generalized daubechies wavelets and applications to image processing

GENERALIZED DAUBECHIES WAVELETSAND APPLICATIONS TO IMAGE

PROCESSING

Hermann RenderUniversity College Dublin (Ireland)

Joint work withOgnyan Kounchev (Bulgarian Academy of Sciences)

Nira Dyn (Tel-Aviv University)David Levin (Tel-Aviv University)

University College Dublin

Overview

Mathematical tools in Signal processing and Image Processing

Daubechies wavelets and Multiresolutional Analysis

Subdivision schemes

Polyharmonic subdivision schemes

Generalized Daubechies wavelets

Regularity of Generalized Daubechies wavelets

Mathematical tools in Signal and Image Processing

A signal is represented by a function f : R→ R of onevariable x (being square integrable).

An image is presented by a function f : R× R→ R of twovariables. Image processing is much more complex than signalprocessing.

A classical way to process periodic signals is to compute theFourier series

f (x) = a0 + a1 cos x + b1 sin x + a2 cos 2x + b2 sin 2x + ....

Processing the Fourier coefficients an and bn is much easierthan all function values f (x) .

Expanding a function according to suitable basis functions(like cos kx and sin kx for k ∈ Z) simplifies the way ofprocessing data.

For analysing non-periodic signals f : R→ R one uses theFourier transform f̂ defined by

f̂ (x) :=

∫ ∞−∞

f (t) e−itxdt.

In wavelet analysis new basis expansions

f (x) =∞∑

j=−∞

∞∑k=−∞

ck,jψj ,k (x)

for non-periodic signals f : R→ R are developed.

Wavelets are used for data compression in JPEG 2000, DjVuand ECW for still images, REDCODE, CineForm, the BBC’sDirac, and Ogg Tarkin for video.

f̂ (x) :=

∫ ∞−∞

f (t) e−itxdt.

f (x) =∞∑

j=−∞

∞∑k=−∞

ck,jψj ,k (x)

f̂ (x) :=

∫ ∞−∞

f (t) e−itxdt.

f (x) =∞∑

j=−∞

∞∑k=−∞

ck,jψj ,k (x)

f̂ (x) :=

∫ ∞−∞

f (t) e−itxdt.

f (x) =∞∑

j=−∞

∞∑k=−∞

ck,jψj ,k (x)

Wavelets

The Fourier transform f̂ (x) :=∫∞−∞ f (t) e−itxdt is an

important tool in physics and mathematics.

Alternative Transforms have been introduced: e.g. ”Gabortransform” (time-frequency analysis) or ”Calderon transform”.

In 1970’s J. Morlet (a geophysical engineer) introduced thewavelet transform

Wψf (x , s) = |s|−d/2∫

f (t)ψ

(t − x

for a function f ∈ L2(Rd).

There are many choices for a ”wavelet function” ψ ∈ L2(Rd).

The variable s is a scaling parameter (or dilation) andchanging s is like zooming in/out.

Two important ingredients: translation and dilation (scaling).

Wavelets

Wψf (x , s) = |s|−d/2∫

f (t)ψ

(t − x

Wavelets

Wψf (x , s) = |s|−d/2∫

f (t)ψ

(t − x

Wavelets

Wψf (x , s) = |s|−d/2∫

f (t)ψ

(t − x

Wavelets

Wψf (x , s) = |s|−d/2∫

f (t)ψ

(t − x

Wavelets

Wψf (x , s) = |s|−d/2∫

f (t)ψ

(t − x

Wavelets

Wψf (x , s) = |s|−d/2∫

f (t)ψ

(t − x

Wavelets

We now give the definition of a (orthonormal) wavelet.

Definition

A function ψ ∈ L2 (R) is an orthonormal wavelet if the system

ψj ,k (x) := 2j/2ψ(2jx − k

)= 2j/2ψ

(2j(x − 2−jk

is an orthonormal basis.

Hence each signal f ∈ L2 (R) can be expanded in a wavelet series

f (x) =∞∑

j=−∞

∞∑k=−∞

ck,jψj ,k (x)

and ck,j are called the wavelet coefficients.

Wavelets

Definition

ψj ,k (x) := 2j/2ψ(2jx − k

)= 2j/2ψ

(2j(x − 2−jk

is an orthonormal basis.Hence each signal f ∈ L2 (R) can be expanded in a wavelet series

f (x) =∞∑

j=−∞

∞∑k=−∞

ck,jψj ,k (x)

Wavelets

Definition

ψj ,k (x) := 2j/2ψ(2jx − k

)= 2j/2ψ

(2j(x − 2−jk

is an orthonormal basis.Hence each signal f ∈ L2 (R) can be expanded in a wavelet series

f (x) =∞∑

j=−∞

∞∑k=−∞

ck,jψj ,k (x)

The breakthrough of Ingrid Daubechies in 1988

TheoremFor each natural number m one can construct a waveletψ ∈ Cm (R) with compact support.

Wavelets can be constructed in a schematic way using thefollowing result of S. Mallat and Y. Meyer in 1988:

Theorem

Let Vj , j ∈ Z, be a Multiresolutional Analysis of L2 (R) with scalingfunction ϕ. Then there exists a wavelet ψ which is explicitlyconstructed in terms of the scaling function ϕ.

The breakthrough of Ingrid Daubechies in 1988

TheoremFor each natural number m one can construct a waveletψ ∈ Cm (R) with compact support.

