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"An Introduction to Wavelet Transforms and Data Compression Using ",
StyleBox["Mathematica",
FontSlant->"Italic"],
"."
}], "Subtitle",
FontSize->12],
Cell["\<\
Alkiviadis G. Akritas
University of Thessaly
Department of Computer and Communication Engineering
GR-38221, Volos
Greece\
\>", "Subsubtitle",
FontSize->12],
Cell[CellGroupData[{
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FontSize->12],
Cell["\<\
Wavelet Transforms trace their origin both to Signal Processing and \
Theoretical Mathematics. Since their introduction they have found \
applications in many areas \[LongDash] most notably by the FBI, where they \
are used to compress fingerprint data before storing it. We present an \
introduction to these transforms and their application and demonstrate the \
main ideas with a picture of Pedro. The book by George Nakos and the papers \
by Colm Mulcahy and Gilbert Strang were inspirational.\
\>", "Text"]
}, Open ]],
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FontSize->12],
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"Plotting various functions with a computer algebra system like ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" is an activity quite similar to signal processing; to wit, in both cases \
we take samples. In this section we review standard ways of plotting \
discrete data sets and two dimensional digital images. The inherent \
difficulties of plotting functions by uniform sampling will lead us to \
adoptive plotting techniques (the main idea of which is at the heart of \
wavelet transforms) and to techniques based on wavelets (see the section on \
Image Compression with the Haar Transform). We also indicate the need for \
data compression."
}], "Text"],
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FontSize->12],
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to plot f(x) from this set of points is to plot these points individually and \
then join adjacent points with line segments. This can be achieved with the \
",
StyleBox["Mathematica",
FontSlant->"Italic"],
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"ListPlot \[LongDash]",
StyleBox[" ",
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" "
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Cell[TextData[{
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StyleBox["Mathematica",
FontSlant->"Italic"],
" used \[LongDash] in their ",
"Plot",
" function \[LongDash] a fixed number of (just 25) points ",
Cell[BoxData[
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below we see a \"darn good\" plot of sin(x) using ",
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"."
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Consider, first, the function f(x) = x + sin(2\[Pi]x), in the interval 0 \
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discrete samples are the same.",
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The reason for aliasing is that we did ",
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As a second example of aliasing consider the function sin(x), in the \
(storied) interval 0 \[LessEqual] x \[LessEqual] 48.01\[Pi]. Its true nature \
is shown in Figure 3. \
\>", "Text"],
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However, using only 25 uniformly sampled points in the interval [ 0, 48.01\
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A final example of aliasing is obtained when we try to plot the function \
sin(90x) using only 32 uniformly spaced sample points. Instead of a \
horizontally telescoped version of the sin(x) function we obtain the pulsing \
pattern shown in Figure 5.\
\>", "Text"],
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\(\(ListPlot[Thread[{xPoints3, yPoints3}], PlotJoined \[Rule] True,
PlotLabel \[Rule] "\"];\)\), "\[IndentingNewLine]",
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Cell[BoxData[
\("Aliased version of the function Sin[90x] in the interval [0, 1]; 32 \
uniformly sampled points were used"\)], "Print"]
}, Open ]],
Cell[TextData[{
"Some definitions are needed now in order to state the famous theorem \
explaining alias. Consider the sinusoid x(t) = sin(\[Omega]t). This repeats \
whenever the time t is increased by T = ",
Cell[BoxData[
\(TraditionalForm\`\(2 \[Pi]\)\/\[Omega]\)]],
", its ",
StyleBox["period",
FontSlant->"Italic"],
". Suppose T is measured in seconds. Then the numbers of cycles in one \
second is f = ",
Cell[BoxData[
\(TraditionalForm\`1\/T\)]],
". Thus, the frequency in Hertz is f = ",
Cell[BoxData[
\(TraditionalForm\`1\/T\)]],
" = ",
Cell[BoxData[
\(TraditionalForm\`\[Omega]\/\(2 \[Pi]\)\)]],
". We can now state the following:"
}], "Text"],
Cell[TextData[{
StyleBox["Shannon's Sampling Theorem: ",
FontWeight->"Bold"],
"Every analog signal x(t) with frequencies not exceeding ",
Cell[BoxData[
\(TraditionalForm\`\[Omega]\_max\)]],
" can be perfectly reconstructed from its discrete samples x(nT), provided \
the sampling rate ",
Cell[BoxData[
\(TraditionalForm\`1\/T\)]],
" exceeds ",
Cell[BoxData[
\(TraditionalForm\`2 f\_max\)]],
" = ",
Cell[BoxData[
FormBox[
FractionBox[
FormBox[\(\[Omega]\_max\),
"TraditionalForm"], "\[Pi]"], TraditionalForm]]],
" (Nyquist rate)."
