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## PowerPoint Slideshow about ' Introduction to Wavelets -part 2' - muncel

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Presentation Transcript

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.
- Wavelets advantages.
- Continuous Wavelet Transform.

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

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

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

Example*:

* 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?

Wavelet Analysis

- Windowing technique with variable size window:
- Long time intervals - Low frequency
- Shorter intervals - High frequency

The main advantage:Local Analysis

- 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

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

Zoom on Details

DWTDCT

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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