Research interests:
Research links at Columbia
University:
Below are brief descriptions of some research areas.
Optical tomography (see below) and Electrical Impedance Tomography are
useful modalities thanks to the large contrast often observed between
the optical and electrical properties of healthy and non-healthy
tissues. However, because of multiple scattering, these are
low-resolution modalities.
Ultrasound tomography enjoys high
resolution capabilities. However, because sound speeds vary little between
healthy and non-healthy tissues, it is low-contrast.
One way to
combine large contrast and high resolution is to use
the photo-acoustic effect: as (near-infra-red or
electromagnetic) radiation propagates through a medium of interest, partial
absorption causes thermal expansion and the generation of
ultrasound. Such ultrasonic waves propagate to the domain's boundary
where they are measured. See the
Wikipedia
web page. The amount of absorbed radiation is first reconstructed by
solving an inverse wave source problem. Next comes the problem
of Quantitative Photo-Acoustic Tomography (QPAT) where the optical
parameters are reconstructed from knowledge of the absorbed radiation
map.
Mathematically, these inverse problems involve parameter reconstructions from
knowledge of Internal Functionals. Such inverse problems are
often referred to as Hybrid Inverse Problems, or
alternatively Coupled-Physics Inverse Problems
or Multi-Waves Inverse Problems.
For recent results on QPAT and the related quantitative
Thermo-acoustic Tomography (QTAT), see the papers
Inverse Diffusion Theory of Photoacoustics (with G. Uhlmann), Inverse Problems, 26(8), 085010, 2010;
Inverse Transport Theory of
Photoacoustics (with A. Jollivet and V. Jugnon), Inverse Problems,
26, 025011, 2010; On
Multi-spectral quantitative photoacoustic tomography (with
K. Ren), Inverse Problems, 27(7), 075003,
2011; Quantitative
Thermo-acoustics and related problems (with K. Ren, G. Uhlmann
and T. Zhou), Inverse Problems 27(5), 055007, 2011; as
well as Multiple-source
quantitative photoacoustic tomography (with K. Ren), Inverse
Problems 28(1), 025010, 2012.
More generally, QPAT, QTAT, as well as the elasticity-based modalities
Transient Elastography (TE) and Magnetic Resonance Elastography (MRE)
all involve reconstructions of parameters in the elliptic equation
\[\nabla\cdot a\nabla u+b\cdot\nabla u+cu=0\]
from knowledge of internal functionals of the form \[H(x)=u(x) \quad
\mbox{ or } \quad H(x) = \Gamma(x) u(x) \] for some typically unknown
function \(\Gamma(x)\). The analysis of what can and cannot be
reconstructed in the possibly complex-valued \((a,b,c)\) for \(a\)
possibly tensor-valued from knowledge of functionals of the form
\(H(x)\) is analyzed in Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions (with
G. Uhlmann), arXiv:1111.5051.
Another hybrid imaging methodology is Ultrasound Modulated Optical
Tomography, also called acousto-optics. In this setting, acoustic waves are emitted to change the index of refraction and the spatial density of the absorbers and scatterers. Light propagating through the medium is influenced by these changes and provides local information about the optical parameters. See the model described in
Inverse Scattering and Acousto-Optic Imaging (with J.C. Schotland), Phys. Rev. Letters, 104, 043902, 2010.
Mathematically, these inverse problems correspond to solving
0-Laplacians from possibly redundant power density measurements of
the form \[H_{ij}(x)=\gamma(x) \nabla u_i(x)\cdot\nabla u_j(x)\] for
\(u_i(x)\) solution of the elliptic equation \(\nabla\cdot
\gamma\nabla u_i=0\) with appropriate conditions \(u_i=g_i\) at the
boundary of the domain of interest. Because the direction of \(\nabla
u_j\) is not available in such measurements, the analysis of such
hybrid inverse problems is significantly more complex than for
functionals of the form \(H(x) = \Gamma(x) u(x)\).
Recent results on the problem are
reported in Cauchy problem
for Ultrasound Modulated EIT , To Appear in Analysis and
PDE; Inverse
diffusion from knowledge of power densities (with E. Bonnetier,
F. Monard and F. Triki), submitted;
and Inverse diffusion
problem with redundant internal information (with
F. Monard), submitted.
The reconstruction of an anisotropic diffusion tensor
\(\gamma\) from such functionals was recently analyzed in two space
dimensions in
Inverse anisotropic diffusion from power density measurements in two dimensions (with
F. Monard), submitted.
A recent review paper on Hybrid Inverse Problems is
available at Hybrid inverse
problems and internal information (review
paper), to appear in Inside-Out, Cambridge University Press, 2012
(G. Uhlmann editor).
The Calderón Prize lecture given during AIP 2011 at Texas A& M is available at Inverse Problems with Internal Functionals. From Calderón's problem to Hybrid Inverse Problems .
