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# Introduction to Wavelets -part 2 - PowerPoint PPT Presentation

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 toWavelets -part 2

By Barak Hurwitz

Wavelets seminar

with Dr’ Hagit Hal-or

• Reminder

• 1D signals

• Wavelet Transform

• CWT,DWT

• Wavelet Decomposition

• Wavelet Analysis

• 2D signals

• Wavelet Pyramid

• some Examples

Reminder – from last week

• Why transform?

• Why wavelets?

• Wavelets like basis components.

• Wavelets examples.

• Continuous Wavelet Transform.

Reminder –Noise in Fourier spectrum

Noise

Coefficient * sinusoid of appropriate frequency

The original signal

• Short time localized waves

• 0 integral value.

• Possibility of time shifting.

• Flexibility.

Coefficient * appropriatelyscaled and shiftedwavelet

The original signal

Step 1

Step 2

Step 3

Step 4

Step 5

Repeat steps 1-4 for all scales

Example –A simulated lunar landscape

CWT of the “Lunar landscape”

1/46

scale

time

mother

• 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

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

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

• 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

Input Signal

HPF

Approximations and Details:

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

• Details: low-scale, high-frequency components

• 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

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

high pass filter

Low pass filter

• 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

* Wavelet used: db2

• We loose the time information

• STFT - Based on the FT and using windowing :

• between time-based and frequency-based.

• limited precision.

• Precision <= size of the window.

• Time window - same for all frequencies.

What’s wrong with Gabor?

• Windowing technique with variable size window:

• Long time intervals - Low frequency

• Shorter intervals - High frequency

• To analyze a localized area of a larger signal.

• For example:

low frequency

• Fourier analysis Vs. Wavelet analysis:

scale

Discontinuity effect

time

High frequency

NOTHING!

exact location

in time of the discontinuity.

more

)

)

(

Y

=

Y

-

x

b

1

a

,

b

x

a

a

2D SIGNAL

Wavelet function

• b– shift coefficient

• a – scale coefficient

• 2D function

1D function

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

LENNA

high pass

high pass

more

Original @ 8bpp

DWT

@0.5bpp

DCT

@0.5 bpp

DWTDCT

0.15bpp 0.18bpp 0.2bpp

DCT

DWT

• 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