Wavelets can be constructed in a schematic way using thefollowing result of S. Mallat and Y. Meyer in 1988:

Theorem

Let Vj , j ∈ Z, be a Multiresolutional Analysis of L2 (R) with scalingfunction ϕ. Then there exists a wavelet ψ which is explicitlyconstructed in terms of the scaling function ϕ.

Multiresolutional Analysis

Definition

A multiresolution analysis (briefly MRA) is a sequence of closedsubspaces Vj , j ∈ Z,of L2 (R) such that

Vj ⊂ Vj+1 for all j ∈ Z,The union ∪j∈ZVj is dense in L2 (R) ,

The intersection ∩j∈ZVj = {0} ,f ∈ Vj if and only if f

(2j ·)∈ V0,

There exists a function ϕ ∈ L2 (R) such that t 7−→ ϕ (t − k) ,for k ∈ Z, is an orthonormal basis of V0.

The function ϕ in the last Definition is usually called the scalingfunction; it is also called the father wavelet.

Definition

Vj ⊂ Vj+1 for all j ∈ Z,

The union ∪j∈ZVj is dense in L2 (R) ,

(2j ·)∈ V0,

Definition

(2j ·)∈ V0,

Definition

The intersection ∩j∈ZVj = {0} ,

f ∈ Vj if and only if f(2j ·)∈ V0,

Definition

(2j ·)∈ V0,

Definition

(2j ·)∈ V0,

Definition

(2j ·)∈ V0,

Nonstationary MRA

DefinitionA nonstationary MRA is obtained if we replace

There exists a function ϕ ∈ L2 (R) such that t 7−→ ϕ (t − k) ,for k ∈ Z, is an orthonormal basis of V0,

by the requirement

For each j ∈ Z there exists a function ϕj ∈ Vj such thatt 7−→ ϕm

(2j t − k

), for k ∈ Z, is an orthonormal basis of Vj .

Fact: Most of the results of MRA carry over in a rather obviousway to non-stationary MRA (by adding just an additional index jto the scaling function ϕ) .

Nonstationary MRA

by the requirement

(2j t − k

Nonstationary MRA

by the requirement

(2j t − k

Subdivison Schemes

Subdivision schemes are used in 3D Computer graphics forrepresenting smooth surfaces.

In the simplest case the data are given by a functionf : Zm → R and one computes

)inductively for k ∈ Z (and j ∈ Zm) from finitely many

function values on level k , hence from f(

)by a simple

subdivision rule.

An important Example of Deslaurier/Dubuc: The

intermediate value f(

)is determined by taking the value

of a cubic polynomial at j/2k+1 interpolating four neighboringpoints at the previous level k .

Subdivison Schemes

)by a simple

subdivision rule.

Subdivison Schemes

)by a simple

subdivision rule.

Subdivison Schemes

)by a simple

subdivision rule.

Subdivison Schemes

The limit function

f (x) = limk→∞

jk2k→ x

is called the limit function of the data.

Typical question: is f continuous/differentiable function?More general, how regular is f ?

Higher regularity leads to better visual presentation.

There is an important connection between Subdivisionschemes and Daubechies wavelets: The autocorrelationfunction of the Daubechies scaling function is equal to thebasis limit function of Deslaurier/Dubuc with data

f0 = 1 and fj = 0 for j ∈ Z, j 6= 0.

Subdivison Schemes

The limit function

f (x) = limk→∞

jk2k→ x

f0 = 1 and fj = 0 for j ∈ Z, j 6= 0.

Subdivison Schemes

The limit function

f (x) = limk→∞

jk2k→ x

f0 = 1 and fj = 0 for j ∈ Z, j 6= 0.

Subdivison Schemes

The limit function

f (x) = limk→∞

jk2k→ x

f0 = 1 and fj = 0 for j ∈ Z, j 6= 0.

Subdivison Schemes

The limit function

f (x) = limk→∞

jk2k→ x

f0 = 1 and fj = 0 for j ∈ Z, j 6= 0.

New methods in Subdivision

Advanced methods: PDE techniques in computer graphics,image processing and data analysis.

In our approach we use polyharmonic functions f : U → R oforder p on an open set U ⊂ Rd defined by the condition

∆P f (x) = 0 for all x ∈ U,

where ∆P is the p-th iteration of the Laplace operator

∆ =∂2

∂x21

+ ...+∂2

∂x2d

Biharmonic functions (hence p = 2) are well known inelasticity problems.

We consider polyharmonic functions of order p as amultivariate generalization of polynomials of degree p.

∆ =∂2

∂x21

+ ...+∂2

∂x2d

∆ =∂2

∂x21

+ ...+∂2

∂x2d

∆ =∂2

∂x21

+ ...+∂2

∂x2d

∆ =∂2

∂x21

+ ...+∂2

∂x2d

∆ =∂2

∂x21

+ ...+∂2

∂x2d

New Ansatz: Polyharmonic Subdivision

Define a subdivision scheme in R2 where the data are givenon the parallel lines

j × R for j ∈ Z.

Use interpolation with polyharmonic functions on equidistantlines j × R for defining the value for the function for the“half-integers lines”

{k + 1

}× R.

By iteration we obtain a ”polyharmonic subdivision scheme”.

We can reduce the ”polyharmonic subdivision scheme” via afamily of non-stationary univariate subdivision schemes.

In these univariate schemes the interpolation with”polynomials of degree 2n − 1” is modified to interpolationwith certain classes of exponential polynomials, so functionsof the form