819 research outputs found
Conjugate-Gradient Preconditioning Methods for Shift-Variant PET Image Reconstruction
Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e., for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shift-variant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shift-variant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85979/1/Fessler85.pd
Fast minimum variance wavefront reconstruction for extremely large telescopes
We present a new algorithm, FRiM (FRactal Iterative Method), aiming at the
reconstruction of the optical wavefront from measurements provided by a
wavefront sensor. As our application is adaptive optics on extremely large
telescopes, our algorithm was designed with speed and best quality in mind. The
latter is achieved thanks to a regularization which enforces prior statistics.
To solve the regularized problem, we use the conjugate gradient method which
takes advantage of the sparsity of the wavefront sensor model matrix and avoids
the storage and inversion of a huge matrix. The prior covariance matrix is
however non-sparse and we derive a fractal approximation to the Karhunen-Loeve
basis thanks to which the regularization by Kolmogorov statistics can be
computed in O(N) operations, N being the number of phase samples to estimate.
Finally, we propose an effective preconditioning which also scales as O(N) and
yields the solution in 5-10 conjugate gradient iterations for any N. The
resulting algorithm is therefore O(N). As an example, for a 128 x 128
Shack-Hartmann wavefront sensor, FRiM appears to be more than 100 times faster
than the classical vector-matrix multiplication method.Comment: to appear in the Journal of the Optical Society of America
Preconditioning Methods for Shift-Variant Image Reconstruction
Preconditioning methods can accelerate the convergence of gradient-based iterative methods for tomographic image reconstruction and image restoration. Circulant preconditioners have been used extensively for shift-invariant problems. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. For inverse problems that are approximately shift-invariant (i.e. approximately block-Toeplitz or block-circulant Hessians), circulant or Fourier-based preconditioners can provide remarkable acceleration. However, in applications with nonuniform noise variance (such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging), the Hessian of the (penalized) weighted least-squares objective function is quite shift-variant, and the Fourier preconditioner performs poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that more accurately approximate the Hessian matrices of shift-variant imaging problems. Compared to diagonal or Fourier preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. Applications to position emission tomography (PET) illustrate the method.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85856/1/Fessler147.pd
An Iterative Shrinkage Approach to Total-Variation Image Restoration
The problem of restoration of digital images from their degraded measurements
plays a central role in a multitude of practically important applications. A
particularly challenging instance of this problem occurs in the case when the
degradation phenomenon is modeled by an ill-conditioned operator. In such a
case, the presence of noise makes it impossible to recover a valuable
approximation of the image of interest without using some a priori information
about its properties. Such a priori information is essential for image
restoration, rendering it stable and robust to noise. Particularly, if the
original image is known to be a piecewise smooth function, one of the standard
priors used in this case is defined by the Rudin-Osher-Fatemi model, which
results in total variation (TV) based image restoration. The current arsenal of
algorithms for TV-based image restoration is vast. In the present paper, a
different approach to the solution of the problem is proposed based on the
method of iterative shrinkage (aka iterated thresholding). In the proposed
method, the TV-based image restoration is performed through a recursive
application of two simple procedures, viz. linear filtering and soft
thresholding. Therefore, the method can be identified as belonging to the group
of first-order algorithms which are efficient in dealing with images of
relatively large sizes. Another valuable feature of the proposed method
consists in its working directly with the TV functional, rather then with its
smoothed versions. Moreover, the method provides a single solution for both
isotropic and anisotropic definitions of the TV functional, thereby
establishing a useful connection between the two formulae.Comment: The paper was submitted to the IEEE Transactions on Image Processing
on October 22nd, 200
Sparse approximate inverse preconditioners on high performance GPU platforms
Simulation with models based on partial differential equations often requires the solution of (sequences of) large and sparse algebraic linear systems. In multidimensional domains, preconditioned Krylov iterative solvers are often appropriate for these duties. Therefore, the search for efficient preconditioners for Krylov subspace methods is a crucial theme. Recent developments, especially in computing hardware, have renewed the interest in approximate inverse preconditioners in factorized form, because their application during the solution process can be more efficient. We present here some experiences focused on the approximate inverse preconditioners proposed by Benzi and Tůma from 1996 and the sparsification and inversion proposed by van Duin in 1999. Computational costs, reorderings and implementation issues are considered both on conventional and innovative computing architectures like Graphics Programming Units (GPUs)
Fast Solvers for Cahn-Hilliard Inpainting
We consider the efficient solution of the modified Cahn-Hilliard equation for binary image inpainting using convexity splitting, which allows an unconditionally gradient stable time-discretization scheme. We look at a double-well as well as a double obstacle potential. For the latter we get a nonlinear system for which we apply a semi-smooth Newton method combined with a Moreau-Yosida regularization technique. At the heart of both methods lies the solution of large and sparse linear systems. We introduce and study block-triangular preconditioners using an efficient and easy to apply Schur complement approximation. Numerical results indicate that our preconditioners work very well for both problems and show that qualitatively better results can be obtained using the double obstacle potential
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
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