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

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.

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 

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

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)

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

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

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

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?

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: ?

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

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

Research In Progress Presentation 2003 Principal Planes of a matrix

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

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)

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

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

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

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

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

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

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

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

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

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

Research In Progress Presentation 2003 Questions? AU VTVT 

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

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

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

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

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