Many results of homogenization for
equations with random coefficients have been obtained over the past
decades. In such theories, random solutions are shown to be accurately
approximated by deterministic, effective medium solutions. Of
considerable interest in many applications, including in solving
inverse problems and/or parameter estimation problem, is the characterization
of the random fluctuations in the solution. Such a characterization is understood in an extremely limited number of cases.
What the random fluctuations look like and what they depend upon is analyzed in a series of papers
Central limits and homogenization in random media, (long version with more detailed
proofs arXiv:0710.0363 ), Multiscale Model. Simul., 7(2), pp. 677-702, 2008;
Random integrals and correctors in homogenization (with J. Garnier, S. Motsch, and V. Perrier), Asymptotic Analysis, 59(1-2), pp. 1-26, 2008;
Corrector theory for elliptic equations in random media with
singular Green's function. Application to random boundaries (with
W. Jing), Comm. Math. Sci., 9(2), pp. 383-411, 2011;
Corrector Theory for Elliptic Equations with Long-range Correlated
Random Potential (with J. Garnier, W. Jing, and Y. Gu), to appear in Asymptotic Analysis.
These papers show that the size of the random fluctuations of the
elliptic solution strongly depend on whether the random parameters are
short-range or long-range. The random fluctuations also need not be asymptotically Gaussian as is shown in
Random Homogenization and
Convergence to Integrals with respect to the Rosenblatt Process
(with Y. Gu), submitted.
Consider the parabolic Anderson model
\[ \partial_t u_\varepsilon + (-\Delta)^{\frac m2} u_\varepsilon +
\frac{1}{\varepsilon^\alpha}q(\frac x\varepsilon) u_\varepsilon =0\]
for \(\alpha\) chosen so that the large but rapidly oscillating
(Gaussian) potential \( q \) has a strong influence on
\(u_\varepsilon\). Then the behavior of \(u_\varepsilon\) as
\(\varepsilon\to0\) depends on spatial dimension.
In dimensions \(d\geq m\), we obtain in the limit a homogenized,
deterministic equation as is shown in:
Homogenization with large spatial random potential, Multiscale Model. Simul., 8, pp. 1484-1510, 2010.
In smaller dimensions \(d < m\), the solution to the equation with
heterogeneous coefficients does not
converge to a deterministic equation but
rather to a stochastic partial
differential equation (SPDE) with multiplicative noise
as is shown in:
Convergence to SPDEs in Stratonovich
form, Comm. Math. Phys., 212(2),
pp. 457-477, 2009.
The generalization to long-range, possibly time-dependent, potentials
is analyzed in:
Convergence to Homogenized or
Stochastic Partial Differential Equations, Appl Math Res
Express, 2011(2), pp. 215-241, 2011.
There are several applications to the macroscopic modeling of the random
fluctuations of PDE solutions. One such application is the derivation
of physics-based noise models in Inverse Problems;
see below.
An other application pertains to the testing of numerical multi-scale
algorithms. We can then ask ourselves whether in a well-understood
environment, the random fluctuations generated by the algorithm are
accurate discretizations of the random fluctuations of the continuous
model; see Corrector theory for MsFEM
and HMM in random media (with W. Jing), Multiscale
Model. Simul., 9, pp. 1549-1587, 2011, for an analysis in
the (already technically challenging) one-dimensional setting.
As waves
propagate through heterogeneous media, they scatter and their energy may change
direction. An accurate macroscopic description of such phenomena
is obtained by using kinetic models to represent the energy density of
the propagating waves. Kinetic models more generally may be used to
track the correlation of two wave fields propagating in possibly
different heterogeneous media. Here is the paper Kinetics of scalar wave fields in
random media on the subject. Kinetic models have also been
validated numerically (with Olivier Pinaud) in Accuracy of transport models for
waves in random media.
In statistically stable random media, i.e., in media in which the
energy density can be shown to depend weakly on the specific
realization of the random media, the kinetic models may be used to
detect and image buried inclusions in a cluttered environment. I
refer, e.g., to the paper Self-averaging in time reversal for
the parabolic wave equation (written with George Papanicolaou and
Lenya Ryzhik) for work on the statistical stability of waves in random
media. [Note that waves are not always stable as was shown in the
paper: On the self-averaging of
wave energy in random media.]
References on Imaging in random media using kinetic models may be
found in a series of papers: Kinetic Models for Imaging in
Random Media
(with Olivier Pinaud) and Transport-based imaging in
random media (with Kui Ren). Results on kinetic-based
reconstructions from experimental data (collected in Larry Carin's
group at Duke University) may be found Experimental validation
of a transport-based imaging method in highly scattering environments
(written with Larry Carin, Dehong Liu, and Kui Ren).
