A Theory for Multiresolution Signal Decomposition: The Wavelet Representation

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A Theory for Multiresolution Signal
Decomposition The Wavelet Representation

• Author Stephane G. Mallat
• Presented by Yuelong Jiang

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Content

• Multiresolution Approximation and Transform
• Wavelet Representation in Multiresolution
• Laplacian pyramid and FWT
• Extension of Wavelet Representation to Image
• Some applications
• Conclusion

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Multiresolution Approximation

• A measurable, square-integrable
  one-dimensional function f(x) can be approximated
  in multiresolution.
• Let A2j be the operator which approximates
  a signal f(x) at a resolution 2j. We expect it
  has following properties
• ?g(x) ? V2j, g(x) f(x) ? A2jf(x) f(x)
• ?j ? Z, V2j ? V2j1
• ?j ? Z, f(x) ? V2j ? f(2x) ? V2j1
• ?Isomorphism I from V1 to I2(Z)
• ?k ? Z, A1fk(x) A1f(x-k), where fk(x) f(x-k)
• I(A1f(x)) (?i)i?Z ? I(A1fk(x)) (?i-k)i?Z

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is dense in L2(R)
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Multiresolution Approximation
We call any set of vector spaces (V2j)j?Z
which satisfies the properties 2-8 a
multiresolution approximation of L2(R). The
associated set of operators A2j satisfying 1-6
give the approximation of any L2(R) function at a
resolution 2j.
How to numerically characterize the operator
A2j? Theorem 1 Let (V2j)j?Z be a multiresolution
approximation of L2(R). There exists a unique
function ?(x) ? L2(R), called a scaling function,
such that is we set ?2j(x) 2j?(2jx) for j ? Z,
(the dilation of ?(x) by 2j), then
is an orthonormal basis of V2j
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Multiresolution Transform
?j ? Z, V2j ? V2j1
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Fourier Transform of a scaling function
Therem 2 Let ?(x) be a scaling function, and let
H be a discrete filter with impulse response
h(n). Let H(?) be the Fourier series defined by
H(?) satisfies the following two
properties H(0) 1 and h(n) O(n-2) at
infinity H(?)2 H(??)2 1 Conversely let
H(?) be a Fourier series satisfying above two
equations and such that H(?) ? 0 for ??0, ?/2
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Wavelet Representation
The difference of information between 2j1 and 2j
is called the detail signal at the resolution 2j.
Let O2j be orthogonal complement of V2j, i.e.
O2j is orthogonal to V2j O2j ? V2j
V2j1 Theorem 3 Let (V2j)j?Z be a
multiresolution vector space sequence, ?(x) the
scaling function, and H the corresponding
conjugate filter. Let ?(x) be a function whose
Fourier transform is given by
Let ?2j(x) 2j?(2jx) denote the dilation of ?(x)
by 2j. Then
is an orthonormal basis of L2(R) and ?(x) is
called an orthogonal wavelet.
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Wavelet Representation
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Wavelet Representation
For any J gt 0, the original discrete signal A1f
measured at the resolution 1 can be represented
by
This set of discrete signals is called an
orthogonal wavelet representation, and consists
of the reference signal at a coarse resolution
A2-Jf and the detail signals at the resolution 2j
for J ? j ? -1.
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Wavelet Representation
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Wavelet Representation
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Laplacian Pyramid and FWT
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FWT
O2j is the orthogonal complement of V2j in V2j1,
is an orthonormal basis of V2j1, so
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FWT
g(n) (-1)1-nh(1-n) In signal processing, G and
H are called quadrature mirror filters. Generally,
we design the H first and then calculate the G,
? and ?.
n0
n0,1
Others
n1
Others
?(x) is Haar function.
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Extension to Image
forms an orthonormal basis of V2j
is an orthonormal basis of O2j
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Extension to Image
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Extension to Image
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Application

• Compact Coding of Wavelet Image Representation
• Texture Discrimination and Fractal Analysis

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Conclusion

• Wavelet representation can indicate the signal
  without information loss.
• Through two pass filters, wavelet representation
  can reconstruct the original signal efficiently.
• Compared with Fourier transform, wavelet is
  localizable in both frequency domain and space
  domain.
• Wavelet representation provides a new way to
  compress or modify images.