Wavelet Analysis in Signal and Image Processing

Wavelet Analysis in Signal and Image Processing

Jean-Pierre Gazeau

Laboratoire Astroparticules et Cosmologie CNRS–Universit´e Diderot Paris 7,

[email protected]

University of Palermo January 14, 2010

1. Introduction to wavelet analysis (a) Hilbert and Fourier : notations

(b) Time-frequency representation : the windowed Fourier or continuous Gabor transform (1D CGT)

(c) One-dimensional continuous wavelet transform (1D CWT) (d) Implementation and interpretation

(e) About the discretization problem

(f) One-dimensional discrete wavelet transform (1D DWT) (g) Multiresolution analysis

2. Wavelet analysis and image processing

(a) Two-dimensional continuous wavelet transform (2D CWT) (b) Two-dimensional discrete wavelet transform (2D DWT)

Lab session 1. 1-D Transform : time-frequency, time-scale 2. 2-D Transform : space parameters, angle-scale

REFERENCES

1. St´ephane Mallat A Wavelet Tour of Signal Processing Academic Press; 2nd edition (1999)

2. Ingrid Daubechies Ten Lectures on Wavelets SIAM 1992

3. S.T. Ali, J.P. Antoine, and JPG, Coherent states and wavelets, a mathemat- ical overview ,

Graduate Textbooks in Contemporary Physics (Springer, New York) (2000) 4. Matlab Wavelet Toolbox

FOURIER SIGNAL ANALYSIS

Signal :

t

“time”

→ s(t) ∈ C

Finite energy signal :

∞

Z

−∞

|s(t)|

²

dt < ∞,

i.e.

s ∈ L

²

(R)

L

²

(R)

: Hilbert space with scalar product

hs

₁

|s

₂

i =

∞

Z

−∞

s

₁

(t)s

₂

(t) dt

Fourier transform : frequency content of signal

ˆ

s(ω)(≡ (F s)(ω)) = 1

√ 2π

∞

Z

−∞

e

^−iωt

s(t) dt

RECONSTRUCTION

s(t)(≡ (F

⁻¹

s)(t)) = ˆ 1

√ 2π

∞

Z

−∞

e

^iωt

s(ω) dω ˆ

PLANCHEREL

hs

₁

|s

₂

i =

∞

Z

−∞

s

₁

(t)s

₂

(t) dt =

∞

Z

−∞

ˆ

s

₁

(ω)ˆ s

₂

(ω) dω = hˆ s

₁

|ˆ s

₂

i

⇒

ENERGY CONSERVATION

ksk² ≡ hs|si =

∞

Z

−∞

|s(t)|²dt =

∞

Z

−∞

|ˆs(ω)|²dω = kˆsk²

LIKE IN EUCLIDEAN GEOMETRY ...

Signal :

s(t) ≡

vector

|si

ou

|s(t)i

Elementary signal or atom : ^√1_2π

e

^iωt

≡

continuous orthonormal basis vector

h 1

√ 2π e

^iωt

| 1

√ 2π e

^iω0t

i = 1 2π

∞

Z

−∞

e

^i(ω0−ω)t

dt

= δ(ω

⁰

− ω)

[orthonormality]

I =

∞

Z

−∞

| 1

√ 2π e

^iωt

ih 1

√ 2π e

^iωt

| dω

[basis]

Euclidean decomposition in elementary signals :

|s(t)i =

∞

Z

−∞

h 1

√ 2π e

^iωt

|s(t)i

| {z }

Fourier transform

| 1

√ 2π e

^iωt

i dω

Example of signal

0 500 1000 1500 2000 2500 3000 3500

−10

−8

−6

−4

−2 0 2 4

TIME-FREQUENCY REPRESENTATIONS (“Gaborets”) Ingredients: translation + modulation.

It is also called the windowed Fourier transform.

One chooses a probe or window

ψ

which is well localized in time and frequency at once, and which is normalized,

kψk = 1

. The probe is then translated in time and frequency, but its size is not modified (in modulus)

ψ(t) → ψ

_b,ω=¹

a

(t) = e

^iat

ψ(t − b) = e

^iωt

ψ(t − b)

The time-frequency transform is then :

s(t) → S(b, ω) = hψ

_b,ω

|si =

Z

+∞

−∞

e

^−iωt

ψ(t − b)s(t) dt.

