A new steplength selection for scaled gradient methods with application to image deblurring

Gradient methods are frequently used in large scale image deblurring problems since they avoid the onerous computation of the Hessian matrix of the objective function. Second order information is typically sought by a clever choice of the steplength parameter defining the descent direction, as in the case of the well-known Barzilai and Borwein rules. In a recent paper, a strategy for the steplength selection approximating the inverse of some eigenvalues of the Hessian matrix has been proposed for gradient methods applied to unconstrained minimization problems. In the quadratic case, this approach is based on a Lanczos process applied every m iterations to the matrix of the most recent m back gradients but the idea can be extended to a general objective function. In this paper we extend this rule to the case of scaled gradient projection methods applied to non-negatively constrained minimization problems, and we test the effectiveness of the proposed strategy in image deblurring problems in both the presence and the absence of an explicit edge-preserving regularization term

Inverse problems in astronomical and general imaging

The resolution and the quality of an imaged object are limited by four contributing factors. Firstly, the primary resolution limit of a system is imposed by the aperture of an instrument due to the effects of diffraction. Secondly, the finite sampling frequency, the finite measurement time and the mechanical limitations of the equipment also affect the resolution of the images captured. Thirdly, the images are corrupted by noise, a process inherent to all imaging systems. Finally, a turbulent imaging medium introduces random degradations to the signals before they are measured. In astronomical imaging, it is the atmosphere which distorts the wavefronts of the objects, severely limiting the resolution of the images captured by ground-based telescopes. These four factors affect all real imaging systems to varying degrees. All the limitations imposed on an imaging system result in the need to deduce or reconstruct the underlying object distribution from the distorted measured data. This class of problems is called inverse problems. The key to the success of solving an inverse problem is the correct modelling of the physical processes which give rise to the corresponding forward problem. However, the physical processes have an infinite amount of information, but only a finite number of parameters can be used in the model. Information loss is therefore inevitable. As a result, the solution to many inverse problems requires additional information or prior knowledge. The application of prior information to inverse problems is a recurrent theme throughout this thesis. An inverse problem that has been an active research area for many years is interpolation, and there exist numerous techniques for solving this problem. However, many of these techniques neither account for the sampling process of the instrument nor include prior information in the reconstruction. These factors are taken into account in the proposed optimal Bayesian interpolator. The process of interpolation is also examined from the point of view of superresolution, as these processes can be viewed as being complementary. Since the principal effect of atmospheric turbulence on an incoming wavefront is a phase distortion, most of the inverse problem techniques devised for this seek to either estimate or compensate for this phase component. These techniques are classified into computer post-processing methods, adaptive optics (AO) and hybrid techniques. Blind deconvolution is a post-processing technique which uses the speckle images to estimate both the object distribution and the point spread function (PSF), the latter of which is directly related to the phase. The most successful approaches are based on characterising the PSF as the aberrations over the aperture. Since the PSF is also dependent on the atmosphere, it is possible to constrain the solution using the statistics of the atmosphere. An investigation shows the feasibility of this approach. Bispectrum is also a post-processing method which reconstructs the spectrum of the object. The key component for phase preservation is the property of phase closure, and its application as prior information for blind deconvolution is examined. Blind deconvolution techniques utilise only information in the image channel to estimate the phase which is difficult. An alternative method for phase estimation is from a Shack-Hartmann (SH) wavefront sensing channel. However, since phase information is present in both the wavefront sensing and the image channels simultaneously, both of these approaches suffer from the problem that phase information from only one channel is used. An improved estimate of the phase is achieved by a combination of these methods, ensuring that the phase estimation is made jointly from the data in both the image and the wavefront sensing measurements. This formulation, posed as a blind deconvolution framework, is investigated in this thesis. An additional advantage of this approach is that since speckle images are imaged in a narrowband, while wavefront sensing images are captured by a charge-coupled device (CCD) camera at all wavelengths, the splitting of the light does not compromise the light level for either channel. This provides a further incentive for using simultaneous data sets. The effectiveness of using Shack-Hartmann wavefront sensing data for phase estimation relies on the accuracy of locating the data spots. The commonly used method which calculates the centre of gravity of the image is in fact prone to noise and is suboptimal. An improved method for spot location based on blind deconvolution is demonstrated. Ground-based adaptive optics (AO) technologies aim to correct for atmospheric turbulence in real time. Although much success has been achieved, the space- and time-varying nature of the atmosphere renders the accurate measurement of atmospheric properties difficult. It is therefore usual to perform additional post-processing on the AO data. As a result, some of the techniques developed in this thesis are applicable to adaptive optics. One of the methods which utilise elements of both adaptive optics and post-processing is the hybrid technique of deconvolution from wavefront sensing (DWFS). Here, both the speckle images and the SH wavefront sensing data are used. The original proposal of DWFS is simple to implement but suffers from the problem where the magnitude of the object spectrum cannot be reconstructed accurately. The solution proposed for overcoming this is to use an additional set of reference star measurements. This however does not completely remove the original problem; in addition it introduces other difficulties associated with reference star measurements such as anisoplanatism and reduction of valuable observing time. In this thesis a parameterised solution is examined which removes the need for a reference star, as well as offering a potential to overcome the problem of estimating the magnitude of the object

