a brief introduction to wavelets - zimmer web...

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Fourier Series and Fourier TransformWindowed Fourier Transform

The Wavelet Transform

A Brief Introduction to Wavelets

Dr. Doreen De LeonMATH 191T, Spring 2019

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Outline

1 Motivation

2 Fourier Series and Fourier Transform

3 Windowed Fourier Transform

4 The Wavelet TransformContinuous Wavelet TransformDiscrete Wavelet Transform and MRA

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Motivation

Wavelets have a number of applications across diverse fields,including:

quantum physics,geology,electrical engineering, andastronomy.

Specific examples of wavelet usage:denoising noisy data; andimage compression.

Wavelets are needed to solve problems that cannot be handledwith “classical” methods.

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In the Beginning

Define the set of functions on R that satisfy∫ ∞−∞|f (x)|2 dx <∞

as L2(R).So, L2(R) is the set of square integrable functions.

Consider a function f ∈ L2(R).Goal: write this (possibly complicated) function as a sum ofwell-known functions.

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The Purpose of Writing Functions as Sums

Suppose we want to analyze and synthesize a signal.

Consider a musical score, such as the one below:

One method of analysis is to separate the music into pureharmonic waves – this is the Fourier series.

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The Fourier Series - Preliminary

Let f be a function whose domain is R and whose range is thecomplex plane that

has period T , i.e.,f (t + T) = f (t),

and

is square integrable, and so∫ t0+T

t0|f (t)|2dt <∞.

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The Fourier Series

The Fourier series of f can be defined as an infinite sum of sineand cosine terms, or it can be defined as

∞∑n=−∞

cne2πiωnt,

where ωn =nT

.

The coefficients are given by

cn =1T

∫ t0+T

t0e−2πiωntf (t) dt.

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The Fourier Transform

The Fourier Transform can be thought of as the continuousanalogue of the Fourier series.To obtain the Fourier transform from the Fourier series:

Let t0 = T2 and take the limit as T →∞.

Denote by f the Fourier transform of f .Then f is given by

f (ω) =∫ ∞−∞

e−2πıωtf (t)dt.

The inverse Fourier transform is then defined by

f (t) =∫ ∞−∞

e2πıωt f (ω)dω.

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Problems with Fourier Transform

The Fourier transform and its inverse reproduce f (t) by asuperposition of complex exponentials, e2πıωt.

The Fourier transform gives the coefficient function needed sothat the (continuous) superposition,

∫e2πıωt f (ω)dω equals f .

Observation: e2πıωt = cos(2πωt) + i sin(2πωt) is spread over alltime.

So, for a high-fidelity signal, infinitely many frequencies areneeded (so we get destructive interference in the right places).

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Problem: Reproducing a “sharp" function

Example: the Dirac delta “function" (OK, distribution)

Then δ(t) =∫ ∞−∞

e2πıωtδ(ω) dω =

∫ ∞−∞

e2πıωt dω.

Idea: the narrower the “spike" in a function, the more high-frequencyexponentials needed to get the desired sharpness, but then we needmore exponentials to cancel the effects of these functions elsewhere.

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Improving the Representation of a Signal

Instead of transforming the entire score of music into pureharmonics, we could try to change short segments of the scoreseparately.

Then each part of the signal is separated into a superposition ofknown functions of different frequency.

But, each section of the signal has a different set of “amplitudes"associated with these functions.

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The Windowed Fourier Transform (WFT)

Let g(u) be a square integrable function that is nonzero on theinterval [−T, 0] and zero for all other t ∈ R.

For each t ∈ R, define

ft(u) = g(u− t)f (u).

Then ft will be nonzero only on the interval [t − T, t].

Define the windowed Fourier transform (WFT) of f as

f (ω, t) = ft(ω) =

∫ ∞−∞

e−2πıωuft(u)du

=

∫ ∞−∞

e−2πıωug(u− t)f (u)du

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Some Properties of WFT

The WFT is symmetric with respect to the time and frequencydomains.

Inverse WFT is defined by:

f (u) = C−1∫ ∫

gω,t(u)f (ω, t) dω dt,

where C =∫|g(u)|2 du.

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What WFT Does

WFT observes the signal f (t) over the length of the window,g(u− t) such that the time parameter t in f (ω, t) is no longersharp – it represents a time window.

