# Workshop Programme

## for period 25 - 29 July 2011

### Introductory Workshop

25 - 29 July 2011

Timetable

Monday 25 July | ||||

10:00-10:55 | Registration | |||

10:55-11:00 | Welcome from John Toland (INI Director Designate) | |||

11:00-11:45 | Uhlmann, G (Washington & UC, Irvine) |
|||

C1 - Photoacoustic and Thermoacoustic Tomography I | Sem 1 | |||

Photoacoustic Acoustic Tomography (PAT) and Thermoacoustic Tomography (TAT) are examples of hybrid inverse methods arising in medical imaging that combine a high resolution modality with another one that can image high contrast between tissues. PAT and TAT combine the hight resolution of ultrasound with the high contrast capabilities of electromagnetic waves. In these lectures we will describe the mathematical model for PAT and TAT and some of the mathematical progress that has been done in understanding these modalities. |
||||

11:45-12:30 | Nachman, A (Toronto) |
|||

C2 - Reconstructions from Partial Boundary Data I | Sem 1 | |||

The classical inverse boundary value problem of Calderón consists in determining the conductivity inside a body from the Dirichlet-to-Neumann map. The problem is of interest in medical imaging and geophysics, where one seeks to image the conductivity of a body by making voltage and current measurements at its surface. Bukhgeim-Uhlmann and Kenig-Sjöstrand-Uhlmann have shown that (in dimensions three and higher) uniqueness in the above problem holds even if measurements are available on possibly very small subsets of the boundary. This course will explain in detail a constructive proof of these results, obtained in joint work with Brian Street. The topics I hope to cover are: 1. Review of the reconstruction method for Calderón's problem with full data. 2. New Green's functions for the Laplacian. 3. Boundedness properties of the corresponding single layer operators. 4. New solutions of the Schrödinger equation. 5. Unique solvability of the main boundary integral equation involving only the partial Cauchy data. |
||||

12:30-13:30 | Lunch at Wolfson Court | |||

14:30-15:15 | Uhlmann, G (Washington & UC, Irvine) |
|||

C1 - Photoacoustic and Thermoacoustic Tomography II | Sem 1 | |||

Photoacoustic Acoustic Tomography (PAT) and Thermoacoustic Tomography (TAT) are examples of hybrid inverse methods arising in medical imaging that combine a high resolution modality with another one that can image high contrast between tissues. PAT and TAT combine the hight resolution of ultrasound with the high contrast capabilities of electromagnetic waves. In these lectures we will describe the mathematical model for PAT and TAT and some of the mathematical progress that has been done in understanding these modalities. |
||||

15:15-16:00 | Nachman, A (Toronto) |
|||

C2 - Reconstructions from Partial Boundary Data II | Sem 1 | |||

The classical inverse boundary value problem of Calderón consists in determining the conductivity inside a body from the Dirichlet-to-Neumann map. The problem is of interest in medical imaging and geophysics, where one seeks to image the conductivity of a body by making voltage and current measurements at its surface. Bukhgeim-Uhlmann and Kenig-Sjöstrand-Uhlmann have shown that (in dimensions three and higher) uniqueness in the above problem holds even if measurements are available on possibly very small subsets of the boundary. This course will explain in detail a constructive proof of these results, obtained in joint work with Brian Street. The topics I hope to cover are: 1. Review of the reconstruction method for Calderón's problem with full data. 2. New Green's functions for the Laplacian. 3. Boundedness properties of the corresponding single layer operators. 4. New solutions of the Schrödinger equation. 5. Unique solvability of the main boundary integral equation involving only the partial Cauchy data. |
||||

16:00-16:30 | Coffee and Tea | |||

17:00-17:30 | Contributed talks | |||

17:30-18:30 | Welcome Drinks Reception |

Tuesday 26 July | ||||

09:00-09:45 | Uhlmann, G (Washington & UC, Irvine) |
|||

C1 - Photoacoustic and Thermoacoustic Tomography III | Sem 1 | |||

Photoacoustic Acoustic Tomography (PAT) and Thermoacoustic Tomography (TAT) are examples of hybrid inverse methods arising in medical imaging that combine a high resolution modality with another one that can image high contrast between tissues. PAT and TAT combine the hight resolution of ultrasound with the high contrast capabilities of electromagnetic waves. In these lectures we will describe the mathematical model for PAT and TAT and some of the mathematical progress that has been done in understanding these modalities. |
||||

