Curtis R. Vogel

Iterative methods for total variation denoising

Total variation (TV) methods are very effective for recovering “blocky,” possibly discontinuous, images from noisy data. A fixed point algorithm for minimizing a TV penalized least squares functional is presented and compared with existing minimization schemes. A variant of the cell-centered finite difference multigrid method of Ewing and Shen is implemented for solving the (large, sparse) linear subproblems. Numerical results are presented for one- and two-dimensional examples; in particular, the algorithm is applied to actual data obtained from confocal microscopy.

Fast, robust total variation-based reconstruction of noisy, blurred images

Tikhonov regularization with a modified total variation regularization functional is used to recover an image from noisy, blurred data. This approach is appropriate for image processing in that it does not place a priori smoothness conditions on the solution image. An efficient algorithm is presented for the discretized problem that combines a fixed point iteration to handle nonlinearity with a new, effective preconditioned conjugate gradient iteration for large linear systems. Reconstructions, convergence results, and a direct comparison with a fast linear solver are presented for a satellite image reconstruction application.

Non-convergence of the L-curve regularization parameter selection method

The L-curve method was developed for the selection of regularization parameters in the solution of discrete systems obtained from ill-posed problems. An analysis of this method is given for selecting a parameter for Tikhonov regularization. This analysis, which is carried out in a semi-discrete, semi-stochastic setting, shows that the L-curve approach yields regularized solutions which fail to converge for a certain class of problems. A numerical example is also presented which indicates that this lack of convergence can arise in practical applications.

Retinal motion estimation in adaptive optics scanning laser ophthalmoscopy

We apply a novel computational technique known as the map-seeking circuit algorithm to estimate the motion of the retina of eye from a sequence of frames of data from a scanning laser ophthalmoscope. We also present a scheme to dewarp and co-add frames of retinal image data, given the estimated motion. The motion estimation and dewarping techniques are applied to data collected from an adaptive optics scanning laser ophthalmoscopy.

Retinally stabilized cone-targeted stimulus delivery

We demonstrate projection of highly stabilized, aberration-corrected stimuli directly onto the retina by means of real-time retinal image motion signals in combination with high speed modulation of a scanning laser. In three subjects with good fixation stability, stimulus location accuracy averaged 0.26 arcminutes or approximately 1.3 microns, which is smaller than the cone-to-cone spacing at the fovea. We also demonstrate real-time correction for image distortions in adaptive optics scanning laser ophthalmoscope (AOSLO) with an intraframe accuracy of about 7 arcseconds.

Quasi-Newton approach to nonnegative image restorations

Abstract Image restoration, or deblurring, is the process of attempting to correct for degradation in a recorded image. Typically the blurring system is assumed to be linear and spatially invariant, and fast Fourier transform (FFT) based schemes result in efficient computational image restoration methods. However, real images have properties that cannot always be handled by linear methods. In particular, an image consists of positive light intensities, and thus a nonnegativity constraint should be enforced. This constraint and other ways of incorporating a priori information have been suggested in various applications, and can lead to substantial improvements in the reconstructions. Nevertheless, such constraints are rarely implemented because they lead to nonlinear problems which require demanding computations. We suggest efficient implementations for three nonnegatively constrained restorations schemes: constrained least squares, maximum likelihood and maximum entropy. We show that with a certain parameterization, and using a Quasi-Newton scheme, these methods are very similar. In addition, our formulation reveals a connection between our approach for maximum likelihood and the expectation–maximization (EM) method used extensively by astronomers. Numerical experiments illustrate that our approach is superior to EM both in terms of accuracy and efficiency.

Iterative SVD-based methods for ill-posed problems

Very large matrices with rapidly decaying singular values commonly arise in the numerical solution of ill-posed problems. The singular value decomposition (SVD) is a basic tool for both the analysis and computation of solutions to such problems. In most applications, it suffices to obtain a partial SVD consisting of only the largest singular values and their corresponding singular vectors. In this paper, two separate approaches—one based on subspace iteration and the other based on the Lanczos method—are considered for the efficient iterative computation of partial SVDs. In the context of ill-posed problems, an analytical and numerical comparison of these two methods is made and the role of the regularization operator in convergence acceleration is explored.

Modeling, simulation, and open-loop control of a continuous facesheet MEMS deformable mirror

We introduce a coupled system of nonlinear partial differential equations (PDEs) to model a particular microelectromechanical systems (MEMS) deformable mirror consisting of a continuous facesheet coupled with electrostatic plate-driven actuators. To make the problem computationally tractable, we reduced the system to a single linear PDE for the facesheet coupled with nonlinear algebraic constraints for each of the actuators. We also introduce a nonlinearly constrained quadratic optimization problem for open-loop control of the MEMS mirror. Numerical simulation and control results are presented, and shortcomings of the model are discussed.

Two-level preconditioners for regularized inverse problems I: Theory

Summary. We compare additive and multiplicative Schwarz preconditioners for the iterative solution of regularized linear inverse problems, extending and complementing earlier results of Hackbusch, King, and Rieder. Our main findings are that the classical convergence estimates are not useful in this context: rather, we observe that for regularized ill-posed problems with relevant parameter values the additive Schwarz preconditioner significantly increases the condition number. On the other hand, the multiplicative version greatly improves conditioning, much beyond the existing theoretical worst-case bounds.We present a theoretical analysis to support these results, and include a brief numerical example. More numerical examples with real applications will be given elsewhere.

A Multigrid Method for Total Variation-Based Image Denoising

Byimage reconstruction we mean obtaining the solution u of an operator equation of the form