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Introduction to Wavelets -part 2. By Barak Hurwitz. Wavelets seminar with Dr ’ Hagit Hal-or. List of topics. Reminder 1D signals Wavelet Transform CWT,DWT Wavelet Decomposition Wavelet Analysis 2D signals Wavelet Pyramid some Examples. Reminder – from last week.

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Introduction to wavelets part 2

Introduction toWavelets -part 2

By Barak Hurwitz

Wavelets seminar

with Dr’ Hagit Hal-or


List of topics
List of topics

  • Reminder

  • 1D signals

    • Wavelet Transform

    • CWT,DWT

    • Wavelet Decomposition

    • Wavelet Analysis

  • 2D signals

    • Wavelet Pyramid

    • some Examples


Reminder from last week
Reminder – from last week

  • Why transform?

  • Why wavelets?

  • Wavelets like basis components.

  • Wavelets examples.

  • Wavelets advantages.

  • Continuous Wavelet Transform.



Reminder noise in fourier spectrum
Reminder –Noise in Fourier spectrum

Noise


1D SIGNAL

Coefficient * sinusoid of appropriate frequency

The original signal


Wavelet Properties

  • Short time localized waves

  • 0 integral value.

  • Possibility of time shifting.

  • Flexibility.



Wavelet Transform

Coefficient * appropriatelyscaled and shiftedwavelet

The original signal


CWT

Step 1

Step 2

Step 3

Step 4

Step 5

Repeat steps 1-4 for all scales


Example a simulated lunar landscape
Example –A simulated lunar landscape


Cwt of the lunar landscape
CWT of the “Lunar landscape”

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scale

time

mother


Scale and frequency
Scale and Frequency

  • Higher scale correspond to the most “stretched” wavelet.

  • The more stretched the wavelet–

    the coarser the signal features being measured by the wavelet coefficient.

Low scale High scale


Scale and frequency cont d
Scale and Frequency (Cont’d)

  • Low scale a : Compressed wavelet :Fine details (rapidly changing) : High frequency

  • High scale a : Stretched wavelet: Coarse details (Slowly changing): Low frequency



The dwt
The DWT

  • Calculating the wavelets coefficients at every possible scale is too much work

  • It also generates a very large amount of data

Solution: choose only a subset of scales and positions, based on power of two (dyadic choice)

Discrete Wavelet Transform


LPF

Input Signal

HPF

Approximations and Details:

  • Approximations: High-scale, low-frequency components of the signal

  • Details: low-scale, high-frequency components


Decimation

  • The former process produces twice the data

  • To correct this, we Down sample(or: Decimate) the filter output by two.

A complete one stage block :

A*

LPF

Input Signal

D*

HPF


Multi-level Decomposition

  • Iterating the decomposition process, breaks the input signal into many lower-resolution components: Wavelet decomposition tree:

high pass filter

Low pass filter


Wavelet reconstruction

  • Reconstruction (or synthesis) is the process in which we assemble all components back

Up sampling

(or interpolation) is done by zero inserting between every two coefficients


Example
Example*:

* Wavelet used: db2


What was wrong with Fourier?

  • We loose the time information


Short Time Fourier Analysis

  • STFT - Based on the FT and using windowing :


STFT

  • between time-based and frequency-based.

  • limited precision.

  • Precision <= size of the window.

  • Time window - same for all frequencies.

What’s wrong with Gabor?


Wavelet analysis
Wavelet Analysis

  • Windowing technique with variable size window:

  • Long time intervals - Low frequency

  • Shorter intervals - High frequency


The main advantage local analysis
The main advantage:Local Analysis

  • To analyze a localized area of a larger signal.

  • For example:


Local analysis cont d
Local Analysis (Cont’d)

low frequency

  • Fourier analysis Vs. Wavelet analysis:

scale

Discontinuity effect

time

High frequency

NOTHING!

exact location

in time of the discontinuity.

more


2d signal

(

)

)

(

Y

=

Y

-

x

b

1

a

,

b

x

a

a

2D SIGNAL

Wavelet function

  • b– shift coefficient

  • a – scale coefficient

  • 2D function

1D function


Time and space definition
Time and Space definition

1D

  • Time– for one dimension waves we start point shifting from source to end in time scale .

  • Space– for image point shifting is two dimensional .

2D




Wavelet decomposition another example
Wavelet Decomposition- Another Example

LENNA


high pass

high pass

high pass

more


Coding example
Coding Example

Original @ 8bpp

DWT

@0.5bpp

DCT

@0.5 bpp



Another example
Another Example

0.15bpp 0.18bpp 0.2bpp

DCT

DWT


Where do we use wavelets
Where do we use Wavelets?

  • Everywhere around us are signals that can be analyzed

  • For example:

    • seismic tremors

    • human speech

    • engine vibrations

    • medical images

    • financial data

    • Music

Wavelet analysis is a new and promising set of tools for analyzing these signals





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