09:45-10:30 | Nachman, A (Toronto) |
|||

C2 - Reconstructions from Partial Boundary Data III | Sem 1 | |||

The classical inverse boundary value problem of Calderón consists in determining the conductivity inside a body from the Dirichlet-to-Neumann map. The problem is of interest in medical imaging and geophysics, where one seeks to image the conductivity of a body by making voltage and current measurements at its surface. Bukhgeim-Uhlmann and Kenig-Sjöstrand-Uhlmann have shown that (in dimensions three and higher) uniqueness in the above problem holds even if measurements are available on possibly very small subsets of the boundary. This course will explain in detail a constructive proof of these results, obtained in joint work with Brian Street. The topics I hope to cover are: 1. Review of the reconstruction method for Calderón's problem with full data. 2. New Green's functions for the Laplacian. 3. Boundedness properties of the corresponding single layer operators. 4. New solutions of the Schrödinger equation. 5. Unique solvability of the main boundary integral equation involving only the partial Cauchy data. |
||||

10:30-11:00 | Morning coffee | |||

11:00-11:45 | Uhlmann, G (Washington & UC, Irvine) |
|||

C1 - Photoacoustic and Thermoacoustic Tomography IV | Sem 1 | |||

11:45-12:30 | Nachman, A (Toronto) |
|||

C2 - Reconstructions from Partial Boundary Data IV | Sem 1 | |||

12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Kress, R (Göttingen) |
|||

Highlighted lecture 1 - Iterative methods in inverse obstacle scattering revisited | Sem 1 | |||

The inverse problem we consider is to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic plane waves. For the sake of simplicity, we will concentrate on the case of scattering from a sound-soft obstacle or a perfect conductor. After reviewing some basics, we will interpret Huygens' principle as a system of two integral equations, named data and field equation, for the unknown boundary of the scatterer and the induced surface flux. Reflecting the ill-posedness of the inverse obstacle scattering problem these integral equations are ill-posed. They are linear with respect to the unknown flux and nonlinear with respect to the unknown boundary and offer, in principle, three immediate possibilities for their iterative solution via linearization and regularization. We will discuss the mathematical foundations of these algorithms and describe the main ideas of their numerical implementation. Further, we will illuminate various relations between them and exhibit connections and differences to the traditional regularized Newton type iterations as applied to the boundary to far field map. Numerical results in 3D are presented. |
||||

15:00-15:30 | Afternoon tea | |||

16:00-18:00 | Poster session |

Wednesday 27 July | ||||

09:00-09:45 | Kirsch, A (KIT) |
|||

C3 - The Factorization Method for Inverse Problems I | Sem 1 | |||

In this talk we introduce the Factorization Method for solving certain inverse problems. We will mainly consider inverse scattering problems but indicate the applicability of this method to other types of inverse problems at the end of the course. First, we explain the Factorization Method for a simple finite dimensional example of an inverse scattering problem (scattering by point sources). Then we turn to a scattering problem for time-harmonic acoustic waves where plane waves are scattered by an inhomogeneous medium. We will briefly discuss the direct problem with respect to uniqueness and existence and derive the Born approximation. In the inverse scattering problem one tries to determine the index of refraction from the knowledge of the far field patterns. First we consider the Born approximation which linearizes the inverse problem. We apply the Factorization Method to this approximation for the determination of the support of the refractive contrast before we, finally, investigate this method for the full nonlinear problem. This talk will be rather elementary. Knowledge of some basic facts on Hilbert spaces (including the space L2(D) and the notion of compactness) is sufficient for understanding this talk. |
||||

