WAVES

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WAVES WAVELETS

• Wayne M. Lawton
• Department of Mathematics
• National University of Singapore
• 2 Science Drive 2
• Singapore 117543

Email matwml_at_nus.edu.sg Tel [phone]
This Lecture is Posted on my Homepage at
http//www.math.nus.edu.sg/matwml/courses/Undergr
aduate/USC/2006/USC3002
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WAVELETS
are functions that oscillate (wiggle) they
sometimes model the real world (visual system
filters)
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WAVELETS
but more often are figments of the mathematical
imagination
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WAVELETS
although they sometimes look like waves.
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WAVES
Waves are dynamic (changing in time) wavelets
that describe the real world. Their dynamics
is determined by differential equations that
express physical reality such as the following
wave equation
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WAVES
which Jean Le Rond DAlembert (1717-1783) solved
thus every wave is a superposition of two waves,
one moving to the right and the other moving to
the left
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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
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REAL WORLD WAVES
Waves 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 the propagation 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
the prism effect discovered by Newton and
chromatic aberration that limits the resolution
of imaging devices such as microscopes, cameras,
and telescopes
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CLASSICAL HARMONIC OSCILLATOR
is a spring with stiffness 1 that has one end
fixed and the other end attached to an object
with mass 1
displacement 0
displacement u(t)
Newtons 2nd and Hooks Laws ?
The state vector
satisfies
therefore
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WAVE PROPAGATION IN A CHAIN OF CHOs
provides a simple model for the propagation of
waves in matter that explains exactly how
dispersion arises
Newton Hook tell us that the displacement u(t,n)
of the k-th object satisfies the
differential-difference Eqn
where the second difference operator
is defined by
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TRAVELLING WAVE SOLUTIONS
to this discrete wave equation are given by
sinusoids
where
is the spatial frequency
and
is the temporal frequency
and the speed
depends on
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GENERAL SOLUTIONS
If we can find an operator
with
then the equation
that describes the propagation of u(t,k) becomes
and we can seek a decomposition
where
Dirac developed an his electron equation by
factoring
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
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FOURIER SERIES
The Fourier transform of the sequence u(t,n) is
therefore
therefore the operator
defined by
satisfies
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FOURIER SERIES
The inverse Fourier transform gives
convolution kernel
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DALEMBERTIAN DECOMPOSITION
of a general solution
into
where
uses the initial value sequences
and
Step 1
Step 2
Step 3
Invert Fourier transform of
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INTERPOLATED WAVES
We first use the Nyquist-Shannon-Borel-Whittaker-
Kotelnikov-Krishan-Raabe-Someya sampling theorem
to define the interpolation operator
then observe that
hence
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MATLAB SIMULATION
at times t 0, t 100, t 1000 of a wave
moving with velocity 1 was computed using
Fourier methods and a 220 1,048,576 point grid
The initial (discrete) wave consisted of samples
of a Gaussian 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 effects
The Fourier transform of the initial wave is, by
Poissons Summation Formula, a theta function
(gt 0) and at time t the Fourier transform (of
the left translated wave) is multiplied by exp
it(w(y)-y )
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DISPERSION
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FOURIER PICTURE
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INHOMOGENEOUS WAVE PROPAGATION
occurs if the masses (and/or stiffnesses) are
random
by defining
we obtain
with self-adjoint
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INHOMOGENEITY ? LOCALIZATION
The spectral theorem gives the general solution
where
We will illustrate the localization property of
the eigenfunctions by computing
them for 512 periodic waves and ms uniform on
2,3.
Then
is an oscillation matrix (with total positivity
properties related to splines) AND a random
matrix.
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LOCALIZATION?REDUCED PROPAGATION
since the high frequency eigenvectors are
localized, they can help propagation beyond their
support.
The high frequency components of waves that
impact a 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 localized eigenvector it can possibly be split
at that local.
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NONLINEARITIES
since Hooks Law only approximates the real world
The Lennard-Jones potential gives a realistic
model for the interatomic forces.
The resulting approximate wave equation is
This is the KdV Equation - it has soliton
solutions.
Solitons describe important biophysical processes
including growth of microtubules during mitosis.
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STATE OF THE ART BIOIMAGING
demands methods based on quantum mechanics and
includes MRI (magnetic resonance imaging), which
utilizes electron spin (predicted by Diracs
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 microalgal photobioreactor cultures.
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QUANTUM HARMONIC OSCILLATOR
is described by solutions of Schrodingers
Equation
where
and
represent the probability densities for the
objects position and momentum (mass x velocity).
He shared the 1933 Nobel Prize with Dirac. He
also found that
where
are the position and momentum for the CHO, are
the solutions of the QHO that have minimal and
equal uncertainly ( ½) in both position and
momentum.
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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.
Quantum mechanics shows that all measurements
are inherently noisy the energy in the coherent
state is
but a single measurement will yield an
energy n with probability
this is a Poisson Distribution hence has variance
E
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REFERENCES WITH COMMENTS
2nd derivative of gaussian in vision edge
detection
http//iria.math.pku.edu.cn/jiangm/courses/dip/ht
ml/node91.html
Marr, David, Vision a computational
investigation into the human representation and
processing of visual information, W.H. Freeman,
New York,1982.
and a more mathematical treatment in
Hurt, Norman, Phase Retrieval and Zero Crossings
mathematical methods in image reconstruction,
Kluwer, Dordrecht, 1989.
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REFERENCES WITH COMMENTS
general introduction to optics
Jenkins, Francis and White, Harvey Fundamentals
of Optics, McGraw-Hill, Singapore, 1976.
Goodman, Joseph, Introduction to Fourier
optics, New York McGraw-Hill, NY, 1996.
oscillation matrices total positivity
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
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REFERENCES WITH COMMENTS
localization, products of random matrices, and
Lyapunov exponents
Kazushige Ishii, Localization of eigenstates and
transport phenomena in the one-dimensional
disordered system, Supplement of the Progress
of Theoretical Physics, No. 53, 1973.
Harry Furstenberg, Noncommuting random products,
Transactions of the American Mathematical
Society, Vol. 108, p. 377-429, 1963.