16,030 research outputs found
Bounds for discrete tomography solutions
We consider the reconstruction of a function on a finite subset of
if the line sums in certain directions are prescribed. The real
solutions form a linear manifold, its integer solutions a grid. First we
provide an explicit expression for the projection vector from the origin onto
the linear solution manifold in the case of only row and column sums of a
finite subset of . Next we present a method to estimate the
maximal distance between two binary solutions. Subsequently we deduce an upper
bound for the distance from any given real solution to the nearest integer
solution. This enables us to estimate the stability of solutions. Finally we
generalize the first mentioned result to the torus case and to the continuous
case
Study of noise effects in electrical impedance tomography with resistor networks
We present a study of the numerical solution of the two dimensional
electrical impedance tomography problem, with noisy measurements of the
Dirichlet to Neumann map. The inversion uses parametrizations of the
conductivity on optimal grids. The grids are optimal in the sense that finite
volume discretizations on them give spectrally accurate approximations of the
Dirichlet to Neumann map. The approximations are Dirichlet to Neumann maps of
special resistor networks, that are uniquely recoverable from the measurements.
Inversion on optimal grids has been proposed and analyzed recently, but the
study of noise effects on the inversion has not been carried out. In this paper
we present a numerical study of both the linearized and the nonlinear inverse
problem. We take three different parametrizations of the unknown conductivity,
with the same number of degrees of freedom. We obtain that the parametrization
induced by the inversion on optimal grids is the most efficient of the three,
because it gives the smallest standard deviation of the maximum a posteriori
estimates of the conductivity, uniformly in the domain. For the nonlinear
problem we compute the mean and variance of the maximum a posteriori estimates
of the conductivity, on optimal grids. For small noise, we obtain that the
estimates are unbiased and their variance is very close to the optimal one,
given by the Cramer-Rao bound. For larger noise we use regularization and
quantify the trade-off between reducing the variance and introducing bias in
the solution. Both the full and partial measurement setups are considered.Comment: submitted to Inverse Problems and Imagin
A Deficiency Problem of the Least Squares Finite Element Method for Solving Radiative Transfer in Strongly Inhomogeneous Media
The accuracy and stability of the least squares finite element method (LSFEM)
and the Galerkin finite element method (GFEM) for solving radiative transfer in
homogeneous and inhomogeneous media are studied theoretically via a frequency
domain technique. The theoretical result confirms the traditional understanding
of the superior stability of the LSFEM as compared to the GFEM. However, it is
demonstrated numerically and proved theoretically that the LSFEM will suffer a
deficiency problem for solving radiative transfer in media with strong
inhomogeneity. This deficiency problem of the LSFEM will cause a severe
accuracy degradation, which compromises too much of the performance of the
LSFEM and makes it not a good choice to solve radiative transfer in strongly
inhomogeneous media. It is also theoretically proved that the LSFEM is
equivalent to a second order form of radiative transfer equation discretized by
the central difference scheme
Experience with light dynamic penetration in a landslide locality
Because of vital prevention and redevelopment measures in problem localities affected by slope instability, geotechnical investigations must be carried out. One of the applicable methods is Light Dynamic Penetration (LDP). Although it was designed for easy field investigations in planning line structures to identify parameters, such load bearing capacity, and compactness, its use proves to be much wider. Using LDP it is possible to detect the interfaces between the discrete layers, potential slip surfaces, or groundwater level. However, it is important to note that the results of LDP are always related to one point only and to investigate a whole area, a complementary survey is needed. In the paper we report research where we used a geophysical method of Electrical Resistivity Tomography (ERT) to validate the results rendered by Light Dynamic Penetration in a redeveloped post-mining locality characteristic of landslides. The results of both measurement methods were unified for better interpretation and clarification to find out whether LDP is suitable to investigate a landslide locality. It shows that LDP provides relevant information on the massif structure in a given locality, but as opposed to other geophysical methods LDP informs only about one point only. The authors thus recommend carrying out surveys in landslide areas combining LDP and ERT as complementary measurements of groundwater level in the openings made using LDP. Mutually combining these two methods, it is possible to obtain a number of valuable information on landslide conditions (depth and course of shear plane, groundwater level, structure of layers, etc.) and physical-mechanical properties of soils
Quantifying admissible undersampling for sparsity-exploiting iterative image reconstruction in X-ray CT
Iterative image reconstruction (IIR) with sparsity-exploiting methods, such
as total variation (TV) minimization, investigated in compressive sensing (CS)
claim potentially large reductions in sampling requirements. Quantifying this
claim for computed tomography (CT) is non-trivial, because both full sampling
in the discrete-to-discrete imaging model and the reduction in sampling
admitted by sparsity-exploiting methods are ill-defined. The present article
proposes definitions of full sampling by introducing four sufficient-sampling
conditions (SSCs). The SSCs are based on the condition number of the system
matrix of a linear imaging model and address invertibility and stability. In
the example application of breast CT, the SSCs are used as reference points of
full sampling for quantifying the undersampling admitted by reconstruction
through TV-minimization. In numerical simulations, factors affecting admissible
undersampling are studied. Differences between few-view and few-detector bin
reconstruction as well as a relation between object sparsity and admitted
undersampling are quantified.Comment: Revised version that was submitted to IEEE Transactions on Medical
Imaging on 8/16/201
3D particle tracking velocimetry using dynamic discrete tomography
Particle tracking velocimetry in 3D is becoming an increasingly important
imaging tool in the study of fluid dynamics, combustion as well as plasmas. We
introduce a dynamic discrete tomography algorithm for reconstructing particle
trajectories from projections. The algorithm is efficient for data from two
projection directions and exact in the sense that it finds a solution
consistent with the experimental data. Non-uniqueness of solutions can be
detected and solutions can be tracked individually
Joint Reconstruction of Absorbed Optical Energy Density and Sound Speed Distribution in Photoacoustic Computed Tomography: A numerical Investigation
Photoacoustic computed tomography (PACT) is a rapidly emerging bioimaging
modality that seeks to reconstruct an estimate of the absorbed optical energy
density within an object. Conventional PACT image reconstruction methods assume
a constant speed-of-sound (SOS), which can result in image artifacts when
acoustic aberrations are significant. It has been demonstrated that
incorporating knowledge of an object's SOS distribution into a PACT image
reconstruction method can improve image quality. However, in many cases, the
SOS distribution cannot be accurately and/or conveniently estimated prior to
the PACT experiment. Because variations in the SOS distribution induce
aberrations in the measured photoacoustic wavefields, certain information
regarding an object's SOS distribution is encoded in the PACT measurement data.
Based on this observation, a joint reconstruction (JR) problem has been
proposed in which the SOS distribution is concurrently estimated along with the
sought-after absorbed optical energy density from the photoacoustic measurement
data. A broad understanding of the extent to which the JR problem can be
accurately and reliably solved has not been reported. In this work, a series of
numerical experiments is described that elucidate some important properties of
the JR problem that pertain to its practical feasibility. To accomplish this,
an optimization-based formulation of the JR problem is developed that yields a
non-linear iterative algorithm that alternatingly updates the two image
estimates. Heuristic analytic insights into the reconstruction problem are also
provided. These results confirm the ill-conditioned nature of the joint
reconstruction problem that will present significant challenges for practical
applications.Comment: 13 pages, submitted to IEEE Transactions on Computational Imagin
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