}], "Text"],
Cell["\<\
Note that the critical Nyquist rate has two samples per oscillation! Also, \
reconstruction requires a strict inequality -- the Nyquist rate cuts it too \
close. \
\>", "Text"],
Cell[TextData[{
"According to Shannon's theorem, the first two examples fail because we \
have one sample per oscillation. In the third example, we observe that we \
have enough sample points for the function sin(90), since 32 > 2",
Cell[BoxData[
\(TraditionalForm\`f\_max\)]],
" = ",
Cell[BoxData[
\(TraditionalForm\`\[Omega]\_max\/\[Pi]\)]],
" = ",
Cell[BoxData[
\(TraditionalForm\`90\/\[Pi]\)]],
" = 28.6479 (Nyquist rate), but nonetheless aliasing appears. This is not \
unusual, and using a few more points yields the anticipated plot. "
}], "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell["Adoptive plotting", "Subsection",
FontSize->12],
Cell["\<\
From the above examples it is obvious that uniformly spaced sample points is \
not the right way to go, unless we are prepared to use a lot of points; this \
is done in the next section where we use a large number of uniformly spaced \
sample points together with wavelet transforms for \"data compression\". \
\>", "Text"],
Cell["\<\
To correct the errors described in the above examples, adoptive plotting (or \
sampling) techniques are used with points clustered where the function \
exhibits great variation. These routines initially generate an internal plot \
based on uniform sampling and they examine the angles between the connecting \
line segments in order to identify regions of great variation. Having \
identified regions of great variation, they next subdivide the corresponding \
intervals further before producing the visible plot.\
\>", "Text"],
Cell[TextData[{
"This is precisely what happens when the current version ",
StyleBox["Mathematica ",
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Cell[BoxData[
\("Step function version of the function Sin[6\[Pi]x] in the interval [0, \
1]; 32 sample points were used"\)], "Print"]
}, Open ]],
Cell["\<\
This should come as no surprise since we know from Calculus that continuous \
functions on [0, 1] can be approximated to any degree of precision with \
linear functions or by step functions. These staircase approximations lead to \
better and better plots as the number of sample points increases, although a \
lot more are needed to obtain the continuous effect. In this context \
questions of data storage and transmission arise. These questions become \
crucial when we explore digital images which are higher-dimensional analogues \
of data points in the plane.\
\>", "Text"],
Cell[TextData[{
"Black and white digital images are derived from two-dimensional arrays of \
",
StyleBox["pixels. ",
FontSlant->"Italic"],
"A pixel is the smallest unit of space in a digital image. Associated with \
each pixel there is a number, in the range [0, 1], representing gray levels \
ranging from black (0) to white (1). So now our data points are triplets {x, \
y, z}, where z measures the gray level at pixel position {x, y}. Viewed from \
above we have two-dimensional step functions, where the steps are shaded \
according to their height. (Color images can be dealt with by decomposing \
them into their red, green and blue components and treating each of these \
like grayscales.) When 256 (= ",
Cell[BoxData[
\(TraditionalForm\`2\^8\)]],
") levels of gray are used, we are talking about 8-",
StyleBox["bit images.",
FontSlant->"Italic"],
"\n\nDigital images come in various resolutions. An image with resolution \
1600x1200 pixels is derived from a matrix of 1600x1200 = 1,920,000 pieces of \
data, each representing a gray level. Clearly, images with such high \
resolution are difficult to store and time-consuming to transmit over low \
band line. Therefore, we should try to come up with a more economical way to \
store the matrices that represent these images. This is achieved by taking \
advantage of regions in the image where there is little or no variation, (for \
example, a black coat in the picture). "
}], "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["Signals and Wavelet Transforms", "Section",
FontSize->12],
Cell[TextData[{
"Most signals start their lives in analog form. They become digital by \
sampling at equal time intervals, ",
Cell[BoxData[
\(TraditionalForm\`\(x\_digital\)(n)\ = \ \(x\_analog\)(nT)\)]],
", where n = 0, \[PlusMinus]1, \[PlusMinus]2, ... and T is the ",
StyleBox["sampling interval.",
FontSlant->"Italic"]
}], "Text"],
Cell[TextData[{
"The Discrete Time Fourier Transform turns the samples ",
Cell[BoxData[
\(TraditionalForm\`\(x\_digital\)(n)\ = \ x(n)\)]],
" into the coefficients of a 2\[Pi] - periodic function \[Chi](\[Omega]) = \
",
Cell[BoxData[
\(TraditionalForm\`\[Sum]\+\(n = \(-\[Infinity]\)\)\%\[Infinity]\( x(
n)\)\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ n\ \[Omega]\)\)]],
". We refer to \[Omega] as the ",
StyleBox["frequency",
FontSlant->"Italic"],
", and we plot the transform \[Chi](\[Omega]) in the interval \[Omega] = -\
\[Pi] and \[Omega] = \[Pi]. Then \"low frequencies\" refer to frequencies \
near zero, and \"high frequencies\" have \[VerticalSeparator]\[Omega]\
\[VerticalSeparator] \[TildeTilde] \[Pi]."
}], "Text"],
Cell[TextData[{
"Notice that the signal ",
Cell[BoxData[
\(TraditionalForm\`x(n)\)]],
" = {..., 1, 1, 1, 1, 1, ...} has exactly zero frequency (\[Omega] = 0, the \
lowest frequency), whereas the alteration between 1 and -1 gives the signal \
",
Cell[BoxData[
\(TraditionalForm\`x(n)\)]],
" = {..., 1, -1, 1, -1, 1, ...} with the highest frequency \[Omega] = \
\[PlusMinus]\[Pi]. In between these two extremes we have the family of pure \
sinusoidal signals ",
Cell[BoxData[
\(TraditionalForm\`x(n)\)]],
" = ",
Cell[BoxData[
\(TraditionalForm\`\[ExponentialE]\^\(\[ImaginaryI]\ n\ \[Omega]\)\)]],
" with frequency 0 < \[VerticalSeparator]\[Omega]\[VerticalSeparator] < \
\[Pi]."
}], "Text"],
Cell[CellGroupData[{
Cell["Lowpass and Highpass Filters", "Subsection",
FontSize->12],
Cell[TextData[{
StyleBox["Sampling",
FontSlant->"Italic"],
", ",
StyleBox["transforming",
FontSlant->"Italic"],
" and ",
StyleBox["filtering ",
FontSlant->"Italic"],
"constitute the basic signal processing operations. ",
"Of the three, filtering (and filters) will be of importance to us."
