THEMATIC PROGRAMS

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For the first three weeks there will be short courses and one hour lectures on new results by senior researchers to be invited around the week’s theme in addition to the graduate course.

July 3-31, 2012
Summer Research School on the Mathematics of Medical Imaging (schedule of courses)
Organizers:
Guillaume Bal, Columbia University
Allan Greenleaf, University of Rochester
Adrian Nachman, University of Toronto
Todd Wittman, UCLA
Luminita Vese, UCLA

The program will open to applications and about 40 participants will be selected. They will be organized into small teams of up to 5 graduate students and postdocs, based on the project they choose. Some of these will be alumni of the AMS MRC 2009 conference on "Inverse Problems". The groups will work on a range of research problems.
For the first three weeks a number of graduate courses and short courses will be offered. In addition, there will be lectures by senior researchers to be invited around the week's theme.

At the end of the month, each team will present their work at a presentation session. Additionally, each team will prepare a technical report, describing their problem, their proposed ideas, and any preliminary results.

August 13-17, 2012
Workshop on Microlocal Methods in Medical Imaging

August 20-24, 2012
Industrial Problem-Solving Workshop on Medical Imaging

Schedule of Courses

Week 1 July 3-6 *Note: All lectures and tutorials to be held in Fields Institute, Room 230.

Course on Medical Image Registration, July 3-6, 2012

Tuesday, July 3, 2012
9:30-10:00	Introduction to the Summer School Adrian Nachman, Guillaume Bal
10:10-12:00	Medical Image Registration, Lecture 1 Jan Modersitzki, University of Lübeck
12:00-2:00	Lunch Break
2:10-3:30	Research in Mathematical Image Processing, Lecture 1 Todd Wittman, UCLA
3:30-4:00	Tea Break
4:10-6:00	Medical Image Registration, Lecture 2 Jan Modersitzki, University of Lübeck
Wednesday, July 4, 2012
9:10-11:00	Medical Image Registration, Lecture 3 Jan Modersitzki, University of Lübeck
11:10-12:30	Research in Mathematical Image Processing, Computer Lab 1 Todd Wittman, UCLA
12:30-2:00	Lunch Break
2:10-3:00	Medical Image Registration, Tutorial Jan Modersitzki, University of Lübeck
3:00-3:30	Tea Break
3:30-6:00	Medical Image Registration, Computer Lab 1 Jan Modersitzki, University of Lübeck
Thursday, July 5, 2012
9:10-11:00	Medical Image Registration, Lecture 4 Jan Modersitzki, University of Lübeck
11:10-12:30	Research in Mathematical Image Processing, Lecture 2 Todd Wittman, College of Charleston
12:30-2:00	Lunch Break
2:10-6:00	Medical Image Registration, Computer Lab 2 Jan Modersitzki, University of Lübeck
Friday, July 6, 2012
9:10-11:00	Medical Image Registration, Lecture 5 Jan Modersitzki, University of Lübeck
11:10-12:30	Medical Image Registration, Tutorial Jan Modersitzki, University of Lübeck
12:30-2:00	Lunch Break
2:10-3:30	Research in Mathematical Image Processing, Computer Lab 1 Todd Wittman, College of Charleston
3:30-4:00	Tea Break
4:00-5:30	Meeting to Discuss Projects

