40 research outputs found
A Fast Alternating Minimization Algorithm for Total Variation Deblurring Without Boundary Artifacts
Recently, a fast alternating minimization algorithm for total variation image
deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang [{\em SIAM
J. Imaging Sci.}, 1 (2008), pp. 248--272]. The method in a nutshell consists of
a discrete Fourier transform-based alternating minimization algorithm with
periodic boundary conditions and in which two fast Fourier transforms (FFTs)
are required per iteration. In this paper, we propose an alternating
minimization algorithm for the continuous version of the total variation image
deblurring problem. We establish convergence of the proposed continuous
alternating minimization algorithm. The continuous setting is very useful to
have a unifying representation of the algorithm, independently of the discrete
approximation of the deconvolution problem, in particular concerning the
strategies for dealing with boundary artifacts. Indeed, an accurate restoration
of blurred and noisy images requires a proper treatment of the boundary. A
discrete version of our continuous alternating minimization algorithm is
obtained following two different strategies: the imposition of appropriate
boundary conditions and the enlargement of the domain. The first one is
computationally useful in the case of a symmetric blur, while the second one
can be efficiently applied for a nonsymmetric blur. Numerical tests show that
our algorithm generates higher quality images in comparable running times with
respect to the Fast Total Variation deconvolution algorithm
Diffusion equations and inverse problems regularization.
The present thesis can be split into two dfferent parts:
The first part mainly deals with the porous and fast diffusion equations.
Chapter 2 presents these equations in the Euclidean setting highlighting the technical issues that arise when trying to extend results in a Riemannian setting. Chapter 3 is devoted to the construction of exhaustion and cut-o_ functions with controlled gradient and Laplacian, on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the distance from a fixed reference point, and without any assumptions on the topology or the injectivity radius. The cut-offs are then applied to the study of the fast and porous media diffusion, of Lq-properties of the gradient and of the selfadjointness of Schrödinger-type operators.
The second part is concerned with inverse problems regularization applied to image deblurring. In Chapter 5 new variants of the Tikhonov filter method, called fractional and weighted Tikhonov, are presented alongside their saturation properties and converse results on their convergence rates. New iterated fractional Tikhonov regularization methods are then introduced. In Chapter 6 the modified linearized Bregman algorithm is investigated. It is showed that the standard approach based on the block circulant circulant block preconditioner may provide low quality restored images and different preconditioning strategies are then proposed, which improve the quality of the restoration
FAST PRECONDITIONERS FOR TOTAL VARIATION DEBLURRING WITH ANTIREFLECTIVE BOUNDARY CONDITIONS
Natural Science Foundation of Fujian Province of China for Distinguished Young Scholars [2010J06002]; NCET; U. Insubria; MIUR [20083KLJEZ]In recent works several authors have proposed the use of precise boundary conditions (BCs) for blurring models, and they proved that the resulting choice (Neumann or reflective, antireflective) leads to fast algorithms both for deblurring and for detecting the regularization parameters in presence of noise. When considering a symmetric point spread function, the crucial fact is that such BCs are related to fast trigonometric transforms. In this paper we combine the use of precise BCs with the total variation (TV) approach in order to preserve the jumps of the given signal (edges of the given image) as much as possible. We consider a classic fixed point method with a preconditioned Krylov method (usually the conjugate gradient method) for the inner iteration. Based on fast trigonometric transforms, we propose some preconditioning strategies that are suitable for reflective and antireflective BCs. A theoretical analysis motivates the choice of our preconditioners, and an extensive numerical experimentation is reported and critically discussed. Numerical tests show that the TV regularization with antireflective BCs implies not only a reduced analytical error, but also a lower computational cost of the whole restoration procedure over the other BCs
Symmetrization Techniques in Image Deblurring
This paper presents a couple of preconditioning techniques that can be used
to enhance the performance of iterative regularization methods applied to image
deblurring problems with a variety of point spread functions (PSFs) and
boundary conditions. More precisely, we first consider the anti-identity
preconditioner, which symmetrizes the coefficient matrix associated to problems
with zero boundary conditions, allowing the use of MINRES as a regularization
method. When considering more sophisticated boundary conditions and strongly
nonsymmetric PSFs, the anti-identity preconditioner improves the performance of
GMRES. We then consider both stationary and iteration-dependent regularizing
circulant preconditioners that, applied in connection with the anti-identity
matrix and both standard and flexible Krylov subspaces, speed up the
iterations. A theoretical result about the clustering of the eigenvalues of the
preconditioned matrices is proved in a special case. The results of many
numerical experiments are reported to show the effectiveness of the new
preconditioning techniques, including when considering the deblurring of sparse
images
Diffusion equations and inverse problems regularization.
