(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 159833, 4160]*) (*NotebookOutlinePosition[ 160532, 4184]*) (* CellTagsIndexPosition[ 160488, 4180]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Mat 240\tWheeling Jesuit University\t\tDr. Kim Roth Linear Algebra\tLab on Wavelets\t\tFebruary 3, 2004\ \>", "Subsubtitle", FontWeight->"Bold", FontSlant->"Plain", Background->RGBColor[1, 0, 0]], Cell[TextData[{ "Name: ", StyleBox[" ", Background->GrayLevel[1]], "\n", StyleBox["Type in your name(s) above and then do a Save As ", FontWeight->"Plain"] }], "Text", FontWeight->"Bold", Background->RGBColor[1, 1, 0]], Cell[CellGroupData[{ Cell["References:", "Subsubsection"], Cell[TextData[{ "Stephen Wolfram \"Mathematica\[RegisteredTrademark] : A System for Doing \ Mathematics by Computer, 3nd Edition\", Addison-Wesley, 1991. \n\nKolman & \ Hill \" Introductory Linear Algebra with Applications, 7th Ed.\" Prentice \ Hall, 2001.\nSection 2.5 Introduction to Wavelets. Implemented into ", StyleBox["Mathematica", FontSlant->"Italic"], " by Dr. Ted Erickson" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Goals:", "Subsubsection"], Cell[TextData[{ "\[Bullet]\tBecome familiar with wavelets using ", StyleBox["Mathematica\[RegisteredTrademark]", FontWeight->"Bold", FontSlant->"Italic"], " Version 5.01 \n\[Bullet]\tPrerequisite knowledge the inverse of a \ matrix..\n\[Bullet]\tJPEG2000 (Joint Photographic Experts Group) compresses \ data using wavelets.\n\[Bullet]\tAdvantages are that the data files are 20% \ smaller and flexible image size without creating a separate data file\n\ \[Bullet]\tData transformation produces an equivalent set of data.\n\[Bullet]\ \tData compression may give a good approximation of the original data" }], "Text"], Cell[TextData[{ StyleBox["General ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" usage instrucitons", FontWeight->"Bold"], ":\n1. Double click each collapsed section below to open.\n2.Hit the Enter \ key near the cursor keys or use Shift Return to activate each cell.)" }], "Text", FontSize->14, Background->RGBColor[1, 1, 0]] }, Open ]], Cell[CellGroupData[{ Cell["Example 1: Sparse data ", "Subsubsection"], Cell[BoxData[ \(\(coins = {{0, 0, 0, 0, 0}, {0, 0, 1, 0, 0}, {4, 2, 1, 0, 1}, {0, 5, 1, 0, 1}, {0, 0, 0, 1, 1}, {0, 0, 15, 95, 10}};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[coins, TableHeadings \[Rule] {{"\", "\", \ "\", "\", "\", "\"}, None}]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"Dollar coin\"\>", "0", "0", "0", "0", "0"}, {"\<\"Half Dollar\"\>", "0", "0", "1", "0", "0"}, {"\<\"Quarter\"\>", "4", "2", "1", "0", "1"}, {"\<\"Dime\"\>", "0", "5", "1", "0", "1"}, {"\<\"Nickel\"\>", "0", "0", "0", "1", "1"}, {"\<\"Penny\"\>", "0", "0", "15", "95", "10"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {{"Dollar coin", "Half Dollar", "Quarter", "Dime", "Nickel", "Penny"}, None}]]]], "Output"] }, Open ]], Cell["\<\ Note that there are a lot of zeros that we could compress \ out.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Example 2: Average-difference representation. ", "Subsubsection"], Cell["We need some statistical commands.", "Text"], Cell[BoxData[ \(<< Statistics`DescriptiveStatistics`\)], "Input", InitializationCell->True], Cell["\<\ We begin with just 2 data values {a,b} and wish to transform it \ into {c,d} where c is the average of a and b while d is the distance from a \ to c. We wish to form a matrix that gives us those values.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t = Array[s, {2, 2}]\)], "Input"], Cell[BoxData[ \({{s[1, 1], s[1, 2]}, {s[2, 1], s[2, 2]}}\)], "Output"] }, Open ]], Cell["This is not the form we expected but we can fix that.