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

asu-logo Outline Introduction Deblurring

Inverse Problems in Image Reconstruction

Wolfgang Stefan

Arizona State University

April 24, 2006

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo Outline Introduction Deblurring

Introduction Introductory Example Deblurring Forward Model Inverse Problem PET Examples Properties and Problems Seismology Example Room for improvement Thanks and Acknowledgment

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo Outline Introduction Deblurring Introductory Example

Schema of a PET acquisition process

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo Outline Introduction Deblurring Introductory Example

Example of a PET scan

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Example of typical PET scan

Typical PET Images show

◮ High noise content ◮ High blurring ◮ Reconstruction artifacts

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo Outline Introduction Deblurring Forward Model Inverse Problem PET Examples Properties and Problems 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 =

• i

fihk−i+1

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Forward Model Example

g = f ∗ h + n

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Estimation of the Point Spread Function (PSF)

Estimations for the PSF come from:

◮ Phantom scans

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo Outline Introduction Deblurring Forward Model Inverse Problem PET Examples Properties and Problems 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

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo Outline Introduction Deblurring Forward Model Inverse Problem PET Examples Properties and Problems 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

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Inverse Problem

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

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo Outline Introduction Deblurring Forward Model Inverse Problem PET Examples Properties and Problems 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 = arg min

f {g − f ∗ h2 2}

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo Outline Introduction Deblurring Forward Model Inverse Problem PET Examples Properties and Problems 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 = arg min

f {g − f ∗ h2 2} ◮ Reconstruction with n normal distr. with σ = 10−7

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

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

Regularization

◮ Add more information about the signal

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Regularization

◮ Add more information about the signal ◮ e.g. statistical properties

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo Outline Introduction Deblurring Forward Model Inverse Problem PET Examples Properties and Problems 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)

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo Outline Introduction Deblurring Forward Model Inverse Problem PET Examples Properties and Problems 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

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo Outline Introduction Deblurring Forward Model Inverse Problem PET Examples Properties and Problems 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 = arg min

f {g − f ∗ h2 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 Outline Introduction Deblurring Forward Model Inverse Problem PET Examples Properties and Problems Seismology Example Room for improvement Thanks and Acknowledgment

Regularization Methods

◮ Common methods are Tikhonov (TK).

R(f ) = TK(f ) =

• Ω

|∇f (x)|2dx.

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Regularization Methods

◮ Common methods are Tikhonov (TK).

R(f ) = TK(f ) =

• Ω

|∇f (x)|2dx.

◮ Total Variation (TV)

R(f ) = TV(f ) =

• Ω

|∇f (x)|dx.

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Regularization Methods

◮ Common methods are Tikhonov (TK).

R(f ) = TK(f ) =

• Ω

|∇f (x)|2dx.

◮ Total Variation (TV)

R(f ) = TV(f ) =

• Ω

|∇f (x)|dx.

◮ Sparse deconvolution (L1)

R(f ) = f 1 =

• Ω

|f (x)|dx.

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

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

Simulated PET

◮ Segmented data from an MRI scan is blurred using a

Gaussian PSF

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo Outline Introduction Deblurring Forward Model Inverse Problem PET Examples Properties and Problems 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.

Wolfgang Stefan Inverse Problems in Image Reconstruction

asu-logo Outline Introduction Deblurring Forward Model Inverse Problem PET Examples Properties and Problems 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. ◮ Note: The PSF is exactly known in this example, TV

regularization

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Real PET data

◮ Reconstruction done using Filtered Back Projection ◮ PSF estimated by a Gaussian ◮ TV regularization

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Notes

◮ Image improvement is possible even with a rough estimation

• f the PSF

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Notes

◮ Image improvement is possible even with a rough estimation

• f the PSF

◮ Total Variation regularization (piecewise constant solution) is

appropriate since the intensity levels depend on the tissue type.

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Notes

◮ Image improvement is possible even with a rough estimation

• f the PSF

◮ Total Variation regularization (piecewise constant solution) is

appropriate since the intensity levels depend on the tissue type.

◮ Improvement of these preliminary results when a better

approximation of the PSF is available

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Notes

◮ Image improvement is possible even with a rough estimation

• f the PSF

◮ Total Variation regularization (piecewise constant solution) is

appropriate since the intensity levels depend on the tissue type.

