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Ana Carpio, Applied Mathematics - Optimization, inverse problems and imaging
PDE constrained optimization methods
in conjunction with topological sensitivity based
techniques provide a powerful tool to reconstruct the
geometry and properties of objects buried in a
medium by scattering of electromagnetic,
acoustic or thermal waves. The idea is to exploit
the topological sensitivity of cost functionals
defined in terms of the data measured at a set of
receptors to produce first guesses of the objects.
The procedure is robust to noise. When enough
incident directions or frequencies are employed, a
topological derivative based iteration may provide a
reasonable description of the objects
geometry. The method works with both harmonic
or general time dependent incident radiations by
either selecting different incident directions or
recording information at different times. We have
applied this strategy to acoustic sounding of
bodies, electrical impedance tomography and
photothermal imaging, combining it with gradient
techniques to identify material parameters. We are
able to generate good first approximations of the
number, size and location of the scatterers, and
improve the description of their shape and material
parameters in a few steps. Making use of iteratively
regularized Gauss-Newton techniques, we can invert
holographic data in noninvasive light imaging of 3D
biological samples. Resorting to bayesian
approaches, and either linearized posteriors about
an optimal point or Markov Chain Monte Carlo
sampling, we are able to quantify uncertainty in the
inversion process for increaing noise magnitudes.
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 Imaged with 405nm light impinging in the z direction, balls of radius 1 |
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- Seeing the invisible: digital holography, EMS Magazine 125, 4-12, 2022 [pdf]
- Processing the 2D and 3D Fresnel experimental databases via topological derivative methods (with M. Pena, M.L. Rapún), Inverse Problems 37, 105012, 2021 [pdf]
- Multifrequency
topological
derivative
approach to
inverse
scattering
methods in
attenuating
media
(with M.L.
Rapún), Symmetry -
Issue
on Advanced
Mathematical
and Simulation
Methods
for Inverse
Problems 13,
1702, 2021 [pdf]
- Bayesian approach to inverse scattering with topological priors (with S. Iakunin, G. Stadler), Inverse Problems 36, 105001, 2020 [pdf] [arxiv]
- When topological derivatives met regularized Gauss-Newton iterations in holographic 3D imaging (with T.G. Dimiduk, F. Le Louer, M.L. Rapún), Journal of Computational Physics 388, 224-251, 2019 [pdf] [arxiv]
- Optimization methods for in-line holography (with T.G. Dimiduk, V. Selgas, P. Vidal), SIAM Journal on Imaging Sciences 11(2), 923-956, 2018 [pdf] [arxiv]
- Noninvasive imaging of
three-dimensional micro and nanostructures by
topological methods (with T.G. Dimiduk, M.L.
Rapún, V. Selgas), SIAM Journal on Imaging
Sciences 9(3), 1324-1354, 2016 [pdf]
[arxiv]
- Parameter identification in photothermal imaging (with M.L. Rapún), Journal of mathematical imaging and vision 49(2), 273-288, 2014 [pdf] [archivo]
- Hybrid topological
derivative-gradient based methods for
nondestructive testing (with M.L. Rapún),
Abstract and Applied Analysis, 816134, 2013 [pdf]
- Hybrid topological
derivative and gradient-based
methods for electrical impedance tomography
(with M.L. Rapún), Inverse
Problems 28(9), 095010, 2012 [pdf]
[archivo]
- Determining
planar multiple sound-soft obstacles from
scattered acoustic fields (with B.T.
Johansson, M.L. Rapún),
Journal of Mathematical Imaging
and Vision 36(2),
185-199, 2010 [pdf]
[archivo]
