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This is my dissertation for my PhD in Applied Mathematics at the University of Arizona. My thesis is about deblurring images in the Bayesian framework with spatially varying hyperparameters and mixed regularization types.
Photographic image deblurring method, using image forming holograph and amplitude weighting transparency
Dissertation defense presentation dealing with deblurring Cygnus images.
High resolution holographic image deblurring methods involving coherent optical analog processing and Fourier transform division filter
Many scientific experiments such as those found in astronomy, geology, microbiology, and X-ray radiography require the use of high-energy instruments to capture images. Since blur and noise are inevitably present in any imaging system, the images must be \deblurred" to extract the full information content. Mathematically, image deblurring is an ill-posed inverse problem that requires regularization. The regularization, in turn, has a large effect on the deblurred image: different regularization strengths, and types, lead to drastically different reconstructions. Moreover, many images contain a mixture of smooth and sharp features which suggests the use of multi-regularization, i.e., varying the type of regularization (e.g. Tikhonov or total variation) across the image. We address these issues by formulating the image deblurring problem within a hierarchical Bayesian framework in which we spatially adapt the strength of the regularization and also vary the regularization type across the image. In this way, the image itself, along with corresponding regularization strength at each pixel, are described jointly by a posterior distribution which we can sample by Markov chain Monte Carlo (MCMC) methods. We illustrate our techniques on simplified test problems and apply them to high-energy X-ray images taken at the Nevada National Security Site. Numerical tests show that our new method is robustly applicable and increases the quality of the image reconstruction when compared to other (Bayesian) methods.
Image deblurring and aperture synthesis using a posteriori processing by Fourier transform holographic spatial filtering
This is my dissertation.
Holographic optical computing image restoration method for a posteriori deblurring of photographs
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A statistical inference method based on the principle of maximum entropy is developed for the purpose of enhancing and restoring satellite images. The proposed maximum entropy image restoration method is shown to overcome the difficulties associated with image restoration and provide the smoothest and most appropriate solution consistent with the measured data. An implementation of the method on the MP-1 computer is described, and results of tests on simulated data are presented.
Presentation to be presented during the MSTS Science and Technology Work-in-Progress Seminar (Webex) scheduled for November 16, 2021.
Abstract We propose a regularization-based deblurring method that works efficiently for galaxy images. The spatial resolution of a ground-based telescope is generally limited by seeing conditions and is much worse than space-based telescopes. This circumstance has generated considerable research interest in the restoration of spatial resolution. Since image deblurring is a typical inverse problem and often ill-posed, solutions tend to be unstable. To obtain a stable solution, much research has adopted regularization-based methods for image deblurring, but the regularization term is not necessarily appropriate for galaxy images. Although galaxies have an exponential or Sérsic profile, the conventional regularization assumes the image profiles to behave linearly in space. The significant deviation between the assumption and real situations leads to blurring of the images and smoothing out the detailed structures. Clearly, regularization on logarithmic domain, i.e., magnitude domain, should provide a more appropriate assumption, which we explore in this study. We formulate a problem of deblurring galaxy images by an objective function with a Tikhonov regularization term on a magnitude domain. We introduce an iterative algorithm minimizing the objective function with a primal–dual splitting method. We investigate the feasibility of the proposed method using simulation and observation images. In the simulation, we blur galaxy images with a realistic point spread function and add both Gaussian and Poisson noise. For the evaluation with the observed images, we use galaxy images taken by the Subaru HSC-SSP. Both of these evaluations show that our method successfully recovers the spatial resolution of the deblurred images and significantly outperforms the conventional methods. The code is publicly available from the GitHub 〈https://github.com/kzmurata-astro/PSFdeconv_amag〉.
Deblurring is one of the most common linear inverse problems in imaging, and most methods for deblurring an image assume that the convolution kernel, the so-called point spread function (PSF), is known or attempt to reconstruct the PSF simultaneously with the image. For many experimental imaging systems, like high energy X-ray radiography systems, the PSF must be computed from calibration imagery. The associated inverse problem is large-scale, so, in order to compute meaningful uncertainties with the reconstruction, it is common to use hierarchical Bayesian models and Markov chain Monte Carlo methods for sampling from the posterior. Classical Gibbs samplers can suffer from long autocorrelation lengths in the Markov chain when the reconstructed quantity in the inverse problem is finely discretized, resulting in poor computational efficiency of the Markov chain. In this work we present partially collapsed Gibbs samplers for linear inverse problems that greatly decrease correlations in the Markov chain, thus increasing the computational efficiency of the algorithms, and we present the results of the enhanced samplers on the problem of reconstructing the point spread function for an X-ray imaging system at the U.S. Department of Energy's Nevada National Security Site.
