asu-logo

OutlineIntroduction

Deblurring

Inverse Problems in Image Reconstruction

Wolfgang Stefan

Arizona State University

April 24, 2006

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

IntroductionIntroductory Example

DeblurringForward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

DeblurringIntroductory Example

Schema of a PET acquisition process

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

DeblurringIntroductory Example

Example of a PET scan

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Example of typical PET scan

Typical PET Images show

I High noise content

I High blurring

I Reconstruction artifacts

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Forward Model

I Signal degradation is modeled as a convolution

g = f ∗ h + n

I where g is the blurred signalI f is the unknown signalI h is the point spread function (PSF) or kernelI n is noiseI Discrete Convolution

(f ∗ h)k =∑

i

fihk−i+1

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Forward Model Example

g = f ∗ h + n

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Estimation of the Point Spread Function (PSF)

Estimations for the PSF come from:

I Phantom scans

I Rough estimation by a Gaussian

I Blind Deconvolution

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Estimation of the Point Spread Function (PSF)

Estimations for the PSF come from:

I Phantom scans

I Rough estimation by a Gaussian

I Blind Deconvolution

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Estimation of the Point Spread Function (PSF)

Estimations for the PSF come from:

I Phantom scans

I Rough estimation by a Gaussian

I Blind Deconvolution

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Inverse Problem

I Find f from g = f ∗ h + n given g and h with unknown n.

I Assuming normal distributed n yields the estimator

f = arg minf‖g − f ∗ h‖2

2

I Reconstruction with n normal distr. with σ = 10−7

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Inverse Problem

I Find f from g = f ∗ h + n given g and h with unknown n.I Assuming normal distributed n yields the estimator

f = arg minf‖g − f ∗ h‖2

2

I Reconstruction with n normal distr. with σ = 10−7

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Inverse Problem

I Find f from g = f ∗ h + n given g and h with unknown n.I Assuming normal distributed n yields the estimator

f = arg minf‖g − f ∗ h‖2

2

I Reconstruction with n normal distr. with σ = 10−7

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Regularization

I Add more information about the signal

I e.g. statistical properties

I or information about the structure (e.g. sparse decon, or totalvariation decon)

I in latter case use a penalty term

I findf = arg min

f‖g − f ∗ h‖2

2 + λR(f ),

where R(f ) is the penalty term and λ is a penalty parameter.

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Regularization

I Add more information about the signal

I e.g. statistical properties

I or information about the structure (e.g. sparse decon, or totalvariation decon)

I in latter case use a penalty term

I findf = arg min

f‖g − f ∗ h‖2

2 + λR(f ),

where R(f ) is the penalty term and λ is a penalty parameter.

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Regularization

I Add more information about the signal

I e.g. statistical properties

I or information about the structure (e.g. sparse decon, or totalvariation decon)

I in latter case use a penalty term

I findf = arg min

f‖g − f ∗ h‖2

2 + λR(f ),

where R(f ) is the penalty term and λ is a penalty parameter.

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Regularization

I Add more information about the signal

I e.g. statistical properties

I or information about the structure (e.g. sparse decon, or totalvariation decon)

I in latter case use a penalty term

I findf = arg min

f‖g − f ∗ h‖2

2 + λR(f ),

where R(f ) is the penalty term and λ is a penalty parameter.

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Regularization

I Add more information about the signal

I e.g. statistical properties

I or information about the structure (e.g. sparse decon, or totalvariation decon)

I in latter case use a penalty term

I findf = arg min

f‖g − f ∗ h‖2

2 + λR(f ),

where R(f ) is the penalty term and λ is a penalty parameter.

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Regularization Methods

I Common methods are Tikhonov (TK).

R(f ) = TK(f ) =

∫Ω|∇f (x)|2dx .

I Total Variation (TV)

R(f ) = TV(f ) =

∫Ω|∇f (x)|dx .

I Sparse deconvolution (L1)

R(f ) = ‖f ‖1 =

∫Ω|f (x)|dx .

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Regularization Methods

I Common methods are Tikhonov (TK).

R(f ) = TK(f ) =

∫Ω|∇f (x)|2dx .