Accelerated Sparse Recovery via Gradient Descent with Nonlinear Conjugate Gradient Momentum

This paper applies an idea of adaptive momentum for the nonlinear conjugate gradient to accelerate optimization problems in sparse recovery. Specifically, we consider two types of minimization problems: a (single) differentiable function and the sum of a non-smooth function and a differentiable function. In the first case, we adopt a fixed step size to avoid the traditional line search and establish the convergence analysis of the proposed algorithm for a quadratic problem. This acceleration is further incorporated with an operator splitting technique to deal with the non-smooth function in the second case. We use the convex $\ell_1$ and the nonconvex $\ell_1-\ell_2$ functionals as two case studies to demonstrate the efficiency of the proposed approaches over traditional methods

Bayesian reconstruction of the cosmological large-scale structure: methodology, inverse algorithms and numerical optimization

We address the inverse problem of cosmic large-scale structure reconstruction from a Bayesian perspective. For a linear data model, a number of known and novel reconstruction schemes, which differ in terms of the underlying signal prior, data likelihood, and numerical inverse extra-regularization schemes are derived and classified. The Bayesian methodology presented in this paper tries to unify and extend the following methods: Wiener-filtering, Tikhonov regularization, Ridge regression, Maximum Entropy, and inverse regularization techniques. The inverse techniques considered here are the asymptotic regularization, the Jacobi, Steepest Descent, Newton-Raphson, Landweber-Fridman, and both linear and non-linear Krylov methods based on Fletcher-Reeves, Polak-Ribiere, and Hestenes-Stiefel Conjugate Gradients. The structures of the up-to-date highest-performing algorithms are presented, based on an operator scheme, which permits one to exploit the power of fast Fourier transforms. Using such an implementation of the generalized Wiener-filter in the novel ARGO-software package, the different numerical schemes are benchmarked with 1-, 2-, and 3-dimensional problems including structured white and Poissonian noise, data windowing and blurring effects. A novel numerical Krylov scheme is shown to be superior in terms of performance and fidelity. These fast inverse methods ultimately will enable the application of sampling techniques to explore complex joint posterior distributions. We outline how the space of the dark-matter density field, the peculiar velocity field, and the power spectrum can jointly be investigated by a Gibbs-sampling process. Such a method can be applied for the redshift distortions correction of the observed galaxies and for time-reversal reconstructions of the initial density field.Comment: 40 pages, 11 figure

Iterative CT reconstruction from few projections for the nondestructive post irradiation examination of nuclear fuel assemblies

The core components (e.g. fuel assemblies, spacer grids, control rods) of the nuclear reactors encounter harsh environment due to high temperature, physical stress, and a tremendous level of radiation. The integrity of these elements is crucial for safe operation of the nuclear power plants. The Post Irradiation Examination (PIE) can reveal information about the integrity of the elements during normal operations and off‐normal events. Computed tomography (CT) is a tool for evaluating the structural integrity of elements non-destructively. CT requires many projections to be acquired from different view angles after which a mathematical algorithm is adopted for reconstruction. Obtaining many projections is laborious and expensive in nuclear industries. Reconstructions from a small number of projections are explored to achieve faster and cost-efficient PIE. Classical reconstruction algorithms (e.g. filtered back projection) cannot offer stable reconstructions from few projections and create severe streaking artifacts. In this thesis, conventional algorithms are reviewed, and new algorithms are developed for reconstructions of the nuclear fuel assemblies using few projections. CT reconstruction from few projections falls into two categories: the sparse-view CT and the limited-angle CT or tomosynthesis. Iterative reconstruction algorithms are developed for both cases in the field of compressed sensing (CS). The performance of the algorithms is assessed using simulated projections and validated through real projections. The thesis also describes the systematic strategy towards establishing the conditions of reconstructions and finds the optimal imaging parameters for reconstructions of the fuel assemblies from few projections. --Abstract, page iii

On the convergence of a linesearch based proximal-gradient method for nonconvex optimization

We consider a variable metric linesearch based proximal gradient method for the minimization of the sum of a smooth, possibly nonconvex function plus a convex, possibly nonsmooth term. We prove convergence of this iterative algorithm to a critical point if the objective function satisfies the Kurdyka-Lojasiewicz property at each point of its domain, under the assumption that a limit point exists. The proposed method is applied to a wide collection of image processing problems and our numerical tests show that our algorithm results to be flexible, robust and competitive when compared to recently proposed approaches able to address the optimization problems arising in the considered applications