◮ Improvement of these preliminary results when a better

approximation of the PSF is available

◮ Increased Artifacts and noise. (More post processing can

improve this)

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

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

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Seismology Data Set

◮ Deep focus earthquakes (events) in South America

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Seismology Data Set

◮ Deep focus earthquakes (events) in South America ◮ Picked up at broad band stations in Europe

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Seismology Data Set

◮ Deep focus earthquakes (events) in South America ◮ Picked up at broad band stations in Europe ◮ Seismic energy reflects at the core-mantle boundary under the

Atlantic ocean

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Seismology Data Set

◮ Deep focus earthquakes (events) in South America ◮ Picked up at broad band stations in Europe ◮ Seismic energy reflects at the core-mantle boundary under the

Atlantic ocean

◮ Each event-station pair produces a seismogram

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

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

D” Evidence

◮ Seismograms sometimes show evidence a reflecting layer at

the core-mantle boundary

◮ An additional seismic phase is visible

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

D” Evidence

◮ Seismograms sometimes show evidence a reflecting layer at

the core-mantle boundary

◮ An additional seismic phase is visible ◮ The additional phase is usually very weak

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

D” Evidence

◮ Seismograms sometimes show evidence a reflecting layer at

the core-mantle boundary

◮ An additional seismic phase is visible ◮ The additional phase is usually very weak ◮ And only visible in very view traces (due to locality of the D”

layer)

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

D” Evidence

◮ Seismograms sometimes show evidence a reflecting layer at

the core-mantle boundary

◮ An additional seismic phase is visible ◮ The additional phase is usually very weak ◮ And only visible in very view traces (due to locality of the D”

layer)

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

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

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

Problem Formulation

◮ Invert the blurring Effect (attenuation) of Earth’s mantle

and core to ...

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Problem Formulation

◮ Invert the blurring Effect (attenuation) of Earth’s mantle

and core to ...

◮ (a) get clearer evidence of the existence of structures like

the ULVS

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Problem Formulation

◮ Invert the blurring Effect (attenuation) of Earth’s mantle

and core to ...

◮ (a) get clearer evidence of the existence of structures like

the ULVS

◮ (b) get timing information to make quantitative estimates

like the height of a structure.

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Estimation of the PSF

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

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Estimation of the PSF

◮ Ideal goal of seismic deconvolution is to produce a spike train ◮ The corresponding PSF is unknown (if it exists)

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Estimation of the PSF

◮ Ideal goal of seismic deconvolution is to produce a spike train ◮ The corresponding PSF is unknown (if it exists) ◮ Estimations of this PSF (in seismology wavelet) come from

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

difficult due to inhomogeneities)

◮ use a very basic (common) shape, like a Gaussian (very rough

estimate)

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Gaussian Wavelet

◮ also called a Ricker Wavelet

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Gaussian Wavelet

◮ also called a Ricker Wavelet ◮ h(t) = 1 σ √ 2πe− t2

2σ2 Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Gaussian Wavelet

◮ also called a Ricker Wavelet ◮ h(t) = 1 σ √ 2πe− t2

2σ2

◮ σ is a width parameter, chosen such that the wavelet

approximates the phase of interest.

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

◮ use synthetic data from 1d model

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

◮ use synthetic data from 1d model ◮ at a critical angle of about 110 deg SKS starts to diffract

along the core

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

SKS at 112 deg deconvolved with SKS from 99 deg

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

SKS at 112 deg deconvolved with a Gaussian

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

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

Real Data (SV) from an earthquake in South America

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Real Data (SH) from an earthquake in South America

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Evidence of the ultra low velocity zone (ULVZ)

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Seismology Conclusions

◮ TV regularized deconvolution is more robust then established

methods

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Seismology Conclusions

◮ TV regularized deconvolution is more robust then established

methods

◮ Automatic travel time picking is more accurate then hand

picking

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Seismology Conclusions

◮ TV regularized deconvolution is more robust then established

methods

◮ Automatic travel time picking is more accurate then hand

picking

◮ TV deconvolution yields usable results even for rough

estimates of the wavelet

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Seismology Conclusions

◮ TV regularized deconvolution is more robust then established

methods

◮ Automatic travel time picking is more accurate then hand

picking

◮ TV deconvolution yields usable results even for rough

estimates of the wavelet

◮ Better estimates of the wavelet e.g. two-sided Gaussian will

improve results further

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Blind deconvolution

◮ recall forward model

g = f ∗ h + n

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Blind deconvolution

◮ recall forward model

g = f ∗ h + n

◮ h is usually unknown

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Blind deconvolution

◮ recall forward model

g = f ∗ h + n

◮ h is usually unknown ◮ Blind deconvolution solves

minf ,hf ∗ h − g2

2 + λ1L1(f − f0)p1 + λ2L2(h − h0)p2

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Blind deconvolution

◮ recall forward model

g = f ∗ h + n

◮ h is usually unknown ◮ Blind deconvolution solves

minf ,hf ∗ h − g2

2 + λ1L1(f − f0)p1 + λ2L2(h − h0)p2 ◮ very limited uniqueness results for p1 = p2 = 2 and