Week 2 July 9-13

Monday, July 9, 2012
9:10-10:30	Research in Mathematical Image Processing, Lecture 3 Todd Wittman, College of Charleston
10:30-12:30	Variational Regularization Methods for Image Analysis and Inverse Problems, Lecture 1 Otmar Scherzer, University of Vienna
12:30-2:00	Lunch Break
2:10-3:30	Numerical Methods for Distributed Parameter Identification, Lecture 1 Eldad Haber, University of British Columbia
3:30-4:00	Tea Break
4:00-5:30	Meeting to Discuss Projects
Tuesday, July 10, 2012
9:10-11:00	Variational Regularization Methods for Image Analysis and Inverse Problems, Lecture 2 Otmar Scherzer, University of Vienna
11:10-12:30	Numerical Methods for Distributed Parameter Identification, Lecture 2 Eldad Haber, University of British Columbia
12:30-2:00	Lunch Break
2:10-3:00
3:00-3:30	Tea Break
3:30-5:30	Research in Mathematical Image Processing, Computer Lab 3 Todd Wittman, College of Charleston
Wednesday, July 11, 2012
9:10-11:00	Variational Regularization Methods for Image Analysis and Inverse Problems, Lecture 3 Otmar Scherzer, University of Vienna
11:10-12:30	Numerical Methods for Distributed Parameter Identification, Lecture 3 Eldad Haber, University of British Columbia
12:30-2:00	Lunch Break
2:10-3:30	Frontiers in Rapid MRI, from Parallel Imaging to Compressed Sensing and Back, Lecture 1 Michael Lustig, UC Berkeley
3:30-4:00	Tea Break
4:10-5:00	Fields Institute Colloquium in Applied Mathematics Momentum Maps, Image Analysis & Solitons Darryl D Holm, Imperial College London
Thursday, July 12, 2012
9:10-10:30	Numerical Methods for Distributed Parameter Identification, Lecture 4 Eldad Haber, University of British Columbia
10:45-12:15	Geometry of Image Registration -Diffeomorphism group and Momentum Maps, Lecture 1 Martins Bruveris, Imperial College London
12:15-2:00	Lunch Break
2:10-3:30	Frontiers in Rapid MRI, from Parallel Imaging to Compressed Sensing and Back, Lecture 2 Michael Lustig, UC Berkeley
3:30-4:00	Tea Break
4:10-5:30	Research in Mathematical Image Processing, Lecture 4 Todd Wittman, College of Charleston
Friday, July 13, 2012
9:10-10:30	Frontiers in Rapid MRI, from Parallel Imaging to Compressed Sensing and Back, Lecture 3 Michael Lustig, UC Berkeley
12:00-2:00	Lunch Break
2:10-3:30	Geometry of Image Registration -Diffeomorphism group and Momentum Maps, Lecture 2 Martins Bruveris, Imperial College London
3:30-4:00	Tea Break
4:00-6:00	Research in Mathematical Image Processing, Computer Lab 4 Todd Wittman, College of Charleston

Week 3 July 16-20

Monday, July 16, 2012
9:10-10:30	Research in Mathematical Image Processing, Lecture 5 Todd Wittman, College of Charleston
10:40-12:30	Variational Regularization Methods for Image Analysis and Inverse Problems, Lecture 4 Otmar Scherzer, University of Vienna
12:30-2:00	Lunch Break
2:10-3:30	Microlocal Approach to Photoacoustic and Thermoacoustic Tomography Lecture 1 Plamen Stefanov, Purdue University
3:30-4:00	Tea Break
4:10-5:30	Geometry of Image Registration -Diffeomorphism group and Momentum Maps, Lecture 3 Martins Bruveris, Imperial College London
Tuesday, July 17, 2012
9:10-11:00	Research in Mathematical Image Processing, Computer Lab 3 Todd Wittman, College of Charleston
11:10-12:30	Meeting to Discuss Projects
12:30-2:00	Lunch Break
2:10-3:00	Electromagnetics, Theory and Practice Lecture 1 Charles Epstein, University of Pennsylvania
3:00-3:30	Tea Break
3:30-5:30	Geometry of Image Registration -Diffeomorphism group and Momentum Maps, Lecture 3 Martins Bruveris, Imperial College London
Wednesday, July 18, 2012
9:10-11:00	Variational Regularization Methods for Image Analysis and Inverse Problems, Lecture 5 Otmar Scherzer, University of Vienna
11:10-12:30	Microlocal Approach to Photoacoustic and Thermoacoustic Tomography Lecture 2 Plamen Stefanov, Purdue University
12:30-2:00	Lunch Break
2:10-3:30	Microlocal Approach to Photoacoustic and Thermoacoustic Tomography Lecture 3 Plamen Stefanov, Purdue University
3:30-4:00	Tea Break
4:10-5:30	The inverse conductivity problem from knowledge of power densities in dimensions two and three, Tutorial 1 Francois Monard, Columbia University
Thursday, July 19, 2012
9:10-10:00	Hybrid inverse problems, Lecture 1 Guillaume Bal, Columbia University
10:15-11:40	Microlocal Approach to Photoacoustic and Thermoacoustic Tomography Lecture 4 Plamen Stefanov, Purdue University
11:40-1:30	Lunch Break
1:30-3:00	Electromagnetics, Theory and Practice Lecture 2 Charles Epstein, University of Pennsylvania
3:00-3:30	Tea Break
3:30-4:30	Microlocal Approach to Photoacoustic and Thermoacoustic Tomography Lecture 5 Plamen Stefanov, Purdue University
4:30-6:00	Meeting to Discuss Projects
Friday, July 20, 2012
9:10-10:00	Hybrid inverse problems, Lecture 2 Guillaume Bal, Columbia University
10:10-11:30	Electromagnetics, Theory and Practice Lecture 3 Charles Epstein, University of Pennsylvania
11:30-1:40	Lunch Break
1:40-3:00	Electromagnetics, Theory and Practice Lecture 3 Charles Epstein, University of Pennsylvania
3:00-3:30	Tea Break
3:30-4:50	Microlocal Approach to Photoacoustic and Thermoacoustic Tomography Lecture 5 Plamen Stefanov, Purdue University
5:00-6:30	The inverse conductivity problem from knowledge of power densities in dimensions two and three, Tutorial 2 Francois Monard, Columbia University

