Research interests:
Some Research links at Columbia
University :
Below are brief (and subjective) descriptions of some of the research
areas I am currently involved with.
As waves
propagate through heterogeneous media, they scatter and energy changes
direction. A very 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).
Time reversal is a
fascinating and very active research area. In a nutshell, it 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. See Mathias Fink's web
page for pioneering work and experiments on the time
reversal I refer to ("time reversal" can have several other
meanings).
Theoretical understanding. 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
many ways. I have worked on a modeling based on the radiative transfer
equations. One of the main advantages of such a model is that it is
intrinsically multi-dimensional and accounts very well for the spatial
diversity observed in physical experiments. One of its main
disadvantages is that it is almost impossible to rigorously justify
anything (from the mathematical point of view). The interpretation
Lenya Ryzhik and I have of time reversal can be found in the
paper: Time Reversal and refocusing in Random
Media. Many ideas in that paper have been influenced by the
pioneering works on the subject by George
Papanicolaou.
Changing media. In many practical applications, the media
during the forward and backward stages of the time reversal experiment
may slightly (at best) differ. With Ramon Verastegui, 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 loose 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 striking 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
are different worlds. In the former, the time reversed signal
backpropagates 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) I gave 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 I gave 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. Whereas a lot is known in the
scattering-free as well as in the highly scattering -modeled by
diffusion equations- regimes, comparatively little is understood (at
least theoretically) in the intermediate regime where both scattering
and advection have a comparable effect. Here is a sequence of three
talks ( Lecture 1, Lecture 2, Lecture 3) I gave 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 excellent
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). Following pioneering works by M. Vogelius, 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.
To learn more about Optical Tomography at Columbia University, you may
want to look at A. Hielscher's web page.