}], "Text"],
Cell[TextData[{
"Filters are also called ",
StyleBox["linear time-invariant ",
FontSlant->"Italic"],
"systems. When presented with the input ",
Cell[BoxData[
\(TraditionalForm\`\(x\_\[Omega]\)(n)\)]],
" these systems produce the output ",
Cell[BoxData[
\(TraditionalForm\`\(H(\[Omega])\) \(\(x\_\[Omega]\)(n)\)\)]],
", where the amplifying factor ",
Cell[BoxData[
\(TraditionalForm\`H(\[Omega])\)]],
" is the ",
StyleBox["frequency response.",
FontSlant->"Italic"],
" Hence, filters act on signals to produce modified signals and this is \
the most important operation in signal processing. Below we examine filters \
that are: (a) ",
StyleBox["finite impulse response",
FontSlant->"Italic"],
" (FIR), meaning each output is a linear combination of a ",
StyleBox["finite",
FontSlant->"Italic"],
" number of input samples, (b) ",
StyleBox["time-invariant, ",
FontSlant->"Italic"],
"meaning their coefficients are constant for all time, and (c) ",
StyleBox["causal",
FontSlant->"Italic"],
", meaning they involve no future samples like ",
Cell[BoxData[
\(TraditionalForm\`x(n + 1)\)]],
". (Note that for images the situation is different because ",
StyleBox["n ",
FontSlant->"Italic"],
"refers to to position, not time. Therefore, once the complete image is \
available, filters in image processing need not be causal.)"
}], "Text"],
Cell[TextData[{
StyleBox["Lowpass Filter: ",
FontWeight->"Bold"],
"The first example is the moving average filter"
}], "Text"],
Cell[BoxData[
\(yLP[n_] :=
1\/2\ x\[LeftDoubleBracket]n - 1\[RightDoubleBracket] +
x\[LeftDoubleBracket]n\[RightDoubleBracket]\/2\)], "Input"],
Cell[TextData[{
"We are going to consider three special inputs, the impulse, the constant \
and the alternating signal. For the impulse ",
Cell[BoxData[
\(TraditionalForm\`x(n)\)]],
" = {..., 0, 0, 1 ,0 , 0, ...} the output is ",
Cell[BoxData[
\(TraditionalForm\`\(y\_IM\)(n)\)]],
" = {..., 0, 0, ",
Cell[BoxData[
\(TraditionalForm\`1\/2\)]],
", ",
Cell[BoxData[
\(TraditionalForm\`1\/2\)]],
", 0, ...}; that is, the impulse response contains the filter \
coefficients."
}], "Text"],
Cell[CellGroupData[{
Cell[BoxData[
\(x = Insert[Table[0, {n, 1, 19}], 1, 10]\)], "Input"],
Cell[BoxData[
\({0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0}\)], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
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}], "Text"],
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input signal ",
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Cell[BoxData[
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" = ",
Cell[BoxData[
\(TraditionalForm\`\(1\/2\) \[ExponentialE]\^\(\[ImaginaryI]\ n\ \
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\[Omega]\)\ = \ \((1\/2\ + \ \(1\/2\) \[ExponentialE]\^\(-\[ImaginaryI]\
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StyleBox["frequency response",
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Cell[BoxData[
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Cell[TextData[{
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the constant signal passes unchanged through the filter. At \[Omega] = \[Pi] \
the frequency response is H = 0; that is, the alternating signal is \
completely blacked out. Writing the response function as \n\t\n\t",
Cell[BoxData[
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\(TraditionalForm\`\(\(=\)\(\ \)\(\(\[ExponentialE]\^\(\(-\[ImaginaryI]\
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\ + \ \(1\/2\) \[ExponentialE]\^\(\(-\[ImaginaryI]\[Omega]\)/2\))\ = \ \(\(\
\[ExponentialE]\^\(\(-\[ImaginaryI]\[Omega]\)/
2\)\) \(cos\)\(\ \)\(\[Omega]\/2\)\(\ \)\)\)\)\)]],
"\n\t\nwe see that the amplitude is ",
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}], "Text"],
Cell[TextData[{
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FontWeight->"Bold"],
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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change."
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\(TraditionalForm\`x(n)\)]],
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Cell[BoxData[
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" = ",
Cell[BoxData[
\(TraditionalForm\`\(1\/2\) \[ExponentialE]\^\(\[ImaginaryI]\ n\ \
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\[Omega]\)\ = \ \((1\/2\ - \ \(1\/2\) \[ExponentialE]\^\(-\[ImaginaryI]\
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Cell[BoxData[
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Cell[TextData[{
"\t\t",
Cell[BoxData[
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\ - \ \(1\/2\) \[ExponentialE]\^\(\(-\[ImaginaryI]\[Omega]\)/2\))\ = \ \(\(\
\[ExponentialE]\^\(\(-\[ImaginaryI]\[Omega]\)/
2\)\) \(\[ImaginaryI]\)\(\ \)\(sin\)\(\ \)\(\[Omega]\/2\)\(\
\ \)\)\)\)\)]],
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Cell[BoxData[
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Cell[TextData[{
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" part of a signal, whereas a highpass filter preserves the ",
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FontSlant->"Italic"],
"part. In wavelet language, lowpass gives averages and highpass gives \
details. In some applications those details are important to keep (like \
edges in an image) whereas in other applications the high frequencies are \
mostly noise from measurements.",
" "
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Cell[CellGroupData[{
Cell["Convolutions", "Subsection",
FontSize->12],
Cell[TextData[{
"A good lowpass filter has many uses, so we look now at the (FIR, linear \
time-invariant and causal) filter with ",
StyleBox["N+",
FontSlant->"Italic"],
"1",
" coefficients ",
Cell[BoxData[
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FontSlant->"Italic"],
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FontSlant->"Italic"],
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
\(TraditionalForm\`N + 1\)]],
" samples from the input signal \[Dash] starting with the current sample ",
Cell[BoxData[
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Cell[BoxData[
\(TraditionalForm\`\(\(x(n - N)\)\(\ \)\(.\)\)\)]],
" Hence, the putput is:\n\n\t\t",
Cell[BoxData[
\(TraditionalForm\`y(
n)\ = \ \(h(0)\) \(x(n)\)\ + \ \(h(1)\) \(x(
n - 1)\)\ + \[CenterEllipsis] + \(h(
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"\n\t\t\nWe may write the action of the filter in the time domain compactly \
as the sum:\n\n\t\t",
Cell[BoxData[
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FontSlant->"Italic"],
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FontSlant->"Italic"],
"This convolution is easily implemented in hardware using three building \
blocks: ",
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FontSlant->"Italic"]
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FontWeight->"Bold"],
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StyleBox["Mathematica ",
FontSlant->"Italic"],
"(see also the Help Browser for this function). For example, the filter ",
Cell[BoxData[
\(TraditionalForm\`h\ = \ {1\/4, \ 1\/2, \ 1\/4}\)]],
" is lowpass, since the alternating signal \[Omega] = \[Pi] is eliminated \
whereas the constant signal is preserved. (Note that the filter coefficients \
sum up to 1.)"