In WFT, the frequency in f (ω, t) is not sharp either – itrepresents a frequency band associated with the band of thefrequency window g(ν − ω) obtained from the Fourier transformof g(u− t).

The product of the time and frequency windows is constant.

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Example of Windowed Fourier Transform

Ex. Gaussian window:

g(t) = (2a)14 e−πat2 .

Let a = 12π .

Then, g looks like

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t

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Problems with Windowed Fourier Transform

For the inverse WFT, f (ω, t) is the coefficient function used inthe superposition of the functions e2πıωug(u− t).

Since the window function g has width T, e2πıωug(u− t) does,too.

So, a spike in a function, with width much smaller than T andlocated near u = u0, can only be reproduced by superimposinge2πıωug(u− t) with t ≈ u0 and very many frequencies, ω.

If a function is very smooth, so its width is much larger than T ,then many low-frequency functions e2πıωug(u− t) must be usedover a wider time interval.

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Problems with Windowed Fourier Transform (cont.)

We see that the WFT introduces a scale into the analysis (T).

This means that, in terms of a musical score, there will be suddenbreaks between the segments of the score.

In image processing, this could appear as an artificial edge.

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Improving the Representation of a Signal

Observe that there is localization in both the time and frequencydomains.

For tones with a high frequency, less time is required to producethem.For tones with a low frequency, more time is required to producethem.

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Continuous Wavelet TransformDiscrete Wavelet Transform and MRA


The continuous wavelet transform is defined by

(Twavf )(a, b) = |a|−12

∫f (t)ψ

(t − b

a

)dt.

Assumption: ψ satisfies∫ψ(t)dt = 0.

The functions ψa,b(s) = |a|−12ψ( s−b

a ) are called wavelets.

And ψ is called the mother wavelet.

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The Wavelet Transform (cont.)

In the definition of the wavelet transform, as the parameter achanges, the ψa,0(s) = |a|−

12ψ( s

a) cover different frequencyranges.

Large values of a correspond to low frequencies.Small values of a correspond to high frequencies.

Changing the parameter b changes the time-localization center(since ψa,b(s) is localized around s = b).

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The Major Benefit of Wavelet Transform

By the definition of the wavelet functions, they have time-widths(or space-widths) related to their frequency:

high frequency = narrow function,low frequency = broad function.

So, wavelets give us a mathematical way to represent a signal(e.g., a radio signal) by:

decomposing according to its frequency components, andanalyzing each component with a resolution matching its scale.

This means wavelets can “zoom in” on short frequency burstsand “zoom out” on longer frequency signals.

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Types of Wavelet Transforms

The continuous wavelet transformThe discrete wavelet transform

Discrete systems with redundancy (frames)Orthonormal (or other) bases of wavelets

We will briefly touch upon the continuous wavelet transform andorthonormal bases of wavelets (multiresolution analysis).

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Continuous Wavelet Transform

Here, ψa,b(x) = |a|−12ψ( x−b

a ) and

(Twav)(a, b) =∫

f (t)ψa,b.

A function can be reconstructed from its wavelet transform bymeans of the following formula:

f = C−1ψ

∫ ∞−∞

∫ ∞−∞

1a2 ((Twav)(a, b))ψa,bdb da, (1)

where C−1ψ = 2π

∫∞−∞ |ψ(ξ)|

2|ξ|−1dξ.

For (1) to make sense, we need C−1ψ <∞.

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An Example of a Wavelet: ψ(t) = (1− t2)e−t2

2

(Also known as the Mexican hat function)Note: This wavelet is often used in analyzing vision. It is scaled

(changing the maximum and minimum values) and dilated (changingthe width) to analyze a signal.

−5 −4 −3 −2 −1 0 1 2 3 4 5−0.5

0

0.5

1

t

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Another Example: Morlet Wavelet

ψ(t) = π−14 (e−ıω0t − e−

ω20

2 )e−t22 , with

ω0 = π√

2ln 2

Real part of ψ(t) Imaginary part of ψ(t)

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Example Application – the Radio

When we listen to music on the radio, what do we hear?

First, the sounds must be compressed, sent over the airwaves,then uncompressed.

