Blind image deconvolution (BID) is a classic yet challenging problem in the field of image processing. Recent advances in deep image prior (DIP) have motivated a series of DIP-based approaches, demonstrating remarkable success in BID. However, due to the high non-convexity of the inherent optimization process, these methods are notorious for their sensitivity to the initialized kernel. To alleviate this issue and further improve their performance, we propose a new framework for BID that better considers the prior modeling and the initialization for blur kernels, leveraging a deep generative model. The proposed approach pre-trains a generative adversarial network-based kernel generator that aptly characterizes the kernel priors and a kernel initializer that facilitates a well-informed initialization for the blur kernel through latent space encoding. With the pre-trained kernel generator and initializer, one can obtain a high-quality initialization of the blur kernel, and enable optimization within a compact latent kernel manifold. Such a framework results in an evident performance improvement over existing DIP-based BID methods. Extensive experiments on different datasets demonstrate the effectiveness of the proposed method.
Blind deconvolution is an ill-posed problem arising in various fields ranging from microscopy to astronomy. The ill-posed nature of the problem requires adequate priors to arrive to a desirable solution. Recently, it has been shown that deep learning architectures can serve as an image generation prior during unsupervised blind deconvolution optimization, however often exhibiting a performance fluctuation even on a single image. We propose to use Wiener-deconvolution to guide the image generator during optimization by providing it a sharpened version of the blurry image using an auxiliary kernel estimate starting from a Gaussian. We observe that the high-frequency artifacts of deconvolution are reproduced with a delay compared to low-frequency features. In addition, the image generator reproduces low-frequency features of the deconvolved image faster than that of a blurry image. We embed the computational process in a constrained optimization framework and show that the proposed method yields higher stability and performance across multiple datasets. In addition, we provide the code.
In recent years, algorithm unrolling has emerged as a powerful technique for designing interpretable neural networks based on iterative algorithms. Imaging inverse problems have particularly benefited from unrolling-based deep network design since many traditional model-based approaches rely on iterative optimization. Despite exciting progress, typical unrolling approaches heuristically design layer-specific convolution weights to improve performance. Crucially, convergence properties of the underlying iterative algorithm are lost once layer-specific parameters are learned from training data. We propose an unrolling technique that breaks the trade-off between retaining algorithm properties while simultaneously enhancing performance. We focus on image deblurring and unrolling the widely-applied Half-Quadratic Splitting (HQS) algorithm. We develop a new parametrization scheme which enforces layer-specific parameters to asymptotically approach certain fixed points. Through extensive experimental studies, we verify that our approach achieves competitive performance with state-of-the-art unrolled layer-specific learning and significantly improves over the traditional HQS algorithm. We further establish convergence of the proposed unrolled network as the number of layers approaches infinity, and characterize its convergence rate. Our experimental verification involves simulations that validate the analytical results as well as comparison with state-of-the-art non-blind deblurring techniques on benchmark datasets. The merits of the proposed convergent unrolled network are established over competing alternatives, especially in the regime of limited training.
This paper presents a new accelerated proximal Markov chain Monte Carlo methodology to perform Bayesian inference in imaging inverse problems with an underlying convex geometry. The proposed strategy takes the form of a stochastic relaxed proximal-point iteration that admits two complementary interpretations. For models that are smooth or regularised by Moreau-Yosida smoothing, the algorithm is equivalent to an implicit midpoint discretisation of an overdamped Langevin diffusion targeting the posterior distribution of interest. This discretisation is asymptotically unbiased for Gaussian targets and shown to converge in an accelerated manner for any target that is $\kappa$-strongly log-concave (i.e., requiring in the order of $\sqrt{\kappa}$ iterations to converge, similarly to accelerated optimisation schemes), comparing favorably to [M. Pereyra, L. Vargas Mieles, K.C. Zygalakis, SIAM J. Imaging Sciences, 13,2 (2020), pp. 905-935] which is only provably accelerated for Gaussian targets and has bias. For models that are not smooth, the algorithm is equivalent to a Leimkuhler-Matthews discretisation of a Langevin diffusion targeting a Moreau-Yosida approximation of the posterior distribution of interest, and hence achieves a significantly lower bias than conventional unadjusted Langevin strategies based on the Euler-Maruyama discretisation. For targets that are $\kappa$-strongly log-concave, the provided non-asymptotic convergence analysis also identifies the optimal time step which maximizes the convergence speed. The proposed methodology is demonstrated through a range of experiments related to image deconvolution with Gaussian and Poisson noise, with assumption-driven and data-driven convex priors. Source codes for the numerical experiments of this paper are available from https://github.com/MI2G/accelerated-langevin-imla.