Week 4 July 23-27

Monday, July 23, 2012
9:10-10:30	Hybrid inverse problems, Lecture 3 Guillaume Bal, Columbia University
10:45-12:30	Work on Projects
12:30-2:00	Lunch Break
2:10-3:30	Quantitative thermo-acoustics and related problems Ting Zhou, MIT
3:30-3:45	Tea Break
3:45-6:00	Work on Projects
Tuesday, July 24, 2012
9:10-10:00	A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data Amir Moradifam, University of Toronto
Friday July 27, 2012
	Project presentations

 

Graduate Courses

Hybrid Inverse Problems
Guillaume Bal, Columbia University

Several coupled-physics modalities, such as Photo-acoustic tomography or Transient elastography, have been proposed and analyzed recently to obtain high contrast, high resolution, reconstructions of constitutive properties of tissues. These inverse problems, called hybrid, coupled-physics, or multi-wave inverse problems, typically involve two steps. The first step is an inverse boundary value problem, which provides internal information about the parameters. The second step, called the quantitative step, aims to reconstruct the parameters from the knowledge of the internal information obtained during the first step. These lectures will review several recent results of uniqueness, stability, and explicit reconstruction procedures obtained for the second step.

Geometry of Image Registration -Diffeomorphism group and Momentum Maps
Martins Bruveris, Imperial College London

Lecture 1: Computational Anatomy - Methods and Mathematical Challenges

Computational anatomy uses the paradigm of pattern theory to study anatomical data obtained via medical imaging methods like CT and MRI. The complexity of this data, the high inter-patient variability and the presence of noise make this task mathematically very challenging. Beginning from the problem of registration - finding point-to-point correspondences between two sets of data - the methods of Riemannian geometry and statistics on manifolds are used to analyse, compare and classify data. This talk will give an overview of the questions studied in computational anatomy and how Riemannian geometry, the diffeomorphism group and geometric mechanics can help answering them.

Lecture 2: Diffeomorphism Group in Computational Anatomy and Hydrodynamics

The diffeomorphism group stands at the intersection of two otherwise unrelated fields. It is used in computational anatomy and is at the heart of the registration process. On the other hand many PDEs of hydrodynamic type can be formulated as geodesic equations on diffeomorphism groups. Both areas are related via Euler-Poincar\'e reduction and share the same geometric framework.

Lecture 3: Geodesics on the diffeomorphism group - the EPDiff equation

In this lecture I will talk about the EPDiff equation, which governs the behaviour of solutions to the registration problem. It can be derived either from a variational principle or as a consequence of momentum preservation. The EPDiff equation is actually a family of equations, parametrized by the choice of the metric, which contains various PDEs known in physics. This lecture will explain how to geometrically derive the EPDiff equation and show some of its mathematical properties.