In uncompression, external noise must be removed from thesignal, so we hear only the music.

Amplifiers are responsible for this noise removal.

Wavelets give a way to compress a signal and then uncompress itwithout keeping high-frequency noise, and without losing anydetails of the original signal.

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Why Do We Need Discrete Wavelets?

Now, looking at a digital recording (or image), information is notin continuous form.

Discrete wavelets allow analysis of sequences of information inthe same way that continuous wavelets analyze continuousfunctions.

Of course, discrete wavelets can be used to analyze continuousfunctions also, by sampling the function at appropriate points,then transforming it.

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Multiresolution Analysis (MRA): Idea

Typically, there are several levels of decomposition of a signal.

The signal is broken up into “approximation” parts and “detail”parts, which are then used to reconstruct the signal.

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Multiresolution Analysis (MRA): Some MathematicalDetail

Consists of sequence of closed subspaces Vj of L2(R)

· · · ⊂ Vj+1 ⊂ Vj ⊂ Vj−1 ⊂ · · · ⊂ L2(R)

satisfying certain properties, including:For every integer j,

Vj+1 ⊂ Vj.

A function f is in one of the sets Vj if and only if, for everyinteger j,

f (2j·) ∈ V0.

The only function contained in all Vj’s is f (x) = 0.

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Other Properties of MRA

If f ∈ V0, then f (x + n) ∈ V0 for every integer n.

There exist functions φ ∈ V0, called scaling functions, such that

{φ(x− n) | n ∈ Z}

forms an orthonormal basis of V0.

This means that {φj,n = 2−j2 φ(2−jx− n) | n ∈ Z} forms an

orthonormal basis for Vj, for all j, so any function in Vj can bewritten as a weighted sum of the φj,n.

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Multiresolution Analysis (cont.)

For each integer j, define the set Wj so that every function in Wj

is orthogonal to every function in Vj, and every function in Vj−1can be written as a sum of two functions, one in Vj and one in Wj.

We can find ψ ∈ W0, (mother wavelet) such that

{ψ(· − k) | k ∈ Z}

is an orthonormal basis for W0.

Then {ψj,n} form an orthonormal basis of Wj, for all j.

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Example of Multiresolution Analysis: the Haar MRA

φ(x) =

{1 if 0 ≤ x < 1,0 otherwise

ψ(x) =

1 if 0 ≤ x < 1

2 ,

−1 if 12 ≤ x < 1,

0 otherwise

(So, Vj for Haar MRA is the space of piecewise constant functions.)

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One Way to Use an MRA

We may represent the discrete wavelet and scaling operators asmatrices.

Store the data from a signal (or function) in a vector, and thenmultiply by the appropriate matrix to get the “approximation”parts and the “detail” parts.

In two dimensions, the operators are constructed in a certain wayas products of the one dimensional operators; the principleproperties of the operators do not change.

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Example: Compression of an Image

The idea behind image compression is to take an image of agiven number of pixels and store it using the smallest amount ofmemory possible, e.g., in a smart phone.

A simple way to compress an image is to decompose the imageusing wavelets, then use thresholding to choose which values todelete.

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Thresholding

Choose the smallest components from the waveletdecomposition based on a given tolerance and remove them.

The threshold may also be defined based on the percentcompression desired.

Due to the properties of wavelets, significant compression can berealized with little effect on the reconstructed image.

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Reconstruction of an Image

The procedure used to decompose the image is in some sense“reversed” based on established mathematical formulas.

We’ll look at examples of images compressed and reconstructedusing different wavelet families, including Haar wavelets andwavelets constructed by Ingrid Daubechies.

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Some References

[1] Albert Boggess and Francis J. Narcowich, A First Course inWavelets with Fourier Analysis, Prentice Hall, Upper SaddleRiver, NJ, 2001.

[2] Ingrid Daubechies, Ten Lectures on Wavelets, CBMS-NSFSeries in Applied Mathematics 61, SIAM, Philadelphia, PA,1992.

[3] Gerald Kaiser, A Friendly Guide to Wavelets, BirkhäuserBoston, Inc., Boston, MA, 1994.

[4] Gilbert Strang and Truong Nguyen, Wavelets and Filter Banks,Wellesley-Cambridge Press, Wellesley, MA, 1996.

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