Radio astronomy is currently thriving with new large ground-based radio telescopes coming online in preparation for the upcoming Square Kilometre Array (SKA). Facilities like LOFAR, MeerKAT/SKA, ASKAP/SKA, and the future SKA-LOW bring tremendous sensitivity in time and frequency, improved angular resolution, and also high-rate data streams that need to be processed. They enable advanced studies of radio transients, volatile by nature, that can be detected or missed in the data. These transients are markers of high-energy accelerations of electrons and manifest in a wide range of temporal scales. Usually studied with dynamic spectroscopy of time series analysis, there is a motivation to search for such sources in large interferometric datasets. This requires efficient and robust signal reconstruction algorithms. To correctly account for the temporal dependency of the data, we improve the classical image deconvolution inverse problem by adding the temporal dependency in the reconstruction problem. Then, we introduce two novel neural network architectures that can do both spatial and temporal modeling of the data and the instrumental response. Then, we simulate representative time-dependent image cubes of point source distributions and realistic telescope pointings of MeerKAT to generate toy models to build the training, validation, and test datasets. Finally, based on the test data, we evaluate the source profile reconstruction performance of the proposed methods and classical image deconvolution algorithm CLEAN applied frame-by-frame. In the presence of increasing noise level in data frame, the proposed methods display a high level of robustness compared to frame-by-frame imaging with CLEAN. The deconvolved image cubes bring a factor of 3 improvement in fidelity of the recovered temporal profiles and a factor of 2 improvement in background denoising.
We study the problem of approximate sampling from non-log-concave distributions, e.g., Gaussian mixtures, which is often challenging even in low dimensions due to their multimodality. We focus on performing this task via Markov chain Monte Carlo (MCMC) methods derived from discretizations of the overdamped Langevin diffusions, which are commonly known as Langevin Monte Carlo algorithms. Furthermore, we are also interested in two nonsmooth cases for which a large class of proximal MCMC methods have been developed: (i) a nonsmooth prior is considered with a Gaussian mixture likelihood; (ii) a Laplacian mixture distribution. Such nonsmooth and non-log-concave sampling tasks arise from a wide range of applications to Bayesian inference and imaging inverse problems such as image deconvolution. We perform numerical simulations to compare the performance of most commonly used Langevin Monte Carlo algorithms.
Deconvolution is a widely used strategy to mitigate the blurring and noisy degradation of hyperspectral images~(HSI) generated by the acquisition devices. This issue is usually addressed by solving an ill-posed inverse problem. While investigating proper image priors can enhance the deconvolution performance, it is not trivial to handcraft a powerful regularizer and to set the regularization parameters. To address these issues, in this paper we introduce a tuning-free Plug-and-Play (PnP) algorithm for HSI deconvolution. Specifically, we use the alternating direction method of multipliers (ADMM) to decompose the optimization problem into two iterative sub-problems. A flexible blind 3D denoising network (B3DDN) is designed to learn deep priors and to solve the denoising sub-problem with different noise levels. A measure of 3D residual whiteness is then investigated to adjust the penalty parameters when solving the quadratic sub-problems, as well as a stopping criterion. Experimental results on both simulated and real-world data with ground-truth demonstrate the superiority of the proposed method.
Removing optical and atmospheric blur from galaxy images significantly improves galaxy shape measurements for weak gravitational lensing and galaxy evolution studies. This ill-posed linear inverse problem is usually solved with deconvolution algorithms enhanced by regularisation priors or deep learning. We introduce a so-called "physics-informed deep learning" approach to the Point Spread Function (PSF) deconvolution problem in galaxy surveys. We apply algorithm unrolling and the Plug-and-Play technique to the Alternating Direction Method of Multipliers (ADMM), in which a neural network learns appropriate hyperparameters and denoising priors from simulated galaxy images. We characterise the time-performance trade-off of several methods for galaxies of differing brightness levels as well as our method's robustness to systematic PSF errors and network ablations. We show an improvement in reduced shear ellipticity error of 38.6% (SNR=20)/45.0% (SNR=200) compared to classic methods and 7.4% (SNR=20)/33.2% (SNR=200) compared to modern methods.
Conventional deconvolution methods utilize hand-crafted image priors to constrain the optimization. While deep-learning-based methods have simplified the optimization by end-to-end training, they fail to generalize well to blurs unseen in the training dataset. Thus, training image-specific models is important for higher generalization. Deep image prior (DIP) provides an approach to optimize the weights of a randomly initialized network with a single degraded image by maximum a posteriori (MAP), which shows that the architecture of a network can serve as the hand-crafted image prior. Different from the conventional hand-crafted image priors that are statistically obtained, it is hard to find a proper network architecture because the relationship between images and their corresponding network architectures is unclear. As a result, the network architecture cannot provide enough constraint for the latent sharp image. This paper proposes a new variational deep image prior (VDIP) for blind image deconvolution, which exploits additive hand-crafted image priors on latent sharp images and approximates a distribution for each pixel to avoid suboptimal solutions. Our mathematical analysis shows that the proposed method can better constrain the optimization. The experimental results further demonstrate that the generated images have better quality than that of the original DIP on benchmark datasets. The source code of our VDIP is available at https://github.com/Dong-Huo/VDIP-Deconvolution.
