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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.
slide6

1D SIGNAL

Coefficient * sinusoid of appropriate frequency

The original signal

slide7

Wavelet Properties

  • Short time localized waves
  • 0 integral value.
  • Possibility of time shifting.
  • Flexibility.
slide9

Wavelet Transform

Coefficient * appropriatelyscaled and shiftedwavelet

The original signal

slide10

CWT

Step 1

Step 2

Step 3

Step 4

Step 5

Repeat steps 1-4 for all scales

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

slide17

LPF

Input Signal

HPF

Approximations and Details:

  • Approximations: High-scale, low-frequency components of the signal
  • Details: low-scale, high-frequency components
slide18

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

slide19

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

slide20

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

slide22

What was wrong with Fourier?

  • We loose the time information
slide23

Short Time Fourier Analysis

  • STFT - Based on the FT and using windowing :
slide24

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

slide35

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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