
JanuaryAugust
2012
Thematic Program on Inverse Problems and Imaging 
July–August 2012
Summer Thematic Program on the Mathematics of Medical
Imaging
Organizers:
Charles Epstein, University of Pennsylvania
Allan Greenleaf, University of Rochester
Jan Modersitzki, University of Lübeck 
Adrian Nachman, University of
Toronto
Gunther Uhlmann, University of Washington
Hongmei Zhu, York University




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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 331, 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 1317, 2012
Workshop on Microlocal Methods in
Medical Imaging
August 2024, 2012
Industrial ProblemSolving Workshop on
Medical Imaging
Schedule of Courses
Week 1 July 36 *Note: All
lectures and tutorials to be held in Fields Institute, Room 230.
Course
on Medical Image Registration, July 36, 2012
Tuesday, July 3, 2012 
9:3010:00 
Introduction to the Summer
School
Adrian Nachman, Guillaume Bal 
10:1012:00 
Medical
Image Registration, Lecture 1
Jan Modersitzki, University of Lübeck 
12:002:00 
Lunch
Break 
2:103:30 
Research
in Mathematical Image Processing, Lecture 1
Todd Wittman, UCLA 
3:304:00 
Tea
Break 
4:106:00 
Medical
Image Registration, Lecture 2
Jan Modersitzki, University of Lübeck 
Wednesday, July 4, 2012 
9:1011:00 
Medical
Image Registration, Lecture 3
Jan Modersitzki, University of Lübeck 
11:1012:30 
Research
in Mathematical Image Processing, Computer Lab
1
Todd Wittman, UCLA 
12:302:00 
Lunch Break 
2:103:00 
Medical
Image Registration, Tutorial
Jan Modersitzki, University of Lübeck 
3:003:30 
Tea Break 
3:306:00 
Medical
Image Registration, Computer Lab 1
Jan Modersitzki, University of Lübeck 
Thursday, July 5, 2012 
9:1011:00 
Medical
Image Registration, Lecture 4
Jan Modersitzki, University of Lübeck 
11:1012:30 
Research
in Mathematical Image Processing, Lecture 2
Todd Wittman, College of Charleston 
12:302:00 
Lunch Break 
2:106:00 
Medical
Image Registration, Computer Lab 2
Jan Modersitzki, University of Lübeck 
Friday, July 6, 2012 
9:1011:00 
Medical
Image Registration, Lecture 5
Jan Modersitzki, University of Lübeck 
11:1012:30 
Medical
Image Registration, Tutorial
Jan Modersitzki, University of Lübeck 
12:302:00 
Lunch Break 
2:103:30 
Research
in Mathematical Image Processing, Computer Lab
1
Todd Wittman, College of Charleston 
3:304:00 
Tea Break 
4:005:30 
Meeting to Discuss Projects 
Week 2 July
913
Monday, July 9, 2012 
9:1010:30 
Research
in Mathematical Image Processing, Lecture 3
Todd Wittman, College of Charleston 
10:3012:30 
Variational Regularization
Methods for Image Analysis and Inverse Problems,
Lecture 1
Otmar Scherzer, University of Vienna 
12:302:00 
Lunch
Break 
2:103:30 
Numerical Methods for
Distributed Parameter Identification, Lecture
1
Eldad Haber, University of British Columbia 
3:304:00 
Tea
Break 
4:005:30 
Meeting to Discuss Projects 
Tuesday, July 10, 2012 
9:1011:00 
Variational
Regularization Methods for Image Analysis and Inverse Problems,
Lecture 2
Otmar Scherzer, University of Vienna 
11:1012:30 
Numerical
Methods for Distributed Parameter Identification,
Lecture 2
Eldad Haber, University of British Columbia 
12:302:00 
Lunch Break 
2:103:00 

3:003:30 
Tea Break 
3:305:30 
Research
in Mathematical Image Processing, Computer
Lab 3
Todd Wittman, College of Charleston 
Wednesday, July 11, 2012 
9:1011:00 
Variational Regularization
Methods for Image Analysis and Inverse Problems,
Lecture 3
Otmar Scherzer, University of Vienna 
11:1012:30 
Numerical
Methods for Distributed Parameter Identification,
Lecture 3
Eldad Haber, University of British Columbia 
12:302:00 
Lunch Break 
2:103:30 
Frontiers in Rapid MRI, from
Parallel Imaging to Compressed Sensing and Back, Lecture
1
Michael Lustig, UC Berkeley 
3:304:00 
Tea Break 
4:105:00 
Fields
Institute Colloquium in Applied Mathematics
Momentum Maps, Image Analysis &
Solitons
Darryl D Holm, Imperial College London 
Thursday, July 12, 2012 
9:1010:30 
Numerical Methods for
Distributed Parameter Identification, Lecture
4
Eldad Haber, University of British Columbia 
10:4512:15 
Geometry
of Image Registration Diffeomorphism group and Momentum Maps,
Lecture 1
Martins Bruveris, Imperial College London 
12:152:00 
Lunch Break 
2:103:30 
Frontiers in Rapid MRI, from
Parallel Imaging to Compressed Sensing and Back, Lecture
2
Michael Lustig, UC Berkeley 
3:304:00 
Tea Break 
4:105:30 
Research
in Mathematical Image Processing, Lecture
4
Todd Wittman, College of Charleston 
Friday, July 13, 2012 
9:1010:30 
Frontiers in Rapid MRI, from
Parallel Imaging to Compressed Sensing and Back, Lecture
3
Michael Lustig, UC Berkeley 
12:002:00 
Lunch Break 
2:103:30 
Geometry
of Image Registration Diffeomorphism group and Momentum Maps,
Lecture 2
Martins Bruveris, Imperial College London 
3:304:00 
Tea Break 
4:006:00 
Research
in Mathematical Image Processing, Computer
Lab 4
Todd Wittman, College of Charleston 
Week 3 July
1620
Monday, July 16, 2012 
9:1010:30 
Research
in Mathematical Image Processing, Lecture 5
Todd Wittman, College of Charleston 
