Bayesian Image Restoration Research Articles

Bayesian Image Super-Resolution With Deep Modeling of Image Statistics.

Modeling statistics of image priors is useful for image super-resolution, but little attention has been paid from the massive works of deep learning-based methods. In this work, we propose a Bayesian image restoration framework, where natural image statistics are modeled with the combination of smoothness and sparsity priors. Concretely, first we consider an ideal image as the sum of a smoothness component and a sparsity residual, and model real image degradation including blurring, downscaling, and noise corruption. Then, we develop a variational Bayesian approach to infer their posteriors. Finally, we implement the variational approach for single image super-resolution (SISR) using deep neural networks, and propose an unsupervised training strategy. The experiments on three image restoration tasks, i.e., ideal SISR, realistic SISR, and real-world SISR, demonstrate that our method has superior model generalizability against varying noise levels and degradation kernels and is effective in unsupervised SISR. The code and resulting models are released via https://zmiclab.github.io/projects.html.

Variational Bayesian image restoration with multi-structured model of wavelet transform coefficients

Recent works have demonstrated that using a properly structured prior for model-based compressive sensing (CS), can improve the recovery performance. However, there exists the low prior utilisation in previous works. Therefore, in this study, we introduce a multi-structured-wavelet Bayesian CS framework (MSW-BCS) that works for image compression with fully employing the prior knowledge. Our work can be briefly summarised by the following three aspects. Firstly, we explore the multi-structured prior knowledge, including the cluster structure and tree structure in the wavelet transform coefficients of images via statistical analyses. Secondly, a multi-variable joint recovery model is designed to describe this multi-structured prior. Finally, the detailed learning algorithm of model parameters based on variational Bayesian inference is given. The simulation experiments show that the proposed recovery model can effectively merge the different priors and achieve the superior performance compared with that of the other well-known CS algorithms under both noise-free and noisy measurement environments.

Statistical analysis of loopy belief propagation in random fields.

Loopy belief propagation (LBP), which is equivalent to the Bethe approximation in statistical mechanics, is a message-passing-type inference method that is widely used to analyze systems based on Markov random fields (MRFs). In this paper, we propose a message-passing-type method to analytically evaluate the quenched average of LBP in random fields by using the replica cluster variation method. The proposed analytical method is applicable to general pairwise MRFs with random fields whose distributions differ from each other and can give the quenched averages of the Bethe free energies over random fields, which are consistent with numerical results. The order of its computational cost is equivalent to that of standard LBP. In the latter part of this paper, we describe the application of the proposed method to Bayesian image restoration, in which we observed that our theoretical results are in good agreement with the numerical results for natural images.

Variational Bayesian image restoration with group-sparse modeling of wavelet coefficients

In this work, we present a recent wavelet-based image restoration framework based on a group-sparse Gaussian scale mixture model. A hierarchical Bayesian estimation is derived using a combination of variational Bayesian inference and a subband-adaptive majorization–minimization method that simplifies computation of the posterior distribution. We show that both of these iterative methods can converge together without needing nested loops, and thus good solutions can be found rapidly in the non-convex search space. We also integrate our method, variational Bayesian with majorization minimization (VBMM), with tree-structured modeling of the wavelet coefficients. This extension achieves significant gains in performance over the coefficient-sparse version of the algorithm. The experimental results demonstrate that the proposed method and its tree-structured extensions are effective for various imaging applications such as image deconvolution, image superresolution and compressive sensing magnetic resonance imaging (MRI) reconstruction, and that they outperform more conventional sparsity-inducing methods based on the l1-norm.

Bayesian Image Restoration for Poisson Corrupted Image Using a Latent Variational Method with Gaussian MRF

Bayesian Image Restoration for Poisson Corrupted Image Using a Latent Variational Method with Gaussian MRF

Bayesian Image Restoration for Poisson Corrupted Image Using a Latent Variational Method with Gaussian MRF

We treat an image restoration problem with a Poisson noise chan- nel using a Bayesian framework. The Poisson randomness might be appeared in observation of low contrast object in the field of imaging. The noise observation is often hard to treat in a theo- retical analysis. In our formulation, we interpret the observation through the Poisson noise channel as a likelihood, and evaluate the bound of it with a Gaussian function using a latent variable method. We then introduce a Gaussian Markov random field (GMRF) as the prior for the Bayesian approach, and derive the posterior as a Gaussian distribution. The latent parameters in the likelihood and the hyperparameter in the GMRF prior could be treated as hid- den parameters, so that, we propose an algorithm to infer them in the expectation maximization (EM) framework using loopy belief propagation(LBP). We confirm the ability of our algorithm in the computer simulation, and compare it with the results of other im- age restoration frameworks.