It is easy to prove that there is conservation of the energy :

ksk

²

=

Z

+∞

−∞

|s(t)|

²

dt =

Z

+∞

−∞

Z

+∞

−∞

|S(b, ω)|

²

db dω 2π

def

= kSk

²

,

and so the reciprocity or reconstruction formula:

s(t)) =

Z

+∞

Z

+∞

S(b, ω)e

^iωt

ψ(t − b) db dω

.

Vostok temperatures

500 1000 1500 2000 2500 3000

−10

−5 0 5

Time variations of Vostok temperatures

0.1 0.2 0.3 0.4

0 5

10

x 10⁵

Fourier transform

Time−frequency representation of Vostok temperature

500 1000 1500 2000 2500 3000

20 40 60 80 100 120

Gabor Transform of the signal superpos.mat

500 1000 1500 2000

−2

−1 0 1 2

0.02 0.04 0.06 0.08 0.1 0.12

0 2

4 6

x 10⁴

500 1000 1500 2000

20 40 60 80 100 120

CONTINUOUS WAVELET TRANSFORM Ingredients (transport + zoom)

1. A mother wavelet or probe

ψ(t) ∈ L

²

(R)

(a) well localized

(b) zero average :

∞

R

−∞

ψ(t) dt = 0

more precisely :

0 < c

_ψ

≡

∞

Z

−∞

| ˆ ψ(ω)|

²

dω

|ω| =

∞

Z

−∞

| ˆ ψ(−ω)|

²

dω

|ω| < ∞

2. The (continuous) family of translated-dilated-contracted versions of the probe

ψ

:

 1

√ a ψ  t − b a



a∈R^?₊,b∈R

Then :

1. ^n√1_aψ ^t−b_a
o

a∈R^?+,b∈R forms an overcomplete family in

L

²

(R)

, which means that any signal decomposes as

s(t) = 1 c

_ψ

∞

Z

−∞

db

∞

Z

0

da

a

²

S(b, a) 1

√ a ψ  t − b a



2. The “coefficient”

S(b, a)

, as a function of the two continuous variables

b

(time) and

a

(scale), is the wavelet transform of the signal :

S(b, a) = h 1

√ a ψ  t − b a



|si

=

∞

Z

−∞

√ 1

a ψ  t − b a



s(t) dt

=

∞

Z

−∞

√ ae

^iωb

ψ(aω)ˆ ˆ s(ω) dω

1. Equivalent to 1 and 2 : resolution of the unity

I = 1 c

_ψ

∞

Z

−∞

db

∞

Z

0

da a

²

   

√ 1

a ψ  t − b a

  1

√ a ψ  t − b a

   

2. Equivalent to 1 and 2 and 3 : energy conservation

ksk

²

= 1 c

_ψ

∞

Z

−∞

db

∞

Z

0

da

a

²

|S(b, a)|

²

In practice, one imposes additional constraints on

ψ

: 1. restrictions on the supports of

ψ

or

ψ ˆ

2. vanishing of higher order moments :

∞

Z

−∞

t

^p

ψ(t) dt = 0, p = 0, 1, . . . , p

_max

Then the wavelet transform ignores polynomial components of the signal (i.e. most regular or smoother parts ) in order to enhance the most singular aspects.

WAVELET = SINGULARITY DETECTOR

EXAMPLES OF WAVELETS 1. Morlet wavelet : modulated gaussian







ψ

_M

(t) = π

⁻¹⁴



e

^iωmt

− e

^−ω22m



e

^−t22

, ψ ˆ

_M

(ω) = π

⁻¹⁴



e

^{−(ω−ωm)22}

− e

^−ω22

e

^−ω22m

 ,

2. Mexican hat : second derivative of the gaussian, two first moments vanish

(

ψ

_H

(t) = (1 − t

²

)e

^−t22

= −

_dt^d2₂

e

^−t22

,

ψ ˆ

_H

(ω) = ω

²

e

^−ω22

.