I Total Variation (TV)

R(f ) = TV(f ) =

∫Ω|∇f (x)|dx .

I Sparse deconvolution (L1)

R(f ) = ‖f ‖1 =

∫Ω|f (x)|dx .

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Regularization Methods

I Common methods are Tikhonov (TK).

R(f ) = TK(f ) =

∫Ω|∇f (x)|2dx .

I Total Variation (TV)

R(f ) = TV(f ) =

∫Ω|∇f (x)|dx .

I Sparse deconvolution (L1)

R(f ) = ‖f ‖1 =

∫Ω|f (x)|dx .

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Simulated PET

I Segmented data from an MRI scan is blurred using aGaussian PSF

I Simulated PET image also includes Gauss distributed noise.

I Note: The PSF is exactly known in this example, TVregularization

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Simulated PET

I Segmented data from an MRI scan is blurred using aGaussian PSF

I Simulated PET image also includes Gauss distributed noise.

I Note: The PSF is exactly known in this example, TVregularization

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Simulated PET

I Segmented data from an MRI scan is blurred using aGaussian PSF

I Simulated PET image also includes Gauss distributed noise.

I Note: The PSF is exactly known in this example, TVregularization Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Real PET data

I Reconstruction done using Filtered Back Projection

I PSF estimated by a Gaussian

I TV regularization

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Notes

I Image improvement is possible even with a rough estimationof the PSF

I Total Variation regularization (piecewise constant solution) isappropriate since the intensity levels depend on the tissuetype.

I Improvement of these preliminary results when a betterapproximation of the PSF is available

I Increased Artifacts and noise. (More post processing canimprove this)

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Notes

I Image improvement is possible even with a rough estimationof the PSF

I Total Variation regularization (piecewise constant solution) isappropriate since the intensity levels depend on the tissuetype.

I Improvement of these preliminary results when a betterapproximation of the PSF is available

I Increased Artifacts and noise. (More post processing canimprove this)

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Notes

I Image improvement is possible even with a rough estimationof the PSF

I Total Variation regularization (piecewise constant solution) isappropriate since the intensity levels depend on the tissuetype.

I Improvement of these preliminary results when a betterapproximation of the PSF is available

I Increased Artifacts and noise. (More post processing canimprove this)

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Notes

I Image improvement is possible even with a rough estimationof the PSF

I Total Variation regularization (piecewise constant solution) isappropriate since the intensity levels depend on the tissuetype.

I Improvement of these preliminary results when a betterapproximation of the PSF is available

I Increased Artifacts and noise. (More post processing canimprove this)

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

I Earth radius 6378 kmI Core-Mantle Boundary at 2890 kmI ULVZ 5-20km thick

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Seismology Data Set

I Deep focus earthquakes (events) in South America

I Picked up at broad band stations in Europe

I Seismic energy reflects at the core-mantle boundary under theAtlantic ocean

I Each event-station pair produces a seismogram

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Seismology Data Set

I Deep focus earthquakes (events) in South America

I Picked up at broad band stations in Europe

I Seismic energy reflects at the core-mantle boundary under theAtlantic ocean

I Each event-station pair produces a seismogram

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Seismology Data Set

I Deep focus earthquakes (events) in South America

I Picked up at broad band stations in Europe

I Seismic energy reflects at the core-mantle boundary under theAtlantic ocean

I Each event-station pair produces a seismogram

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Seismology Data Set

I Deep focus earthquakes (events) in South America

I Picked up at broad band stations in Europe

I Seismic energy reflects at the core-mantle boundary under theAtlantic ocean

I Each event-station pair produces a seismogram

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

D” Evidence

I Seismograms sometimes show evidence a reflecting layer atthe core-mantle boundary

I An additional seismic phase is visible

I The additional phase is usually very weak

I And only visible in very view traces (due to locality of the D”layer)

I i.e. even traces with very poor SNR have to be considered

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

D” Evidence

I Seismograms sometimes show evidence a reflecting layer atthe core-mantle boundary

I An additional seismic phase is visible

I The additional phase is usually very weak

I And only visible in very view traces (due to locality of the D”layer)