09:45-10:30 | Siltanen, S (Helsinki) |
|||

C4 - Introduction to computational inversion I | Sem 1 | |||

Inverse problems arise from indirect measurements of physical quantities. Examples include recovering the internal structure of objects from boundary measurements, for example X-ray attenuation from projection images or electric conductivity distribution from current-to-voltage measurements at the boundary. A defining feature of inverse problems is "ill-posedness", or extreme sensitivity to measurement and modeling errors: two quite different objects may produce almost exactly the same data. This is why specially regularized reconstruction methods are needed for the practical solution of inverse problems. This course explains how to detect ill-posedness in practical measurements and how to design noise-robust computational inversion algorithms. X-ray tomography is used as a guiding example, and Tikhonov regularization is the basic numerical methodology. Matlab software is provided for the participants to enable numerical experiments. The zip archive contains Matlab files relating to Tomography with explicitly constructed measurement matrix. Feel free to experiment with these files. If you use them as part of your research, please include a reference to the origin and the author of the files. |
||||

10:30-11:00 | Morning coffee | |||

11:00-11:45 | Kirsch, A (KIT) |
|||

C3 - The Factorization Method for Inverse Problems II | Sem 1 | |||

In this talk we introduce the Factorization Method for solving certain inverse problems. We will mainly consider inverse scattering problems but indicate the applicability of this method to other types of inverse problems at the end of the course. First, we explain the Factorization Method for a simple finite dimensional example of an inverse scattering problem (scattering by point sources). Then we turn to a scattering problem for time-harmonic acoustic waves where plane waves are scattered by an inhomogeneous medium. We will briefly discuss the direct problem with respect to uniqueness and existence and derive the Born approximation. In the inverse scattering problem one tries to determine the index of refraction from the knowledge of the far field patterns. First we consider the Born approximation which linearizes the inverse problem. We apply the Factorization Method to this approximation for the determination of the support of the refractive contrast before we, finally, investigate this method for the full nonlinear problem. This talk will be rather elementary. Knowledge of some basic facts on Hilbert spaces (including the space L2(D) and the notion of compactness) is sufficient for understanding this talk. |
||||

11:45-12:30 | Siltanen, S (Helsinki) |
|||

C4 - Introduction to computational inversion II | Sem 1 | |||

Inverse problems arise from indirect measurements of physical quantities. Examples include recovering the internal structure of objects from boundary measurements, for example X-ray attenuation from projection images or electric conductivity distribution from current-to-voltage measurements at the boundary. A defining feature of inverse problems is "ill-posedness", or extreme sensitivity to measurement and modeling errors: two quite different objects may produce almost exactly the same data. This is why specially regularized reconstruction methods are needed for the practical solution of inverse problems. This course explains how to detect ill-posedness in practical measurements and how to design noise-robust computational inversion algorithms. X-ray tomography is used as a guiding example, and Tikhonov regularization is the basic numerical methodology. Matlab software is provided for the participants to enable numerical experiments. The zip archive contains Matlab files relating to Tomography with explicitly constructed measurement matrix. Feel free to experiment with these files. If you use them as part of your research, please include a reference to the origin and the author of the files. |
||||

12:30-13:30 | Lunch at Wolfson Court | |||

13:30-17:30 | Free afternoon | |||

19:30-22:00 | Conference dinner at Peterhouse |

Thursday 28 July | ||||

09:00-09:45 | Kirsch, A (KIT) |
|||

C3 - The Factorization Method for Inverse Problems III | Sem 1 | |||

In this talk we introduce the Factorization Method for solving certain inverse problems. We will mainly consider inverse scattering problems but indicate the applicability of this method to other types of inverse problems at the end of the course. First, we explain the Factorization Method for a simple finite dimensional example of an inverse scattering problem (scattering by point sources). Then we turn to a scattering problem for time-harmonic acoustic waves where plane waves are scattered by an inhomogeneous medium. We will briefly discuss the direct problem with respect to uniqueness and existence and derive the Born approximation. In the inverse scattering problem one tries to determine the index of refraction from the knowledge of the far field patterns. First we consider the Born approximation which linearizes the inverse problem. We apply the Factorization Method to this approximation for the determination of the support of the refractive contrast before we, finally, investigate this method for the full nonlinear problem. This talk will be rather elementary. Knowledge of some basic facts on Hilbert spaces (including the space L2(D) and the notion of compactness) is sufficient for understanding this talk. |
||||