Theoretical understanding. Time reversal consists
of understanding why time-reversed waves propagating in highly
heterogeneous media refocus much more tightly at their original
location than when they propagate in a homogeneous medium. A numerical simulation demonstrating
the effects of time reversed wave pulses propagating in random media
can be seen on this web page. The
reason for the very good refocusing observed in heterogeneous media
and absent in homogeneous media is multiple scattering. Multiple
interactions of waves with the underlying structure can be modeled in
several ways. Radiative transfer models are multi-dimensional models that accounts very well for the spatial multiple scattering observed in physical experiments. Our quantitative analysis of time reversal refocusing can be found in the
paper: Time Reversal and refocusing in Random
Media.
Changing media. In many practical applications, the media
during the forward and backward stages of the time reversal experiment
may slightly (at best) differ. We have shown in
the paper Time Reversal in Changing
Environment, how the refocusing of the time reversed signal is
affected by changes in the underlying media. How much can media change
before time reversed waves lose their enhanced refocusing properties?
Our answer is: very little. A mathematical analysis in the simplified
setting of paraxial approximations may be found in the following paper
with L. Ryzhik: Stability of time
reversed waves in changing media.
Detection and Imaging. Several studies have recently been
conducted to understand how the refocusing properties of time
reversed waves may be used in detection and imaging in heterogeneous
media. Unless the underlying propagating media is known (or the
available approximation correlates very closely with the exact media,
which is seldom the case in applications) time reversal and imaging
live in different worlds. In the former, the time reversed signal
back-propagates through the real physical media whereas in the latter,
its propagation must be simulated on the computer, whence in a very
different manner. In a recent paper with Olivier Pinaud, Time reversal based detection in
random media, we show that time reversal may still be used in
detection and imaging in highly heterogeneous media provided that the
underlying media is known statistically (which is much easier to get
than full knowledge) and statistically stable, which is
arguably the main hindrance to successful detection and imaging in
heterogeneous media. Reconstructions based on experimental data
(collected by Larry Carin's group at Duke University) may be found in
Electromagnetic Time-Reversal Imaging in
Changing Media: Experiment and Analysis.
Links to Talks and Lecture Notes
Here is a link to the presentation (20Mb
file with movies) given at the AIP 2007 in Vancouver in the summer of
2007.
Here is a two-hour talk given at the
end of a course on "Waves in Random Media", two tutorial lectures ( Lecture 1; Lecture 2) given in September
of 2005 at the CIRM, Luminy, France, and a fairly long presentation
given in the fall of 2003 at IPAM, on wave propagation in random
media and time reversal.
Here is a link to Lecture Notes for
a course on waves in random media given at Columbia in the fall of 2005.
Transport equations are ubiquitous in medical and geophysical imaging.
Scattering-free transport equations model high-energy particles
propagating through human tissues, as in X-Ray
Tomography. Lower frequency (near-infra-red) photons as used in Optical Tomography are modeled by transport equations
with large scattering coefficients.
Inverse transport theories strongly depend on the amount of scattering
in the forward transport equation. In highly scattering environments, inverse transport is replaced by inverse diffusion problems, which are well-studied. In the absence of scattering, inverse transport is an integral geometry problem, which is also well understood. Inverse transport bridges the gap between the two regimes. See the paper Inverse Transport Theory and applications (review paper), Inverse Problems,
25, 055006, 2009, for a review of recent results obtained in inverse transport theory.
A major difficulty in inverse transport theory is to understand which optical properties of the underlying medium may be stably reconstructed. This crucially depends on the source (isotropic versus angularly resolved) and on the measurements (time dependent or not, angularly resolved or angularly averaged). Here is a sequence of recent papers on this topic. Time-dependent angularly
averaged inverse transport (with A. Jollivet),
Inverse Problems, 25, 075010, 2009. (long version with additional
proofs and results arXiv:0902.3432 ) ; Approximate
stability in inverse transport (with A. Jollivet), in Biomedical Mathematics, Ed. Y. Censor, M. Jiang, G. Wang, Medical Physics Publishing, Wisconsin, 2010;
Stability for time dependent
inverse transport (with A. Jollivet), SIAM J. Math. Anal., 42(2), pp. 679-700, 2010;
Inverse transport with
isotropic sources and angularly averaged measurements , (with
I. Langmore and F. Monard), Inverse Probl. Imaging,
2(1), pp. 23-42, 2008
Here is a sequence of three
talks ( Lecture 1, Lecture 2, Lecture 3) given on the subject at
the University of Washington during the Summer School organized in
August 2005 by
Gunther Uhlmann.
X-Ray tomography (or computerized tomography) is ubiquitous in medical
imaging. Not surprisingly, it uses X-ray beams, composed of
highly-energetic photons. The image reconstruction from X-ray
measurements is based on the Radon transform; see the
books by Frank Natterer on the subject.
