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A Theory For Multiresolution Signal Decomposition: The Wavelet Representation
Stephane Mallat, IEEE Transactions on Pattern Analysis and Machine Intelligence, July 1989
Presented by:Randeep Singh GakhalCMPT 820, Spring 2004
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Outline
Introduction History Multiresolution Decomposition Wavelet Representation Extension to Images Conclusions
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Introduction
What is a wavelet? It’s just a signal that acts like a variable
strength magnifying glass.
What does wavelet analysis give us? Localized analysis of a signal For analysis of low frequencies, we want a
large frequency resolution For analysis of high frequencies, we want
a small frequency resolution Wavelets makes both possible.
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History
Wavelets originated as work done by engineers, not mathematicians.
Mallat saw wavelets being used in many disciplines and tried to tie things together and formalize the science.
This paper was co-authored by Yves Meyer who let Mallat have all the credit as he was already a full professor.
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Multiresolution Decomposition
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We want to decompose f(x): Finite energy and measurable
The approximation operator at resolution 2j :
The operator is an orthogonal projection onto a vector space :
The Multiresolution Operator
jA2
)()( 2 RLxf
jj VLxfA2
2
2(R):)(
jV2
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Approximation is the most accurate possible at that resolution:
Lower resolution approximations can be extracted from higher resolution approximations:
Properties of the Multiresolution Approximation I
)()()()(,)(22
xfxfAxfxgVxg jj
122, jj VVZj
1
2
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Operator at different resolutions:
an isomorphism: I2 is the vector space of square
summable sequences Given I, we can move freely between
the two domains without loss of information
Properties of the Multiresolution Approximation II
122)2()(, jj VxfVxfZj
)(: 21 ZIVI 4
3
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Properties of the Multiresolution Approximation III
Translation in the approximation domain:
Translation in the sample domain:
where:
)()(, 11 kxfAxfAZk k
Zikik
Zii
xfAI
xfAI
)())((
)())((
1
1
)()( kxfxfk
5
6
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Properties of the Multiresolution Approximation IV
As the resolution increases (j∞), the approximation converges to the original:
And vice versa:
j
jjj VV22
lim is dense
}0{lim22
jj
jj VV
7
8
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Multiresolution Transformation
Conditions 1-8 define the requirements for a vector space to be a multiresolution approximation
The operator projects the signal onto the vector space
Before we can compute the projection, we need an orthonormal basis for the vector space
jA2jV2
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For the multiresolution approximation for :
Orthonormal Basis Theorem
(R)2L
ZjjV 2
(R))( 2Lx a unique scaling function:
)2(2)(2
xx jjj
such that Zn
jj nxj 22
2
is an orthonormal basis of ZjjV 2
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Projecting onto the Vector Space
n
jjj nxnuufxfA j )2()2(),(2)(2
Zn
j
Zn
jd
nuuf
nuuffA
j
jj
2)(*)(
)2(),(
2
22
The discrete approximation:
The continuous approximation:
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Implementation of the Multiresolution Transform
Let the signal that is of highest resolution be at 1 (j=0) andsuccessively coarser approximations be at decreasing j (j<0).
We can express the inner product at a resolution j based on the higher resolution j+1:
)2(),()2(~
)2(),( 1
22 1 kuufknhnuuf j
k
jjj
where )(),()( 12nuunh
With this form, we can compute all discrete approximations (j<0) from the original (j=0).
This is the Pyramid Transform
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The Scaling Function TheoremThe scaling function essentially characterizes the entiremultiresolution approximation.
The calculation of the discrete approximation does NOT require the scaling function explicitly – we use h(n).
If H() satisfies:
Then, we define the scaling function as:
We can define the filter first and the scaling function is the result!!!
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An Example of Multiresolution Decomposition
Successive discrete (a) and continuous (b) approximations of a function f(x).
(a) (b)
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Wavelet Representation
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Projecting onto the Orthogonal Complement
Wavelet representation is derived from the detail signal.
At resolution 2j, detail signal is difference in informationbetween approximation j and j+1.