}], "Text"],
Cell[BoxData[
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x[\([n - 2]\)]\)], "Input"],
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Note again how the impulse response contains the filter coefficients.\
\>", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
The interesting part starts when the input is the linear signal x = {0, 1, 2, \
3, 4}. Padding the signal with two zeros on both ends we have:\
\>", "Text"],
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Cell[BoxData[
\(x = {0, 0, 0, 1, 2, 3, 4, 0, 0}; Table[yLP[n], {n, 3, 9}]\)], "Input"],
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}, Open ]],
Cell["\<\
Note that the signal is preserved at the center! Moreover, the result is the \
same as the one obtained using ListConvolve!\
\>", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
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Cell[TextData[{
"bur we observe that a unit ",
StyleBox["delay ",
FontSlant->"Italic"],
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with three terms produce seven terms. We convolved an input signal of length \
",
Cell[BoxData[
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", with a second order filter, ",
Cell[BoxData[
\(TraditionalForm\`N\ = \ 2\)]],
". The output is of length ",
Cell[BoxData[
\(TraditionalForm\`L\ + \ N\ = \ 7\)]],
", which is reasonable given that a fourth degree polynomial, ",
Cell[BoxData[
\(TraditionalForm\`0 + x + 2 x\^2 + 3 x\^3 + 4 x\^4\)]],
", times a second degree polynomial, ",
Cell[BoxData[
\(TraditionalForm\`1\/4 + \(1\/2\) x + \(1\/4\) x\^2\)]],
" yields a sixth degree polynomial, ",
Cell[BoxData[
\(TraditionalForm\`0 + \(1\/4\) x + x\^2 + 2 x\^3 +
3 x\^4 + \(11\/4\) x\^5 + x\^6\)]],
"."
}], "Text"],
Cell[TextData[{
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FontSlant->"Italic"],
" and padded with ",
Cell[BoxData[
\(TraditionalForm\`N\ = \ 2\)]],
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Starting from the right end we form the dot products ",
Cell[BoxData[
\(TraditionalForm\`{1\/4, \ 1\/2, \ 1\/4}\)]],
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Cell[BoxData[
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".{1, 0, 0} = ",
Cell[BoxData[
\(TraditionalForm\`1\/4\)]],
", ",
Cell[BoxData[
\(TraditionalForm\`{1\/4, \ 1\/2, \ 1\/4}\)]],
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Cell[BoxData[
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Cell[BoxData[
\(TraditionalForm\`{1\/4, \ 1\/2, \ 1\/4}\)]],
".{4, 3, 2} = 3, ",
Cell[BoxData[
\(TraditionalForm\`{1\/4, \ 1\/2, \ 1\/4}\)]],
".{0, 4, 3} = ",
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Cell[BoxData[
\(TraditionalForm\`{1\/4, \ 1\/2, \ 1\/4}\)]],
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polynomial. The thing to note here is that inversion is associated with \
convolution!"
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Cell[TextData[{
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" and ",
Cell[BoxData[
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" on the left and ",
Cell[BoxData[
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" and ",
Cell[BoxData[
\(TraditionalForm\`x(6)\)]],
" for ",
Cell[BoxData[
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Cell[BoxData[
\(TraditionalForm\`y(6)\)]],
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Cell[TextData[{
"The above remarks apply when the input is the sinusoidal signal x = {0, 1, \
0, -1, 0} with frequency \[Omega] = ",
Cell[BoxData[
\(TraditionalForm\`\[Pi]\/2\)]],
". We observe that it, too, is preserved with a unit delay between input \
and output; additionally, the amplitude of the output has changed to ",
Cell[BoxData[
\(TraditionalForm\`1\/2\)]],
". "
}], "Text"],
Cell[CellGroupData[{
Cell[BoxData[
\(x = {0, 0, 0, 1, 0, \(-1\), 0, 0, 0};
Table[yLP[n], {n, 3, 9}]\)], "Input"],
Cell[BoxData[
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Cell["or", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
\(ListConvolve[{1\/4, 1\/2, 1\/4}, {0, 1, 0, \ \(-1\), \ 0}, {1, \(-1\)},
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Cell[BoxData[
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Cell[TextData[{
"To see that the amplitude is reduced by a half let us look at the \
frequency response function ",
Cell[BoxData[
\(TraditionalForm\`H(\[Omega])\ = \
H(\[ExponentialE]\^\[ImaginaryI]\[Omega])\)]],
". For the filter under consideration we have \n\n\t\t",
Cell[BoxData[
\(TraditionalForm\`H(\[Omega])\ = \
1\/4 + \(1\/2\) \[ExponentialE]\^\(-\[ImaginaryI]\[Omega]\) + \
\(1\/4\) \[ExponentialE]\^\(\(-2\) \[ImaginaryI]\[Omega]\)\)]],
".\n\t\t\nEvaluating at \[Omega] = ",
Cell[BoxData[
\(TraditionalForm\`\(\(\[Pi]\/2\)\(\ \)\)\)]],
"we obtain ",
Cell[BoxData[
\(TraditionalForm\`H\ = \ \(\(-1\)\/2\) \[ImaginaryI]\)]],
", from which we have that the magnitude of the frequency is ",
Cell[BoxData[
\(TraditionalForm\`\(\(1\/2\)\(.\)\)\)]],
" (Recall that from Fourier transforms we have that convolutions in the \
time domain correspond to multiplications in the frequency domain.)"
