In the presented work we study methods of image deblurring using two images of the same scene with different exposure times, focusing on two main approach categories, the so called deconvolution and non-deconvolution methods. We present theoretical backgrounds on both categories and evaluate their limitations and advantages. We dedicate one section to a comparison of both method categories on test data (images) for which we use a MATLAB implementation of the methods. We also compare the effectiveness of said methods against the results of a selected single- image de-noising algorithm. We do not focus at computational efficiency of algorithms and work with grayscale images only.

abstract: Magnetic Resonance Imaging (MRI) is limited in speed and resolution by the inherently low Signal to Noise Ratio (SNR) of the underlying signal. Advances in sampling efficiency are required to support future improvements in scan time and resolution. SNR efficiency is improved by sampling data for a larger proportion of total imaging time. This is challenging as these acquisitions are typically subject to artifacts such as blurring and distortions. The current work proposes a set of tools to help with the creation of different types of SNR efficient scans. An SNR efficient pulse sequence providing diffusion imaging data with full brain coverage and minimal distortion is first introduced. The proposed method acquires single-shot, low resolution image slabs which are then combined to reconstruct the full volume. An iterative deblurring algorithm allowing the lengthening of spiral SPoiled GRadient echo (SPGR) acquisition windows in the presence of rapidly varying off-resonance fields is then presented. Finally, an efficient and practical way of collecting 3D reformatted data is proposed. This method constitutes a good tradeoff between 2D and 3D neuroimaging in terms of scan time and data presentation. These schemes increased the SNR efficiency of currently existing methods and constitute key enablers for the development of SNR efficient MRI. / Dissertation/Thesis / Ph.D. Electrical Engineering 2011

The Blind Image Deconvolution/Deblurring (BID) problem was realised in the early 1960s but it still remains a challenging task for the image processing research community to find an efficient, reliable and most importantly a diversely applicable deblurring scheme. The main challenge arises from little or no prior information about the image or the blurring process as well as the lack of optimal restoration filters to reduce or completely eliminate the blurring effect. Moreover, restoration can be marred by the two common side effects of deblurring; namely the noise amplification and ringing artefacts that arise in the deblurred image due to an unrealizable or imperfect restoration filter. Also, developing a scheme that can process different types of blur, especially for real images, is yet to be realized to a satisfactory level. This research is focused on the development of blind restoration schemes for real life blurred images. The primary objective is to design a BID scheme that is robust in term of Point Spread Function (PSF) estimation, efficient in terms of restoration speed, and effective in terms of restoration quality. A desired scheme will require a deblurring measure to act as a feedback of quality regarding the deblurred image and lead the estimation of the blurring PSF. The blurred image and the estimated PSF can then be passed on to any classical restoration filter for deblurring. The deblurring measures presented in this research include blind non-Gaussianity measures as well as blind Image Quality Measures (IQMs). These measures are blind in the sense that they are able to gauge the quality of an image directly from it without the need to reference a high quality image. The non-Gaussianity measures include spatial and spectral kurtosis measures; while the image quality analysers include the Blind/Reference-less Image Spatial QUality Evaluator (BRISQUE), Natural Image Quality Evaluator (NIQE) index and Reblurring based Peak Signal to Noise Ratio (RPSNR) measure. BRISQUE, NIQE and spectral kurtosis, are introduced for the first time as deblurring measures for BID. RPSNR is a novel full reference yet blind IQM designed and used in this research work. Experiments were conducted on different image datasets and real life blurred images. Optimization of the BID schemes has been achieved using a gradient descent based scheme and a Genetic Algorithm (GA). Quantitative results based on full-reference and non-reference IQMs, present BRISQUE as a robust and computationally efficient blind feedback quality measure. Also, parametric and arbitrarily shaped (non-parametric or generic) PSFs were treated for the blind deconvolution of images. The parametric forms of PSF include uniform Gaussian, motion and out-of-focus blur. The arbitrarily shaped PSFs comprise blurs that have a much more complex blur shape which cannot be easily modelled in the parametric form. A novel scheme for arbitrarily shaped PSF estimation and blind deblurring has been designed, implemented and tested on artificial and real life blurred images. The scheme provides a unified base for the estimation of both parametric and arbitrarily shaped PSFs with the BRISQUE quality measure in conjunction with a GA. Full-reference and non-reference IQMs have been utilised to gauge the quality of deblurred images for the BID schemes. In the real BID case, only non-reference IQMs can be employed due to the unavailability of the reference high quality image. Quantitative results of these images depict the restoration ability of the BID scheme. The significance of the research work lies in the BID scheme‘s ability to handle parametric and arbitrarily shaped PSFs using a single algorithm, for single-shot blurred images, with enhanced optimization through the gradient descent scheme and GA in conjunction with multiple feedback IQMs.

