Introduction to Wavelets -part 2

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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

List of topics
• 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.

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

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)
• Low scale a : Compressed wavelet :Fine details (rapidly changing) : High frequency
• High scale a : Stretched wavelet: Coarse details (Slowly changing): Low frequency
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*:

* 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
• 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:
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

(

)

)

(

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

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

high pass

high pass

high pass

more

Coding Example

Original @ 8bpp

DWT

@0.5bpp

DCT

@0.5 bpp

Another Example

0.15bpp 0.18bpp 0.2bpp

DCT

DWT

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