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}, Closed]],
Cell[CellGroupData[{
Cell["Wavelet Transforms", "Subsection",
FontSize->12],
Cell[TextData[{
"As we said, lowpass filters greatly reduce the high frequency components; \
when these components represent noise in signals, this reduction is the right \
thing to do. However, if we want to ",
StyleBox["reconstruct",
FontSlant->"Italic"],
" the signal, then we use two filters, a highpass as well as a lowpass \
filter. This generates a ",
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"structure",
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FontSlant->"Italic"],
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FontSlant->"Italic"],
" transform",
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FontSlant->"Italic"],
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filters that are fast to compute. Between the DWT and the IDWT compression \
and trasmission of the signal can take place, and this is the key to more and \
more applications."
}], "Text"],
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"Wavelet transforms are associated with ",
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FontSlant->"Italic"],
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FontSlant->"Italic"],
" a signal, to wit keeping only its even-numbered components ",
Cell[BoxData[
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\[DownArrow]2)",
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\n\n\t\t\t\t ",
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length ",
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four or five times. \n",
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coefficients of the highpass filter are small enough and not worth the \
additional effort to run them through another filter bank structure. They \
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"The Haar Wavelet Transform and Its ",
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Cell["\<\
Below we choose the simplest of the wavelet transforms, named after Alfred \
Haar. It naturally combines the simplest examples of lowpass and highpass \
filters that we have previously seen (to wit, the moving average filter and \
the moving difference filter) into the most basic example of a filter bank. \
\
\>", "Text"],
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Cell[TextData[{
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Cell[BoxData[
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"\tis the differencing filter\t\t",
Cell[BoxData[
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".\n\t\t\nor as we have already defined in ",
StyleBox["Mathematica",
FontSlant->"Italic"],
":"
}], "Text"],
Cell[BoxData[
\(yLP[n_] := \(1\/2\) x[\([n - 1]\)] + \(1\/2\) x[\([n]\)]\)], "Input"],
Cell[BoxData[
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Cell["\<\
Let the signal be the vector x = {7, 5, 6, 2}. To take care of the left end, \
we prepend 0 to the vector x.\
\>", "Text"],
Cell[BoxData[
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Cell["\<\
We next feed this vector to the \"first stage\" of the Wavelet Transform \
Structure to obtain the following outputs. From the lowpass filter we obtain \
the averages:\
\>", "Text"],
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Cell[BoxData[
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Cell[BoxData[
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Cell["whereas from the highpass filter we obtain the differences:", "Text"],
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Cell["\<\
Notice that the differences tend to be smaller than the averages. For smooth \
inputs this is even more true. \
\>", "Text"],
Cell[TextData[{
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Cell[BoxData[
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". They are redundant, since they came from only four samples of the \
vector x. We now downsample and keep the even-numbered components."
}], "Text"],
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Cell["\<\
This is the first step in our analysis. We have found the four \"first-level\
\" wavelet coefficients of the signal, or we have separated the signal into \
wavelets. As we will see in the synthesis, the inverse transform uses these \
coefficients (that is, adds up wavelets times coefficients) to reconstruct \
the signal vector x. \
\>", "Text"],
Cell[TextData[{
"The thing to note here is that the coefficients {6, 4} can be obtained by \
splitting the signal input vector into pairs x = {{7, 5},{6, 2}} and taking \
the average of each pair; i.e. 6 = ",
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Cell[BoxData[
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Cell[BoxData[
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". This observation is fully exploited in programming the Haar transform."