Detail signal is result of orthogonal projection of f(x) onorthogonal complement of in , .jV2 12 jV jO
2
Analogous to case for multiresolution approximation in , we need an orthonormal basis for to express the detailsignal.
jV2jO2
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The Orthogonal Complement Basis Theorem
For the multiresolution vector space , scalingfunction , and conjugate filter , define:
2ˆ
22)(ˆ
He i
ZjjV 2
)(x )(H
xx jjj 22)(2
Then is an orthonormal basis of Zn
jj nxj 22
2 jO
2
And is an orthonormal basis of 2),(222
Zjn
jj nxj )(2 RL
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Projecting onto the Orthogonal Complement Vector Space
n
jjj
OnxnuufxfP jj )2()2(),(2)(
22
Zn
j
Zn
j
nuuf
nuuffD
j
jj
2)(*)(
)2(),(
2
22
The discrete detail signal:
The continuous detail signal:
Orthogonal wavelet representation consists of a signal at a coarse resolution and a succession of refinements consisting of difference signals:
12
, jJ
dJ fDfA j
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Implementation of the Orthogonal Wavelet Representation
We express the inner product at a resolution j based on the higher resolution j+1:
)2(),()2(~)2(),( 1
22 1 kuufkngnuuf j
k
jjj
where )(),()( 12nuung
We can thus successively decompose the discrete representation to compute the orthogonal wavelet representation.
This is pyramid transform is called the Fast Wavelet Transform (FWT)
fAd1
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Implementation of the Orthogonal Wavelet Representation
Cascading algorithm to compute detail signals successively, generating the wavelet representation.The output becomes the input to calculating thenext (and coarser) detail signal.
fAdj2
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An Example of Wavelet Representation
Successive continuous approximations (a) and discrete detail signals (b) for a function f(x).
(a) (b)
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Extension to Images
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Two dimensional scaling function is separable:
Images require a natural extension to our previously discussed multiresolution analysis to two dimensions.
Orthonormal basis of is now:
Multiresolution Image Decomposition
)()(),( yxyx
jV2
2),(2))2,2(2(
Zmn
jjj mynxj
jV2
So our basis can be expressed as:
2),(22))2()2(2(
Zmn
jjj mynx jj
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At resolution 2j, our discrete approximation has 2jN pixels.
The discrete characterization of our image is:
Multiresolution Image Decomposition Continued
2),(222
)2()2(),,(Zmn
jjd mynxyxffA jjj
An example of an original image (resolution=1) and threeapproximations at 1/2, 1/4, and 1/8.
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For the two-dimensional scaling function with (x) the wavelet associated with (x), the three wavelets
The Multidimensional Orthogonal Complement Basis Theorem
)()(),( yxyx
define the orthonormal basis for
)()(),(1 yxyx )()(),(2 yxyx )()(),(3 yxyx
jO2
2),(
3
2
2
2
1
2
))2,2(2
),2,2(2
),2,2(2(
Zmn
jjj
jjj
jjj
mynx
mynx
mynx
j
j
j
and the orthonormal basis for
3),,(
3
2
2
2
1
2
))2,2(2
),2,2(2
),2,2(2(
Zjmn
jjj
jjj
jjj
mynx
mynx
mynx
j
j
j
)( 22 RL
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The Multideminsional Wavelet Representation
We can now define our wavelet representation of the image:
Or, in a filter form:
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The Three Frequency Channels
We can interpret the decomposition as a breakdown of the signal into spatially oriented frequency channels.
Decomposition of frequency support
Arrangement of wavelet representations
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Applying the Decomposition
(a) Original image
(b) Wavelet representation
(c) Black and white viewof high amplitude coefficients
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Conclusions and Future Work
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Conclusions
We can express any signal as a series of multiresolution approximations.
Using wavelets, we can represent the signal as a coarse approximation and a series of difference signals without any loss.
The multiresolution representation is characterized by the scaling function. The scaling function gives us the wavelet function.
Applied to images, we can get frequency content along each of the dimensions and joint frequency content
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References
[1] S. Mallat, "A Theory for Multiresolution Signal Decomposition: The Wavelet Representation", IEEE Trans. on Pattern Analysis and Machine Intelligence, 11(7):674-693, 1989.
[2] B. B. Hubbard, “The World According to Wavelets”, A.K. Peters, Ltd., 1998.