Bayesian Restoration Research Articles

Unsupervised segmentation of images using bi-dimensional pairwise Markov chains model

<p>The pair-wise Markov chain (PMC) model serves as an extension to the hidden Markov chain (HMC) model and has been widely used in unsupervised restoration tasks associated with reconstructing the hidden data. In fact, the PMC model can treat fairly complicated situations for which application of Bayesian restoration estimators such as maximum <italic>A Posteriori</italic> (MAP), or maximal <italic>Posterior</italic> mode (MPM) remains possible. The major novelty in this work is to construct a PMC model with observational data in two dimensions, and subsequently adapt the estimation algorithms, as well as, image restoration methods for that context. Often, the transformation of an image from a two-dimensional format to a one-dimensional sequence occurs via Hilbert-Peano scan (HPS), whereas in the proposed model, the second component of the observed process takes over this role to exceed the situation of pixel missing information after transformation for a to be segmented image. To reconstruct the hidden process, we used the MPM decision criterion after estimating the model's parameters with two algorithms: Stochastic expectation maximization (SEM) and iterative conditional estimation (ICE). In this study, experimental, numerical, and visual results are shown to demonstrate the superiority of the proposed model over the classical PMC for unsupervised restorations.</p>

Bayesian Restoration of Audio Degraded by Low-Frequency Pulses Modeled via Gaussian Process

A common defect found when reproducing old vinyl and gramophone recordings with mechanical devices are the long pulses with significant low-frequency content caused by the interaction of the arm-needle system with deep scratches or even breakages on the media surface. Previous approaches to their suppression on digital counterparts of the recordings depend on a prior estimation of the pulse location, usually performed via heuristic methods. This paper proposes a novel Bayesian approach capable of jointly estimating the pulse location; interpolating the almost annihilated signal underlying the strong discontinuity that initiates the pulse; and also estimating the long pulse tail by a simple Gaussian Process, allowing its suppression from the corrupted signal. The posterior distribution for the model parameters as well for the pulse is explored via Markov-Chain Monte Carlo (MCMC) algorithms. Controlled experiments indicate that the proposed method, while requiring significantly less user intervention, achieves perceptual results similar to those of previous approaches and performs well when dealing with naturally degraded signals.

Single-image Bayesian Restoration and Multi-image Super-resolution Restoration for B-mode Ultrasound Using an Accurate System Model Involving Correlated Nature of the Speckle Noise.

B-mode ultrasound is an essential part of radiological examinations due to its low cost, safety, and portability, but has the drawbacks of the speckle noise and output of most systems is two-dimensional (2D) cross sections. Image restoration techniques, using mathematical models for image degradation and noise, can be used to boost resolution (deconvolution) as well as to reduce the speckle. In this study, new single-image Bayesian restoration (BR) and multi-image super-resolution restoration (BSRR) methods are proposed for in-plane B-mode ultrasound images. The spatially correlated nature of the speckle was modeled, allowing for examination of two different models for BR and BSRR for uncorrelated Gaussian (BR-UG, BSRR-UG) and correlated Gaussian (BR-CG, BSRR-CG). The performances of these models were compared with common image restoration methods (Wiener filter, bilateral filtering, and anisotropic diffusion). Well-recognized metrics (peak signal-to-noise ratio, contrast-to-noise ratio, and normalized information density) were used for algorithm free-parameter estimation and objective evaluations. The methods were tested using superficial tissue (2D scan data collected from volunteers, tissue-mimicking resolutions, and breast phantoms). Improvement in image quality was assessed by experts using visual grading analysis. In general, BSRR-CG performed better than all other methods. A potential downside of BSRR-CG is increased computation time, which can be addressed by the use of high-performance graphics processing units (GPUs).

Bayesian estimation of Weibull mixture in heavily censored data setting

In reliability or warranty analysis, engineers must often deal with lifetimes data that are non-homogeneous. Most of the time, this variability is unobserved but has to be taken into account for reliability or warranty cost analysis. A further problem is that in reliability analysis, data are heavily censored which makes estimations more difficult. The two-component Weibull mixture is then a highly relevant model to capture heterogeneity for a large majority of operating lifetimes. Unfortunately, the performance of classical estimation methods (maximum of likelihood via EM, Bayes approach via MCMC) is jeopardized due to the high number of parameters and the heavy censoring. In order to overcome the problem of heavy censoring for Weibull mixture parameters estimation, this research proposes a Bayesian bootstrap method, called Bayesian Restoration Maximization. The key is to provide a sampling from the posterior distribution. Thanks to an importance sampling technique, this sample focuses on the posterior mean. Prior distributions elicitation and sensibility analysis are discussed. Simulations results showed that, for heavily censored data, the BRM method outperforms the EM and S-EM algorithms in terms of estimates accuracy. On the other hand, it is a non-iterative method which therefore provides very short computation times. In addition, the BRM method does not suffer from the problem of label switching from Bayesian sampling algorithms such as MCMC methods. Finally, two real data sets are analyzed to illustrate the application of the method.