}], "Text"],
Cell["\<\
Now the downsampled output {6, 4} of the lowpasss filter becomes our new \
input vector x into the \"second stage\" of the Wavelet Transform Structure. \
Proceeding in a similar fashion we obtain the following outputs.\
\>", "Text"],
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Cell[CellGroupData[{
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" and 1 = ",
Cell[BoxData[
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". This completes the iteration of the Haar analysis bank, which is \
graphically represented below:\n\n\t\t\t\t\t\t\t\t\t \t\t",
Cell[BoxData[
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Cell[BoxData[
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"\n\t\t\n\t\tFigure 9: Haar analysis bank.\n\t\t\t\t\n",
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": For the input vector x = {7, 5, 6, 2} the wavelet coefficients are {5, \
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analysis. The sequence of inverse operations is ",
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"To recover (with a unit delay) the input vector x = {7, 5, 6, 2} from the \
wavelet coefficients {5, 1, 1, 2} we proceed as follows: In the first stage \
the inputs are the coefficients {5} and {1} which were computed last in the \
analysis process; we upsample them, and run them through the lowpass and \
highpass filters ",
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StyleBox["Mathematica",
FontSlant->"Italic"],
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Cell["\<\
In this implementation we do not use down/up-sampling. Instead we take \
advantage of the speedups discussed in the analysis/synthesis and compute \
averages and differences.\
\>", "Text"],
Cell[BoxData[{
\(\(discreteWaveletTransform::usage = \*"\"\0. If this is not the case, zeros are appended to fix the \
situation. The averaging-differencing algorithm (LowPass/HighPass Filtering) \
is performed on the (new) list. After i (or i+1)iterations the output is a \
list of the same length.\>\"";\)\[IndentingNewLine]\[IndentingNewLine]\), "\n\
",
\(discreteWaveletTransform[list_List,
flag_Integer: 0] := \[IndentingNewLine]Module[{tempList = list,
list2, tempInteger = \(i = \(Log[2, Length[list]] // N\) //
IntegerPart\), pairedRow, averages, differences, fl = flag,
nextRow, temp}, \[IndentingNewLine]If[i \[Equal] 0,
Print[\*"\"\0\>\""]; Return[Null]]; \[IndentingNewLine]If[
2\^i \[NotEqual] \ Length[list],
list2 = Append[tempList,
Table[0, {i, 1, 2\^\(i + 1\) - Length[list]}]] // Flatten;
tempInteger = \(i = i + 1\); tempList = list2]; nextRow = {};
If[fl \[Equal] 1, \ Print[tempList // N]]; \[IndentingNewLine]While[
tempInteger \[NotEqual] 0, \[IndentingNewLine]pairedRow =
pairedList[tempList]; averages = Apply[aver, pairedRow, 1];
differences = diff[pairedRow, averages];
nextRow = Prepend[nextRow, {averages, differences}] // Flatten; \
If[fl \[Equal] 1, \
Print[nextRow]]; \[IndentingNewLine]tempInteger =
tempInteger - 1;
temp = 2^tempInteger; \[IndentingNewLine]tempList =
Drop[nextRow, \(-2^i\) + temp]; \[IndentingNewLine]nextRow =
Drop[nextRow, temp];\[IndentingNewLine]];
If[fl \[Equal] 1, \(Prepend[nextRow, First[tempList]];\), \
Prepend[nextRow, First[tempList]]]\[IndentingNewLine]]\)}], "Input"],
Cell[BoxData[
\(\(\( (*\ \ \ \ \ \ \ \ \ \ 2
nd\ function\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
*) \)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\(\(\
discreteInverseWaveletTransform::usage = \*"\"\0. If this is not the case, zeros are appended to fix the \
situation. The INVERSE averaging-differencing algorithm (LowPass/HighPass \
Filtering) is performed on the (new) list. After i (or i+1)iterations the \
output is a list of the same length.\>\"";\)\(\n\)
\)\[IndentingNewLine]
discreteInverseWaveletTransform[list_List,
flag_Integer: 0] := \[IndentingNewLine]Module[{tempList = list,
tempList2 = list, list2, av, di, fl = flag,
i = \(Log[2, Length[list]] // N\) // IntegerPart, j, m = 1,
temp = 1}, \[IndentingNewLine]If[i \[Equal] 0,
Print[\*"\"\0\>\""]; Return[Null]]; \[IndentingNewLine]If[
2\^i \[NotEqual] \ Length[list],
list2 = Append[tempList,
Table[0, {i, 1, 2\^\(i + 1\) - Length[list]}]] // Flatten;
i = i + 1;
tempList = \(tempList2 = list2\)]; \[IndentingNewLine]m =
1; \[IndentingNewLine]While[m \[LessEqual] i, j = 0;
k = 1; \[IndentingNewLine]While[k <= temp, \
av = tempList[\([k]\)]; di = tempList[\([k + temp]\)];
tempList2 = ReplacePart[tempList2, av + di, k + j];
tempList2 =
ReplacePart[tempList2, av - di,
k + j + 1]; \[IndentingNewLine]j = j + 1;
k = k + 1\[IndentingNewLine]];
If[fl \[Equal] 1, \ Print[tempList]];
temp = 2*temp; \[IndentingNewLine]tempList = tempList2;
m = m + 1\[IndentingNewLine]];
tempList\[IndentingNewLine]]\)\)\)], "Input"],
Cell[BoxData[
\(\(\( (*\ \ \ \ 3 d\ function; \
computes\ the\ average\ value\ of\ 2\ numbers\ \ \ \ \ *) \)\(\
\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(aver[a_, b_] := \(\(\((a + b)\)/
2. \)\(\[IndentingNewLine]\)
\)\)\)\)], "Input"],
Cell[BoxData[
\(\(\( (*\ \ \ \ 4 th\ function; \
splits\ a\ list\ in\ pairs\ \ \ \ *) \)\(\[IndentingNewLine]\)\(\
\[IndentingNewLine]\)\(\(pairedList[list_] :=
Partition[list, 2]\)\(\[IndentingNewLine]\)
\)\)\)], "Input"],
Cell[BoxData[
\(\(\( (*\ \ \ \ \ \ \ \ \ \ 5
th\ function\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ *) \)\(\n\
\)\(\(\(diff::usage = "\";\)\(\
\[IndentingNewLine]\)
\)\[IndentingNewLine]
diff[list1_, list2_] := \[IndentingNewLine]Fold[
Append[#1, #2[\([1, 1]\)] - #2[\([2]\)]] &, {},
Thread[{list1, list2}]]\)\)\)], "Input"],
Cell[TextData[{
StyleBox["Example",
FontWeight->"Bold"],
"."
