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.
Confocal laser-scanning microscopy (CLSM) is one of the most popular optical architectures for fluorescence imaging. In CLSM, a focused laser beam excites the fluorescence emission from a specific specimen position. Some actuators scan the probed region across the sample and a photodetector collects a single intensity value for each scan point, building a two-dimensional image pixel-by-pixel. Recently, new fast single-photon array detectors have allowed the recording of a full bi-dimensional image of the probed region for each scan point, transforming CLSM into image scanning microscopy (ISM). This latter offers significant improvements over traditional imaging but requires an optimal processing tool to extract a super-resolved image from the four-dimensional dataset. Here we describe the image formation process in ISM from a statistical point of view, and we use the Bayesian framework to formulate a multi-image deconvolution problem. Notably, the single-photon detector suffers exclusively from the photon shot noise, enabling the development of an effective likelihood model. We derive an iterative likelihood maximization algorithm and test it on experimental and simulated data. Furthermore, we demonstrate that the ISM dataset is redundant, enabling the possibility of obtaining reconstruction sampled at twice the scanning step. Our results prove that in ISM, under appropriate conditions, the Nyquist-Shannon sampling criterium is effectively relaxed. This finding can be exploited to speed up the acquisition process by a factor of four, further improving the versatility of ISM systems.
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.
We present a model for non-blind image deconvolution that incorporates the classic iterative method into a deep learning application. Instead of using large over-parameterised generative networks to create sharp picture representations, we build our network based on the iterative Landweber deconvolution algorithm, which is integrated with trainable convolutional layers to enhance the recovered image structures and details. Additional to the data fidelity term, we also add Hessian and sparse constraints as regularization terms to improve the image reconstruction quality. Our proposed model is \textit{self-supervised} and converges to a solution based purely on the input blurred image and respective blur kernel without the requirement of any pre-training. We evaluate our technique using standard computer vision benchmarking datasets as well as real microscope images obtained by our enhanced depth-of-field (EDOF) underwater microscope, demonstrating the capabilities of our model in a real-world application. The quantitative results demonstrate that our approach is competitive with state-of-the-art non-blind image deblurring methods despite having a fraction of the parameters and not being pre-trained, demonstrating the efficiency and efficacy of embedding a classic deconvolution approach inside a deep network.
Recently, Optimization-Derived Learning (ODL) has attracted attention from learning and vision areas, which designs learning models from the perspective of optimization. However, previous ODL approaches regard the training and hyper-training procedures as two separated stages, meaning that the hyper-training variables have to be fixed during the training process, and thus it is also impossible to simultaneously obtain the convergence of training and hyper-training variables. In this work, we design a Generalized Krasnoselskii-Mann (GKM) scheme based on fixed-point iterations as our fundamental ODL module, which unifies existing ODL methods as special cases. Under the GKM scheme, a Bilevel Meta Optimization (BMO) algorithmic framework is constructed to solve the optimal training and hyper-training variables together. We rigorously prove the essential joint convergence of the fixed-point iteration for training and the process of optimizing hyper-parameters for hyper-training, both on the approximation quality, and on the stationary analysis. Experiments demonstrate the efficiency of BMO with competitive performance on sparse coding and real-world applications such as image deconvolution and rain streak removal.
Image production tools do not always create a clear image, noisy and blurry images are sometimes created. Among these cases, Poissonian noise is one of the most famous noises that appear in medical images and images taken in astronomy. Blurred image with Poissonian noise obscures important details that are of great importance in medicine or astronomy. Therefore, studying and increasing the quality of images that are affected by this type of noise is always considered by researchers. In this paper, in the first step, based on framelet transform, a local minimal prior is introduced, and in the next step, this tool together with fractional calculation is used for Poissonian blurred image deconvolution. In the following, the model is generalized to the blind case. To evaluate the performance of the presented model, several images such as real images have been investigated.
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.
Synthetic aperture sonar (SAS) image resolution is constrained by waveform bandwidth and array geometry. Specifically, the waveform bandwidth determines a point spread function (PSF) that blurs the locations of point scatterers in the scene. In theory, deconvolving the reconstructed SAS image with the scene PSF restores the original distribution of scatterers and yields sharper reconstructions. However, deconvolution is an ill-posed operation that is highly sensitive to noise. In this work, we leverage implicit neural representations (INRs), shown to be strong priors for the natural image space, to deconvolve SAS images. Importantly, our method does not require training data, as we perform our deconvolution through an analysis-bysynthesis optimization in a self-supervised fashion. We validate our method on simulated SAS data created with a point scattering model and real data captured with an in-air circular SAS. This work is an important first step towards applying neural networks for SAS image deconvolution.