L1 = L2 = I e.g. by Scherzer and Justen

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Blind deconvolution

◮ recall forward model

g = f ∗ h + n

◮ h is usually unknown ◮ Blind deconvolution solves

minf ,hf ∗ h − g2

2 + λ1L1(f − f0)p1 + λ2L2(h − h0)p2 ◮ very limited uniqueness results for p1 = p2 = 2 and

L1 = L2 = I e.g. by Scherzer and Justen

◮ For general L and p, no uniqueness, Stefan

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Blind deconvolution

◮ recall forward model

g = f ∗ h + n

◮ h is usually unknown ◮ Blind deconvolution solves

minf ,hf ∗ h − g2

2 + λ1L1(f − f0)p1 + λ2L2(h − h0)p2 ◮ very limited uniqueness results for p1 = p2 = 2 and

L1 = L2 = I e.g. by Scherzer and Justen

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

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Total least squares (TLS)

◮ Idea rewrite convolution as matrix vector product:

g = Hf + n

◮ where H is a Toeplitz matrix

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Total least squares (TLS)

◮ Idea rewrite convolution as matrix vector product:

g = Hf + n

◮ where H is a Toeplitz matrix ◮ and allow noise in H and g i.e.

g = (H + E)f + n

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Total least squares (TLS)

◮ Idea rewrite convolution as matrix vector product:

g = Hf + n

◮ where H is a Toeplitz matrix ◮ and allow noise in H and g i.e.

g = (H + E)f + n

◮ Total least squares solution fTLS solves

minE|nF subject to g = (H + E)f + n

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

TLS cont.

◮ The TLS solution satisfies:

minf Hf − g2

2

1 + f 2

2

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

TLS cont.

◮ The TLS solution satisfies:

minf Hf − g2

2

1 + f 2

2 ◮ include regularization:

minf Hf − g2

2

1 + f 2

2

+ λL(f − f0)p

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

TLS cont.

◮ The TLS solution satisfies:

minf Hf − g2

2

1 + f 2

2 ◮ include regularization:

minf Hf − g2

2

1 + f 2

2

+ λL(f − f0)p

◮ p=2 Renaut, Guo

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

TLS cont.

◮ The TLS solution satisfies:

minf Hf − g2

2

1 + f 2

2 ◮ include regularization:

minf Hf − g2

2

1 + f 2

2

+ λL(f − f0)p

◮ p=2 Renaut, Guo ◮ p=1 ??

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

TLS cont.

◮ Generalize TLS problem to:

minE|nF subject to g = (H + γE)f + n

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

TLS Open Questions

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

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

TLS Open Questions

◮ What happens if we impose a structure on E e.g. Toeplitz? ◮ What is the relation to blind deconvolution?

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

TLS Open Questions

◮ What happens if we impose a structure on E e.g. Toeplitz? ◮ What is the relation to blind deconvolution? ◮ How does the performance of LBFGS compare to methods

like RQ iterations Renaut, Guo

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

TLS Open Questions

◮ What happens if we impose a structure on E e.g. Toeplitz? ◮ What is the relation to blind deconvolution? ◮ How does the performance of LBFGS compare to methods

like RQ iterations Renaut, Guo

◮ Is there a practical advantage in applications like Seismology

• r PET scans?

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

TLS Open Questions

◮ What happens if we impose a structure on E e.g. Toeplitz? ◮ What is the relation to blind deconvolution? ◮ How does the performance of LBFGS compare to methods

like RQ iterations Renaut, Guo

◮ Is there a practical advantage in applications like Seismology

• r PET scans?

◮ Preliminary results on the seismic data show faster

convergence and smoother, more reasonable reconstructions, why?

Wolfgang Stefan Inverse Problems in Image Reconstruction

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

Thanks to

◮ My Advisor Rosemary Renaut and Ed Garnero from Geology ◮ Sebastian Rost and Matthew Fouch for discussions and data ◮ This study was partly supported by the grant NSF CMG-02223 ◮ Haewon Nam and Kewei Chen for the data ◮ Svetlana Roudenko for discussion

Wolfgang Stefan Inverse Problems in Image Reconstruction