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In this age, there are many applications of inverse problems to lots of areas ranging from astronomy, geoscience and so on. For example, image reconstruction and deblurring require the use of methods to solve inverse problems. Since the problems are subject to many factors and noise, we can't simply apply general inversion methods. Furthermore in the problems of interest, the number of unknown variables is huge, and some may depend nonlinearly on the data, such that we must solve nonlinear problems. It is quite different and significantly more challenging to solve nonlinear problems than linear inverse problems, and we need to use more sophisticated methods to solve these kinds of problems. / Master of Science / In various research areas, there are many required measurements which can't be observed due to physical and economical reasons. Instead, these unknown measurements can be recovered by known measurements. This phenomenon can be modeled and be solved by mathematics.

In this digital age, it is more important than ever to have good methods for processing images. We focus on the removal of blur from a captured image, which is called the image deblurring problem. In particular, we make no assumptions about the blur itself, which is called a blind deconvolution. We approach the problem by miniming an energy functional that utilizes total variation norm and a fidelity constraint. In particular, we extend the work of Chan and Wong to use a reference image in the computation. Using the shock filter as a reference image, we produce a superior result compared to existing methods. We are able to produce good results on non-black background images and images where the blurring function is not centro-symmetric. We consider using a general Lp norm for the fidelity term and compare different values for p. Using an analysis similar to Strong and Chan, we derive an adaptive scale method for the recovery of the blurring function. We also consider two numerical methods in this disseration. The first method is an extension of Picards method for PDEs in the discrete case. We compare the results to the analytical Picard method, showing the only difference is the use of the approximation versus exact derivatives. We relate the method to existing finite difference schemes, including the Lax-Wendroff method. We derive the stability constraints for several linear problems and illustrate the stability region is increasing. We conclude by showing several examples of the method and how the computational savings is substantial. The second method we consider is a black-box implementation of a method for solving the generalized eigenvalue problem. By utilizing the work of Golub and Ye, we implement a routine which is robust against existing methods. We compare this routine against JDQZ and LOBPCG and show this method performs well in numerical testing.

<p>An image that has been subject to the out-of-focus phenomenon has reducedsharpness, contrast and level of detail depending on the amount of defocus. Torestore out-of-focused images is a complex task due to the information loss thatoccurs. However there exist many restoration algorithms that attempt to revertthis defocus by estimating a noise model and utilizing the point spread function.The purpose of this thesis, proposed by FLIR Systems, was to find a robustalgorithm that can restore focus and from the customer’s perspective be userfriendly. The thesis includes three implemented algorithms that have been com-pared to MATLABs built-in. Three image series were used to evaluate the limitsand performance of each algorithm, based on deblurring quality, implementationcomplexity, computation time and usability.Results show that the Alternating Direction Method for total variation de-convolution proposed by Tao et al. [29] together with its the modified discretecosines transform version restores the defocused images with the highest qual-ity. These two algorithms include features such as, fast computational time, fewparameters to tune and a powerful noise reduction.</p>

This thesis investigates the removal of spatially-variant blur from photographs degraded by camera shake, and the removal of large occluding objects from photographs of popular places. Spatially-variant blur caused by camera shake is modelled using a weighted set of camera poses, which induce homographies on the image. The blur in an image is parameterised by the set of weights, which fully describe the spatially-variant blur at all pixels. We demonstrate direct estimation of the blur weights from single and multiple images captured by conventional cameras, by adapting existing (spatially-invariant) deblurring algorithms. This permits a sharp image to be recovered from a blurry "shaken" image without any user interaction. To reduce the computational cost of our model, we introduce an approximation based on local-uniformity of the blur. By grouping pixels into local regions which share a single PSF, we can use fast 2D convolutions to perform the blur computation. For deblurring images with saturated pixels, we modify the forward model to include this non-linearity, and re-derive the Richardson-Lucy algorithm. To prevent ringing appearing in the output, we propose separate updates for pixels affected/not affected by saturation. In order to remove large occluders from photos, we automatically retrieve a set of exemplar images of the same scene from the Internet. We extract homographies between each of these images and the target image to provide pixel correspondences. Finally we combine pixels from several exemplars in a seamless manner to replace the occluded pixels, by solving an energy minimisation problem.

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