Fourier series is a
useful orthonormal
representation on
Wavelets, discovered in the last 15 years, are another
kind of basis for
Summary: This module gives an overview of wavelets and their usefulness as a basis in image processing. In particular we look at the properties of the Haar wavelet basis.
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Fourier series is a
useful orthonormal
representation on
Wavelets, discovered in the last 15 years, are another
kind of basis for
Fourier series 
Wavelets  basis functions give frequency info but are local in time.
In Fourier basis, the basis functions are harmonic
multiples of
In Haar wavelet basis, the
basis functions are scaled and
translated versions of a "mother wavelet"
Basis functions
Let
Let
Larger
Check: each
Any two basis functions are orthogonal.

Also,
Using what we know about Hilbert spaces: For any
This demonstration lets you create a signal by combining Haar basis functions, illustrating the synthesis equation of the Haar Wavelet Transform. See here for instructions on how to use the demo.
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