In this work, we study the problem of non-blind image deconvolution and propose a novel recurrent network architecture that leads to very competitive restoration results of high image quality. Motivated by the computational efficiency and robustness of existing large scale linear solvers, we manage to express the solution to this problem as the solution of a series of adaptive non-negative least-squares problems. This gives rise to our proposed Recurrent Least Squares Deconvolution Network (RLSDN) architecture, which consists of an implicit layer that imposes a linear constraint between its input and output. By design, our network manages to serve two important purposes simultaneously. The first is that it implicitly models an effective image prior that can adequately characterize the set of natural images, while the second is that it recovers the corresponding maximum a posteriori (MAP) estimate. Experiments on publicly available datasets, comparing recent state-of-the-art methods, show that our proposed RLSDN approach achieves the best reported performance both for grayscale and color images for all tested scenarios. Furthermore, we introduce a novel training strategy that can be adopted by any network architecture that involves the solution of linear systems as part of its pipeline. Our strategy eliminates completely the need to unroll the iterations required by the linear solver and, thus, it reduces significantly the memory footprint during training. Consequently, this enables the training of deeper network architectures which can further improve the reconstruction results.
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.
This paper introduces a new Plug-and-Play (PnP) alternating direction of multipliers (ADMM) scheme based on a recently proposed denoiser using the Schroedinger equation's solutions of quantum physics. The efficiency of the proposed algorithm is evaluated for Poisson image deconvolution, which is very common for imaging applications, such as, for example, limited photon acquisition. Numerical results show the superiority of the proposed scheme compared to recent state-of-the-art techniques, for both low and high signal-to-noise-ratio scenarios. This performance gain is mostly explained by the flexibility of the embedded quantum denoiser for different types of noise affecting the observations.
The Shack-Hartmann wavefront sensor is widely used to measure aberrations induced by atmospheric turbulence in adaptive optics systems. However if there exists strong atmospheric turbulence or the brightness of guide stars is low, the accuracy of wavefront measurements will be affected. In this paper, we propose a compressive Shack-Hartmann wavefront sensing method. Instead of reconstructing wavefronts with slope measurements of all sub-apertures, our method reconstructs wavefronts with slope measurements of sub-apertures which have spot images with high signal to noise ratio. Besides, we further propose to use a deep neural network to accelerate wavefront reconstruction speed. During the training stage of the deep neural network, we propose to add a drop-out layer to simulate the compressive sensing process, which could increase development speed of our method. After training, the compressive Shack-Hartmann wavefront sensing method can reconstruct wavefronts in high spatial resolution with slope measurements from only a small amount of sub-apertures. We integrate the straightforward compressive Shack-Hartmann wavefront sensing method with image deconvolution algorithm to develop a high-order image restoration method. We use images restored by the high-order image restoration method to test the performance of our the compressive Shack-Hartmann wavefront sensing method. The results show that our method can improve the accuracy of wavefront measurements and is suitable for real-time applications.
In this paper, we solve blind image deconvolution problem that is to remove blurs form a signal degraded image without any knowledge of the blur kernel. Since the problem is ill-posed, an image prior plays a significant role in accurate blind deconvolution. Traditional image prior assumes coefficients in filtered domains are sparse. However, it is assumed here that there exist additional structures over the sparse coefficients. Accordingly, we propose new problem formulation for the blind image deconvolution, which utilizes the structural information by coupling Student's-t image prior with overlapping group sparsity. The proposed method resulted in an effective blind deconvolution algorithm that outperforms other state-of-the-art algorithms.
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.
Image deconvolution is the process of recovering convolutional degraded images, which is always a hard inverse problem because of its mathematically ill-posed property. On the success of the recently proposed deep image prior (DIP), we build an image deconvolution model with deep image and kernel priors (DIKP). DIP is a learning-free representation which uses neural net structures to express image prior information, and it showed great success in many energy-based models, e.g. denoising, super-resolution, inpainting. Instead, our DIKP model uses such priors in image deconvolution to model not only images but also kernels, combining the ideas of traditional learning-free deconvolution methods with neural nets. In this paper, we show that DIKP improve the performance of learning-free image deconvolution, and we experimentally demonstrate this on the standard benchmark of six standard test images in terms of PSNR and visual effects.
Deconvolution microscopy has been extensively used to improve the resolution of the wide-field fluorescent microscopy, but the performance of classical approaches critically depends on the accuracy of a model and optimization algorithms. Recently, the convolutional neural network (CNN) approaches have been studied as a fast and high performance alternative. Unfortunately, the CNN approaches usually require matched high resolution images for supervised training. In this paper, we present a novel unsupervised cycle-consistent generative adversarial network (cycleGAN) with a linear blur kernel, which can be used for both blind- and non-blind image deconvolution. In contrast to the conventional cycleGAN approaches that require two deep generators, the proposed cycleGAN approach needs only a single deep generator and a linear blur kernel, which significantly improves the robustness and efficiency of network training. We show that the proposed architecture is indeed a dual formulation of an optimal transport problem that uses a special form of the penalized least squares cost as a transport cost. Experimental results using simulated and real experimental data confirm the efficacy of the algorithm.