I i.e. even traces with very poor SNR have to be considered

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

D” Evidence

I Seismograms sometimes show evidence a reflecting layer atthe core-mantle boundary

I An additional seismic phase is visible

I The additional phase is usually very weak

I And only visible in very view traces (due to locality of the D”layer)

I i.e. even traces with very poor SNR have to be considered

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

D” Evidence

I Seismograms sometimes show evidence a reflecting layer atthe core-mantle boundary

I An additional seismic phase is visible

I The additional phase is usually very weak

I And only visible in very view traces (due to locality of the D”layer)

I i.e. even traces with very poor SNR have to be considered

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Problem Formulation

I Invert the blurring Effect (attenuation) of Earth’s mantleand core to ...

I (a) get clearer evidence of the existence of structures likethe ULVS

I (b) get timing information to make quantitative estimateslike the height of a structure.

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Problem Formulation

I Invert the blurring Effect (attenuation) of Earth’s mantleand core to ...

I (a) get clearer evidence of the existence of structures likethe ULVS

I (b) get timing information to make quantitative estimateslike the height of a structure.

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Problem Formulation

I Invert the blurring Effect (attenuation) of Earth’s mantleand core to ...

I (a) get clearer evidence of the existence of structures likethe ULVS

I (b) get timing information to make quantitative estimateslike the height of a structure.

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Estimation of the PSF

I Ideal goal of seismic deconvolution is to produce a spike train

I The corresponding PSF is unknown (if it exists)I Estimations of this PSF (in seismology wavelet) come from

I stacking traces (problem, traces are very different)I estimating Earth’s filter (basically a low pass filter, very

difficult due to inhomogeneities)I use a very basic (common) shape, like a Gaussian (very rough

estimate)

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Estimation of the PSF

I Ideal goal of seismic deconvolution is to produce a spike train

I The corresponding PSF is unknown (if it exists)

I Estimations of this PSF (in seismology wavelet) come fromI stacking traces (problem, traces are very different)I estimating Earth’s filter (basically a low pass filter, very

difficult due to inhomogeneities)I use a very basic (common) shape, like a Gaussian (very rough

estimate)

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Estimation of the PSF

I Ideal goal of seismic deconvolution is to produce a spike train

I The corresponding PSF is unknown (if it exists)I Estimations of this PSF (in seismology wavelet) come from

I stacking traces (problem, traces are very different)I estimating Earth’s filter (basically a low pass filter, very

difficult due to inhomogeneities)I use a very basic (common) shape, like a Gaussian (very rough

estimate)

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Gaussian Wavelet

I also called a Ricker Wavelet

I h(t) = 1σ√

2πe−

t2

2σ2

I σ is a width parameter, chosen such that the waveletapproximates the phase of interest.

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Gaussian Wavelet

I also called a Ricker Wavelet

I h(t) = 1σ√

2πe−

t2

2σ2

I σ is a width parameter, chosen such that the waveletapproximates the phase of interest.

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Gaussian Wavelet

I also called a Ricker Wavelet

I h(t) = 1σ√

2πe−

t2

2σ2

I σ is a width parameter, chosen such that the waveletapproximates the phase of interest.

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

I use synthetic data from 1d model

I at a critical angle of about 110 deg SKS starts to diffractalong the core

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

I use synthetic data from 1d model

I at a critical angle of about 110 deg SKS starts to diffractalong the core

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

SKS at 112 deg deconvolved with SKS from 99 deg

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

SKS at 112 deg deconvolved with a Gaussian

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Real Data (SV) from an earthquake in South America

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Real Data (SH) from an earthquake in South America

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Evidence of the ultra low velocity zone (ULVZ)

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Seismology Conclusions

I TV regularized deconvolution is more robust then establishedmethods

I Automatic travel time picking is more accurate then handpicking

I TV deconvolution yields usable results even for roughestimates of the wavelet

I Better estimates of the wavelet e.g. two-sided Gaussian willimprove results further

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Seismology Conclusions

I TV regularized deconvolution is more robust then establishedmethods

I Automatic travel time picking is more accurate then handpicking

I TV deconvolution yields usable results even for roughestimates of the wavelet

I Better estimates of the wavelet e.g. two-sided Gaussian willimprove results further

