A Theory For Multiresolution Signal Decomposition: The Wavelet Representation

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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 Gakhal
• CMPT 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?
• Its 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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The Multiresolution Operator

• 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

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Properties of the Multiresolution Approximation I

• Approximation is the most accurate possible at
  that resolution
• Lower resolution approximations can be extracted
  from higher resolution approximations

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2
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Properties of the Multiresolution Approximation II

• 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

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Properties of the Multiresolution Approximation
III

• Translation in the approximation domain
• Translation in the sample domain
• where

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Properties of the Multiresolution Approximation IV

• As the resolution increases (j?8), the
  approximation converges to the original
• And vice versa

is dense
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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

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Orthonormal Basis Theorem

• For the multiresolution approximation
  
• for

? a unique scaling function
such that
is an orthonormal basis of
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Projecting onto the Vector Space
The continuous approximation
The discrete approximation
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Implementation of the Multiresolution Transform
Let the signal that is of highest resolution be
at 1 (j0) and successively coarser
approximations be at decreasing j (jlt0).
We can express the inner product at a resolution
j based on the higher resolution j1
where
With this form, we can compute all discrete
approximations (jlt0) from the original (j0).
This is the Pyramid Transform
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The Scaling Function Theorem
The scaling function essentially characterizes
the entire multiresolution 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
(a)
(b)
Successive discrete (a) and continuous (b)
approximations of a function f(x).
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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 information between approximation j
and j1. Detail signal is result of orthogonal
projection of f(x) on orthogonal complement of
in , .
Analogous to case for multiresolution
approximation in , we need an orthonormal
basis for to express the detail signal.
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The Orthogonal Complement Basis Theorem
For the multiresolution vector space ,
scaling function , and conjugate filter
, define
Then is an
orthonormal basis of
And is an
orthonormal basis of
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Projecting onto the Orthogonal Complement Vector
Space
The continuous detail signal
The discrete detail signal
Orthogonal wavelet representation consists of a
signal at a coarse resolution and a succession of
refinements consisting of difference signals
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Implementation of the Orthogonal Wavelet
Representation
We express the inner product at a resolution j
based on the higher resolution j1
where
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)
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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 the next (and coarser)
detail signal.
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An Example of Wavelet Representation
(a)
(b)
Successive continuous approximations (a) and
discrete detail signals (b) for a function f(x).
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Extension to Images
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Multiresolution Image Decomposition
Images require a natural extension to our
previously discussed multiresolution analysis to
two dimensions. Orthonormal basis of is now
Two dimensional scaling function is separable
So our basis can be expressed as
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Multiresolution Image Decomposition Continued
The discrete characterization of our image is
At resolution 2j, our discrete approximation has
2jN pixels.
An example of an original image (resolution1)
and three approximations at 1/2, 1/4, and 1/8.
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The Multidimensional Orthogonal Complement Basis
Theorem
For the two-dimensional scaling function
with ?(x) the wavelet
associated with ?(x), the three wavelets
define the orthonormal basis for
and the orthonormal basis for
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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 view of 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.