Inferring model‐based probability of occurrence from preferentially sampled data with uncertain absences using expert knowledge

Summary For several data bases in species distribution modelling (SDM), presences are known but absences are uncertain and can be non‐detections. The environment influences both the non‐detections and the species occurrences (preferential sampling), affecting model predictions. The objective is to estimate model‐based probability of occurrence when absences are uncertain and sampling is preferential. Our Bayesian image restoration (BIR) model has an environmental, spatial and non‐detection component. The latter is based on the number of ‘control species’ (selected by expert knowledge to have wide geographic range and occupy several habitats) observed. Control species are assumed present in all grid cells so an observed absence of the control species is a non‐detection. The observed absence of a focal species is likely a non‐detection when the environment is conducive, the focal species was observed in the neighbourhood, and the number of control species is small. Simulating an artificial species (true prevalence 0·31) and preferential sampling (observed prevalence 0·18), the BIR modelled prevalence was 0·32 (0·30–0·34). The estimation was robust to failure of assumptions with a mis‐specified environmental model. The restored map was close to the true map, and the true realized niche of the species was inferred (with uncertainty). The link between the non‐detection and control species is crucial, and a sensitivity analysis is recommended. BIR restored the simulated atlas map more accurately than a covariate adjustment method. When applied to the vascular plant Asarum europaeum L., BIR changed the interpretation. The uncertainty maps assisted decision‐making. A BIR implementation new to the SDM field is presented. BIR using control species is most fruitfully applied at coarse scale and intermediate extent, with limited environmental extremes and sufficient expert knowledge (e.g. many Central European countries). BIR is a framework, and we studied one implementation. Any proxy (road density, distance to city, direct expert knowledge, detectability traits, etc.) associated with non‐detection can be used so BIR can be applied in diverse settings. Including non‐detections reduces model indeterminacy but often requires expert knowledge. However, the benefits of using a priori knowledge and Bayesian estimation are expected to outweigh the risks of model‐based inference.

Bayesian image restoration for mosaic active imaging

In this paper, we focus on the restoration of images acquired with a new active imaging concept. This new instrument generates a mosaic of active imaging acquisitions. We first describe a simplified forward model of this so-called ''mosaic active imaging''. We also assume a prior on the distribution of images, using the \ac{TV}, and deduce a restoration algorithm. This algorithm iterates one step for the estimation of the restored image and one step for the estimation of the acquisition parameters. We then provide the details useful to the implementation of these two steps. In particular, we show that the image estimation can be performed with graph-cuts. This allows a fast resolution of this image estimation step. Finally, we detail numerical experiments showing that acquisitions made with a mosaic active imaging device can be restored even under severe noise levels, with few acquisitions.

Combining Poisson singular integral and total variation prior models in image restoration

In this paper, a novel Bayesian image restoration method based on a combination of priors is presented. It is well known that the Total Variation (TV) image prior preserves edge structures while imposing smoothness on the solutions. However, it tends to oversmooth textured areas. To alleviate this problem we propose to combine the TV and the Poisson Singular Integral (PSI) models, which, as we will show, preserves the image textures. The PSI prior depends on a parameter that controls the shape of the filter. A study on the behavior of the filter as a function of this parameter is presented. Our restoration model utilizes a bound for the TV image model based on the majorization–minimization principle, and performs maximum a posteriori Bayesian inference. In order to assess the performance of the proposed approach, in the experimental section we compare it with other restoration methods.

Semi-Huber quadratic function and comparative study of some MRFs for Bayesian image restoration

The present work introduces an alternative method to deal with digital image restoration into a Bayesian framework, particularly, the use of a new half-quadratic function is proposed which performance is satisfactory compared with respect to some other functions in existing literature. The bayesian methodology is based on the prior knowledge of some information that allows an efficient modelling of the image acquisition process. The edge preservation of objects into the image while smoothing noise is necessary in an adequate model. Thus, we use a convexity criteria given by a semi-Huber function to obtain adequate weighting of the cost functions (half-quadratic) to be minimized. The principal objective when using Bayesian methods based on the Markov Random Fields (MRF) in the context of image processing is to eliminate those effects caused by the excessive smoothness on the reconstruction process of image which are rich in contours or edges. A comparison between the new introduced scheme and other three existing schemes, for the cases of noise filtering and image deblurring, is presented. This collection of implemented methods is inspired of course on the use of MRFs such as the semi-Huber, the generalized Gaussian, the Welch, and Tukey potential functions with granularity control. The obtained results showed a satisfactory performance and the effectiveness of the proposed estimator with respect to other three estimators.

