Look Closer to Inverse Problem

1. 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 Research In Progress Presentation 2003

2. Outline Numerical Methods Linearization Singular Value Decomposition What is MATRIX? Singular Matrices Inverse problem Solving inverse problem Multi-Freq Recon. Improve the solutions Time-Domain Recon. Conclusions Research In Progress Presentation 2003

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

4. from movie The Matrix, WarnerBros,1999, What is MATRIX Unfortunately no one can be told what the matrix is, you have to see it for yourself Research In Progress Presentation 2003

5. 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) Research In Progress Presentation 2003

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

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

8. 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 Research In Progress Presentation 2003

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

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

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

12. 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 Research In Progress Presentation 2003

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

14. [A] is an ill-posed matrix -> very thin hyper-ellipsoid -> decreasing spectrum 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 a singular matrix -> degenerated ellipsoid -> 0 singular value Research In Progress Presentation 2003

15. 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) • Tikhonov regularization (standard) Truncation level  Research In Progress Presentation 2003

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

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

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

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

20. 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 Research In Progress Presentation 2003

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

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

23. 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 Research In Progress Presentation 2003

24. 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 Research In Progress Presentation 2003

25. 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 Research In Progress Presentation 2003

26. Questions? A U  VT Research In Progress Presentation 2003

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

28. Outline details • Numerical Methods and linearization • What is MATRIX? 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 Research In Progress Presentation 2003

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

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

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