SPECT (Single Photon Emission Computed [or Computerized] Tomography)
is an inverse source problem. One is interested in reconstructing the
spatial distribution of a source of radiation from boundary
measurements. Mathematically, because the photons are absorbed by
human tissues before they can be measured, one has to invert an
attenuated Radon transform . Exact inversion formulas have only
obtained very recently and independently by A.L. Bukhgeim and
collaborators using the technique of A-analytic functions and by
R.G. Novikov using techniques borrowed from inverse
scattering. I have worked on extension of the latter work for more
general source terms (with applications in Doppler tomography for
instance) and for incomplete angular measurements. See the following
paper On the attenuated Radon transform
with full and partial measurements. An extension to scattering
problems may be found in the joint paper with A. Tamasan: Inverse source problems in transport
equations. One of the main advantages of explicit inversion
formulas is that they allow us to construct fast inversion
algorithms. With Philippe Moireau, we have generalized the
Fast Slant Stack method to the inversion of the attenuated Radon
transform with full or partial measurements. See the following paper:
Fast numerical inversion of the
attenuated Radon transform with full and partial measurements.
Ray transforms also appear in geophysical imaging. The main difference
with respect to most imaging techniques is that rays are now curved.
In the simplified (yet practically useful) case where the rays are the
geodesics of a hyperbolic metric, I have extended the explicit
formulas obtained in Euclidean geometry to the scalar and vectorial
source problem in hyperbolic geometry; see the paper:
Ray Transforms in Hyperbolic Geometry.
Ray transforms also find applications in the reconstruction of
concentration profiles in the atmosphere. In the non-scattering
framework, frequency-dependent radiation emitted by different gases
may be measured by satellites. In the one-dimensional setting, there
is only one ray (radiation moves up). The z-dependent concentration
profiles may thus be reconstructed from the frequency dependency in
the measurements. Unlike the ray transform problems mentioned above,
the concentration profile reconstruction is a severely ill-posed
problem (so that noise is severely amplified during the reconstruction
process). With Kui Ren, we show this and provide methods to recover
thin layers with sharp contrast (dust layers or ozone layers) in the
following paper Atmospheric Concentration
Profile Reconstructions from Radiation Measurements.
Here is a presentation I gave at IPAM in
the fall of 2003 on SPECT problems.
Optical tomography (OT) uses near infra red (NIR) photons to probe
human tissues. NIR photons are very low energy and thus are quite
harmless. The corollary of their low energy is that they scatter a
lot with human tissues, which significantly reduces their use in
medical imaging. Despite its relatively poor spatial resolution, OT
has a however great advantage: it can image tissue properties (such as
absorption) that other imaging techniques cannot do.
Motivated by works by S. Arridge, I
have worked on the modeling of photon propagation in domains that are
highly scattering everywhere except in small-volume non-scattering
layers. The understanding of such domains is important if photons are
to be used to image properties of the human head.
With Kui Ren, we have shown
that a macroscopic (hence computationally cheap) generalized
diffusion equation including singular interfaces could be used to
model such a propagation; see
Generalized diffusion model in optical tomography with clear layers
. A theoretical and numerical analysis of such models can also
be found in the papers Transport through
diffusive and non-diffusive regions, embedded objects, and clear
layers and Particle transport
through scattering regions with clear layers and inclusions .
Once the clear layers have been modeled by singular interfaces, it
remains an interesting problem to understand whether they (and the
other relevant physical parameters involved) can be reconstructed from
boundary measurements. Some answers are provided in the paper Reconstructions in impedance and
optical tomography with singular interfaces. The latter paper
adapts to singular layers the factorization method developed by
A. Kirsch .
Inverse Problems in OT are known to be extremely ill-posed, with an
error in the reconstruction of the diffusion and absorption properties
logarithmic in the error level on the measured data (in the sense that
an accuracy of 10E-10 in the data may result in an accuracy of 1/10 in
the reconstruction). Here is a way to somewhat overcome this
stability constraint by using asymptotic expansions
of small volume
inclusions. Another promising approach to increase the amount of
available data is to use time harmonic sources. The following paper in
collaboration with with K. Ren and A. Hielscher deals with this
issue: Frequency Domain Optical Tomography
Based on the Equation of Radiative Transfer.
In that paper, we show that increasing the source modulation frequency
improves the reconstruction of the optial coefficients. Theoretical
explanations based on careful stationary phase analyses can be found
in the two papers:
Angular average of time-harmonic transport solutions (with A.Jollivet, I.Langmore, and F.Monard)
Comm. Partial Differential Equations, 36(6),
pp. 1044-1070, 2011; and
Inverse Transport with isotropic time harmonic sources (with F. Monard), SIAM J. Math. Anal., 2012.