Discussion of the physical and mathematical fundamentals, performance capabilities and potential applications of image deblurring by holography. It is shown that a blurred photograph may be deblurred and a sharpened image extracted from it because the 'blurring' in the original photograph did not irretrievably lose the imaging information, notably in the spatial-frequency regions required for faithful imaging and for high resolution. The principles of optical image deblurring are highly mathematical in nature. One of the most important applications of holographic image deblurring is shown to be the sharpening of electron micrographs of biological specimens beyond the ultimate that can be achieved with even the most powerful instruments available.
The speed of the holographic image deblurring method has recently been further enhanced by a new speed in the realization of the powerful holographic image-deblurring filter. The filter makes it possible to carry out the deblurring, in the optical computer used, in times of the order of one second. The experimental achievements using the holographic image-enhancement method are illustrated with examples ranging from out-of-focus or motion-blurred photographs, including 'amateur' photos recorded on Polaroid film, to the sharpening of the best available electron micrographs of viruses. Images recorded with X-rays, notably from rocket-borne photos of the sun, and out-of-focus photographs from cameras in NASA satellites have been similarly deblurred.
Applications of the optical computer include an approach for increasing the sharpness of images obtained from the most powerful electron microscopes and fingerprint/credit card identification. The information-handling capability of the various optical computing processes is very great. Modern synthetic-aperture radars scan upward of 100,000 resolvable elements per second. Fields which have assumed major importance on the basis of optical computing principles are optical image deblurring, coherent side-looking synthetic-aperture radar, and correlative pattern recognition. Some examples of the most dramatic image deblurring results are shown.
Purpose: Deconvolution is a widely used tool in the field of image reconstruction algorithm when the linear imaging system has been blurred by the imperfect system transfer function. However, due to the nature of Gaussian-liked distribution for point spread function (PSF), the components with coherent high frequency in the image are hard to restored in most of the previous scanning imaging system, even the relatively accurate PSF is acquired. We propose a novel method for deconvolution of images which are obtained by using shape-modulated PSF. Methods: We use two different types of PSF - Gaussian shape and donut shape - to convolute the original image in order to simulate the process of scanning imaging. By employing deconvolution of the two images with corresponding given priors, the image quality of the deblurred images are compared. Then we find the critical size of the donut shape compared with the Gaussian shape which has similar deconvolution results. Through calculation of tightened focusing process using radially polarized beam, such size of donut is achievable under same conditions. Results: The effects of different relative size of donut and Gaussian shapes are investigated. When the full width at half maximum (FWHM) ratio of donut and Gaussian shape is set about 1.83, similar resolution results are obtained through our deconvolution method. Decreasing the size of donut will favor the deconvolution method. A mask with both amplitude and phase modulation is used to create a donut-shaped PSF compared with the non-modulated Gaussian PSF. Donut with size smaller than our critical value is obtained. Conclusion: The utility of donutshaped PSF are proved useful and achievable in the imaging and deconvolution processing, which is expected to have potential practical applications in high resolution imaging for biological samples.
We present a new multigrid preconditioner for the linear systems arising in the semismooth Newton method solution of certain control-constrained, quadratic distributed optimal control problems. Using a piecewise constant discretization of the control space, each semismooth Newton iteration essentially requires inverting a principal submatrix of the matrix entering the normal equations of the associated unconstrained optimal control problem, the rows (and columns) of the submatrix representing the constraints deemed inactive at the current iteration. Previously developed multigrid preconditioners for the aforementioned submatrices were based on constructing a sequence of conforming coarser spaces and proved to be of suboptimal quality for the class of problems considered. Instead, the multigrid preconditioner introduced in this work uses nonconforming coarse spaces, and it is shown that, under reasonable geometric assumptions on the constraints that are deemed inactive, the preconditioner approximates the inverse of the desired submatrix to optimal order. The preconditioner is tested numerically on a classical elliptic-constrained optimal control problem and further on a constrained image-deblurring problem.