Departamento de Ciencia e Ingeniería de los Materiales e Ingeniería Química
http://hdl.handle.net/10016/14126
Fri, 30 Sep 2016 22:01:00 GMT2016-09-30T22:01:00ZDiffuse optical tomography using the one-way radiative transfer equation
http://hdl.handle.net/10016/23561
Diffuse optical tomography using the one-way radiative transfer equation
González Rodríguez, Pedro; Kim, Arnold Dongkyoon
We present a computational study of diffuse optical tomography using the one-way radiative transfer equation. The one-way radiative transfer is a simplification of the radiative transfer equation to approximate the transmission of light through tissues. The major simplification of this approximation is that the intensity satisfies an initial value problem rather than a boundary value problem. Consequently, the inverse problem to reconstruct the absorption and scattering coefficients from transmission measurements of scattered light is simplified. Using the initial value problem for the one-way radiative transfer equation to compute the forward model, we are able to quantitatively reconstruct the absorption and scattering coefficients efficiently and effectively for simple problems and obtain reasonable results for complicated problems.
Fri, 08 May 2015 00:00:00 GMThttp://hdl.handle.net/10016/235612015-05-08T00:00:00ZLaurent expansion of the inverse of perturbed, singular matrices
http://hdl.handle.net/10016/23560
Laurent expansion of the inverse of perturbed, singular matrices
González Rodríguez, Pedro; Moscoso Castro, Miguel Ángel; Kindelan Segura, Manuel
In this paper we describe a numerical algorithm to compute the Laurent expansion of the inverse of singularly perturbed matrices. The algorithm is based on the resolvent formalism used in complex analysis to study the spectrum of matrices. The input of the algorithm are the matrix coefficients of the power series expansion of the perturbed matrix. The matrix coefficients of the Laurent expansion of the inverse are computed using recursive analytical formulae. We show that the computational complexity of the proposed algorithm grows algebraically with the size of the matrix, but exponentially with the order of the singularity. We apply this algorithm to several matrices that arise in applications. We make special emphasis to interpolation problems with radial basis functions.
Thu, 15 Oct 2015 00:00:00 GMThttp://hdl.handle.net/10016/235602015-10-15T00:00:00ZIllumination Strategies for Intensity-Only Imaging
http://hdl.handle.net/10016/23559
Illumination Strategies for Intensity-Only Imaging
Novikov, Alexei; Moscoso Castro, Miguel Ángel; Papanicolaou, George
We propose a new strategy for narrow band, active array imaging of weak localized scatterers when only the intensities are recorded and measured at the array. We consider a homogeneous medium so that wave propagation is fully coherent. We show that imaging with intensity-only measurements can be carried out using the time reversal operator of the imaging system, which can be obtained from intensity measurements using an appropriate illumination strategy and the polarization identity. Once the time reversal operator has been obtained, we show that the images can be formed using its singular value decomposition (SVD). We use two SVD-based methods to image the scatterers. The proposed approach is simple and efficient. It does not need prior information about the sought image, and it guarantees exact recovery in the noise-free case. Furthermore, it is robust with respect to additive noise. Detailed numerical simulations illustrate the performance of the proposed imaging strategy when only the intensities are captured.
Thu, 30 Jul 2015 00:00:00 GMThttp://hdl.handle.net/10016/235592015-07-30T00:00:00ZImaging strong localized Scatterers with Sparsity promoting optimization
http://hdl.handle.net/10016/23558
Imaging strong localized Scatterers with Sparsity promoting optimization
Chai, Anwei; Moscoso Castro, Miguel Ángel; Papanicolaou, George
We study active array imaging of small but strong scatterers in homogeneous media when multiple
scattering between them is important. We use the Foldy-Lax equations to model wave propagation
with multiple scattering when the scatterers are small relative to the wavelength. In active array
imaging we seek to locate the positions and reflectivities of the scatterers, that is, to determine
the support of the reflectivity vector and the values of its nonzero elements from echoes recorded
on the array. This is a nonlinear inverse problem because of the multiple scattering. We show
in this paper how to avoid the nonlinearity and form images non-iteratively through a two-step
process which involves 1 norm minimization. However, under certain illuminations imaging may be
affected by screening, where some scatterers are obscured by multiple scattering. This problem can
be mitigated by using multiple and diverse illuminations. In this case, we determine solution vectors
that have a common support. The uniqueness and stability of the support of the reflectivity vector
obtained with single or multiple illuminations are analyzed, showing that the errors are proportional
to the amount of noise in the data with a proportionality factor dependent on the sparsity of the
solution and the mutual coherence of the sensing matrix, which is determined by the geometry of the
imaging array. Finally, to filter out noise and improve the resolution of the images, we propose an
approach that combines optimal illuminations using the singular value decomposition of the response
matrix together with sparsity promoting optimization jointly for all illuminations. This work is an
extension of our previous paper [5] on imaging using optimization techniques where we now account
for multiple scattering effects.
Thu, 12 Jun 2014 00:00:00 GMThttp://hdl.handle.net/10016/235582014-06-12T00:00:00Z