Imaging tissues and cells beyond the diffraction limit with structured illumination microscopy and Bayesian image reconstruction

BackgroundStructured illumination microscopy (SIM) is a family of methods in optical fluorescence microscopy that can achieve both optical sectioning and super-resolution effects. SIM is a valuable method for high-resolution imaging of fixed cells or tissues labeled with conventional fluorophores, as well as for imaging the dynamics of live cells expressing fluorescent protein constructs. In SIM, one acquires a set of images with shifting illumination patterns. This set of images is subsequently treated with image analysis algorithms to produce an image with reduced out-of-focus light (optical sectioning) and/or with improved resolution (super-resolution).FindingsFive complete, freely available SIM datasets are presented including raw and analyzed data. We report methods for image acquisition and analysis using open-source software along with examples of the resulting images when processed with different methods. We processed the data using established optical sectioning SIM and super-resolution SIM methods and with newer Bayesian restoration approaches that we are developing.ConclusionsVarious methods for SIM data acquisition and processing are actively being developed, but complete raw data from SIM experiments are not typically published. Publically available, high-quality raw data with examples of processed results will aid researchers when developing new methods in SIM. Biologists will also find interest in the high-resolution images of animal tissues and cells we acquired. All of the data were processed with SIMToolbox, an open-source and freely available software solution for SIM.

Variational Bayesian blind restoration reconstruction based on shear wave transform for low-dose medical CT image

A variational Bayesian blind restoration reconstruction based on shear wave transform for low-dose medical computed tomography (CT) image is proposed. The proposed algorithm eliminates the effects of the point spread function in the process of low-dose medical CT image reconstruction and improves the reconstructed image quality. The shear wave transform is used to sparsely represent the CT image, which can speed up the efficiency of image processing. In the Bayesian framework, a posteriori probability objective function with unknown parameters is constructed. These unknown parameters include the shear wave coefficients, the point spread function, and the hyper parameters. It specified a Laplacian distribution model for the prior probability distribution of the shear wave coefficients. The autoregressive model is used as the prior model of the point spread function. All of hyper parameters follow the gamma distribution. The variational Bayesian method is used to estimate all of unknown parameters and solve the above posteriori probability objective function. These generalized parameter estimators are used to realize the low-dose CT image blind restoration reconstruction. Computer simulation results indicate that a good performance-reconstructed image can be obtained and some metrics such as peak signal-to-noise ratio (PSNR), universal image quality index (UIQI), structural similarity index metric (SSIM), and sum of square differences error (SSDE) are improved.

Modeling and Pre-Treatment of Photon-Starved CT Data for Iterative Reconstruction

An increasing number of X-ray CT procedures are being conducted with drastically reduced dosage, due at least in part to advances in statistical reconstruction methods that can deal more effectively with noise than can traditional techniques. As data become photon-limited, more detailed models are necessary to deal with count rates that drop to the levels of system electronic noise. We present two options for sinogram pre-treatment that can improve the performance of photon-starved measurements, with the intent of following with model-based image reconstruction. Both the local linear minimum mean-squared error (LLMMSE) filter and pointwise Bayesian restoration (PBR) show promise in extracting useful, quantitative information from very low-count data by reducing local bias while maintaining the lower noise variance of statistical methods. Results from clinical data demonstrate the potential of both techniques.

Phasic Triplet Markov Chains.

Hidden Markov chains have been shown to be inadequate for data modeling under some complex conditions. In this work, we address the problem of statistical modeling of phenomena involving two heterogeneous system states. Such phenomena may arise in biology or communications, among other fields. Namely, we consider that a sequence of meaningful words is to be searched within a whole observation that also contains arbitrary one-by-one symbols. Moreover, a word may be interrupted at some site to be carried on later. Applying plain hidden Markov chains to such data, while ignoring their specificity, yields unsatisfactory results. The Phasic triplet Markov chain, proposed in this paper, overcomes this difficulty by means of an auxiliary underlying process in accordance with the triplet Markov chains theory. Related Bayesian restoration techniques and parameters estimation procedures according to the new model are then described. Finally, to assess the performance of the proposed model against the conventional hidden Markov chain model, experiments are conducted on synthetic and real data.

