Numerical Methods for Large-Scale Ill-Posed Inverse Problems - PowerPoint PPT Presentation

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks Numerical Methods for Large-Scale Ill-Posed Inverse Problems Julianne Chung University of Maryland Collaborators: James G. Nagy (Emory) Eldad Haber (Emory) Dianne O’Leary (University of Maryland) Ioannis Sechopoulos (Emory) Chao Yang (Lawrence Berkeley National Laboratory) Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks What is an inverse problem? Forward Model Physical System Input Signal Output Signal Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks What is an inverse problem? Forward Model Physical System Input Signal Output Signal Inverse Problem Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks Application: Image Deblurring Given: Blurred image and some information about the blurring Goal: Compute approximation of true image Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks Application: Super-Resolution Imaging 1−th low resolution image Given: LR images and some information about the motion parameters Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks Application: Super-Resolution Imaging Given: LR images and some information about the motion parameters Goal: Improve parameters and approximate HR image Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks Application: Tomographic Imaging Given: 2D projection images Goal: Approximate 3D volume Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks What is an Ill-Posed Inverse Problem? Hadamard (1923): A problem is ill-posed if the solution does not exist, is not unique, or does not depend continuously on the data. Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks What is an Ill-Posed Inverse Problem? Hadamard (1923): A problem is ill-posed if the solution does not exist, is not unique, or does not depend continuously on the data. True image: Blurred & noisy image: Forward Problem Invers e Problem Inverse Solution: Naive inverse solution is corrupted with noise! Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks Outline Regularization for Least Squares Systems 1 High Performance Implementation 2 Polyenergetic Tomosynthesis 3 Concluding Remarks 4 Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation The Linear Problem: b = Ax + ε Polyenergetic Tomosynthesis The Nonlinear Problem: b = A ( y ) x + ε Concluding Remarks Outline Regularization for Least Squares Systems 1 High Performance Implementation 2 Polyenergetic Tomosynthesis 3 Concluding Remarks 4 Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation The Linear Problem: b = Ax + ε Polyenergetic Tomosynthesis The Nonlinear Problem: b = A ( y ) x + ε Concluding Remarks The Linear Problem b = Ax + ε where x ∈ R n - true data A ∈ R m × n - large, ill-conditioned matrix ε ∈ R m - noise, statistical properties may be known b ∈ R m - known, observed data Goal: Given b and A , compute approximation of x Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation The Linear Problem: b = Ax + ε Polyenergetic Tomosynthesis The Nonlinear Problem: b = A ( y ) x + ε Concluding Remarks Regularization Tikhonov Regularization � b � A � � � � � � || b − Ax || 2 2 + λ 2 || Lx || 2 � � min ⇔ min − x � � 2 0 λ L x x � � 2 Selecting a good regularization parameter, λ , is difficult Discrepancy Principle Generalized Cross-Validation L-curve Difficult for large-scale problems Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation The Linear Problem: b = Ax + ε Polyenergetic Tomosynthesis The Nonlinear Problem: b = A ( y ) x + ε Concluding Remarks Illustration of Semi-convergence Behavior min x � b − Ax � 2 Iteration 0 Typical Behavior for Ill-Posed Problems C G 1.1 Relative Error 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 20 40 60 80 100 120 140 Iteration Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation The Linear Problem: b = Ax + ε Polyenergetic Tomosynthesis The Nonlinear Problem: b = A ( y ) x + ε Concluding Remarks Illustration of Semi-convergence Behavior min x � b − Ax � 2 Iteration 0 Typical Behavior for Ill-Posed Problems C G 1.1 1 Relative Error 0.9 0.8 0.7 0.6 0.5 0.4 Iteration 10 0.3 20 40 60 80 100 120 140 Iteration Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation The Linear Problem: b = Ax + ε Polyenergetic Tomosynthesis The Nonlinear Problem: b = A ( y ) x + ε Concluding Remarks Illustration of Semi-convergence Behavior min x � b − Ax � 2 Iteration 0 Typical Behavior for Ill-Posed Problems C G 1.1 1 Relative Error 0.9 0.8 0.7 0.6 0.5 0.4 Iteration 10 0.3 20 40 60 80 100 120 140 Iteration Iteration 28 Solution gets better Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation The Linear Problem: b = Ax + ε Polyenergetic Tomosynthesis The Nonlinear Problem: b = A ( y ) x + ε Concluding Remarks Illustration of Semi-convergence Behavior min x � b − Ax � 2 Iteration 0 Typical Behavior for Ill-Posed Problems C G 1.1 Relative Error 1 0.9 0.8 0.7 0.6 0.5 0.4 Iteration 10 Iteration 85 0.3 20 40 60 80 100 120 140 Iteration Iteration 28 Solution gets better Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Numerical Methods for Large-Scale Ill-Posed Inverse Problems - PowerPoint PPT Presentation

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks Numerical Methods for Large-Scale Ill-Posed Inverse Problems Julianne Chung University of Maryland Collaborators: James

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