Sub-diffraction Limit Localization of Proteins in Volumetric Space Using Bayesian Restoration of Fluorescence Images from Ultrathin Specimens

Photon diffraction limits the resolution of conventional light microscopy at the lateral focal plane to 0.61λ/NA (λ = wavelength of light, NA = numerical aperture of the objective) and at the axial plane to 1.4nλ/NA2 (n = refractive index of the imaging medium, 1.51 for oil immersion), which with visible wavelengths and a 1.4NA oil immersion objective is ∼220 nm and ∼600 nm in the lateral plane and axial plane respectively. This volumetric resolution is too large for the proper localization of protein clustering in subcellular structures. Here we combine the newly developed proteomic imaging technique, Array Tomography (AT), with its native 50–100 nm axial resolution achieved by physical sectioning of resin embedded tissue, and a 2D maximum likelihood deconvolution method, based on Bayes' rule, which significantly improves the resolution of protein puncta in the lateral plane to allow accurate and fast computational segmentation and analysis of labeled proteins. The physical sectioning of AT allows tissue specimens to be imaged at the physical optimum of modern high NA plan-apochormatic objectives. This translates to images that have little out of focus light, minimal aberrations and wave-front distortions. Thus, AT is able to provide images with truly invariant point spread functions (PSF), a property critical for accurate deconvolution. We show that AT with deconvolution increases the volumetric analytical fidelity of protein localization by significantly improving the modulation of high spatial frequencies up to and potentially beyond the spatial frequency cut-off of the objective. Moreover, we are able to achieve this improvement with no noticeable introduction of noise or artifacts and arrive at object segmentation and localization accuracies on par with image volumes captured using commercial implementations of super-resolution microscopes.

BAYESIAN IMAGE RESTORATION, USING CONFIGURATIONS

In this paper, we develop a Bayesian procedure for removing noise from images that can be viewed as noisy realisations of random sets in the plane. The procedure utilises recent advances in configuration theory for noise free random sets, where the probabilities of observing the different boundary configurations are expressed in terms of the mean normal measure of the random set. These probabilities are used as prior probabilities in a Bayesian image restoration approach. Estimation of the remaining parameters in the model is outlined for salt and pepper noise. The inference in the model is discussed in detail for 3 X 3 and 5 X 5 configurations and examples of the performance of the procedure are given.

Astronomical image restoration using variational methods and model combination

In this work we develop a variational framework for the combination of several prior models in Bayesian image restoration and apply it to astronomical images. Since each combination of a given observation model and a prior model produces a different posterior distribution of the underlying image, the use of variational posterior distribution approximation on each posterior will produce as many posterior approximations as priors we want to combine. A unique approximation is obtained here by finding the distribution on the unknown image given the observations that minimizes a linear convex combination of the Kullback–Leibler divergences associated with each posterior distribution. We find this distribution in closed form and also relate the proposed approach to other prior combination methods in the literature. Experimental results on both synthetic images and on real astronomical images validate the proposed approach.

Bayesian Image Restoration Using a Large‐Scale Total Patch Variation Prior

Edge‐preserving Bayesian restorations using nonquadratic priors are often inefficient in restoring continuous variations and tend to produce block artifacts around edges in ill‐posed inverse image restorations. To overcome this, we have proposed a spatial adaptive (SA) prior with improved performance. However, this SA prior restoration suffers from high computational cost and the unguaranteed convergence problem. Concerning these issues, this paper proposes a Large‐scale Total Patch Variation (LS‐TPV) Prior model for Bayesian image restoration. In this model, the prior for each pixel is defined as a singleton conditional probability, which is in a mixture prior form of one patch similarity prior and one weight entropy prior. A joint MAP estimation is thus built to ensure the iteration monotonicity. The intensive calculation of patch distances is greatly alleviated by the parallelization of Compute Unified Device Architecture(CUDA). Experiments with both simulated and real data validate the good performance of the proposed restoration.

Bayesian Image Restoration for Medical Images Using Radon Transform

We propose an image reconstruction algorithm using Bayesian inference for Radon transformed observation data, which often appears in the field of medical image reconstruction known as computed tomography (CT). In order to apply our Bayesian reconstruction method, we introduced several hyper-parameters that control the ratio between prior information and the fidelity of the observation process. Since the quality of the reconstructed image is influenced by the estimation accuracy of these hyper-parameters, we propose an inference method for them based on the marginal likelihood maximization principle as well as the image reconstruction method. We are able to demonstrate a reconstruction result superior to that obtained using the conventional filtered back projection method.