The present thesis can be split into two dfferent parts:
The first part mainly deals with the porous and fast diffusion equations.
Chapter 2 presents these equations in the Euclidean setting highlighting the technical issues that arise when trying to extend results in a Riemannian setting. Chapter 3 is devoted to the construction of exhaustion and cut-o_ functions with controlled gradient and Laplacian, on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the distance from a fixed reference point, and without any assumptions on the topology or the injectivity radius. The cut-offs are then applied to the study of the fast and porous media diffusion, of Lq-properties of the gradient and of the selfadjointness of Schrödinger-type operators.
The second part is concerned with inverse problems regularization applied to image deblurring. In Chapter 5 new variants of the Tikhonov filter method, called fractional and weighted Tikhonov, are presented alongside their saturation properties and converse results on their convergence rates. New iterated fractional Tikhonov regularization methods are then introduced. In Chapter 6 the modified linearized Bregman algorithm is investigated. It is showed that the standard approach based on the block circulant circulant block preconditioner may provide low quality restored images and different preconditioning strategies are then proposed, which improve the quality of the restoration
Truncated decompositions and filtering methods with Reflective/Anti-Reflective boundary conditions: a comparison
The paper analyzes and compares some spectral filtering methods as truncated
singular/eigen-value decompositions and Tikhonov/Re-blurring regularizations in
the case of the recently proposed Reflective [M.K. Ng, R.H. Chan, and W.C.
Tang, A fast algorithm for deblurring models with Neumann boundary conditions,
SIAM J. Sci. Comput., 21 (1999), no. 3, pp.851-866] and Anti-Reflective [S.
Serra Capizzano, A note on anti-reflective boundary conditions and fast
deblurring models, SIAM J. Sci. Comput., 25-3 (2003), pp. 1307-1325] boundary
conditions. We give numerical evidence to the fact that spectral decompositions
(SDs) provide a good image restoration quality and this is true in particular
for the Anti-Reflective SD, despite the loss of orthogonality in the associated
transform. The related computational cost is comparable with previously known
spectral decompositions, and results substantially lower than the singular
value decomposition. The model extension to the cross-channel blurring
phenomenon of color images is also considered and the related spectral
filtering methods are suitably adapted.Comment: 22 pages, 10 figure
Tikhonov-type iterative regularization methods for ill-posed inverse problems: theoretical aspects and applications
Ill-posed inverse problems arise in many fields of science and engineering. The ill-conditioning and the big dimension make the task of numerically solving this kind of problems very challenging.
In this thesis we construct several algorithms for solving ill-posed inverse problems. Starting from the classical Tikhonov regularization method we develop iterative methods that enhance the performances of the originating method.
In order to ensure the accuracy of the constructed algorithms we insert a priori knowledge on the exact solution and empower the regularization term. By exploiting the structure of the problem we are also able to achieve fast computation even when the size of the problem becomes very big.
We construct algorithms that enforce constraint on the reconstruction, like nonnegativity or flux conservation and exploit enhanced version of the Euclidian norm using a regularization operator and different semi-norms, like the Total Variaton, for the regularization term.
For most of the proposed algorithms we provide efficient strategies for the choice of the regularization parameters, which, most of the times, rely on the knowledge of the norm of the noise that corrupts the data.
For each method we analyze the theoretical properties in the finite dimensional case or in the more general case of Hilbert spaces.
Numerical examples prove the good performances of the algorithms proposed in term of both accuracy and efficiency