Lecture 4: Curve matching

This lecture will give an overview of various approaches to curve matching within the framework of Riemannian geometry. The main questions are how to define Riemannian metrics on the space of curves, which metrics are useful and numerically treatable and how to deal with the problem of point-to-point correspondences

Electromagnetics, Theory and Practice
Charles Epstein, University of Pennsylvania

This short course introduces the fundamental concepts of Electromagnetic theory as embodied in Maxwell's equations. Following a short discussion of Maxwell's equations in free space, and the definition of the time harmonic Maxwell Equations, we will discuss the classical boundary value problems, which arise in scattering theory. After defining these problems and establishing the abstract uniqueness of solutions, we will describe various methods for representing solutions to the time harmonic equations using layer potentials, leading to the so-called, Boundary Integral Equation Method (BIEM). Following a short discussion of classical Fredholm theory, we show how these representations lead to a variety of numerical methods for the approximate solution of scattering problems. The course concludes with a short introduction to the Fast Multipole Method (FMM) of Rokhlin and Greengard, which has made it possible to solve large, interesting problems using the BIEM.

Numerical Methods for Distributed Parameter Identification
Eldad Haber, University of British Columbia

Lectures 1 and 2: An introduction to numerical methods for inverse
problems governed by PDE's
Lecture 3: Design in inverse problems
Lecture 4.1 - Joint inversion and data fusion
Lecture 4.2 - Optimal mass transport and related inverse problems

Medical Image Registration
Jan Modersitzki, University of Lübeck
Topics to be covered:

• Introduction to images and transformations.
• Forward and backward models for image deformations.
• Landmark based registration: landmark detection, parameterized models, regularized models, implementation issues.
• Principal axes based registration: introduction to principal axes, transformation properties, implementation issues.
• Images as functions: embedding, discretization, quantization.
• Image distance measures: sum of squared differences, cross-correlation,mutual information, normalized gradient fields.
• Parametric registration: modeling, numerical and implementation issues.
• Non-parametric registration: well-posedness, regularization, physical models, elasticity.
• Numerical methods for non-parametric registration: discretization, sparse representations, optimization.
  

The inverse conductivity problem from knowledge of power densities in dimensions two and three
Francois Monard, Columbia University

In the context of hybrid medical imaging methods, coupling ultrasonic waves with electrical impedance imaging leads in certain contexts to an inverse conductivity problem with internal data functionals of "power density" type. After presenting how to derive such a problem, we will review inversion techniques that were obtained in the past few years for this problem, first in the isotropic case, and if time allows, in the anisotropic case. In both cases, if a "rich enough" set of functionals is provided, all of the conductivity tensor is uniquely reconstructible with good stability properties. This will be contrasted with the results available when considering the same problem from boundary measurements (i.e. Calderon's problem).

A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data
Amir Moradifam, University of Toronto

We consider the hybrid problem of reconstructing the isotropic electric conductivity of a body $\Omega$ from the knowledge of the magnitude $|J|$ of one current generated by a given voltage $f$ on the boundary $\partial\Omega$. The corresponding voltage potential $u$ in $\Omega$ is a minimizer of the weighted least gradient problem

\[u=\hbox{argmin} \{\int_{\Omega}a(x)|\nabla u|: u \in H^{1}(\Omega), \ \ u|_{\partial \Omega}=f\},\] with $a(x)= |J(x)|$. In this talk I will present an alternating split Bregman algorithm for treating such least gradient problems, for $a\in L^2(\Omega)$ non-negative and $f\in H^{1/2}(\partial \Omega)$.

I will sketch a convergence proof by focusing to a large extent on the dual problem. This leads naturally to the alternating split Bregman algorithm. The dual problem also turns out to yield a novel method to recover the full vector field $J$ from knowledge of its magnitude, and of the voltage $f$ on the boundary. I will present several numerical experiments that illustrate the convergence behavior of the proposed algorithm. This is a joint work with A. Nachman and A. Timonov.

