine]", 
    \(\(Print["\<The flash drive directory is \>", 
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    \(\(imgurl = \
"\<http://cam.mathlab.stthomas.edu/pvf/Math316/TestImages/\>";\)\), "\
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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",
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  ", and click on ",
  StyleBox["AddOns & Links",
    FontSlant->"Italic"],
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  StyleBox["WaveletFunctions",
    FontFamily->"Courier"],
  ".  "
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Cell["Generating the Daubechies Filters", "Section"],

Cell["\<\
There is a simple command in WaveletFunctions that gives you access to the \
Daubechies filters.  The command is called Daub and it takes a single \
argument that is an even integer.  The first three filters (Daub[2] is the \
Haar filter!) are returned symbolically while higher order filters are \
returned numerically.\
\>", "Text"],

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    \(h2 = Daub[2]\), "\[IndentingNewLine]", 
    \(h4 = Daub[4]\), "\[IndentingNewLine]", 
    \(h6 = Daub[6]\)}], "Input"],

Cell["\<\
Let's look at the modulus of the Fourier series of each filter.\
\>", "Text"],

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Cell["Wavelet Transforms With Daubechies Filters", "Section"],

Cell["\<\
The routines WT1D and WT2D can be used with Daubechies filters.  The routines \
accept the input vector or matrix, the filter h (the routines then create g), \
and a directive NumIterations.  The transform is returned.\
\>", "Text"],

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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["\<\
Recall in Computer Session One, we performed image compression using the Haar \
transform.  Actually, we modified the Haar filter so that it would map \
integers to integers and did compression.

In this task, you will re-visit that exercise, but instead of using the Haar \
filter, you'll try the Daub4 filter.  Now this filter is not easy to modify \
so that it maps integers to integers, so we will compute the wavelet \
transform using Daub4 and then Round the results to integers.  Other than \
this small change, the compression scheme works just like it did in the first \
example of the HaarImageCompression notebook.  

You are to cut and paste from that example, but use Daub4 and rounding \
instead of the modified Haar filter (1,1).  Use the same number of iterations \
and the same percentage for quantizing.  Are the results better or worse than \
the Haar filter results?

Repeat the exercise with the Daub6 filter.\
\>", "Text"],

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

Cell["\<\
In this task, you will write a module that, given a vector and a filter, \
computes one iteration of the wavelet transform.  Your module should first \
create the highpass filter from the given lowpass filter and then proceed \
much like the 1D discrete Haar wavelet transform.  

I have included that module below:\
\>", "Text"],

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          V = Partition[V, 2, 2]; \[IndentingNewLine]\t
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  StyleBox["Length[h]",
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  " and skip by two, but if you try this on a generic vector, you don't get \
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If you can get Partition working for the generic vector, then you probably \
have the command you need for your module.\
\>", "Text"],

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Cell[CellGroupData[{

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

Cell["\<\
Write a module that, given a vector and a filter, computes one iteration of \
the inverse wavelet transform.

This is the hardest module the students would write in my wavelets course - I \
often don't assign it, but if I have a particularly bright student, I make it \
an extra credit problem.\
\>", "Text"]
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