Two-page tutorial on wavelets for multiresolution analysis.

1. Wavelets & MRA in two pages © Russell John Childs, PhD, 2017-10-01
Einstein summation convention: = . Eg:
Expansion function: .
Basis function:
Scaling function:
Vector space:
Discrete Wavelet Transform:
where:
Properties of wavelets:
Fast wavelet transform:
where means removing odd samples, means inserting 0s between samples, is time-
reversed and ⨁ signifies: .
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⨁
⨁
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2. Wavelet packets: The Discrete Wavelet Transform only down-samples the low-pass bands:
For this three-scale analysis tree of the DWT, three expansions are possible:
⨁
⨁ ⨁
⨁ ⨁ ⨁
whereas wavelet packets allow for greater control.
Lifting (basic): Allows for creation of wavelets by:
(1) Splitting signal into even and odd samples: ,
(2) Predicting + replacing odd samples with for some chosen .
(3) Updating even samples with for some chosen
2-D Wavelets:
Scaling functions and wavelets:
Discrete Wavelet Transform:
C++: Sold separately.
Topics to be explored:
Colour images: 2-D wavelets on each monochromatic R, G, B or 3-D wavelets?
Inter-frame encoding: 1-D wavelets on each pixel or 3-D wavelets on frames?
Interpolation, extrapolation, lifting transforms.