Wavelets decompose signals into different frequency bands using scaling functions and wavelet basis functions. The discrete wavelet transform (DWT) uses these functions to represent a discrete signal as a sum of wavelet coefficients at different scales. The fast wavelet transform provides an efficient algorithm to compute the DWT by successively applying filters and downsampling. Wavelet packets generalize the DWT by allowing decomposition of both low and high frequency bands at each level. Lifting transforms provide an alternative method to construct wavelets by splitting, predicting, and updating the signal. Two-dimensional wavelets extend the concepts to images by applying separable 1D wavelets along rows and columns.

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.