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Seismology Conclusions

I TV regularized deconvolution is more robust then establishedmethods

I Automatic travel time picking is more accurate then handpicking

I TV deconvolution yields usable results even for roughestimates of the wavelet

I Better estimates of the wavelet e.g. two-sided Gaussian willimprove results further

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Seismology Conclusions

I TV regularized deconvolution is more robust then establishedmethods

I Automatic travel time picking is more accurate then handpicking

I TV deconvolution yields usable results even for roughestimates of the wavelet

I Better estimates of the wavelet e.g. two-sided Gaussian willimprove results further

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Blind deconvolution

I recall forward model

g = f ∗ h + n

I h is usually unknownI Blind deconvolution solves

minf ,h‖f ∗ h − g‖22 + λ1‖L1(f − f0)‖p1 + λ2‖L2(h − h0)‖p2

I very limited uniqueness results for p1 = p2 = 2 andL1 = L2 = I e.g. by Scherzer and Justen

I For general L and p, no uniqueness, StefanI In practical applications p = 1 is often better, Stefan

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Blind deconvolution

I recall forward model

g = f ∗ h + n

I h is usually unknown

I Blind deconvolution solves

minf ,h‖f ∗ h − g‖22 + λ1‖L1(f − f0)‖p1 + λ2‖L2(h − h0)‖p2

I very limited uniqueness results for p1 = p2 = 2 andL1 = L2 = I e.g. by Scherzer and Justen

I For general L and p, no uniqueness, StefanI In practical applications p = 1 is often better, Stefan

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Blind deconvolution

I recall forward model

g = f ∗ h + n

I h is usually unknownI Blind deconvolution solves

minf ,h‖f ∗ h − g‖22 + λ1‖L1(f − f0)‖p1 + λ2‖L2(h − h0)‖p2

I very limited uniqueness results for p1 = p2 = 2 andL1 = L2 = I e.g. by Scherzer and Justen

I For general L and p, no uniqueness, StefanI In practical applications p = 1 is often better, Stefan

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Blind deconvolution

I recall forward model

g = f ∗ h + n

I h is usually unknownI Blind deconvolution solves

minf ,h‖f ∗ h − g‖22 + λ1‖L1(f − f0)‖p1 + λ2‖L2(h − h0)‖p2

I very limited uniqueness results for p1 = p2 = 2 andL1 = L2 = I e.g. by Scherzer and Justen

I For general L and p, no uniqueness, StefanI In practical applications p = 1 is often better, Stefan

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Blind deconvolution

I recall forward model

g = f ∗ h + n

I h is usually unknownI Blind deconvolution solves

minf ,h‖f ∗ h − g‖22 + λ1‖L1(f − f0)‖p1 + λ2‖L2(h − h0)‖p2

I very limited uniqueness results for p1 = p2 = 2 andL1 = L2 = I e.g. by Scherzer and Justen

I For general L and p, no uniqueness, Stefan

I In practical applications p = 1 is often better, Stefan

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Blind deconvolution

I recall forward model

g = f ∗ h + n

I h is usually unknownI Blind deconvolution solves

minf ,h‖f ∗ h − g‖22 + λ1‖L1(f − f0)‖p1 + λ2‖L2(h − h0)‖p2

I very limited uniqueness results for p1 = p2 = 2 andL1 = L2 = I e.g. by Scherzer and Justen

I For general L and p, no uniqueness, StefanI In practical applications p = 1 is often better, Stefan

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Total least squares (TLS)

I Idea rewrite convolution as matrix vector product:

g = Hf + n

I where H is a Toeplitz matrix

I and allow noise in H and g i.e.

g = (H + E )f + n

I Total least squares solution fTLS solves

min‖E |n‖F

subject tog = (H + E )f + n

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Total least squares (TLS)

I Idea rewrite convolution as matrix vector product:

g = Hf + n

I where H is a Toeplitz matrixI and allow noise in H and g i.e.

g = (H + E )f + n

I Total least squares solution fTLS solves

min‖E |n‖F

subject tog = (H + E )f + n

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Total least squares (TLS)

I Idea rewrite convolution as matrix vector product:

g = Hf + n

I where H is a Toeplitz matrixI and allow noise in H and g i.e.

g = (H + E )f + n

I Total least squares solution fTLS solves

min‖E |n‖F

subject tog = (H + E )f + n

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

TLS cont.

