Haar Wavelet Basishttp://cnx.org/contenthttp://cnx.org/content/m10764/latest/m10764Haar Wavelet Basis2.92002/07/292011/09/02 10:09:37.461 GMT-5DanCalderonDan Calderonslycaldron@netscape.netJustinRombergJustin Rombergjrom@ece.gatech.eduMatthewHutchinsonMatthew Hutchinsonmhutch85@gmail.comRoyHaRoy Harha@alumni.rice.eduMariyahPoonawalaMariyah Poonawalamariyah@rice.eduPrashantSinghPrashant Singhprash@ece.rice.edurha jrommariyah prash mhutch slycaldronrha jrombasisexpansionfourierfourier serieshaarhaar wavelethilberthilbert spaceswaveletwaveletsScience and TechnologyThis 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.
enIntroduction
Fourier series is a
useful orthonormal
representation on
L20T
especiallly for inputs into LTI systems. However, it is ill
suited for some applications, i.e. image processing (recall
Gibb's phenomena).
Wavelets, discovered in the last 15 years, are another
kind of basis for
L20T
and have many nice properties.
Basis Comparisons
Fourier series -
cn give frequency information. Basis functions last the entire
interval.
Wavelets - basis functions give frequency info but are
local in time.
In Fourier basis, the basis functions are harmonic
multiples of
ω0t
In Haar wavelet basis, the
basis functions are scaled and
translated versions of a "mother wavelet"
ψt.
Basis functions
ψj,kt
are indexed by a scale j and a
shift k.
Let
0tTφt1
Then
φt2j2ψ2jtkjℤk0,1,2,…,2j-1φt2j2ψ2jtkψt10tT2-10T2T
Let
ψj,kt2j2ψ2jtk
Larger j → "skinnier" basis
function,
j012…,
2j
shifts at each scale:
k0,1,…,2j-1
Check: each
ψj,kt
has unit energy
tψj,kt21∥ψj,k(t)∥21
Any two basis functions are orthogonal.
Also,
ψj,kφ span
L20THaar Wavelet Transform
Using what we know about Hilbert spaces: For any
ftL20T,
we can write
Synthesisftjjkkwj,kψj,ktc0φtAnalysiswj,kt0Tftψj,ktc0t0Tftφtthe
wj,k are real
The Haar transform is super useful especially in
image compressionHaar Wavelet Demonstration