Most existing methods usually formulate the non-blind deconvolution problem into a maximum-a-posteriori framework and address it by manually designing kinds of regularization terms and data terms of the latent clear images. However, explicitly designing these two terms is quite challenging and usually leads to complex optimization problems which are difficult to solve. In this paper, we propose an effective non-blind deconvolution approach by learning discriminative shrinkage functions to implicitly model these terms. In contrast to most existing methods that use deep convolutional neural networks (CNNs) or radial basis functions to simply learn the regularization term, we formulate both the data term and regularization term and split the deconvolution model into data-related and regularization-related sub-problems according to the alternating direction method of multipliers. We explore the properties of the Maxout function and develop a deep CNN model with a Maxout layer to learn discriminative shrinkage functions to directly approximate the solutions of these two sub-problems. Moreover, given the fast-Fourier-transform-based image restoration usually leads to ringing artifacts while conjugate-gradient-based approach is time-consuming, we develop the Conjugate Gradient Network to restore the latent clear images effectively and efficiently. Experimental results show that the proposed method performs favorably against the state-of-the-art ones in terms of efficiency and accuracy.
A new Plug-and-Play (PnP) alternating direction of multipliers (ADMM) scheme is proposed in this paper, by embedding a recently introduced adaptive denoiser using the Schroedinger equation's solutions of quantum physics. The potential of the proposed model is studied for Poisson image deconvolution, which is a common problem occurring in number of imaging applications, such as limited photon acquisition or X-ray computed tomography. Numerical results show the efficiency and good adaptability of the proposed scheme compared to recent state-of-the-art techniques, for both high and low signal-to-noise ratio scenarios. This performance gain regardless of the amount of noise affecting the observations is explained by the flexibility of the embedded quantum denoiser constructed without anticipating any prior statistics about the noise, which is one of the main advantages of this method. The main novelty of this work resided in the integration of a modified quantum denoiser into the PnP-ADMM framework and the numerical proof of convergence of the resulting algorithm.
We propose a general framework for solving inverse problems in the presence of noise that requires no signal prior, no noise estimate, and no clean training data. We only require that the forward model be available and that the noise be statistically independent across measurement dimensions. We build upon the theory of $\mathcal{J}$-invariant functions (Batson & Royer 2019, arXiv:1901.11365) and show how self-supervised denoising \emph{\`a la} Noise2Self is a special case of learning a noise-tolerant pseudo-inverse of the identity. We demonstrate our approach by showing how a convolutional neural network can be taught in a self-supervised manner to deconvolve images and surpass in image quality classical inversion schemes such as Lucy-Richardson deconvolution.
Existing video super-resolution (SR) algorithms usually assume that the blur kernels in the degradation process are known and do not model the blur kernels in the restoration. However, this assumption does not hold for video SR and usually leads to over-smoothed super-resolved images. In this paper, we propose a deep convolutional neural network (CNN) model to solve video SR by a blur kernel modeling approach. The proposed deep CNN model consists of motion blur estimation, motion estimation, and latent image restoration modules. The motion blur estimation module is used to provide reliable blur kernels. With the estimated blur kernel, we develop an image deconvolution method based on the image formation model of video SR to generate intermediate latent images so that some sharp image contents can be restored well. However, the generated intermediate latent images may contain artifacts. To generate high-quality images, we use the motion estimation module to explore the information from adjacent frames, where the motion estimation can constrain the deep CNN model for better image restoration. We show that the proposed algorithm is able to generate clearer images with finer structural details. Extensive experimental results show that the proposed algorithm performs favorably against state-of-the-art methods.
Microscopy is a powerful visualization tool in biology, enabling the study of cells, tissues, and the fundamental biological processes; yet, the observed images typically suffer from blur and background noise. In this work, we propose a unifying framework of algorithms for Gaussian image deblurring and denoising. These algorithms are based on deep learning techniques for the design of learnable regularizers integrated into the Wiener-Kolmogorov filter. Our extensive experimentation line showcases that the proposed approach achieves a superior quality of image reconstruction and surpasses the solutions that rely either on deep learning or on optimization schemes alone. Augmented with the variance stabilizing transformation, the proposed reconstruction pipeline can also be successfully applied to the problem of Poisson image deblurring, surpassing the state-of-the-art methods. Moreover, several variants of the proposed framework demonstrate competitive performance at low computational complexity, which is of high importance for real-time imaging applications.