Bayesian image restoration models for combining expert knowledge on recording activity with species distribution data

Biological atlases are, for many species, the only source of information on their distribution over large geographical areas, and are widely used to inform models of the environmental distribution of species. Such data are not collected using standardized survey techniques, however, and spatial variations in coverage (the relative extent or completeness of records) may lead to variations in the probability that the species will be recorded at locations where it is present (the “recording probability”). If spatial patterns in recording probabilities are correlated with key environmental variables, then biased estimates of the relationships between environmental variables and species distributions may be obtained. We outline a general statistical framework for modelling the environmental distribution of species using, known as Bayesian Image Restoration (BIR). BIR can be used in combination with any species distribution model, but in addition allows us to account for spatial heterogeneity in recording probabilities by utilizing expert knowledge on spatial patterns in coverage. We illustrate the methodology by applying it to maps of the recorded distribution of two plant species in Germany, taken from the German atlas of vascular plants. We find that estimated spatial patterns in recording probabilities for both species are correlated with key environmental variables. Consequently, different relationships between the probability of presence of a species and environmental variables were obtained when the species distribution models were parameterised within a BIR framework. Care must be taken in the application of BIR, since the resulting inferences can depend strongly upon the modelling assumptions that are adopted. Nevertheless, we conclude that BIR has the potential to make better use of uncertain information on species distributions than conventional methods, and can be used to formally investigate the robustness of inferences on the environmental distribution of species to assumptions concerning spatial patterns in recording probabilities.

Deterministic flows of order-parameters in stochastic processes of quantum Monte Carlo method

In terms of the stochastic process of quantum-mechanical version of Markov chain Monte Carlo method (the MCMC), we analytically derive macroscopically deterministic flow equations of order parameters such as spontaneous magnetization in infinite-range (d(= ∞)-dimensional) quantum spin systems. By means of the Trotter decomposition, we consider the transition probability of Glauber-type dynamics of microscopic states for the corresponding (d + 1)-dimensional classical system. Under the static approximation, differential equations with respect to macroscopic order parameters are explicitly obtained from the master equation that describes the microscopic-law. In the steady state, we show that the equations are identical to the saddle point equations for the equilibrium state of the same system. The equation for the dynamical Ising model is recovered in the classical limit. We also check the validity of the static approximation by making use of computer simulations for finite size systems and discuss several possible extensions of our approach to disordered spin systems for statistical-mechanical informatics. Especially, we shall use our procedure to evaluate the decoding process of Bayesian image restoration. With the assistance of the concept of dynamical replica theory (the DRT), we derive the zero-temperature flow equation of image restoration measure showing some 'non-monotonic' behaviour in its time evolution.

A polynomial-time perfect sampler for the Q-Ising with a vertex-independent noise

We present a polynomial-time perfect sampler for the Q-Ising with a vertex-independent noise. The Q-Ising, one of the generalized models of the Ising, arose in the context of Bayesian image restoration in statistical mechanics. We study the distribution of Q-Ising on a two-dimensional square lattice over n vertices, that is, we deal with a discrete state space {1,?,Q} n for a positive integer Q. Employing the Q-Ising (having a parameter β) as a prior distribution, and assuming a Gaussian noise (having another parameter ?), a posterior is obtained from the Bayes' formula. Furthermore, we generalize it: the distribution of noise is not necessarily a Gaussian, but any vertex-independent noise. We first present a Gibbs sampler from our posterior, and also present a perfect sampler by defining a coupling via a monotone update function. Then, we show O(nlog?n) mixing time of the Gibbs sampler for the generalized model under a condition that β is sufficiently small (whatever the distribution of noise is). In case of a Gaussian, we obtain another more natural condition for rapid mixing that ? is sufficiently larger than β. Thereby, we show that the expected running time of our sampler is O(nlog?n).

Variational Bayesian Image Restoration With a Product of Spatially Weighted Total Variation Image Priors

In this paper, a new image prior is introduced and used in image restoration. This prior is based on products of spatially weighted total variations (TV). These spatial weights provide this prior with the flexibility to better capture local image features than previous TV based priors. Bayesian inference is used for image restoration with this prior via the variational approximation. The proposed restoration algorithm is fully automatic in the sense that all necessary parameters are estimated from the data and is faster than previous similar algorithms. Numerical experiments are shown which demonstrate that image restoration based on this prior compares favorably with previous state-of-the-art restoration algorithms.

Variational Bayesian Image Restoration Based on a Product of $t$-Distributions Image Prior

Image priors based on products have been recognized to offer many advantages because they allow simultaneous enforcement of multiple constraints. However, they are inconvenient for Bayesian inference because it is hard to find their normalization constant in closed form. In this paper, a new Bayesian algorithm is proposed for the image restoration problem that bypasses this difficulty. An image prior is defined by imposing Student-t densities on the outputs of local convolutional filters. A variational methodology, with a constrained expectation step, is used to infer the restored image. Numerical experiments are shown that compare this methodology to previous ones and demonstrate its advantages.