I The TLS solution satisfies:

minf‖Hf − g‖2

2

1 + ‖f ‖22

I include regularization:

minf‖Hf − g‖2

2

1 + ‖f ‖22

+ λ‖L(f − f0)‖p

I p=2 Renaut, Guo

I p=1 ??

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

TLS cont.

I The TLS solution satisfies:

minf‖Hf − g‖2

2

1 + ‖f ‖22

I include regularization:

minf‖Hf − g‖2

2

1 + ‖f ‖22

+ λ‖L(f − f0)‖p

I p=2 Renaut, Guo

I p=1 ??

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

TLS cont.

I The TLS solution satisfies:

minf‖Hf − g‖2

2

1 + ‖f ‖22

I include regularization:

minf‖Hf − g‖2

2

1 + ‖f ‖22

+ λ‖L(f − f0)‖p

I p=2 Renaut, Guo

I p=1 ??

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

TLS cont.

I The TLS solution satisfies:

minf‖Hf − g‖2

2

1 + ‖f ‖22

I include regularization:

minf‖Hf − g‖2

2

1 + ‖f ‖22

+ λ‖L(f − f0)‖p

I p=2 Renaut, Guo

I p=1 ??

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

TLS cont.

I Generalize TLS problem to:

min‖E |n‖F

subject tog = (H + γE )f + n

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

TLS Open Questions

I What happens if we impose a structure on E e.g. Toeplitz?

I What is the relation to blind deconvolution?

I How does the performance of LBFGS compare to methodslike RQ iterations Renaut, Guo

I Is there a practical advantage in applications like Seismologyor PET scans?

I Preliminary results on the seismic data show fasterconvergence and smoother, more reasonable reconstructions,why?

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

TLS Open Questions

I What happens if we impose a structure on E e.g. Toeplitz?

I What is the relation to blind deconvolution?

I How does the performance of LBFGS compare to methodslike RQ iterations Renaut, Guo

I Is there a practical advantage in applications like Seismologyor PET scans?

I Preliminary results on the seismic data show fasterconvergence and smoother, more reasonable reconstructions,why?

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

TLS Open Questions

I What happens if we impose a structure on E e.g. Toeplitz?

I What is the relation to blind deconvolution?

I How does the performance of LBFGS compare to methodslike RQ iterations Renaut, Guo

I Is there a practical advantage in applications like Seismologyor PET scans?

I Preliminary results on the seismic data show fasterconvergence and smoother, more reasonable reconstructions,why?

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

TLS Open Questions

I What happens if we impose a structure on E e.g. Toeplitz?

I What is the relation to blind deconvolution?

I How does the performance of LBFGS compare to methodslike RQ iterations Renaut, Guo

I Is there a practical advantage in applications like Seismologyor PET scans?

I Preliminary results on the seismic data show fasterconvergence and smoother, more reasonable reconstructions,why?

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

TLS Open Questions

I What happens if we impose a structure on E e.g. Toeplitz?

I What is the relation to blind deconvolution?

I How does the performance of LBFGS compare to methodslike RQ iterations Renaut, Guo

I Is there a practical advantage in applications like Seismologyor PET scans?

I Preliminary results on the seismic data show fasterconvergence and smoother, more reasonable reconstructions,why?

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo

OutlineIntroduction

Deblurring

Forward ModelInverse ProblemPET ExamplesProperties and ProblemsSeismology ExampleRoom for improvementThanks and Acknowledgment

Thanks to

I My Advisor Rosemary Renaut and Ed Garnero from Geology

I Sebastian Rost and Matthew Fouch for discussions and data

I This study was partly supported by the grant NSF CMG-02223

I Haewon Nam and Kewei Chen for the data

I Svetlana Roudenko for discussion

Wolfgang Stefan Inverse Problems in Image Reconstruction