0 500 1000 1500 2000 2500 3000 3500

−10

−5 0 5

CWT of the Vostok temperature data: 2sd der. of Gaussian, width=0.5

500 1000 1500 2000 2500 3000

20 40 60 80 100 120

INTERPRETATION Efficiency of the wavelet transform due to

•

condition(s) of admissibility :

∞

R

−∞

t

^p

ψ(t) dt = 0

,

•

constraints on support of

ψ

. Indeed, if –

ψ

has support of length

≈ T

around

0

–

ψ ˆ

has support of length

≈ Ω

around

ω

₀ Then

T Ω = cste

(Fourier-Heisenberg) and

–

  

√1

a

ψ(

^t−b_a

)  



has support of length

≈ aT

around

b

, –

  

√ a ˆ ψ(aω)  



a support of length

≈ Ω/a

around

ω

₀

/a

(relative band- width

∆ω/ω ≈

cste).

Consequences

•

if

a  1

, then

– ^√1_a

ψ(

^t−b_a

)

is wide window, –

e

^−iωb

√

a ˆ ψ(aω)

is very sharp,

and the CWT reacts mainly to low frequencies (low-band filter),

•

if

a  1

, then

– ^√1_a

ψ(

^t−b_a

)

is narrow window, –

e

^−iωb

√

a ˆ ψ(aω)

is wide window,

and the CWT reacts mainly to high frequencies (high-band filter) while offering an efficient temporal localization.

CONCLUSION The wavelet transform

s(t) → S(b, a) =



 



 



∞

R

−∞

√1

a

ψ(

^t−b_a

)s(t) dt

∞

R

−∞

√ ae

^iωb

ψ(aω)ˆ ˆ s(ω) dω

acts as a local filter, for time and scale at once : it selects the part of the signal possibly concentrated around instant

b

and scale

a

.

Furthermore, if the wavelet

ψ

is well localized, then the energy density

|S(b, a)|

² of the CWT will be concentrated on those parts of the signal which are the most significant in terms of information.

The CWT acts as a

mathematical microscope :

• ψ ≡

optics

• b ≡

position

• 1/a ≡

global magnification

DISCRETIZATION PROBLEMS Redundancy due to continuum of the wavelet representation

s( t

|{z}

1D

) = 1 c

_ψ

∞

Z

−∞

db

∞

Z

0

da

a

²

S( b, a

|{z}

2D

) 1

√ a ψ  t − b a



The CWT unmixes parts of signal which live at at same instants, but at different scales

Discretization : might eliminate this redundancy through the choice of a minimal grid

Γ = {(b

_k

, a

_j

), j, k ∈ Z}

in the half-plane time-scale

R × R

^?+

= {(b, a)}

s(t) = X

j,k∈Z

hψ

_bkaj

|si ˜ ψ

_bkaj

(t)

| {z }

dual frame

A good grid is that one for which there exists

0 < m ≤ M

s.t.

mksk

²

≤ X

j,k

|hψ

_bkaj

|si|

²

≤ M ksk

²

One speaks of discrete frame, of resolving power

M − m

M + m .

If

m = M

, then the frame is tight. It is orthonormal basis if

m = M = 1

.

EXAMPLE

Choice induced by the non-euclidean geometry (Lobatchevski) of the time-scale half-plane :

a

_j

= a

^j₀

, b

_k

= kb

₀

a

^j₀

, a

₀

> 0, k, j ∈ Z ψ

_bkaj

(t) = a

^−j/2₀

ψ(a

^−j₀

t − kb

₀

)

Dyadic wavelets are obtained with :

a

₀

= 2, b

₀

= 1

(but the approach is totally different from Discrete Wavelet Transform)

1D-DISCRETE WAVELET ANALYSIS

FIRST STEP IN DISCRETE WAVELETS (HAAR) : REFINING PERIODIC SAMPLINGS

Smoothing + sampling of signal

s(t)

at the integer scale

n ∈ Z

(scale “zero”)

(Π

₀

s)(t) ≡ s

₀

(t) = X

n

hϕ(t − n)|si

| {z }

˜ ϕ?s(n)

ϕ(t − n) ∈ V

₀

where

• ϕ(t)

: characteristic function of

[0, 1]

(scaling function or father wavelet)

• ˜ ϕ(t)

^def

= ϕ(−t)

• f ∗ g(t)

^def

= R

+∞

−∞

f (u − t)g(t) dt

(convolution)

•

Also:

hϕ(t − n)|si = R

+∞

−∞

ϕ(t − n)s(t) dt = R

n+1

n

s(t) dt

is the average os the signal on the interval

[n, n + 1]

• {ϕ(t − n)}

_n∈Z : orthonormal system spans subspace

V

₀

' l

²

(Z)

^{of signal} constant on intervals

[n, n + 1]

.