10:4012:30 
Variational Regularization
Methods for Image Analysis and Inverse Problems,
Lecture 4
Otmar Scherzer, University of Vienna 
12:302:00 
Lunch
Break 
2:103:30 
Microlocal
Approach to Photoacoustic and Thermoacoustic Tomography
Lecture 1
Plamen Stefanov, Purdue University 
3:304:00 
Tea
Break 
4:105:30 
Geometry
of Image Registration Diffeomorphism group and Momentum Maps,
Lecture 3
Martins Bruveris, Imperial College London 
Tuesday, July 17, 2012 
9:1011:00 
Research
in Mathematical Image Processing, Computer Lab
3
Todd Wittman, College of Charleston 
11:1012:30 
Meeting to Discuss Projects 
12:302:00 
Lunch Break 
2:103:00 
Electromagnetics,
Theory and Practice Lecture 1
Charles Epstein, University of Pennsylvania 
3:003:30 
Tea Break 
3:305:30 
Geometry of Image
Registration Diffeomorphism group and Momentum Maps,
Lecture 3
Martins Bruveris, Imperial College London 
Wednesday, July 18, 2012 
9:1011:00 
Variational Regularization
Methods for Image Analysis and Inverse Problems,
Lecture 5
Otmar Scherzer, University of Vienna 
11:1012:30 
Microlocal
Approach to Photoacoustic and Thermoacoustic Tomography
Lecture 2
Plamen Stefanov, Purdue University 
12:302:00 
Lunch Break 
2:103:30 
Microlocal
Approach to Photoacoustic and Thermoacoustic Tomography
Lecture 3
Plamen Stefanov, Purdue University 
3:304:00 
Tea Break 
4:105:30 
The inverse
conductivity problem from knowledge of power densities in
dimensions two and three, Tutorial 1
Francois Monard, Columbia University 
Thursday, July 19, 2012 
9:1010:00 
Hybrid inverse problems,
Lecture 1
Guillaume Bal, Columbia University 
10:1511:40 
Microlocal
Approach to Photoacoustic and Thermoacoustic Tomography
Lecture 4
Plamen Stefanov, Purdue University 
11:401:30 
Lunch Break 
1:303:00 
Electromagnetics,
Theory and Practice Lecture 2
Charles Epstein, University of Pennsylvania 
3:003:30 
Tea Break 
3:304:30 
Microlocal Approach
to Photoacoustic and Thermoacoustic Tomography
Lecture 5
Plamen Stefanov, Purdue University 
4:306:00 
Meeting to Discuss Projects 
Friday, July 20, 2012 
9:1010:00 
Hybrid inverse
problems, Lecture 2
Guillaume Bal, Columbia University 
10:1011:30 
Electromagnetics,
Theory and Practice Lecture 3
Charles Epstein, University of Pennsylvania 
11:301:40 
Lunch Break 
1:403:00 
Electromagnetics,
Theory and Practice Lecture 3
Charles Epstein, University of Pennsylvania 
3:003:30 
Tea Break 
3:304:50 
Microlocal
Approach to Photoacoustic and Thermoacoustic Tomography
Lecture 5
Plamen Stefanov, Purdue University 
5:006:30 
The inverse conductivity
problem from knowledge of power densities in dimensions two
and three, Tutorial 2
Francois Monard, Columbia University 
Week 4 July 2327
Graduate Courses
Hybrid Inverse
Problems
Guillaume Bal, Columbia University
Several coupledphysics modalities, such as Photoacoustic 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,
coupledphysics, or multiwave 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 interpatient
variability and the presence of noise make this task mathematically
very challenging. Beginning from the problem of registration 
finding pointtopoint 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 EulerPoincar\'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 pointtopoint 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 socalled, 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, crosscorrelation,mutual
information, normalized gradient fields.
 Parametric registration: modeling, numerical and implementation
issues.
 Nonparametric registration: wellposedness, regularization,
physical models, elasticity.
 Numerical methods for nonparametric 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 $a\in L^2(\Omega)$ nonnegative
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,
 XRay Based Computerized Tomography,
 Thermoacoustic Tomography,
 Schlieren Tomography.
 Image and Noise Models:
Basic Concepts of Statistics,
Digitized (Discrete) Images,
Noise Models,
Priors for Images,
 Maximum APosteriori 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 minicourse is to present a microlocal
approach to
multiwave imaging, including thermo and photoacoustic 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 minicourse
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
Xray 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
Xray and Radon transforms.
Research in
Mathematical Image Processing
Todd Wittman, UCLA
The goal of this course is to give graduate students handson
dataintensive 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 RudinOsherFatemi Total Variation image model
denoising by nonlocal means
The ChanVese 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 thermoacoustics
and related problems
Ting Zhou, MIT
Thermoacoustic tomography is a hybrid multiwaves medical
imaging modality that aims to combine the good optical contrast
observed in tissues with the good resolution properties of ultrasound.
Thermoacoustic 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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