WAVES & WAVELETSWayne M. Lawton

Department of Mathematics

National University of Singapore

2 Science Drive 2

Singapore 117543Email matwml@nus.edu.sg

Tel (65) 6516-2749

http://www.math.nus.edu.sg/~matwml/courses/Undergraduate/USC/2006/USC3002

This Lecture is Posted on my Homepage at

WAVELETSare functions that oscillate (wiggle) ; they

sometimes model the real world (visual system filters)

WAVELETS

but more often are figments of the mathematical imagination

WAVELETS

although they sometimes look like waves.

WAVESWaves are dynamic (changing in time) wavelets

),( xtWthat describe the real world. Their dynamics isdetermined by differential equations that express physical reality such as the following wave equation

),(),( 2 xtWxtW x

0),( xtWxtxt

WAVESwhich Jean Le Rond D’Alembert (1717-1783) solved

),(),(),( xtWxtWxtW

),0(),( txWxtW

ydxxWyWyW

0),0(),0(),0(2

thus every wave is a superposition of two waves, onemoving to the right and the other moving to the left

),(),( xtWxtW x

WAVE-BASED IMAGING

We sense the world through waves : light & sound

Images show the spatial / temporal distribution of physical quantities include reflectivity (everyday images), transmission (X-ray tomography), and

refractive index (wavefront LASIK)

Image quality is determined by resolution that enables discernment of small details

For simple wave propagation - resolution is obtained by using ‘broadband waves’ – such as short pulses used by bats to determine distance

REAL WORLD WAVESWaves propagation in matter is less simple due to discreteness, inhomogeneity, and nonlinearity

Matter is made of atoms, held apart by electric forces, whose coordinated oscillations make waves

The discrete nature of matter not only complicates thepropagation of sound waves in matter but also effects propagation of electromagnetic waves in matter by causing the speed of light to be frequency dependent. This effect is called dispersion and it explains theprism effect discovered by Newton and chromaticaberration that limits the resolution of imaging devices such as microscopes, cameras, and telescopes

CLASSICAL HARMONIC OSCILLATOR

)()( tutu

is a spring with stiffness = 1 that has one end fixed and the other end attached to an object with mass = 1

Newton’s 2nd and Hook’s Laws

displacement =0)(tu

displacement = u(t)

The state vector

)(

)()(

tu

tutv

satisfies )(

01

10)( tvtv

)0(cossinsincos)0(exp)(

01

10 vttttvttv

therefore

WAVE PROPAGATION IN A CHAIN OF CHO’s

provides a simple model for the propagation of waves in matter that explains exactly how dispersion arises

),)((),( 2 ntuDntu

)1,(),(2)1,(),)(( 2 ntuntuntuntuD

where the second difference operator 2D is defined by

)2,( ntu )1,( ntu ),( ntu )1,( ntu

Newton &Hook tell us that the displacement u(t,n) of the k-th object satisfies the differential-difference Eqn

TRAVELLING WAVE SOLUTIONS

to this discrete wave equation are given by sinusoids

))((cos),( tynyAntu

where ],[ y is the spatial frequency

)2,( ntu )1,( ntu ),( ntu )1,( ntu

)2/(sin2)( yy and is the temporal frequency

and the speed 24

1)(

)(2y

y

yyc

depends on y

GENERAL SOLUTIONSIf we can find an operator

),)((),( 2 ktuDktu

uuu

then the equation 1D

that describes the propagation of u(t,k) becomes

with 212 DD

0),(11 ktuDD tt

and we can seek a decompositionwhere 0),(1 ktuDt

Dirac developed an his electron equation by factoring222 t as a product of 1st order differential

operators using a multiwavelet approach – choosing matrix cofficients (in a Clifford Algebra) – leading to electron spin (and MRI), positrons, 1933 Nobel Prize

FOURIER SERIES

The Fourier transform of the sequence u(t,n) is

],[,)exp(),(),)((

yinyntuytFuZn

therefore

satisfies

22 ))((),))((( yytuDF

therefore the operator

22

1 DD

1D defined by

)(),))((( 1 yiytuDF

FOURIER SERIESThe inverse Fourier transform gives

ydyniytuDFntuD )(exp),())((),()( 121

1

ydyniyiytuF )(exp))((),())((21

ydyniyniimymtu

Zm

)))((exp))(((exp)exp(),( 21

21

21

Zm mn

mn

mn

mnmtu

)(

)(sin

)(

)(sin),(

21

21

21

21

),)(*( ntuK convolution kernel14

)1(8)(

2

m

mmK

m

D’ALEMBERTIAN DECOMPOSITION

),)((),( 2 ktuDktu uuuinto

of a general solution

where 0),(1 ktuDt

uses the initial value sequences ),0( nu and ),0( nu

Step 1 ),0(),0)((),0)(( 121

1 nunuDnuD

Step 2)(

),0))(((),0)(( 1

yi

yuDFyFu

Step 3 Invert Fourier transform of ),0)(( yFu

INTERPOLATED WAVES

We first use the Nyquist-Shannon-Borel-Whittaker-Kotelnikov-Krishan-Raabe-Someya sampling theorem to define the interpolation operator I