Smoothing + finer sampling of signal s(t)

at the half-integer scale ( scale 1), i.e. on V

₁

' l

²

(Z/2) : (Π

₁

s)(t) = X

n

h √

2ϕ(2t − n)|si

| {z }

˜

ϕ₁?s(ⁿ₂)

√

2ϕ(2t − n) ∈ V

₁

SECOND STEP IN DISCRETE WAVELETS:

HAAR WAVELET

•

Information at scale one

=

Information at scale zero or Approximation

+

Details :

(Π

₁

s)(t) = (Π

₀

s)(t) + (∆

₀

s)(t) ⇔ V

₁

= V

₀

⊕ W

₀

•

Details :

(∆

₀

s)(t) = X

n

hψ(t − n)|si

| {z }

d_0,n

ψ(t − n) ∈ W

₀

•

Mother wavelet or Haar wavelet :

ψ(t) = ϕ(2t) − ϕ(2t − 1)

• {ψ(t − n)}

_n∈Z

:

orthonormal system spans

W

₀ of details at scale 1, or- thogonal complement of

V

₀ in

V

₁

THIRD STEP IN DISCRETE WAVELETS:

HAAR TRANSFORM AND MULTIRESOLUTION ANALYSIS

Information (or Tendency) at scale

j + 1

=

Tendency at

j +

Fluctuations (Details) :

(Π

_j+1

s)(t) = (Π

_j

s)(t) + (∆

_j

s)(t) ⇔ V

_j+1

= V

_j

⊕ W

_j

(Π

_j

s)(t) = X

n

h2

^j/2

ϕ(2

^j

t − n)|si

| {z }

˜

ϕj?s(n/2^j)

2

^j/2

ϕ(2

^j

t − n) ∈ V

_j

(∆

_j

s)(t) = X

n

h2

^j/2

ψ(2

^j

t − n)|si

| {z }

d_j,n

2

^j/2

ψ(2

^j

t − n) ∈ W

_j

d

_j,n

=

∞

R

−∞

2

^j/2

ψ(2

^j

t − n)s(t) dt

: Wavelet Transform of signal Multiresolution analysis of

L

²

(R)

^:

· · · V

_j−1

⊂ V

_j

⊂ V

_j+1

· · · {2

^j/2

ψ(2

^j

t − n)}

_j,n∈Z : orthonormal basis of

L

²

(R) = L

W

SUMMARY :

Original Signal (or smoothing + sampling on fine grid

Z/2

^N):

(Π

_N

s)(t) ≡ s

_N

(t) ∈ l

²

(Z/2

^N

) ≡ V

_N

V

_N

= V

_{N −1}

⊕ W

_{N −1}

= V

_{N −2}

⊕ [W

_{N −2}

⊕ W

_{N −1}

] =

· · · = V

₀

tendency

|{z}

⊕ [W

₀

⊕ W

₁

+ · · · + ⊕W

_{N −1}

]

| {z }

f luctuation

Corresponds to analysis :

s

_N

= s

₀

+ [r

₀

+ r

₁

+ · · · r

_{N −1}

] ≡ (s

₀

, r

₀

, r

₁

, . . . , r

_{N −1}

) r

_j

= ∆

_j

s = X

n

d

_j,n

2

^j/2

ψ(2

^j

t − n) ≡ (d

_j,n

)

_n∈Z Discrete wavelet transform :

d

_j,n

=

∞

R

−∞

2

^j/2

ψ(2

^j

t − n)s(t) dt =

∞

R

−∞

2

^j/2

ψ(

^t−2₂−j^−jn

)s(t) dt 2

^−j : dilation (

j > 0

) or contraction (

j < 0

) parameter (scale)

