Abstract: This paper is devoted to the numerical solution of
large-scale linear ill-posed systems. A multilevel regularization
method is proposed. This method is based on a synthesis of
the Arnoldi-Tikhonov regularization technique and the multilevel
technique. We show that if the Arnoldi-Tikhonov method is
a regularization method, then the multilevel method is also a
regularization one. Numerical experiments presented in this paper
illustrate the effectiveness of the proposed method.
Abstract: Tikhonov regularization and reproducing kernels are the
most popular approaches to solve ill-posed problems in computational
mathematics and applications. And the Fourier multiplier operators
are an essential tool to extend some known linear transforms
in Euclidean Fourier analysis, as: Weierstrass transform, Poisson
integral, Hilbert transform, Riesz transforms, Bochner-Riesz mean
operators, partial Fourier integral, Riesz potential, Bessel potential,
etc. Using the theory of reproducing kernels, we construct a simple
and efficient representations for some class of Fourier multiplier
operators Tm on the Paley-Wiener space Hh. In addition, we give
an error estimate formula for the approximation and obtain some
convergence results as the parameters and the independent variables
approaches zero. Furthermore, using numerical quadrature integration
rules to compute single and multiple integrals, we give numerical
examples and we write explicitly the extremal function and the
corresponding Fourier multiplier operators.
Abstract: Diffuse Optical Tomography (DOT) is a non-invasive imaging modality used in clinical diagnosis for earlier detection of carcinoma cells in brain tissue. It is a form of optical tomography which produces gives the reconstructed image of a human soft tissue with by using near-infra-red light. It comprises of two steps called forward model and inverse model. The forward model provides the light propagation in a biological medium. The inverse model uses the scattered light to collect the optical parameters of human tissue. DOT suffers from severe ill-posedness due to its incomplete measurement data. So the accurate analysis of this modality is very complicated. To overcome this problem, optical properties of the soft tissue such as absorption coefficient, scattering coefficient, optical flux are processed by the standard regularization technique called Levenberg - Marquardt regularization. The reconstruction algorithms such as Split Bregman and Gradient projection for sparse reconstruction (GPSR) methods are used to reconstruct the image of a human soft tissue for tumour detection. Among these algorithms, Split Bregman method provides better performance than GPSR algorithm. The parameters such as signal to noise ratio (SNR), contrast to noise ratio (CNR), relative error (RE) and CPU time for reconstructing images are analyzed to get a better performance.
Abstract: The numerical analytic continuation of a function f(z) = f(x + iy) on a strip is discussed in this paper. The data are only given approximately on the real axis. The periodicity of given data is assumed. A truncated Fourier spectral method has been introduced to deal with the ill-posedness of the problem. The theoretic results show that the discrepancy principle can work well for this problem. Some numerical results are also given to show the efficiency of the method.
Abstract: We study the problem of reconstructing a three dimensional binary matrices whose interiors are only accessible through few projections. Such question is prominently motivated by the demand in material science for developing tool for reconstruction of crystalline structures from their images obtained by high-resolution transmission electron microscopy. Various approaches have been suggested to reconstruct 3D-object (crystalline structure) by reconstructing slice of the 3D-object. To handle the ill-posedness of the problem, a priori information such as convexity, connectivity and periodicity are used to limit the number of possible solutions. Formally, 3Dobject (crystalline structure) having a priory information is modeled by a class of 3D-binary matrices satisfying a priori information. We consider 3D-binary matrices with periodicity constraints, and we propose a polynomial time algorithm to reconstruct 3D-binary matrices with periodicity constraints from two orthogonal projections.
Abstract: Nondestructive testing in engineering is an inverse
Cauchy problem for Laplace equation. In this paper the problem
of nondestructive testing is expressed by a Laplace-s equation with
third-kind boundary conditions. In order to find unknown values on
the boundary, the method of fundamental solution is introduced and
realized. Because of the ill-posedness of studied problems, the TSVD
regularization technique in combination with L-curve criteria and
Generalized Cross Validation criteria is employed. Numerical results
are shown that the TSVD method combined with L-curve criteria is
more efficient than the TSVD method combined with GCV criteria.