}], "Text"],
Cell[CellGroupData[{
Cell[BoxData[
\(discreteWaveletTransform[{7, 5, 6, 2}, 1]\)], "Input"],
Cell[BoxData[
\({7.`, 5.`, 6.`, 2.`}\)], "Print"],
Cell[BoxData[
\({6.`, 4.`, 1.`, 2.`}\)], "Print"],
Cell[BoxData[
\({5.`, 1.`, 1.`, 2.`}\)], "Print"]
}, Open ]],
Cell["\<\
5 is the general average and the coefficients 1, 1, 2 are the details. This \
transformation is reversible:\
\>", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
\(discreteInverseWaveletTransform[%, 1]\)], "Input"],
Cell[BoxData[
\({5.`, 1.`, 1.`, 2.`}\)], "Print"],
Cell[BoxData[
\({6.`, 4.`, 1.`, 2.`}\)], "Print"],
Cell[BoxData[
\({7.`, 5.`, 6.`, 2.`}\)], "Output"]
}, Open ]]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["Image Compression with the Haar Transform", "Section",
FontSize->12],
Cell["\<\
We saw that in adoptive plotting of functions few samples are taken when \
there is little or no variation. The same idea will be used in image \
processing to take advantage of regions of little or no color variation (as \
in example in the color of a shirt, pants, hair etc). The Haar wavelet \
transform with the averaging and differencing scheme is ideally suited for \
this purpose and results in a method of image compression (storing much less \
information than would otherwise be necessary). At first though we examine \
some of the advantages of these transforms.\
\>", "Text"],
Cell[CellGroupData[{
Cell["Wavelet Transform Benefits", "Subsection",
FontSize->12],
Cell["\<\
Let us look at the advantages of using the Haar wavelet transform. \
\>", "Text"],
Cell[CellGroupData[{
Cell[BoxData[{
\(\(data = {74, 58, 26, 42, 66, 66, 58, 34};\)\), "\[IndentingNewLine]",
\(discreteWaveletTransform[data]\)}], "Input"],
Cell[BoxData[
\({53.`, \(-3.`\), 16.`, 10.`, 8.`, \(-8.`\), 0.`, 12.`}\)], "Output"]
}, Open ]],
Cell["\<\
53 is the general average and the coefficients -3, 16, 10, 8, -8, 0, 12 are \
the details. We have transformed a set of eight numbers into another set of \
eight numbers and, moreover, this transformation is reversible, as we have \
seen. In doing this transformation our advantage is that we can \"play \
around\" with the detail coefficients. By taking advantage of regions of \
little variation, we can change the set of coefficient details and use this \
new set (with the inverse wavelet transform) to approximate the original one. \
This approximation should be very close to the original.\
\>", "Text"],
Cell["\<\
In the above example, one detail coefficient is zero indicating a region of \
little variation (due to the two adjacent numbers 66). The next smallest \
detail coefficient is -3. We change -3 to 0 and applying the inverse wavelet \
transform we obtain:\
\>", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
\(dataApproximated =
discreteInverseWaveletTransform[{53, 0, 16, 10, 8, \(-8\), 0,
12}]\)], "Input"],
Cell[BoxData[
\({77, 61, 29, 45, 63, 63, 55, 31}\)], "Output"]
}, Open ]],
Cell["\<\
The plot of the original (red) against the approximated (green) set of \
numbers is seen in Figure 11a.\
\>", "Text"],
Cell[CellGroupData[{
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\(\(plot2 =
ListPlot[dataApproximated, PlotJoined \[Rule] True,
AxesOrigin \[Rule] {0, 0}, PlotStyle \[Rule] RGBColor[0, 1, 0],
DisplayFunction \[RuleDelayed] Identity];\)\), "\n",
\(\(Show[plot1, plot2, DisplayFunction \[RuleDelayed] $DisplayFunction, \
PlotLabel \[Rule] "\"];\)\)}], "Input"],
Cell[GraphicsData["PostScript", "\<\
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Cell["\<\
Considering that this is based only on 3 detail coefficients, the \
approximation is surprisingly good!\
\>", "Text"],
Cell["\<\
Overall, this scheme can be applied to a number string of any length. We can \
always pad at the end with zeros until the length of the string is a power of \
two. As we are going to see next, the full potential of this transformation \
comes when we apply it to strings of big length.\
\>", "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell["\<\
Plotting with Wavelet Transforms \[LongDash] Lossless and Lossy Compression \
\
\>", "Subsection",
FontSize->12],
Cell[TextData[{
"The compression process works as follows: We start with a data string of \
any length and using the wavelet transform we obtain another string of the \
same length. We can now use the inverse transform algorithm in two ways: (a) \
We can use all the detail coefficients to obtain the original list in which \
case the transform is called ",
StyleBox["lossless compression. ",
FontSlant->"Italic"],
"That is, we lost nothing. (b) We can define a ",
StyleBox["threshold",
FontSlant->"Italic"],
" value \[Epsilon], replace by zero all the detail coefficients with \
magnitude \[LessEqual] \[Epsilon] and, based on the altered version of the \
transformed string, obtain an approximation of the original string. This is \
referred to as",
" ",
StyleBox["lossy compression. ",
FontSlant->"Italic"],
" The surprising thing is that we can zero a sizable proportion of the \
detail coefficients and still get a decent approximation to the original \
string. "
}], "Text"],
Cell[TextData[{
"Let us apply the above in the function ",
Cell[BoxData[
\(TraditionalForm\`\(E\^\(\(-10\) x\)\) Sin[100 x]\)]],
" which we are going to plot with uniform sampling in the interval [0, 1]. \
We picked this particular function because it has a great region of no \
variation (low activity) as can be seen from Figure 12."
