In image deconvolution problems, the diagonalization of the underlying operators by means of the fast Fourier transform (FFT) usually yields very large speedups. When there are incomplete observations (e.g., in the case of unknown boundaries), standard deconvolution techniques normally involve non-diagonalizable operators, resulting in rather slow methods or, otherwise, use inexact convolution models, resulting in the occurrence of artifacts in the enhanced images. In this paper, we propose a new deconvolution framework for images with incomplete observations that allows us to work with diagonalized convolution operators, and therefore is very fast. We iteratively alternate the estimation of the unknown pixels and of the deconvolved image, using, e.g., an FFT-based deconvolution method. This framework is an efficient, high-quality alternative to existing methods of dealing with the image boundaries, such as edge tapering. It can be used with any fast deconvolution method. We give an example in which a state-of-the-art method that assumes periodic boundary conditions is extended, using this framework, to unknown boundary conditions. Furthermore, we propose a specific implementation of this framework, based on the alternating direction method of multipliers (ADMM). We provide a proof of convergence for the resulting algorithm, which can be seen as a “partial” ADMM, in which not all variables are dualized. We report experimental comparisons with other primal–dual methods, where the proposed one performed at the level of the state of the art. Four different kinds of applications were tested in the experiments: deconvolution, deconvolution with inpainting, superresolution, and demosaicing, all with unknown boundaries.
Methods that use exact models have the potential to completely eliminate the occurrence of artifacts. One such method is the one proposed in @cite_46 , which implicitly uses model , although that model is not explicitly mentioned in the paper. That method is limited to the use of quadratic regularizers, which allow a fast implementation but yield relatively low-quality deconvolution results. The method has been extended in @cite_44 , allowing the use of more general regularizers. In the latter form, the method achieves a relatively high speed by limiting the estimation of the boundary zone (an estimation that is time-consuming, in that method's formulation) to a few initial iterations of the optimization procedure. This version of the method improves on the results of @cite_46 due to the use of more appropriate regularizers, but the imperfect estimation of the boundary zone gives rise to artifacts in the deblurred images. Similar approaches for astronomical images can be found in @cite_42 @cite_4 . Another method, proposed in @cite_11 , uses model , and is rather slow due to the use of non-diagonalizable BTTB matrices.
@cite_4 @cite_42 @cite_44 @cite_46 @cite_11
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In this paper we propose a solution to the problem of reducing the boundary effects (ripples) in the deconvolution of astronomical images. The approach applies to the Richardson-Lucy method (RLM), namely the most frequently used de- convolution method in Astronomy, and is based on the idea of using RLM for attempting a reconstruction of the astronomical target in a domain broader than that of the detected image. Even if, in general, the reconstruction outside the image domain is not reliable, this approach, in a sense, is letting RLM to choose the appropriate boundary conditions and, as a consequence, the reconstruction inside the domain is considerably improved. We propose a simple implementation of this approach, allowing a reduction of its computational burden. Numerical experiments indicate that it is possible to obtain excellent results. Extensions and applications of the method are briefly discussed. We propose a solution to the problem of boundary artifacts appearing in several recently published fast deblurring algorithms based on iterated shrinkage thresholding in a sparse domain and Fourier domain deconvolution. Our approach adapts an idea proposed by Reeves for deconvolution by the Wiener filter. The time of computation less than doubles. Fast Fourier transform (FFT)-based restorations are fast, but at the expense of assuming that the blurring and deblurring are based on circular convolution. Unfortunately, when the opposite sides of the image do not match up well in intensity, this assumption can create significant artifacts across the image. If the pixels outside the measured image window are modeled as unknown values in the restored image, boundary artifacts are avoided. However, this approach destroys the structure that makes the use of the FFT directly applicable, since the unknown image is no longer the same size as the measured image. Thus, the restoration methods available for this problem no longer have the computational efficiency of the FFT. We propose a new restoration method for the unknown boundary approach that can be implemented in a fast and flexible manner. We decompose the restoration into a sum of two independent restorations. One restoration yields an image that comes directly from a modified FFT-based approach. The other restoration involves a set of unknowns whose number equals that of the unknown boundary values. By summing the two, the artifacts are canceled. Because the second restoration has a significantly reduced set of unknowns, it can be calculated very efficiently even though no circular convolution structure exists. We propose a total variation based model for simultaneous image inpainting and blind deconvolution. We demonstrate that the tasks are inherently coupled together and that solving them individually will lead to poor results. The main advantages of our model are that (i) boundary conditions for deconvolution required near the interface between observed and occluded regions are naturally generated through inpainting; (ii) inpainting results are enhanced through deconvolution (as opposed to inpainting blurry images). As a result, ringing effects due to imposing improper boundary conditions and errors due to imperfection of inpainting blurry images are reduced. Moreover, our model can also be used to generate boundary conditions for regular deconvolution problems that yields better results than previous methods.© 2005 Wiley Periodicals, Inc. © 2005 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 15, 92–102, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002 ima.20041