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t . {a, b}\ // MatrixForm\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(a\ s[1, 1] + b\ s[1, 2]\)}, {\(a\ s[2, 1] + b\ s[2, 2]\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ a, s[ 1, 1]], Times[ b, s[ 1, 2]]], Plus[ Times[ a, s[ 2, 1]], Times[ b, s[ 2, 2]]]}]]], "Output"] }, Open ]], Cell["\<\ The first row of the product of t and the vector {a,b} is \ found\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((t . {a, b})\)[\([1]\)]\)], "Input"], Cell[BoxData[ \(a\ s[1, 1] + b\ s[1, 2]\)], "Output"] }, Open ]], Cell[TextData[{ "We now assign the values of the first row equal to ", Cell[BoxData[ \(TraditionalForm\`\(\(1\/2\)\(.\)\)\)]] }], "Text"], Cell[BoxData[ \(\(s[1, 1] = \(s[1, 2] = 1/2\);\)\)], "Input"], Cell["\<\ We now recheck what the product of t and {a,b} equals after this \ change.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t . {a, b}\)], "Input"], Cell[BoxData[ \({a\/2 + b\/2, a\ s[2, 1] + b\ s[2, 2]}\)], "Output"] }, Open ]], Cell["\<\ We note that the first row is now the average of a and b. Next we need a variable d that measures the difference of a from the mean \ (average) of a and b.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(d = a - Mean[{a, b}]\)], "Input"], Cell[BoxData[ \(a + 1\/2\ \((\(-a\) - b)\)\)], "Output"] }, Open ]], Cell[TextData[{ "This obviously simplifies so let's try the Mathematica command ", StyleBox["Simplify.", FontWeight->"Bold"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[d]\)], "Input"], Cell[BoxData[ \(\(a - b\)\/2\)], "Output"] }, Open ]], Cell["\<\ We now change to values of the second row so it computes this value \ d from a and b.\ \>", "Text"], Cell[BoxData[{ \(\(s[2, 1] = 1/2;\)\), "\[IndentingNewLine]", \(\(s[2, 2] = \(-1\)/2;\)\)}], "Input"], Cell["Let's see if it worked.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t . {a, b}\)], "Input"], Cell[BoxData[ \({a\/2 + b\/2, a\/2 - b\/2}\)], "Output"] }, Open ]], Cell["\<\ Finally, let's see the matrix version of t after the changes we \ made above.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[t]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/2\), \(1\/2\)}, {\(1\/2\), \(-\(1\/2\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "In order to retreive the data from the encoded form, we need the inverse \ of the transformation ", StyleBox["t.", FontWeight->"Bold"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(int = Inverse[t]\)], "Input"], Cell[BoxData[ \({{1, 1}, {1, \(-1\)}}\)], "Output"] }, Open ]], Cell["Let's check to see if this correct.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(orgin = t . {{1, 1}, {1, \(-1\)}}\)], "Input"], Cell[BoxData[ \({{1, 0}, {0, 1}}\)], "Output"] }, Open ]], Cell["Let's look as this in MatrixForm", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[orgin]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0"}, {"0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["Let's check another way if this correct.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(int . {Mean[{a, b}], a - Mean[{a, b}]}\)], "Input"], Cell[BoxData[ \({a + 1\/2\ \((\(-a\) - b)\) + \(a + b\)\/2, \(-a\) - 1\/2\ \((\(-a\) - b)\) + \(a + b\)\/2}\)], "Output"] }, Open ]], Cell["\<\ Mathematica did not automatically simplify this, so let's do \ it.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[%]\)], "Input"], Cell[BoxData[ \({a, b}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Example 3 4-dimensional data {a,b,c,d}", "Subsubsection"], Cell["\<\ This time we want a 4x4 matrix that t4 that produces the pariwise \ averages and the distances.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t4A = {{1/2, 1/2, 0, 0}, {0, 0, 1/2, 1/2}, {1/2, \(-1\)/2, 0, 0}, {0, 0, 1/2, \(-1\)/2}}; MatrixForm[t4A]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/2\), \(1\/2\), "0", "0"}, {"0", "0", \(1\/2\), \(1\/2\)}, {\(1\/2\), \(-\(1\/2\)\), "0", "0"}, {"0", "0", \(1\/2\), \(-\(1\/2\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["Let's check to see if this correct.