2

^−j

n

: translation parameter (localization)

0 500 1000 1500 2000 2500 3000 3500

−10 0 10

Signal

Temperatures

0 200 400 600 800 1000 1200 1400 1600 1800

−2 0 2

L=3 coef ondel

0 100 200 300 400 500 600 700 800 900

−5 0 5

L =2 coef ondel

0 50 100 150 200 250 300 350 400 450

−5 0 5

L=1 coef ondel

0 50 100 150 200 250

−10 0 10

L=0 coef ondel

−50 0 50

Ingredients of multiresolution analysis : 1. A scaling function

ϕ(t) ∈ L

²

(R)

^s.t.

{ϕ(t − n)}

_n∈Z is orthonormal system 2. Space

V

₀ linear span of

{ϕ(t − n)}

_n∈Z

3. Sequence

· · · V

_j−1

⊂ V

_j

⊂ V

_j+1

· · ·

defined by

f (t) ∈ V

₀

⇔ f (2

^j

t) ∈ V

_j and s.t.

∩

_j

V

_j

= 0

,

∪

_j

V

_j dense in

L

²

(R)

4. A wavelet, i.e. a function

ψ(t)

s. t.

{ψ(t − n)}

_n∈Z spans the orthogonal complement

W

₀ of

V

₀ in

V

₁

= V

₀

⊕ W

₀

Then :

1.

{2

^j/2

ψ(2

^j

t − n)}

_j,n∈Z orthonormal basis of

L

²

(R)

2. Any signal

s(t)

decomposes as

s(t) = P

j,n

d

_j,n

2

^j/2

ψ(2

^j

t − n)

3. The coefficient

d

_j,n

=

∞

R

−∞

2

^j/2

ψ(2

^j

t − n)s(t) dt = h2

^j/2

ψ(2

^j

t − n)|s(t)i

, as function of discrete variables

j

et

n

, is the wavelet transform of the signal.

EXAMPLES OF WAVELETS

1. compact support : Haar

≡

Daubechies 1, Shannon (FT of Haar), Daubechies 2, Daubechies 3, ....

2. noncompact support 3. biorthogonal etc.

Daubechies Wavelet Ordre 2

0 500 1000 1500 2000 2500 3000 3500

−10 0 10

Signal

Temperatures

0 200 400 600 800 1000 1200 1400 1600 1800

−2 0 2

L=5 coefond

0 100 200 300 400 500 600 700 800 900

−2 0 2

L=4 coefond

0 50 100 150 200 250 300 350 400 450

−5 0 5

L=3 coefond

0 50 100 150 200 250

−5 0 5

L=2 coefond

0 20 40 60 80 100 120

−20 0 20

L=1 coefond

0 10 20 30 40 50 60

−20 0 20

L=0 coefond

−100 0 100

0 2 4 6 8 10 12 14 16 18 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

k

I( k )

−18

−16

−14

−12

−10

−8

−6

l/2^j

j

dj,l

0 5 10 15

0 1 2 3 4 5 6 7 8

BASICS OF IMAGE PROCESSING

•

A digital image

a[m, n]

described in a 2D discrete space is derived from an analog image

a(x, y)

in a 2D continuous space through a sampling process that is frequently referred to as digitization.

•

The 2D continuous image

a(x, y)

is divided into

N

rows and

M

columns.

The intersection of a row and a column is termed a pixel. The value assigned to the integer coordinates

[m, n]

with

{m = 0, 1, 2, . . . , M − 1}

and

{n =

0, 1, 2, . . . , N − 1}

is

a[m, n]

. In fact, in most cases

a(x, y)

–which we might consider to be the physical signal that impinges on the face of a 2D sensor–is actually a function of many variables including depth

(z)

, color

(λ)

, and time

(t)

. Unless otherwise stated, we will consider the case of 2D, static images.

Digitization of a continuous image

•

The pixel at coordinates

[m = 10, n = 3]

has the integer brightness value 110.

•

This image has been divided into

N = 16

rows and

M = 16

columns.

•

The value assigned to every pixel is the average brightness in the pixel rounded to the nearest integer value.

