WAVES & WAVELETS

1. WAVES & WAVELETS Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg Tel (65) 6516-2749 This Lecture is Posted on my Homepage at http://www.math.nus.edu.sg/~matwml/courses/Undergraduate/USC/2006/USC3002

2. WAVELETS are functions that oscillate (wiggle) ; they sometimes model the real world (visual system filters)

3. WAVELETS but more often are figments of the mathematical imagination

4. WAVELETS although they sometimes look like waves.

5. 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

6. WAVES which Jean Le Rond D’Alembert (1717-1783) solved thus every wave is a superposition of two waves, one moving to the right and the other moving to the left

7. 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

8. 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

9. 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) Newton’s 2nd and Hook’s Laws  The state vector satisfies therefore

10. 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 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

11. 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

12. 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

13. FOURIER SERIES The Fourier transform of the sequence u(t,n) is therefore therefore the operator defined by satisfies

14. FOURIER SERIES The inverse Fourier transform gives convolution kernel

15. D’ALEMBERTIAN DECOMPOSITION of a general solution into where uses the initial value sequences and Step 1 Step 2 Step 3 Invert Fourier transform of

16. 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

17. MATLAB SIMULATION at times t = 0, t = 100, t = 1000 of a wave moving with velocity = 1 was computed using Fourier methods and a 2^20 = 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 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 )

18. DISPERSION

19. FOURIER PICTURE

20. INHOMOGENEOUS WAVE PROPAGATION occurs if the masses (and/or stiffnesses) are random by defining we obtain with self-adjoint

21. 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 m’s uniform on [2,3]. Then is an oscillation matrix (with total positivity properties related to splines) AND a random matrix.

25. 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.

26. NONLINEARITIES since Hook’s Lawonly approximatesthe 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.

27. STATE OF THE ART BIOIMAGING demands 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 microalgal photobioreactor cultures.

28. QUANTUM HARMONIC OSCILLATOR is described by solutions of Schrodinger’s 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.

29. 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

30. REFERENCES WITH COMMENTS 2nd derivative of gaussian in vision & edge detection http://iria.math.pku.edu.cn/~jiangm/courses/dip/html/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.

31. 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