Variational Regularization Methods for Image Analysis and Inverse Problems
Otmar Scherzer, University of Vienna
Topics to be covered:

• Case Examples of Imaging:
  - Denoising,
  - Image Inpainting,
  - X-Ray Based Computerized Tomography,
  - Thermoacoustic Tomography,
  - Schlieren Tomography.
• Image and Noise Models:
  -Basic Concepts of Statistics,
  -Digitized (Discrete) Images,
  -Noise Models,
  -Priors for Images,
  - Maximum A-Posteriori Estimation, MAP Estimation for Noisy Images.
• Variational Regularization Methods for the Solution of Inverse Problems:
  - Quadratic Tikhonov Regularization in Hilbert Spaces,
  -Variational Regularization Methods in Banach Spaces,
  - Regularization with Sparsity Constraints,
  - Linear Inverse Problems with Convex Constraints,
  - Schlieren Tomography.
• Convex Regularization Methods for Denoising
• Scale Spaces
  

Microlocal Approach to Photoacoustic and Thermoacoustic Tomography
Plamen Stefanov, Purdue University

The purpose of this mini-course is to present a microlocal approach to
multi-wave imaging, including thermo- and photo-acoustic tomography. The mathematical model is an inverse source problem for the acoustic equation. We assume a variable sound speed. We will review first the theory of the wave equation and its microlocal parametrix. Then we will show how to get sharp uniqueness results for full and partial boundary observations using unique continuation. Next, we will study the stability problem with full and partial data.

In brain imaging, the speed is piecewise smooth only. This changes the
propagation of singularities and created new challenges. We will review the recent progress about this case as well.

Numerical simulations will be shown as well. The mini-course is based on joint papers with Gunther Uhlmann and the numerical results are obtained together with Uhlmann, Qian and Zhao

Microlocal Analysis and Inverse Problems
Gunther Uhlmann, University of Washington and UC Irvine

Microlocal Analysis (MA), which is roughly speaking local analysis in phase space, was developed over 40 years ago by H\"ormander, Maslov, Sato and many others in order to understand the propagation of singularities of solutions of partial differential equations. The early roots of MA were in the theory of geometrical optics. MA has been used successfully in determining the singularities of medium parameters in several inverse problems ranging from X-ray tomography to reflection seismology, synthetic aperture radar and electrical impedance tomography, among several others.

We will briefly discuss some basic concepts of microlocal analysis like the wave front set of a distribution, pseudodifferential and Fourier integral operators and conormal distributions. We will also describe how pseudodifferential and Fourier integral operators arise in several inverse problems and concentrate on studying generalized Radon transforms. These consist, roughly speaking, on integrating a function over families of curves, surfaces, and other submanifolds and generalize the standard X-ray and Radon transforms.

Research in Mathematical Image Processing
Todd Wittman, UCLA

The goal of this course is to give graduate students hands-on data-intensive research experience in medical image processing. Students will be encouraged to experiment with techniques found in recent literature on image processing, particularly algorithms involving variational methods, compressive sensing, and machine learning.

Possible Research Projects
i.) Similarity metrics for medical imagery
ii.) Change detection in MR brain images
iii.) Characterization of placental vascular networks
iv.) Sparse reconstruction in computerized tomography
v.) Contrast enhancement in MR images
vi.) Fusion of medical images from different imaging modalities
vii.) Automatic detection and segmentation of cells in bone marrow tissue.

The lectures will give an introduction to the mathematics of Image Processing. Topics will include:
-The Rudin-Osher-Fatemi Total Variation image model
-denoising by nonlocal means
-The Chan-Vese active contours segmentation model
-Introduction to Wavelets
-Introduction to Compressive sensing and L1 minimization by Bregman iteration.

There will be a discussion of medical image formats and programming with the Matlab Image Processing Toolbox. The lectures will alternate with a Matlab computer lab session where the students will be guided on programming the image processing algorithms discussed in the lecture.
Throughout the course, each team will have regular meetings with the instructor to update on their progress and obtain suggestions for further lines of inquiry.

Quantitative thermo-acoustics and related problems
Ting Zhou, MIT

Thermo-acoustic tomography is a hybrid multi-waves medical imaging modality that aims to combine the good optical contrast observed in tissues with the good resolution properties of ultrasound. Thermo-acoustic imaging consists of two steps: first to reconstruct an amount of electromagnetic radiation absorbed by tissues by solving inverse problems of acoustic waves; secondly to quantitatively reconstruct the optical property of the tissues from the absorption (reconstructed from the first step), which is an internal functional. We are mostly interested in the second step and show some uniqueness and stability results for the full Maxwell's system models under the assumption that the parameter is small, and the uniqueness, stability and reconstruction results for the scalar model. The key ingredient in showing the second result is the complex geometric optics (CGO) solutions.

 

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