•

The process of representing the amplitude of the 2D signal at a given coor- dinate as an integer value with

L

different gray levels is usually referred to

There are standard values for the various parameters encountered in digital image processing. These values can be caused by video standards, by algorithmic requirements, or by the desire to keep digital circuitry simple.

Commonly encountered values :

Rows Columns Gray Levels

Parameter

N M Λ

Typical values 256, 512, 256, 512, 2, 64, 525, 625, 768, 1024, 256, 1024,

1024, 1035 1320 16384

Quite frequently we see cases of

M = N = 2K

where

{K = 8, 9, 10}

. This can be motivated by digital circuitry or by the use of certain algorithms such as

the fast Fourier transform.

The number of distinct gray levels is usually a power of 2, that is,

Λ = 2B

where

B

is the number of bits in the binary representation of the brightness levels. When

B > 1

we speak of a gray-level image; when

B = 1

we speak of

a binary image. In a binary image there are just two gray levels which can be referred to, for example, as “black” and “white” or “0” and “1”.

HISTOGRAMS

More and more cameras let you view histograms on the camera’s monitor. The histogram, like those found in most serious photo-editing programs such as

Photoshop and Picture Window, let you evaluate the distribution of tones.

Since most image corrections can be diagnosed by looking at a histogram, it helps to look at it while still in a position to reshoot the image. Each pixel in an

image can be set to any of 256 levels of brightness from pure black (0) to pure white (255). A histogram is a graph that shows how the 256 possible levels of

brightness are distributed in the image.

How to read a Histogram

How to read a Histogram

•

The horizontal axis represents the range of brightness from 0 (shadows) on the left to 255 (highlights) on the right. Think of it as a line with 256 spaces on which to stack pixels of the same brightness. Since these are the only values that can be captured by the camera, the horizontal line also represents the camera’s maximum potential dynamic range.

•

The vertical axis represents the number of pixels that have each one of the 256 brightness values. The higher the line coming up from the horizontal axis, the more pixels there are at that level of brightness.

•

To read the histogram, you look at the distribution of pixels. An image that uses the entire dynamic range of the camera will have a reasonable number of pixels at every level of brightness. An image that has low contrast will

Example

This high-key fog scene has most of its values towards the highlight end of the scale. The distinct vertical line to the left of middle gray shows how many pixels there are in the uniformly gray frame border. You can see that there are

no really dark values in the image. In fact, the image uses only a little more than half the camera’s dynamic range.

CONTINUOUS WAVELET TRANSFORM OF IMAGES

A wavelet or probe

ψ(x)

,

x = (x

₁

, x

₂

) ∈ R

^{2 is chosen}

•

well localized

•

admissible, which means

c

_ψ

=

cste

Z Z

R²

| ˆ ψ(k)|

²

d

²

k

kkk

²

< ∞ ⇔ ˆ ψ(0) = 0

Image analysis is carried out by affine transport of the probe

ψ

in the euclidean plane :

•

Dilation/contraction (

a > 0

)

•

Translations (

b = (b

_x

, b

_y

) ∈ R

²⁾

•

Rotations (

θ ∈ [0, 2π]

)

ψ(x) → ψ

_b,θ,a

(x) ≡

_a¹

ψ R

⁻¹

(θ)

^x−b_a

Wavelet transform of a 2D signal :

L

²

(R

²

) 3 s(x) → S(b, θ, a) = Z Z

R²

ψ

_b,θ,a

(x)s(x) d

²

x

= Z Z

R²

ψ ˆ

_b,θ,a

(k)ˆ s(k) d

²

k

ˆ

s(k) = 1 2π

Z Z

e

^−ik·x

s(x) d

²

x, ˆ ψ

_b,θ,a

(k) = ae

^−ib·k

ψ(aR ˆ

⁻¹

(θ)k)

RECONSTRUCTION

s(x) = 1 c

_ψ

Z Z

R²

d

²

b

∞

Z

0

da a

³

2π

Z

0

dθS(b, θ, a)ψ

_b,θ,a

(x)