09:45-10:30 | Borcea, L (Rice) |
|||

C5 - Coherent interferometric imaging in random media I | Sem 1 | |||

I will describe the mathematical problem of imaging remote sources or reflectors in heterogeneous (cluttered) media with passive and active arrays of sensors. Because the inhomogeneities in the medium are not known and cannot be estimated, we model the uncertainty about the clutter with spatial random perturbations of the wave speed. The goal of the lectures is to carry out analytically a comparative study of the resolution and signal-to-noise ratio (SNR) of two array imaging methods: the widely used Kirchhoff migration (KM) and coherent interferometry (CINT). By noise in the images we mean fluctuations that are due to the random medium. Kirchhoff migration and its variants are widely used in seismic inversion, radar and elsewhere. It forms images by superposing the wave fields received at the array, delayed by the travel times from the array sensors to the imaging points. KM works well in smooth and known media, where there is no wave scattering and the travel times can be estimated accurately. It also works well with data that has additive, uncorrelated measurement noise, provided the array is large, because the noise is averaged out by the summation over the many sensors. KM images in clutter are unreliable and difficult to interpret because of the significant wave distortion by the inhomogeneities. The distortion is very different from additive, uncorrelated noise, and it cannot be reduced by simply summing over the sensors in the array. CINT images efficiently in clutter at ranges that do not exceed one or two transport mean free paths. Beyond such ranges the problem becomes much more difficult, specially in the case of active arrays, because the clutter backscatter overwhelms the echoes from the reflectors that we wish to image. Coherent imaging in such media may work only after pre-processing the data with filters of clutter backscatter. The CINT method forms images by superposing time delayed, local cross-correlations of the wave fields received at the array. The local cross correlations are computed in appropriate time windows and over limited array sensor offsets. It has been shown with analysis and verified with numerical simulations that the time and offset thresholding in the computation of the cross-correlations is essential in CINT, because it introduces a smoothing that is necessary to achieve statistical stability, at the expense of some loss in resolution. By statistical stability we mean negligibly small fluctuations in the CINT image even when cumulative fluctuation effects in the random medium are not small. |
||||

10:30-11:00 | Morning coffee | |||

11:00-11:45 | Kirsch, A (KIT) |
|||

C3 - The Factorization Method for Inverse Problems IV | Sem 1 | |||

11:45-12:30 | Borcea, L (Rice) |
|||

C5 - Coherent interferometric imaging in random media II | Sem 1 | |||

I will describe the the mathematical problem of imaging remote sources or reflectors in heterogeneous (cluttered) media with passive and active arrays of sensors. Because the inhomogeneities in the medium are not known, and they cannot be estimated in detail from the data gathered at the array, we model the uncertainty about the clutter with spatial random perturbations of the wave speed. The goal of the lectures is to carry out analytically a comparative study of the resolution and signal-to-noise ratio (SNR) of two array imaging methods: the widely used Kirchhoff migration (KM) and coherent interferometry (CINT). By noise in the images we mean fluctuations that are due to the random medium. Kirchhoff migration [2, 3] and its variants are widely used in seismic inversion, radar [10] and elsewhere. It forms images by superposing the wave fields received at the array, delayed by the travel times from the array sensors to the imaging points. KM works well in smooth and known media, where there is no wave scattering and the travel times can be estimated accurately. It also works well with data that has additive, uncorrelated measurement noise, provided the array is large, because the noise is averaged out by the summation over the many sensors, as expected from the law of large numbers. KM images in heterogeneous (cluttered) media are unreliable and difficult to interpret because of the significant wave distortion by the inhomogeneities. The distortion is very different from additive, uncorrelated noise, and it cannot be reduced by simply summing over the sensors in the array. CINT images efficiently in clutter [5, 6, 7], at ranges that do not exceed one or two transport mean free paths [11]. Beyond such ranges the problem becomes much more difficult, specially in the case of active arrays, because the clutter backscatter overwhelms the echoes from the reflectors thatwe wish to image. Coherent imaging in such media may work only after pre-processing the data with filters of clutter backscatter, as is done in [9, 1]. The CINT method has been introduced in [5, 6, 7] for mitigating wave distortion effects induced by clutter. It forms images by superposing time delayed, local cross-correlations of the wave fields received at the array. Here local cross correlations means that they are computed in appropriate time windows and over limited array sensor offsets. It has been shown with analysis and verified with numerical simulations [5, 6, 7, 8, 4] that the time and offset thresholding in the computation of the cross-correlations is essential in CINT, because it introduces a smoothing that is necessary to achieve statistical stability, at the expense of some loss in resolution. By statistical stability we mean negligibly small fluctuations in the CINT image even when cumulative fluctuation effects in the random medium are not small. |
||||