Zk kx

kxktuxtIu

)(

)(sin),(),)((

then observe that

),)((),)((),)((),)(( 21

21

1 ktIUktIUxtuDxtIudtd

hence

),]()[(),)(( 3241 xtIuxtIu xxdt

d

MATLAB SIMULATION

at times t = 0, t = 100, t = 1000 of a wave moving with velocity = 1 was computed using Fourier methods anda 2^20 = 1,048,576 point grid

The initial (discrete) wave consisted of samples of aGaussian function with mean = 0 and sigma = 2. The waves a t = 100 and t = 1000 were translated to the

left by 100 and 1000 to compare the dispersive effectsThe Fourier transform of the initial wave is, by Poisson’s Summation Formula, a theta function (> 0) and at time t the Fourier transform (of the left translated wave) is multiplied by exp it(w(y)-y )

DISPERSION

FOURIER PICTURE

INHOMOGENEOUS WAVE PROPAGATION

occurs if the masses (and/or stiffnesses) are random

),)((),( 2 ntuDntumn

by defining we obtain

)2,( ntu )1,( ntu ),( ntu )1,( ntu

2nm 1nm nm 1nm

),(),(~ ntumntu n),)(~~

(),(~2 ntuDntu

11

2

)1,(~)1,(~2)1,(~),)(~~

(

nnnnn mm

ntu

m

ntu

mm

ntuntuD

with self-adjoint 2

~D

INHOMOGENEITY LOCALIZATION

The spectral theorem gives the general solution

where

We will illustrate the localization property of the eigenfunctions by computing them for512 – periodic waves and m’s uniform on [2,3].

dnEtiAntu )())((cos)(),(~

)(nE

)())(~

( 22 nEnED

Then 2

~D is an oscillation matrix (with total positivity

properties related to splines) AND a random matrix.

LOCALIZATIONREDUCED PROPAGATION

since the high frequency eigenvectors are localized,they can help propagation beyond their support.

The high frequency components of waves that impacta random inhomogeneous media are scattered back.

This backscattering can be attributed to impedance.

Backscattering causes extreme image degradation.

But it can be wisely exploited, by radiating a protein molecule at a frequency corresponding to a localizedeigenvector it can possibly be split at that local.

NONLINEARITIES

since Hook’s Law only approximates the real world

The Lennard-Jones potential gives a realistic model for the interatomic forces.

This is the KdV Equation - it has soliton solutions.

The resulting approximate wave equation is

Solitons describe important biophysical processes including growth of microtubules during mitosis.

),(),(),( 3241 xtSxtSxxtS xx

STATE OF THE ART BIOIMAGINGdemands methods based on quantum mechanics and

includes MRI (magnetic resonance imaging), which utilizes electron spin (predicted by Dirac’s Equation,SQUID (super quantum interference device) that can image the firing of single nerve cells, and the work of

Su WW, Li J, Xu NS, State and parameter estimation

of microalgal photobioreactor cultures based on local irradiance measurement, J. Biotechnology, (2003) Oct 9, 105(1-2):165-178. Local photosynthetic photon flux fluence rate determined by a submersible 4pi quantum micro-sensor was used developing a versatile on-line estimator for stirred-tank microalgalphotobioreactor cultures.

QUANTUM HARMONIC OSCILLATOR

),()(),( 22 xtxixtdt

dx

is described by solutions of Schrodinger’s Equation

where

),)(exp())((exp1

),( 02

021

4xtyitxxxt

represent the probability densities for the objectsposition and momentum (mass x velocity). He shared the 1933 Nobel Prize with Dirac. He also found that

2|),(| xt and 2|),(ˆ| xt

)sin()(),cos()( 00 tAtytAtxwhere are the position and momentum for the CHO, are thesolutions of the QHO that have minimal and equal uncertainly ( = ½) in both position and momentum.

COHERENT STATES AND GABOR WAVELETS

R. J. Glauber, Physical Review 131 (1963) 2766 coined the term coherent states for these solutions, proved that they were produced when a classical electrical current interacts with the electromagnetic field, and thus introduced them to quantum optics.

but a single measurement will yield an

Quantum mechanics shows that all measurements areinherently noisy – the energy in the coherent state is

2/2AE

energy = n with probability!

)(exp

n

EE n

this is a Poisson Distribution hence has variance = E

REFERENCES WITH COMMENTS

2nd derivative of gaussian in vision & edge detectionhttp://iria.math.pku.edu.cn/~jiangm/courses/dip/html/node91.html

and a more mathematical treatment in

Hurt, Norman, Phase Retrieval and Zero Crossings – mathematical methods in image reconstruction, Kluwer, Dordrecht, 1989.

Marr, David, Vision : a computational investigation into the human representation and processing of visual information, W.H. Freeman, New York,1982.

REFERENCES WITH COMMENTSgeneral introduction to optics

Goodman, Joseph, Introduction to Fourier optics,

New York : McGraw-Hill, NY, 1996.

oscillation matrices & total positivity

Jenkins, Francis and White, Harvey Fundamentals of Optics, McGraw-Hill, Singapore, 1976.

Gantmacher, F.P. and Krein, M.G., Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, AMS, Providence, RI, 2002.

Karlin, Samuel, Total Positivity, Stanford University Press, Stanford, CA, 1968