Inverse Problems in Image Reconstruction Wolfgang Stefan Arizona - PowerPoint PPT Presentation

Outline Introduction Deblurring Inverse Problems in Image Reconstruction Wolfgang Stefan Arizona State University April 24, 2006 asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Outline Introduction Deblurring Introduction Introductory Example Deblurring Forward Model Inverse Problem PET Examples Properties and Problems Seismology Example Room for improvement Thanks and Acknowledgment asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Outline Introduction Introductory Example Deblurring Schema of a PET acquisition process asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Outline Introduction Introductory Example Deblurring Example of a PET scan asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Example of typical PET scan Typical PET Images show ◮ High noise content ◮ High blurring ◮ Reconstruction artifacts asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Forward Model ◮ Signal degradation is modeled as a convolution g = f ∗ h + n ◮ where g is the blurred signal ◮ f is the unknown signal ◮ h is the point spread function (PSF) or kernel ◮ n is noise ◮ Discrete Convolution � ( f ∗ h ) k = f i h k − i +1 asu-logo i Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Forward Model Example g = f ∗ h + n asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Estimation of the Point Spread Function (PSF) Estimations for the PSF come from: ◮ Phantom scans asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Estimation of the Point Spread Function (PSF) Estimations for the PSF come from: ◮ Phantom scans ◮ Rough estimation by a Gaussian asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Estimation of the Point Spread Function (PSF) Estimations for the PSF come from: ◮ Phantom scans ◮ Rough estimation by a Gaussian ◮ Blind Deconvolution asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Inverse Problem ◮ Find f from g = f ∗ h + n given g and h with unknown n . asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Inverse Problem ◮ Find f from g = f ∗ h + n given g and h with unknown n . ◮ Assuming normal distributed n yields the estimator ˆ f {� g − f ∗ h � 2 f = arg min 2 } asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Inverse Problem ◮ Find f from g = f ∗ h + n given g and h with unknown n . ◮ Assuming normal distributed n yields the estimator ˆ f {� g − f ∗ h � 2 f = arg min 2 } ◮ Reconstruction with n normal distr. with σ = 10 − 7 asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Regularization ◮ Add more information about the signal asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Regularization ◮ Add more information about the signal ◮ e.g. statistical properties asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Regularization ◮ Add more information about the signal ◮ e.g. statistical properties ◮ or information about the structure (e.g. sparse decon, or total variation decon) asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Regularization ◮ Add more information about the signal ◮ e.g. statistical properties ◮ or information about the structure (e.g. sparse decon, or total variation decon) ◮ in latter case use a penalty term asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Regularization ◮ Add more information about the signal ◮ e.g. statistical properties ◮ or information about the structure (e.g. sparse decon, or total variation decon) ◮ in latter case use a penalty term ◮ find ˆ f {� g − f ∗ h � 2 f = arg min 2 + λ R ( f ) } , where R ( f ) is the penalty term and λ is a penalty parameter. asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Regularization Methods ◮ Common methods are Tikhonov (TK). � |∇ f ( x ) | 2 dx . R ( f ) = TK ( f ) = Ω asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Regularization Methods ◮ Common methods are Tikhonov (TK). � |∇ f ( x ) | 2 dx . R ( f ) = TK ( f ) = Ω ◮ Total Variation (TV) � R ( f ) = TV ( f ) = |∇ f ( x ) | dx . Ω asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Regularization Methods ◮ Common methods are Tikhonov (TK). � |∇ f ( x ) | 2 dx . R ( f ) = TK ( f ) = Ω ◮ Total Variation (TV) � R ( f ) = TV ( f ) = |∇ f ( x ) | dx . Ω ◮ Sparse deconvolution ( L 1 ) � R ( f ) = � f � 1 = | f ( x ) | dx . asu-logo Ω Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Simulated PET ◮ Segmented data from an MRI scan is blurred using a Gaussian PSF asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Simulated PET ◮ Segmented data from an MRI scan is blurred using a Gaussian PSF ◮ Simulated PET image also includes Gauss distributed noise . asu-logo Wolfgang Stefan Inverse Problems in Image Reconstruction

Forward Model Inverse Problem Outline PET Examples Introduction Properties and Problems Deblurring Seismology Example Room for improvement Thanks and Acknowledgment Simulated PET ◮ Segmented data from an MRI scan is blurred using a Gaussian PSF ◮ Simulated PET image also includes Gauss distributed noise . asu-logo ◮ Note: The PSF is exactly known in this example, TV regularization Wolfgang Stefan Inverse Problems in Image Reconstruction