12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Kurylev, Y (UCL) |
|||

Highlighted lecture 2 - Stability of Inverse Problems and related Topics from Differential Geometry | Sem 1 | |||

This talk (joint with M. Lassas and T. Yamaguchi) deals with the problem of stabilisation of anisotropic inverse spectral problem. We treat such problems as inverse problems on an unknown Riemannian manifold and formulate conditions for stability in terms of geometric constaints. Analysing the influence of these constraints, we obtain a wider class of objects than Riemannian manifolds, namely orbifolds. We then consider the relation between the stability of inverse problems on manifolds and uniqueness of the inverse problems on the orbifolds. |
||||

15:00-15:30 | Afternoon tea | |||

16:00-16:20 | Bleyer, I (Austrian Academy of Sciences) |
|||

Contributed talks. 1. A Double Regularization Approach for Inverse Problems with Noisy Data and Inexact Operator | Sem 1 | |||

16:20-16:40 | Oksanen, L (University of Helsinki) |
|||

Contributed Talks. 2. Inverse problem for the wave equation with disjoint sources and receivers | Sem 1 | |||

16:40-17:00 | Ha Quang, M (Italian Institute of Technology) |
|||

Contributed Talks. 3. Vector-valued Reproducing Kernel Hilbert Spaces with applications in function extension and image colorization | Sem 1 |

Friday 29 July | ||||

09:00-09:45 | Siltanen, S (Helsinki) |
|||

C4 - Introduction to computational inversion III | Sem 1 | |||

Inverse problems arise from indirect measurements of physical quantities. Examples include recovering the internal structure of objects from boundary measurements, for example X-ray attenuation from projection images or electric conductivity distribution from current-to-voltage measurements at the boundary. A defining feature of inverse problems is "ill-posedness", or extreme sensitivity to measurement and modeling errors: two quite different objects may produce almost exactly the same data. This is why specially regularized reconstruction methods are needed for the practical solution of inverse problems. This course explains how to detect ill-posedness in practical measurements and how to design noise-robust computational inversion algorithms. X-ray tomography is used as a guiding example, and Tikhonov regularization is the basic numerical methodology. Matlab software is provided for the participants to enable numerical experiments. The zip archive contains Matlab files relating to Tomography with explicitly constructed measurement matrix. Feel free to experiment with these files. If you use them as part of your research, please include a reference to the origin and the author of the files. |
||||

09:45-10:30 | Borcea, L (Rice) |
|||

C5 - Coherent interferometric imaging in random media III | Sem 1 | |||

I will describe the the mathematical problem of imaging remote sources or reflectors in heterogeneous (cluttered) media with passive and active arrays of sensors. Because the inhomogeneities in the medium are not known, and they cannot be estimated in detail from the data gathered at the array, we model the uncertainty about the clutter with spatial random perturbations of the wave speed. The goal of the lectures is to carry out analytically a comparative study of the resolution and signal-to-noise ratio (SNR) of two array imaging methods: the widely used Kirchhoff migration (KM) and coherent interferometry (CINT). By noise in the images we mean fluctuations that are due to the random medium. Kirchhoff migration [2, 3] and its variants are widely used in seismic inversion, radar [10] and elsewhere. It forms images by superposing the wave fields received at the array, delayed by the travel times from the array sensors to the imaging points. KM works well in smooth and known media, where there is no wave scattering and the travel times can be estimated accurately. It also works well with data that has additive, uncorrelated measurement noise, provided the array is large, because the noise is averaged out by the summation over the many sensors, as expected from the law of large numbers. KM images in heterogeneous (cluttered) media are unreliable and difficult to interpret because of the significant wave distortion by the inhomogeneities. The distortion is very different from additive, uncorrelated noise, and it cannot be reduced by simply summing over the sensors in the array. CINT images efficiently in clutter [5, 6, 7], at ranges that do not exceed one or two transport mean free paths [11]. Beyond such ranges the problem becomes much more difficult, specially in the case of active arrays, because the clutter backscatter overwhelms the echoes from the reflectors thatwe wish to image. Coherent imaging in such media may work only after pre-processing the data with filters of clutter backscatter, as is done in [9, 1]. The CINT method has been introduced in [5, 6, 7] for mitigating wave distortion effects induced by clutter. It forms images by superposing time delayed, local cross-correlations of the wave fields received at the array. Here local cross correlations means that they are computed in appropriate time windows and over limited array sensor offsets. It has been shown with analysis and verified with numerical simulations [5, 6, 7, 8, 4] that the time and offset thresholding in the computation of the cross-correlations is essential in CINT, because it introduces a smoothing that is necessary to achieve statistical stability, at the expense of some loss in resolution. By statistical stability we mean negligibly small fluctuations in the CINT image even when cumulative fluctuation effects in the random medium are not small. |
||||