ENERGY CONSERVATION

Z Z

R²

|s(x)|

²

d

²

x = 1 c

_ψ

Z Z

R²

d

²

b

∞

Z

0

da a

³

2π

Z

0

dθ|S(b, θ, a)|

²

RESOLUTION OF THE UNITY

I = 1 c

_ψ

Z Z

R²

d

²

b

∞

Z

0

da a

³

2π

Z

0

dθ|ψ

_b,θ,a

ihψ

_b,θ,a

|

EXAMPLES OF 2D WAVELETS 1. Isotropic

•

Mexican Hat (Marr) :

ψ

_H

(x) = −∆e

^−kxk2/2

= (2 − kxk

²

)e

^−kxk2/2

•

Difference wavelets (h(x)regular> 0):

ψ(x) = α

⁻²

h(α

⁻²

x) − h(x), 0 < α < 1

2. Orientational

•

Morlet wavelet :

ψ

_M

(t) = e

^ik0·x

e

^−kAxk22 + small correction terms s. t. ψ^ˆ_H(0) = 0 A =diag[^√1_

1,^√1_

2](anisotropy matrix)

•

Cauchy wavelet :

ψ ˆ

has support in strictly convex cone

Anisotropic mexican hat

Anisotropic mexican hat

INTERPRETATION

If the wavelet

ψ

is well localized in

x

and in

k

at once, then the wavelet analysis acts with constant relative bandwidth : ^∆kkk_kkk

=

cste

⇒

efficiency at

large frequencies or at small scales

⇒

detector of discontinuities in images

•

e.g. point singularities ( contour vertices)

•

e.g. orientational features (borders, edges, segments, mikado)

Consequently The 2D CWT acts as a

orientational mathematical microscope :

• ψ ≡optics

• b ≡position

• 1/a ≡global zoom

• θ ≡orientation parameter

VISUALIZATION

6 possible choices of two-dimensional sections of the CWT

S(b, θ, a)

in space of parameters

(b

_x1

, b

_x2

, θ, a)

,

e.g. :

•

representation position :

(θ, a)

is fixed

•

representation direction-scale :

(b

_x1

, b

_x2

)

is fixed

DISCRETE WAVELET TRANSFORM OF IMAGES

One here comes back to dyadic multiresolution: a wavelet orthonormal basis in

L

²

(R

²

)

is built up from (tensor) products involving

•

a scale function

ϕ

associated to a multiresolution

{V

_j

}

_j∈Z of

L

²

(R)

•

a wavelet

ψ

whose the dilated-translated

2

^j/2

ψ(2

^j

t − n)

form an orthonor- mal basis of

L

²

(R) = L

j

W

_j

For this purpose, one defines three wavelets :

ψ

¹

(x

₁

, x

₂

) = ϕ(x

₁

)ψ(x

₂

)

(horizontal)

ψ

²

(x

₁

, x

₂

) = ψ(x

₁

)ϕ(x

₂

)

(vertical)

, ψ

³

(x

₁

, x

₂

) = ψ(x

₁

)ψ(x

₂

)

(diagonal)

,

and one puts, for

1 ≤ k ≤ 3

,

ψ

_j,n^k

(x) = 2

^j

ψ

^k

(2

^j

x

₁

− n

₁

, 2

^j

x

₂

− n

₂

)

Then,

• {ψ

_j,n¹

, ψ

_j,n²

, ψ

_j,n³

}

form an orthonormal basis of the subspace of details

W

_j²

= (V

_j

⊗ W

_j

) ⊕ (W

_j

⊗ V

_j

) ⊕ (W

_j

⊗ W

_j

)

at scale

j

• L

²

(R

²

) = L

j

W

_j²

•

The whole image

s(x)

decomposes as

s(x) = P

k,j,n

d

^k_j,n

2

^j

ψ

_j,n^k

(x)

•

The coefficient

d

^k_j,n

=

∞

Z

−∞

2

^j

ψ

^k

(2

^j

x

₁

− n

₁

, 2

^j

x

₂

− n

₂

)s(x) d

²

x = hψ

_j,n^k

|s(t)i,

as function of the three discrete variables

k

,

j

and

n

, is the discrete wavelet transform of the image.

EXAMPLES OF WAVELETS

1. compact support : Haar

≡

Daubechies 1, Shannon (TF de Haar), Daubechies 2,

Daubechies 3, ....

2. noncompact support 3. biorthogonal etc.