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Clear[a, b, c, d]; rec = t4A . {a, b, c, d}\)], "Input"], Cell[BoxData[ \({a\/2 + b\/2, c\/2 + d\/2, a\/2 - b\/2, c\/2 - d\/2}\)], "Output"] }, Open ]], Cell["Let's see this in MatrixForm", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[rec] // MatrixForm\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(a + b\)\/2\)}, {\(\(c + d\)\/2\)}, {\(\(a - b\)\/2\)}, {\(\(c - d\)\/2\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Times[ Rational[ 1, 2], Plus[ a, b]], Times[ Rational[ 1, 2], Plus[ c, d]], Times[ Rational[ 1, 2], Plus[ a, Times[ -1, b]]], Times[ Rational[ 1, 2], Plus[ c, Times[ -1, d]]]}]]], "Output"] }, Open ]], Cell["\<\ Next we want another transformation that forms the average of top \ two positions and the distance of the top data value from this new average. \ We will leave the last two positions alone.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t4B = {{1/2, 1/2, 0, 0}, {1/2, \(-1\)/2, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}; MatrixForm[t4B]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/2\), \(1\/2\), "0", "0"}, {\(1\/2\), \(-\(1\/2\)\), "0", "0"}, {"0", "0", "1", "0"}, {"0", "0", "0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["Let's check to see what this does..", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t4B . t4A . {a, b, c, d}\)], "Input"], Cell[BoxData[ \({a\/4 + b\/4 + c\/4 + d\/4, a\/4 + b\/4 - c\/4 - d\/4, a\/2 - b\/2, c\/2 - d\/2}\)], "Output"] }, Open ]], Cell["Let's see it in MatrixFrom", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[%]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(a\/4 + b\/4 + c\/4 + d\/4\)}, {\(a\/4 + b\/4 - c\/4 - d\/4\)}, {\(a\/2 - b\/2\)}, {\(c\/2 - d\/2\)} }], "\[NoBreak]", ")"}], MatrixForm[ { Plus[ Times[ Rational[ 1, 4], a], Times[ Rational[ 1, 4], b], Times[ Rational[ 1, 4], c], Times[ Rational[ 1, 4], d]], Plus[ Times[ Rational[ 1, 4], a], Times[ Rational[ 1, 4], b], Times[ Rational[ -1, 4], c], Times[ Rational[ -1, 4], d]], Plus[ Times[ Rational[ 1, 2], a], Times[ Rational[ -1, 2], b]], Plus[ Times[ Rational[ 1, 2], c], Times[ Rational[ -1, 2], d]]}]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Example 3 data", "Subsubsection"], Cell[BoxData[ \(\(data = {37, 33, 6, 16};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(trans = t4B . t4A . data\)], "Input"], Cell[BoxData[ \({23, 12, 2, \(-5\)}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Compression -- threshold number \[Epsilon]\ \>", "Subsubsection"], Cell["\<\ One simple compression is to let the data positions whose abosolute \ value is below a certain value \[Epsilon] to made equal to 0.\ \>", "Text"], Cell["\<\ For example, if the threshold value is 3 for the data above. then \ the position with 2 in it is set to zero so that the data cna be compressed. \ \ \>", "Text"], Cell[BoxData[ \(compress[m_, e_] := Floor[m/e]*e\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(compress[trans, 3]\)], "Input"], Cell[BoxData[ \({21, 12, 0, \(-6\)}\)], "Output"] }, Open ]], Cell[BoxData[ \(compres2[m_, e_] := Block[{n}, n = m; Do[If[Abs[n[\([i]\)]] <= e, n[\([i]\)] = 0, n[\([i]\)] = m[\([i]\)]], {i, 1, Length[m] - 1}]; n]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(w = compres2[trans, 3]\)], "Input"], Cell[BoxData[ \({23, 12, 0, \(-5\)}\)], "Output"] }, Open ]], Cell["\<\ We need the inverse matrices for t4A and t4B to convert it back \ into the approximation of the original data.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(invtA = Inverse[t4A]; MatrixForm[invtA]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "1", "0"}, {"1", "0", \(-1\), "0"}, {"0", "1", "0", "1"}, {"0", "1", "0", \(-1\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(invtB = Inverse[t4B]; invtB // MatrixForm\)], "Input"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(invtB\\)\\\" is similar to existing symbol \ \\\"\\!\\(invtA\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "1", "0", "0"}, {"1", \(-1\), "0", "0"}, {"0", "0", "1", "0"}, {"0", "0", "0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ Finally we 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