10:30-11:00 | Morning coffee | |||

11:00-11:45 | Siltanen, S (Helsinki) |
|||

C4 - Introduction to computational inversion IV | Sem 1 | |||

11:45-12:30 | Borcea, L (Rice) |
|||

C5 - Coherent interferometric imaging in random media IV | Sem 1 | |||

I will describe the the mathematical problem of imaging remote sources or reflectors in heterogeneous (cluttered) media with passive and active arrays of sensors. Because the inhomogeneities in the medium are not known, and they cannot be estimated in detail from the data gathered at the array, we model the uncertainty about the clutter with spatial random perturbations of the wave speed. The goal of the lectures is to carry out analytically a comparative study of the resolution and signal-to-noise ratio (SNR) of two array imaging methods: the widely used Kirchhoff migration (KM) and coherent interferometry (CINT). By noise in the images we mean fluctuations that are due to the random medium. Kirchhoff migration [2, 3] and its variants are widely used in seismic inversion, radar [10] and elsewhere. It forms images by superposing the wave fields received at the array, delayed by the travel times from the array sensors to the imaging points. KM works well in smooth and known media, where there is no wave scattering and the travel times can be estimated accurately. It also works well with data that has additive, uncorrelated measurement noise, provided the array is large, because the noise is averaged out by the summation over the many sensors, as expected from the law of large numbers. KM images in heterogeneous (cluttered) media are unreliable and difficult to interpret because of the significant wave distortion by the inhomogeneities. The distortion is very different from additive, uncorrelated noise, and it cannot be reduced by simply summing over the sensors in the array. CINT images efficiently in clutter [5, 6, 7], at ranges that do not exceed one or two transport mean free paths [11]. Beyond such ranges the problem becomes much more difficult, specially in the case of active arrays, because the clutter backscatter overwhelms the echoes from the reflectors thatwe wish to image. Coherent imaging in such media may work only after pre-processing the data with filters of clutter backscatter, as is done in [9, 1]. The CINT method has been introduced in [5, 6, 7] for mitigating wave distortion effects induced by clutter. It forms images by superposing time delayed, local cross-correlations of the wave fields received at the array. Here local cross correlations means that they are computed in appropriate time windows and over limited array sensor offsets. It has been shown with analysis and verified with numerical simulations [5, 6, 7, 8, 4] that the time and offset thresholding in the computation of the cross-correlations is essential in CINT, because it introduces a smoothing that is necessary to achieve statistical stability, at the expense of some loss in resolution. By statistical stability we mean negligibly small fluctuations in the CINT image even when cumulative fluctuation effects in the random medium are not small. |
||||

12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Burger, M (Münster) |
|||

Highlighted lecture 3 - Inverse Problems in Biomedical Imaging | Sem 1 | |||

For several decades, medical imaging applications, in particular computerized tomography, was a source for application and development for inverse problems. The recent emergence of the field towards dynamic and multimodal imaging creates a variety of novel challenges for inverse problems, with respect to modelling, analysis, as well as computation. In this talk we shall highlight some mathematical issues of emerging biomedical imaging techniques such as Dynamic PET, SPECT, MRI and Optical Techniques. |