1. Research In Progress Presentation 2003 Look Closer to Inverse Problem Qianqian Fang Thanks to : Paul M. Meaney, Keith D. Paulsen, Dun Li, Margaret Fanning, Sarah A. Pendergrass and all other friends RIP 2003

2. Research In Progress Presentation 2003 Outline Numerical Methods Linearization What is M A TR I X? Inverse problem Singular Value Decomposition Solving inverse problem Improve the solutions Conclusions SingularMatrices Multi-Freq Recon. Time-Domain Recon.

3. Research In Progress Presentation 2003 Numerical Methods and linearization  Modern Numerical Techniques Modern Numerical Techniques Reality Infinitely Complicated, Dynamically Changing, Noisy and Interrelated Model Diff. Equ./Integral Equ. Linear Relation Ax=b Nonlinear methods NN, GA, SA, Monte-Calo Mathematical Numerical Accuracy  Efficiency 

4. Research In Progress Presentation 2003 What is M A TR I X from movie The Matrix, WarnerBros,1999, Unfortunatel y no one can be told what the matrix is, you have to see it for yourself

5. Research In Progress Presentation 2003 What is MATRIX   Linear Transform  Map from one space to another  Stretch, Rotations, Projections   Structural Information- on grid  Simple data structure (comparing with list/tree/object etc)  But not that simple (comparing with single variable)

6. Research In Progress Presentation 2003 Geometric Interpretations  2X2 matrix->Map 2D image to 2D image

7. Research In Progress Presentation 2003 Geometric Interpretations 2  3D matrix 1. Stretching 2. Rotation 3. Projection Diagonal Matrix Orthogonal Matrix Projection Matrix

8. Research In Progress Presentation 2003 Geometric Interpretations 3  N-Dimensional matrix-> Hyper-ellipsoid Orthogonal Basis Singular Matrix Ellipsoid will collapse To a “thin” hyperplane Information along “Singular” direction Will be wiped out After the transform Information losing

9. Research In Progress Presentation 2003 Inverse Problem  Which is inverse? Which is forward?  Information  Sensitivity X domainY domain Transformation The latter discovered? The more difficult one? Integration operator has a smoothing nature Forward? Inverse?

10. Research In Progress Presentation 2003 Inversion: Information Perspective  From damaged information to get all.  From limited # of projected images to recover the full object  Projections -> Related to singular matrix -- From the website of "PHOTOGRAPHY CLUBS in Singapore" Multi-view scheme: ?

11. Research In Progress Presentation 2003 SVD-the way to degeneration  Singular Value Decomposition  What this means  Good/Bad, how good/how bad AU  VTVT AU VTVT  2 miles 4 miles Thin SVD economy

12. Research In Progress Presentation 2003 One step further…  SVE- Singular Value Expansion  Solving Ax=y  Given the knowledge of SVD and noise, we master the fate of the inverse problem Principal Planes

13. Research In Progress Presentation 2003 Principal Planes of a matrix

14. Research In Progress Presentation 2003 Singular Values   - Diagonal Matrix {  i }  Ranking of importance,  Ranking of ill-posedness  How linearly dependent for equations [A] is an orthogonal matrix -> Hyper-sphere -> Perfectly linearly independent [A] is an ill-posed matrix -> very thin hyper-ellipsoid -> decreasing spectrum [A] is a singular matrix -> degenerated ellipsoid -> 0 singular value

15. Research In Progress Presentation 2003 Regularization, the saver  Eliminating the bad effect of small singular values, keep major information  A filter, filter out high frequency noise AND high freq. useful information  Truncated SVD(T-SVD) Truncation level  Tikhonov regularization (standard)

16. Research In Progress Presentation 2003 L -curve: A useful tool Over-smoothed solution “best solution” Under-smoothed solution : Regularization parameter increasing † See reference [1]

17. Research In Progress Presentation 2003 Can we do better?  Adding more linearly independent measurement  More antenna/more receivers  Same antenna, but more frequency points

18. Research In Progress Presentation 2003 Multiple-Frequency Reconstruction Project the object with different Wavelength microwave Low frequency component stabilize the reconstruction High frequency component brings up details

19. Research In Progress Presentation 2003 Reconstruction results I: Simulations  High contrast(1:6)/Large object True object Result from single freq. recon Result from 3 freq. recon Cross cut of reconstruction Background inclusion Large object

20. Research In Progress Presentation 2003 Reconstruction results I: Phantom  Saline Background/Agar Phantom with inclusion Results from Single frequency Reconstructor At 900MHz Results from Multi-frequency Reconstructor 500/700/900MHz

21. Research In Progress Presentation 2003 Time-Domain solver  A vehicle to get full-spectrum by one-run A pulse signal is transmitted From source Interacting with inhomogeneity A distorted pulse is received At receivers FFT Full Spectrum Response retrieved

22. Research In Progress Presentation 2003 Animations Microwave scattered by object Source: Diff Gaussian Pulse Object

23. Research In Progress Presentation 2003 Conclusions  SVD gives us a scale to measure the Difficulties for solving inverse problem  SVD gives us a microscope that shows the very details of how each components affects the inversion  Incorporate noise and a priori information, SVD provide the complete information (in linear sense)  Regularization is necessary to by suppressing noise  Difficulties can be released by adding more linearly independent measurements

24. Research In Progress Presentation 2003 Key Ideas  Decomposing a complex problem into some building blocks, they are simple, invariant to input, but addable, which can create certain degree of complexity, but manageable.  Find out the unchanged part from changing, that are the rules we are looking for  It is impossible to get something from nothing

25. Research In Progress Presentation 2003 References  Rank-Deficient and Discrete Ill- Posed Problems, Per Christian Hansen, SIAM 1998  Regularization Methods for Ill-Posed Problems, Morozov  Matrix Computations, G. Golub, 1989  Linear Algebra and it’s applications, G. Strang

26. Research In Progress Presentation 2003 Questions? AU VTVT 

27. Research In Progress Presentation 2003 Eigen-values vs. Singular value Eigen-vectors Directions: Invariant of rotations Singular Singular-vectors Directions: Maximum span

28. Research In Progress Presentation 2003 Outline details  Numerical Methods and linearization  What is M A TR I X? Geometric interpretations  Inverse Problem  Singular value decomposition and implementations in inverse problem  Solving inverse problem  Improve the solution, can we?  Multiple-Frequency Reconstruction & Time- Domain Reconstruction  Conclusions

29. Research In Progress Presentation 2003 Right Singular Vectors  Eigen-modes for solution  Building blocks for solutions,  if the solution is a image, v i are components of the image  Less variant respect to different y=> a property of the system

30. Research In Progress Presentation 2003 Left Singular Vectors  A group of “basic RHS’s”-> source mode  Arbitrary RHS y can be decomposed with this basis

31. Research In Progress Presentation 2003 Noise  Always Noise  Small perturbation for RHS  Ax=y  y = y +  y † Modified from coca-cola’s patch