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

Outline

Introduction History Multiresolution Decomposition Wavelet Representation Extension to Images Conclusions

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.

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.

Multiresolution Decomposition

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

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

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

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

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

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

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

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:

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

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!!!

An Example of Multiresolution Decomposition

Successive discrete (a) and continuous (b) approximations of a function f(x).

(a) (b)

Wavelet Representation

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

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

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

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

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

An Example of Wavelet Representation

Successive continuous approximations (a) and discrete detail signals (b) for a function f(x).

(a) (b)

Extension to Images

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

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.

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

The Multideminsional Wavelet Representation

We can now define our wavelet representation of the image:

Or, in a filter form:

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

Applying the Decomposition

(a) Original image

(b) Wavelet representation

(c) Black and white viewof high amplitude coefficients

Conclusions and Future Work

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

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.