The abstract goes here.
Abstract: Acoustic Imaging based sound localization using microphone
array is a challenging task in digital-signal processing.
Discrete Fourier transform (DFT) based near-field acoustical holography
(NAH) is an important acoustical technique for sound source
localization and provide an efficient solution to the ill-posed problem.
However, in practice, due to the usage of small curtailed aperture
and its consequence of significant spectral leakage, the DFT could
not reconstruct the active-region-of-sound (AROS) effectively, especially
near the edges of aperture. In this paper, we emphasize the
fundamental problems of DFT-based NAH, provide a solution to
spectral leakage effect by the extrapolation based on linear predictive
coding and 2D Tukey windowing. This approach has been tested to
localize the single and multi-point sound sources. We observe that
incorporating extrapolation technique increases the spatial resolution,
localization accuracy and reduces spectral leakage when small curtail
aperture with a lower number of sensors accounts.
Abstract: Visualizing sound and noise often help us to determine
an appropriate control over the source localization. Near-field acoustic
holography (NAH) is a powerful tool for the ill-posed problem.
However, in practice, due to the small finite aperture size, the discrete
Fourier transform, FFT based NAH couldn-t predict the activeregion-
of-interest (AROI) over the edges of the plane. Theoretically
few approaches were proposed for solving finite aperture problem.
However most of these methods are not quite compatible for the
practical implementation, especially near the edge of the source. In
this paper, a zip-stuffing extrapolation approach has suggested with
2D Kaiser window. It is operated on wavenumber complex space
to localize the predicted sources. We numerically form a practice
environment with touch impact databases to test the localization of
sound source. It is observed that zip-stuffing aperture extrapolation
and 2D window with evanescent components provide more accuracy
especially in the small aperture and its derivatives.
Abstract: Compensating physiological motion in the context
of minimally invasive cardiac surgery has become an attractive
issue since it outperforms traditional cardiac procedures offering
remarkable benefits. Owing to space restrictions, computer vision
techniques have proven to be the most practical and suitable solution.
However, the lack of robustness and efficiency of existing methods
make physiological motion compensation an open and challenging
problem. This work focusses on increasing robustness and efficiency
via exploration of the classes of 1−and 2−regularized optimization,
emphasizing the use of explicit regularization. Both approaches are
based on natural features of the heart using intensity information.
Results pointed out the 1−regularized optimization class as the best
since it offered the shortest computational cost, the smallest average
error and it proved to work even under complex deformations.
Abstract: The electromagnetic imaging of inhomogeneous
dielectric cylinders buried in a slab medium by transverse electric
(TE) wave illumination is investigated. Dielectric cylinders of
unknown permittivities are buried in second space and scattered a
group of unrelated waves incident from first space where the scattered
field is recorded. By proper arrangement of the various unrelated
incident fields, the difficulties of ill-posedness and nonlinearity are
circumvented, and the permittivity distribution can be reconstructed
through simple matrix operations. The algorithm is based on the
moment method and the unrelated illumination method. Numerical
results are given to demonstrate the capability of the inverse
algorithm. Good reconstruction is obtained even in the presence of
additive Gaussian random noise in measured data. In addition, the
effect of noise on the reconstruction result is also investigated.
Abstract: From a set of shifted, blurred, and decimated image , super-resolution image reconstruction can get a high-resolution image. So it has become an active research branch in the field of image restoration. In general, super-resolution image restoration is an ill-posed problem. Prior knowledge about the image can be combined to make the problem well-posed, which contributes to some regularization methods. In the regularization methods at present, however, regularization parameter was selected by experience in some cases and other techniques have too heavy computation cost for computing the parameter. In this paper, we construct a new super-resolution algorithm by transforming the solving of the System stem Є=An into the solving of the equations X+A*X-1A=I , and propose an inverse iterative method.
Abstract: In this paper, we consider the problem for identifying the unknown source in the Poisson equation. A modified Tikhonov regularization method is presented to deal with illposedness of the problem and error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. Numerical examples show that the proposed method is effective and stable.