This paper proposes a novel approach to regularize the ill-posed blind image deconvolution (blind image deblurring) problem using deep generative networks. We employ two separate deep generative models - one trained to produce sharp images while the other trained to generate blur kernels from lower dimensional parameters. To deblur, we propose an alternating gradient descent scheme operating in the latent lower-dimensional space of each of the pretrained generative models. Our experiments show excellent deblurring results even under large blurs and heavy noise. To improve the performance on rich image datasets not well learned by the generative networks, we present a modification of the proposed scheme that governs the deblurring process under both generative and classical priors.
In image deconvolution problems, the diagonalization of the underlying operators by means of the 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, through the use of 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.
While variational methods have been among the most powerful tools for solving linear inverse problems in imaging, deep (convolutional) neural networks have recently taken the lead in many challenging benchmarks. A remaining drawback of deep learning approaches is their requirement for an expensive retraining whenever the specific problem, the noise level, noise type, or desired measure of fidelity changes. On the contrary, variational methods have a plug-and-play nature as they usually consist of separate data fidelity and regularization terms. In this paper we study the possibility of replacing the proximal operator of the regularization used in many convex energy minimization algorithms by a denoising neural network. The latter therefore serves as an implicit natural image prior, while the data term can still be chosen independently. Using a fixed denoising neural network in exemplary problems of image deconvolution with different blur kernels and image demosaicking, we obtain state-of-the-art reconstruction results. These indicate the high generalizability of our approach and a reduction of the need for problem-specific training. Additionally, we discuss novel results on the analysis of possible optimization algorithms to incorporate the network into, as well as the choices of algorithm parameters and their relation to the noise level the neural network is trained on.
We study \emph{TV regularization}, a widely used technique for eliciting structured sparsity. In particular, we propose efficient algorithms for computing prox-operators for $\ell_p$-norm TV. The most important among these is $\ell_1$-norm TV, for whose prox-operator we present a new geometric analysis which unveils a hitherto unknown connection to taut-string methods. This connection turns out to be remarkably useful as it shows how our geometry guided implementation results in efficient weighted and unweighted 1D-TV solvers, surpassing state-of-the-art methods. Our 1D-TV solvers provide the backbone for building more complex (two or higher-dimensional) TV solvers within a modular proximal optimization approach. We review the literature for an array of methods exploiting this strategy, and illustrate the benefits of our modular design through extensive suite of experiments on (i) image denoising, (ii) image deconvolution, (iii) four variants of fused-lasso, and (iv) video denoising. To underscore our claims and permit easy reproducibility, we provide all the reviewed and our new TV solvers in an easy to use multi-threaded C++, Matlab and Python library.
As an integral component of blind image deblurring, non-blind deconvolution removes image blur with a given blur kernel, which is essential but difficult due to the ill-posed nature of the inverse problem. The predominant approach is based on optimization subject to regularization functions that are either manually designed, or learned from examples. Existing learning based methods have shown superior restoration quality but are not practical enough due to their restricted and static model design. They solely focus on learning a prior and require to know the noise level for deconvolution. We address the gap between the optimization-based and learning-based approaches by learning a universal gradient descent optimizer. We propose a Recurrent Gradient Descent Network (RGDN) by systematically incorporating deep neural networks into a fully parameterized gradient descent scheme. A hyper-parameter-free update unit shared across steps is used to generate updates from the current estimates, based on a convolutional neural network. By training on diverse examples, the Recurrent Gradient Descent Network learns an implicit image prior and a universal update rule through recursive supervision. The learned optimizer can be repeatedly used to improve the quality of diverse degenerated observations. The proposed method possesses strong interpretability and high generalization. Extensive experiments on synthetic benchmarks and challenging real-world images demonstrate that the proposed deep optimization method is effective and robust to produce favorable results as well as practical for real-world image deblurring applications.
Most existing non-blind restoration methods are based on the assumption that a precise degradation model is known. As the degradation process can only be partially known or inaccurately modeled, images may not be well restored. Rain streak removal and image deconvolution with inaccurate blur kernels are two representative examples of such tasks. For rain streak removal, although an input image can be decomposed into a scene layer and a rain streak layer, there exists no explicit formulation for modeling rain streaks and the composition with scene layer. For blind deconvolution, as estimation error of blur kernel is usually introduced, the subsequent non-blind deconvolution process does not restore the latent image well. In this paper, we propose a principled algorithm within the maximum a posterior framework to tackle image restoration with a partially known or inaccurate degradation model. Specifically, the residual caused by a partially known or inaccurate degradation model is spatially dependent and complexly distributed. With a training set of degraded and ground-truth image pairs, we parameterize and learn the fidelity term for a degradation model in a task-driven manner. Furthermore, the regularization term can also be learned along with the fidelity term, thereby forming a simultaneous fidelity and regularization learning model. Extensive experimental results demonstrate the effectiveness of the proposed model for image deconvolution with inaccurate blur kernels, deconvolution with multiple degradations and rain streak removal.