Sensibility of Bayesian inference methods for reliability prediction of ageing systems, case of Diesel locomotives

This paper addresses the problem of reliability analysis of in-service identical systems when a limited number of lifetime data is available compared to censored ones. Lifetime (resp. censored) data characterise the life of failed (resp. non-failed) systems in the sample. Because, censored data induce biassed estimators of reliability model parameters, a methodology approach is proposed to overcome this inconvenience and improve the accuracy of the parameter estimation based on Bayesian inference methods. These methods combine, in an effective way, system’s life data and expert opinions learned from failure diagnosis of similar systems. Three Bayesian inference methods are considered: Classical Bayesian, Extended Bayesian and Bayesian Restoration Maximisation methods. Given a sample of lifetime data, simulated according to prior opinions of maintenance expert, a sensibility analysis of each Bayesian method is performed. Reliability analysis of critical subsystems of Diesel locomotives is established under the proposed methodology approach. The relevance of each Bayesian inference methods with respect to collected reliability data of critical subsystems and expert opinions is discussed.

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.

Bayesian Restoration of Audio Signals Degraded by Impulsive Noise Modeled as Individual Pulses

Impulsive noise, also known as clicks, is a very common type of distortion in old gramophone recordings. Existing methods (both heuristic and statistical) for removal of this type of defect usually do not exploit its underlying physical generation. This work proposes a model in which each click is individually modeled, which is more physically meaningful. A Bayesian method based on the reversible-jump Metropolis-Hastings algorithm for joint detection and removal of impulsive noise in audio signals is devised. Simulations with artificial and real audio signals as well as comparisons with competing approaches are presented to illustrate and validate the proposed method.

A Bayesian Restoration Approach for Hyperspectral Images

In this paper, a Bayesian restoration technique for multiple observations of hyperspectral (HS) images is presented. As a prototype problem, we assume that a low-spatial-resolution HS observation and a high-spatial-resolution multispectral (MS) observation of the same scene are available. The proposed approach applies a restoration on the HS image and a joint fusion with the MS image, accounting for the joint statistics with the MS image. The restoration is based on an expectation-maximization algorithm, which applies a deblurring step and a denoising step iteratively. The Bayesian framework allows to include spatial information from the MS image. To keep the calculation feasible, a practical implementation scheme is presented. The proposed approach is validated by simulation experiments for general HS image restoration and for the specific case of pansharpening. The experimental results of the proposed approach are compared with pure fusion and deconvolution results for performance evaluation.

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.

Exact Bayesian Prediction in a Class of Markov-switching Models

Jump-Markov state-space systems (JMSS) are widely used in statistical signal processing. However as is well known Bayesian restoration in JMSS is an NP-hard problem, so in practice all inference algorithms need to resort to some approximations. In this paper we focus on the computation of the conditional expectation of the hidden variable of interest given the available observations, which is optimal from the Bayesian quadratic risk viewpoint. We show that in some stochastic systems, namely the Partially Pairwise Markov-switching Chains (PPMSC) and Trees (PPMST), no approximation scheme is actually needed since the conditional expectation of interest (be it either in a filtering or prediction problem) can be computed exactly and in a number of operations linear in the number of observations.

Effective Image Restorations Using a Novel Spatial Adaptive Prior

Bayesian or Maximum a posteriori (MAP) approaches can effectively overcome the ill-posed problems of image restoration or deconvolution through incorporating a priori image information. Many restoration methods, such as nonquadratic prior Bayesian restoration and total variation regularization, have been proposed with edge-preserving and noise-removing properties. However, these methods are often inefficient in restoring continuous variation region and suppressing block artifacts. To handle this, this paper proposes a Bayesian restoration approach with a novel spatial adaptive (SA) prior. Through selectively and adaptively incorporating the nonlocal image information into the SA prior model, the proposed method effectively suppress the negative disturbance from irrelevant neighbor pixels, and utilizes the positive regularization from the relevant ones. A twostep restoration algorithm for the proposed approach is also given. Comparative experimentation and analysis demonstrate that, bearing high-quality edge-preserving and noise-removing properties, the proposed restoration also has good deblocking property.

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).

