It was shown in [Garnier et al., SIAM J. Imaging Sciences 2 (2009), 396]
that it is possible to image reflectors by backpropagating
cross correlations of signals generated
by ambient noise sources and recorded at passive sensor arrays.
The resolution of the image depends on the directional diversity
of the noise signals relative to the locations of the sensor array and the
When directional diversity is limited it is possible to enhance it
by exploiting the scattering properties of the medium since scatterers
will act as secondary noise sources. However, scattering increases the
fluctuation level of the cross correlations and therefore tends to destabilize the image
by reducing its signal-to-noise ratio.
In this paper we study the trade-off in passive, correlation-based imaging
between resolution enhancement and
signal-to-noise ratio reduction that is due to scattering.
In this paper we study passive sensor imaging with ambient noise
sources by suitably migrating cross correlations of the recorded signals.
We propose and study different imaging functionals.
A new functional is introduced that is an inverse Radon transform applied
to a special function of the cross correlation matrix.
We analyze the properties of the new imaging functional in the high-frequency regime
which shows that it produces sharper images than the usual
Kirchhoff migration functional.
Numerical simulations confirm the theoretical predictions.
We demonstrate that increased power transmission through a random single-mode or multi-mode
channel can be obtained
in the localization regime by optimizing the spatial wave front or the time pulse profile of the source.
The idea is to select and excite the few modes or the few frequencies whose transmission coefficients
are anomalously large compared to the typical exponentially small value.
We prove that time reversal is optimal for maximizing the transmitted intensity
at a given time or space, while iterated time reversal is optimal for maximizing the
total transmitted energy. The statistical stability of the optimal transmitted intensity and energy
is also obtained.
In this paper we analyze a wave-based imaging modality called ghost imaging that
can produce an image of an object illuminated by a partially coherent source. The image of the object
is obtained by correlating the intensities
measured by two detectors, one that does not view the object and another one that does view the object.
More exactly, a high-resolution detector measures the intensity of a wave field emitted by a partially coherent source
which has not interacted with the object to be imaged.
A bucket (or single-pixel) detector collects the total (spatially-integrated) intensity of the wave field emitted by the same source
that has interacted with the object.
The correlation of the intensity measured at the high-resolution detector
with the intensity measured by the bucket detector gives an image of the object.
In this paper we analyze this imaging modality when the medium through which the waves propagate is random.
We discuss the relation with time reversal focusing and with correlation-based imaging using ambient noise sources.
We clarify the role of the partial coherence of the source and we study how scattering affects the resolution properties
of the ghost imaging function in the paraxial regime: the image resolution is all the better as the source is less coherent, and all the worse
as the medium is more scattering.
Pulse propagation in randomly perturbed single-mode waveguides is considered.
By an asymptotic analysis the pulse front propagation is reduced
to an effective equation with diffusion and dispersion.
Apart from a random time shift due to a random total travel time,
two main phenomena can be distinguished.
First, coupling and energy conversion
between forward- and backward-propagating modes is responsible
for an effective diffusion of the pulse front. This attenuation and spreading
is somewhat similar to the one-dimensional case addressed by
the O'Doherty-Anstey theory.
Second, coupling between the forward-propagating mode and the evanescent modes
results in an effective dispersion. In the case of small-scale random fluctuations
we show that the second mechanism is dominant.
Waves can be used to probe and image an unknown medium. Passive imaging uses ambient noise sources to illuminate the medium. This paper considers passive imaging with moving sensors. The motivation is to generate large synthetic apertures, which should result in enhanced resolution. However Doppler effects and lack of reciprocity significantly affect the imaging process. This paper discusses the consequences in terms of resolution and it shows how to design appropriate imaging functions depending on the sensor trajectory and velocity.
The detection, localization, and characterization of a collection
of targets embedded in a medium is an important problem in
multistatic wave imaging. The responses between each pair of
source and receiver are collected and assembled in the form of a
response matrix, known as the multi-static response matrix. When
the data are corrupted by measurement or instrument noise, the
structure of the response matrix is studied by using random matrix
theory. It is shown how the targets can be efficiently detected,
localized and characterized. Both the case of a collection of
point reflectors in which the singular vectors have all the same
form and the case of small-volume electromagnetic inclusions in
which the singular vectors may have different forms depending on
their magnetic or dielectric type are addressed.
We consider the Czirók model for collective motion of locusts along a one-dimensional torus. In the model, each agent's velocity locally interacts with other agents' velocities in the system, and there is also exogenous randomness to each agent's velocity. The interaction tends to create the alignment of collectivemotion. By analyzing the associated nonlinear Fokker-Planck equation, we obtain the condition for the existence of stationary order states and the conditions for their linear stability. These conditions depend on the noise level, which should be strong enough, and on the interaction between the agent's velocities, which should be neither too small, nor too strong. We carry out the fluctuation analysis of the interacting system and describe the large deviation principle to calculate the transition probability from one order state to the other. Numerical simulations confirm our analytical findings.