Introduction to Wavelets part 2 By Barak Hurwitz

Introduction to Wavelets part 2 By Barak Hurwitz Wavelets seminar with Dr’ Hagit Hal-or

List of topics • Reminder • 1 D signals – Wavelet Transform – CWT, DWT – Wavelet Decomposition – Wavelet Analysis • 2 D signals – Wavelet Pyramid – some Examples

Reminder – from last week • Why transform? • Why wavelets? • Wavelets like basis components. • Wavelets examples. • Wavelets advantages. • Continuous Wavelet Transform.

Reminder -Why transform?

Reminder – Noise in Fourier spectrum Noise

1 D SIGNAL Coefficient * sinusoid of appropriate frequency The original signal

Wavelet Properties n n Short time localized waves 0 integral value. Possibility of time shifting. Flexibility.

Wavelets families

Wavelet Transform Coefficient * appropriately scaled and shifted wavelet 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

CWT of the “Lunar landscape” 1/46 scale mother time

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

Shift Smoothly over the analyzed function

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)

Approximations and Details: • Approximations: High-scale, low- frequency components of the signal • Details: low-scale, high-frequency components LPF Input Signal HPF

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 : Input Signal LPF A* HPF D*

Multi-level Decomposition • Iterating the decomposition process, breaks the input signal into many lowerresolution 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: db 2

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

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 NOTHING! High frequency exact location in time of the discontinuity.

2 D SIGNAL Wavelet function • b – shift • coefficient a – scale coefficient • 2 D function Ya , b (x)= 1 D function ( Y a 1 x -b a )

Time and Space definition 1 D • Time – for one dimension waves we start point shifting from source to end in time scale. 2 D • Space – for image point shifting is two dimensional.

Image Pyramids

Wavelet Decomposition

Wavelet Decomposition. Another Example LENNA

high pass

Coding Example Original @ 8 bpp DWT @0. 5 bpp DCT @0. 5 bpp

Zoom on Details DWT DCT

Another Example 0. 15 bpp DCT DWT 0. 18 bpp 0. 2 bpp

Where do we use Wavelets? • Everywhere around us are signals that can • be analyzed For example: – seismic tremors Wavelet – human speechanalysis is a new – engine and vibrations promising set of tools – medical images for analyzing these signals – financial data – Music