Unsupervised segmentation of triplet Markov chains hidden with long-memory noise

The hidden Markov chain (HMC) model is a couple of random sequences ( X, Y), in which X is an unobservable Markov chain, and Y is its observable noisy version. Classically, the distribution p( y| x) is simple enough to ensure the Markovianity of p( x| y), that enables one to use different Bayesian restoration techniques. HMC model has recently been extended to “pairwise Markov chain” (PMC) model, in which one directly assumes the Markovianity of the pair Z=( X,Y), and which still enables one to recover X from Y. Finally, PMC has been extended to “triplet Markov chain” (TMC) model, which is obtained by adding a third chain U and considering the Markovianity of the triplet T=( X, U, Y). When U is not too complex, X can still be recovered from Y. Then U can model different situations, like non-stationarity or semi-Markovianity of ( X, Y). Otherwise, PMC and TMC have been extended to pairwise “partially” Markov chains (PPMC) and triplet “partially” Markov chains (TPMC), respectively. In a PPMC Z=( X, Y) the distribution p( x| y) is a Markov distribution, but p( y| x) may not be and, similarly, in a TPMC T=( X, U, Y) the distribution p( x, u| y) is a Markov distribution, but p( y| x, u) may not be. However, both PPMC and TPMC can enable one to recover X from Y, and TPMC include different long-memory noises. The aim of this paper is to show how a particular Gaussian TPMC can be used to segment a discrete signal hidden with long-memory noise. An original parameter estimation method, based on “Iterative Conditional Estimation” (ICE) principle, is proposed and some experiments concerned with unsupervised segmentation are provided. The particular unsupervised segmentation method used in experiments can also be seen as identification of different stationarities in fractional Brownian noise, which is widely used in different problems in telecommunications, economics, finance, or hydrology.

Bayesian Restoration Using a New Nonstationary Edge-Preserving Image Prior

In this paper, we propose a class of image restoration algorithms based on the Bayesian approach and a new hierarchical spatially adaptive image prior. The proposed prior has the following two desirable features. First, it models the local image discontinuities in different directions with a model which is continuous valued. Thus, it preserves edges and generalizes the on/off (binary) line process idea used in previous image priors within the context of Markov random fields (MRFs). Second, it is Gaussian in nature and provides estimates that are easy to compute. Using this new hierarchical prior, two restoration algorithms are derived. The first is based on the maximum a posteriori principle and the second on the Bayesian methodology. Numerical experiments are presented that compare the proposed algorithms among themselves and with previous stationary and non stationary MRF-based with line process algorithms. These experiments demonstrate the advantages of the proposed prior.

Multisensor triplet Markov chains and theory of evidence

Hidden Markov chains (HMC) are widely applied in various problems occurring in different areas like Biosciences, Climatology, Communications, Ecology, Econometrics and Finances, Image or Signal processing. In such models, the hidden process of interest X is a Markov chain, which must be estimated from an observable Y, interpretable as being a noisy version of X. The success of HMC is mainly due to the fact that the conditional probability distribution of the hidden process with respect to the observed process remains Markov, which makes possible different processing strategies such as Bayesian restoration. HMC have been recently generalized to “Pairwise” Markov chains (PMC) and “Triplet” Markov chains (TMC), which offer similar processing advantages and superior modeling capabilities. In PMC, one directly assumes the Markovianity of the pair ( X, Y) and in TMC, the distribution of the pair ( X, Y) is the marginal distribution of a Markov process ( X, U, Y), where U is an auxiliary process, possibly contrived. Otherwise, the Dempster–Shafer fusion can offer interesting extensions of the calculation of the “a posteriori” distribution of the hidden data. The aim of this paper is to present different possibilities of using the Dempster–Shafer fusion in the context of different multisensor Markov models. We show that the posterior distribution remains calculable in different general situations and present some examples of their applications in remote sensing area.

On a 3D-matrix representation of the tertiary structure of a protein

The main object of the present paper is a preliminary discussion of a way to describe any three-dimensional structure of a folded protein by means of a suitable three-dimensional matrix, as well as of its possible use for the forecast of the natural spatial structure of a protein of which only the amino acid sequence is known. The contents of this paper are confined to the introduction of the language and to its application to some particularly simple (and short) proteins as samples of this kind of description and as a start-point to draw the mathematical tools to be added in this context to express the geometric information in such a way as to apply the methods of inference of computational statistics. As a matter of fact, we hope to be able to show in future papers that such representation, when suitably completed by the introduction of a suitable discrete version of the notion of “deformation field” and of other geometric notions, allows the reconstruction of the folded configuration of a protein by a suitable “abstract” version of some recent and powerful statistical methods usually applied to image recognition, in turn suitably extended from the two-dimensional to the three-dimensional case (cf. [D. Geman, S. Geman, Stochastic relaxation, Gibbs distribution and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intell. 6 (1984)]).