}], "Text"],
Cell[BoxData[
\(\(f[x_] := \(E\^\(\(-20\) x\)\) Sin[100 x];\)\)], "Input"],
Cell[CellGroupData[{
Cell[BoxData[
\(\(Plot[f[x], {x, 0, 1}, PlotRange \[Rule] All,
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We see that the approximation with 256 sample points is excellent, whereas \
the one with 32 sample points is quite poor. In both cases though half of the \
points plotted were essentially wasted!\
\>", "Text"],
Cell[TextData[{
"Consider the string of 256 elements, ",
StyleBox["yPoints6,",
FontSlant->"Italic"],
" used to plot this function. Their values range from -0.39 to 0.73."
}], "Text"],
Cell[CellGroupData[{
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Cell[TextData[{
"After eight cycles of averaging and differencing we get a transformed \
string of the same length, ",
StyleBox["yPoints6WT,",
FontSlant->"Italic"],
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}], "Text"],
Cell[CellGroupData[{
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Cell[BoxData[
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Cell[TextData[{
"We next pick a threshold value of \[Epsilon] = 0.014 and zero all values \
of ",
StyleBox["yPointsWT",
FontSlant->"Italic"],
" (the detail coefficients) \[LessEqual] \[Epsilon]. This way we get an \
altered transformed list, ",
StyleBox["yPoints6WTSparse",
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}], "Text"],
Cell[CellGroupData[{
Cell[BoxData[{
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Cell[BoxData[
\(32\)], "Output"]
}, Open ]],
Cell[TextData[{
"From this sparse list, ",
StyleBox["yPoints6WTSparse",
FontSlant->"Italic"],
", ",
" we obtain an approximation, ",
StyleBox["yPoints6WTApprox",
FontSlant->"Italic"],
",",
" of the original list and we plot it in Figure 15. "
}], "Text"],
Cell[CellGroupData[{
Cell[BoxData[{
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AxesOrigin \[Rule] {0, 0}, PlotRange \[Rule] {\(-1\), 1},
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It should be noted that the threshold values are very low compared to the \
range of values. Yet, despite this fact, we obtain good results using few \
detail coefficients. The reason for this behavior is that there is no \
variation in more than half of the interval of interest. Using 32 detail \
coefficients, out of 256, gives a compression ration of 8:1, whereas using 62 \
gives a compression ratio of 4.1:1. \
\>", "Text"],
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detail coefficients respectively, in sparse strings of 256 elements; however, \
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strings by applying the ",
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\[LongDash] than arbitrary strings of length 256."
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black unless specified otherwise, then we need only say where there is white. \
In the case ",
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We assume that both images of Figures 17 and 18 have resolution \
256\[Cross]256; that is, each image is represented by a 256\[Cross]256 matrix \
with entries 0 or 1. Using the technique of lossy compression we will show \
that to store the first image we need 2 pieces of information whereas for the \
second we need 5. In each case we apply the Haar transform on each row \
separately, and then do the same on each column of the resulting matrix. The \
final result is a new 256\[Cross]256 matrix with an overall average at the \
top left hand corner and a lot of detail coefficients. Regions of little \
variation in the original image manifest themselves as numerous small or zero \
elements in the transformed matrix.\
\>", "Text"],
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We start with the image of Figure 17. Each one of the first 128 rows is made \
up of 128 ones (1\[Rule]white) followed by 128 zeros (0\[Rule]black).\
\>", "Text"],
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When we apply the Haar wavelet transform to each one of these 128 rows the \
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\>", "Text"],
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"Likewise, after the transformation, the ",
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"At this point the columns of the 256\[Cross]256 matrix are all zero except \
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Cell[TextData[{
"Since our wavelet transform is invertible the image of Figure 17 can be \
stored in a 2\[Cross]2 matrix form with only two nonzero pieces of \
information\n\n\t\t",
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Cell["\<\
The image compression process can be summarized as follows: (a) form the \
original matrix, (b) transform all the rows to obtain the transformed matrix, \
(c) to work on the columns of the transformed matrix transpose it to obtain \
the transposed transformed matrix, (d) transform the columns of the \
transformed matrix and transpose it to obtain the final matrix, (e) print the \
first few rows and columns.\
\>", "Text"],
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Cell["Compression of the Image of Figure 18", "Subsubsection"],
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The reader is encouraged to apply the same process to the image of Figure 18. \
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\>", "Text"],
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Compression of arbitrary images (Thanks to my brother M. G. Akritas for \
suggesting this section) \
\>", "Subsection",
FontSize->12],
Cell["\<\
To process an arbitrary digital black-and-white image we have first to get a \
digital version of it. For this purpose, we first read off as a string the \
contents of the file containing the encapsulated postscript form of a picture \
\
\>", "Text"],
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and get rid off of the header and trailer information of the file. Notice \
the length of the string with only image information. (In our case it is \
314!)\
\>", "Text"],
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\( (*\ \ For\ some\ versions\ of\ Mma\ we\ have\ to\ use\ *) \)], "Input"],
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Convert the string into a character list, and then partition it in pairs\
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Turn each character pair, say {7, A}, into the string form \"16^^7A\" and \
that into an expression; this gives us a list of numbers in the range [0, \
255], since (the maximum) 16^^FF is 255. The length of the partitioned list \
is half the length of the original character list.\
\>", "Text"],
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results in the number of rows being ",
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example)! This is achieved by trial and error!"
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We next \"massage\" the resulting list into a square one. In the case of our \
314x164 matrix we remove the excess 58 (= 314-256) rows (the rows can be \
removed from the top or from the bottom \[LongDash] chosen here \[LongDash] \
or as the picture dictates), and since each row has only 164 columns we pad \
them with \"white columns\" accordingly (46 on the left and 46 on the right). \
\
\>", "Text"],
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