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Cell[CellGroupData[{
Cell["Cumulative Energy and PSNR", "Title",
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Cell["\<\
Wavelet Workshop
June 2-5, 2009
University of South Florida\
\>", "Subtitle",
 CellChangeTimes->{{3.421426484890625*^9, 3.42142648821875*^9}, {
  3.4523426437597103`*^9, 3.452342654156664*^9}}],

Cell["\<\
Catherine Beneteau
Caroline Haddad
David Ruch
Patrick Van Fleet\
\>", "Subsubtitle",
 CellChangeTimes->{{3.42144927371875*^9, 3.42144928703125*^9}}],

Cell[CellGroupData[{

Cell["Overview", "Section",
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Cell[CellGroupData[{

Cell["Objectives", "Subsection"],

Cell["\<\
The purpose of this notebook is to give you a brief introduction to the \
software package DiscreteWavelets and show you how to use it to load images.  \
You will learn how to use measures and tools such as cumulative energy and \
PSNR.\
\>", "Text",
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Cell[CellGroupData[{

Cell["DiscreteWavelets", "Subsection"],

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

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

Cell["\<\
If you ever need help with DiscreteWavelets, go to Help, then Documentation \
Center.  On the bottom right, you will find a link for Installed Add-Ons.  \
Click this link and then the link to  DiscreteWavelets to access \
documentation, examples, and applications of the software package.\
\>", "Text",
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Cell[CellGroupData[{

Cell["Cumulative Energy", "Section",
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Cell["\<\
One of the important skills students develop in this course is the ability to \
write modules or functions.  The cumulative energy function is an excellent \
starting point for this development. \
\>", "Text",
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Cell[CellGroupData[{

Cell["Calculating Cumulative Energy", "Subsection",
 CellChangeTimes->{{3.42142954565625*^9, 3.421429551046875*^9}}],

Cell["\<\

To compute the cumulative energy of vector v, we sort the absolute values of \
the elements largest to smallest and then square the components of the \
resulting vector.  Call this vector y.  We find the kth element of the \
cumulative energy vector by summing the first k elements of y and dividing \
the result by the norm of v squared.

The first two steps of this process are easy.  Let's use a small vector as an \
example.\
\>", "Text",
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Cell[TextData[{
 "Next comes the cumulative sums.  We could write a loop for k = 1 through 10 \
and an inner loop j = 1 to k to perform the task, but we instead encourage \
our students to take advantage of built-in functions that accomplish tasks.  \
In most cases, these built-in functions are much faster than attempting to \
extract individual elements from a vector or list.  For cumulative sums, the \
built-in command ",
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point (0), and a list of numbers (y)"
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Cell[TextData[{
 "Notice that there is a small problem.  Our cumulative sum list is close but \
contains an unwanted 0 at the beginning.  We can get rid of the 0 by using \
the ",
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Cell["\<\
Dividing the above list by norm v squared completes the task.\
\>", "Text",
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Cell["The Cumulative Energy Module", "Subsection",
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Cell[TextData[{
 "In ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 ", it is very easy to write modules or functions.  If you have experience \
writing modules or functions, there are three important things to note: (1) \
use a Return statement to indicate what, if anything, is being returned by \
the module, (2) a local variables list is highly recommended, and (3) you \
cannot pass a module command a variable (say x) and then overwrite x within \
the module - you must make a copy of it.\n\nIn the cell below, we will write \
a module for computing the cumulative energy of a vector (or matrix!) \
together."
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Cell["\<\
The cell below creates a vector of 20 random integers and then calls the MyCE \
module and the DiscreteWavelets command CE to compute the cumulative energy.  \
Don't execute the cell until you have completed and executed the cell above.\
\>", "Text",
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Cell["Exercise 1", "Subsubsection",
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Cell[TextData[{
 "The MyCE module above will not work on matrices.  The module is expecting a \
list of numbers and a matrix is a list of lists in ",
 StyleBox["Mathematica",
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(Hint: Check the command ",
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Cell[TextData[{
 "The MyCE module, as it's written, performs all computations symbolically.  \
This is fine for short-length vectors but for longer vectors, the computation \
time increases.  How can you modify the MyCE module so that it will work on \
matrices?  (Hint: Check the command ",
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Cell["Exercise 2", "Subsubsection",
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Cell["\<\
Load the grayscale image of the chess pieces from the DiscreteWavelets \
package, find its cumulative energy using the MyCE module, and then plot the \
cumulative energy of the image.  How many elements of the image constitute \
90% of the energy?  How many zeros are in the image?\
\>", "Text",
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Cell[TextData[{
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(Hint: If you wish, you can generate a constant vector using the ",
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Cell["Exercise 2", "Subsubsection",
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Cell["\<\
How do you suppose we compute the PSNR between two color images?  Load the \
Legos image from the DiscreteWavelets package and add matrices of random \
integers ranging from -10 to 10 to each channel.  Then compute the PSNR \
between the original image and the \"noisy\" one.\
\>", "Text",
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