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WAVES & WAVELETSWayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543Email [email protected]
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