wavelet and multiresolution process
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Wavelet and multiresolution process. Pei Wu 5.Nov 2012. Mathematical preliminaries: Some topology. Open set: any point A in the set must have a open ball O( r,A ) contained in the set. Closed set: complement of open set. - PowerPoint PPT PresentationTRANSCRIPT
Wavelet and multiresolution process
Pei Wu5.Nov 2012
Mathematical preliminaries: Some topology Open set: any point A in the set must have
a open ball O(r,A) contained in the set. Closed set: complement of open set. Intersection of closed set is always closed.
Union of open set is always open Compact: if we put infinite point in the set
it must have infinity point “gather” around some point in the set.
Complete: a “converge” sequence must converge at a point in the set.
Mathematical preliminaries: Hilbert space Hilbert space is a space…
linear complete with norm with inner product
Example: Euclidean space, L2 space, …
Mathematical preliminaries: orthonormal basis f,g is orthogonal iff <f,g>=0 f is normalized iff <f,f>=1 Orthonormal basis: e1, e2, e3,… s.t.
a set of basis is called complete if
equivalent condition for orthonormal
A set of element {ei} is orthonormal if and only if:
A orthonormal set induces isometric mapping between Hilbert space and l2.
Motivation in context of Fourier transform we
suppose the frequency spectrum is invariant across time:
However in many cases we want:
Example: Music
Windowed Fourier Transform
Analyze of Windowed Fourier transform A function cannot be localized in both
time and frequency (uncertainty principle).
High frequency resolution means low time resolving power.
Trade-off between frequency resolution and time resolution
Adaptive resolution Use big ruler to measure big thing,
small ruler to measure small thing.
Wavelet Use scale transform to construct
ruler with different resolution.
CWT(continuous wavelet transform)
Proof (1)
Proof (2)
Discretizing CWT a,b take only discrete number:
And we want them to be orthogonal:
Example for wavelet (a)Meyer (b,c)Battle-Lemarie
Example for wavelet (2) (d) Haar (e,f)Daubechies
Constructing orthogonal wavelet Multiresolution analysis A series of linear subspace {Vi} that:
Example
From scaling function to wavelet Firstly we find a set of orthonormal
basis in V0:
hn would play important role in discrete analysis
Example: Haar wavelet
Relaxing orthogonal condition is linearly independent
but not orthogonal.
is orthonormal basis of V0
Example: Battle-Lemarie Wavelet Use spline to get continuous function
Meyer Wavelet: compact support
Fast Wavelet transform Mallat algorithm : top-down
Given c1 how can we get c0 and d0? Given c0 and d0 how to reconstruct
c1 ?
Mallat algorithm (2)
Mallat algorithm (3):frequency domain perspect Subband coding
Adaptive resolution
2D Wavelet Wavelet expansion of 2D function Basis for 2D function:
Mallet algorithm
Frequency Domain Decomposition
Denoise using wavelet
Wavelet packet We can carry on
decomposition on high-frequency part
Adaptive approach to decide decompose or not.
Demo: finger-print image
Demo: finger-print image
Thank You!!