Variational Bayesian sparse kernel-based blind image deconvolution with Student's-t priors

Spatial-temporal source term estimation using deep neural network prior and its application to Chernobyl wildfires

The source term of atmospheric emissions of hazardous materials is a crucial aspect of the analysis of unintended release. Motivated by wildfires of regions contaminated by radioactivity, the focus is placed on the case of airborne transmission of material from 5 dimensions: spatial location described by longitude and latitude in a given area with potentially many sources, time profiles, height above ground level, and the size of particles carrying the material. Since the atmospheric inverse problem is typically ill-posed and the number of measurements is usually too low to estimate the whole 5D tensor, some prior information is necessary. For the first time in this domain, a method based on deep image prior utilizing the structure of a deep neural network to regularize the inversion is proposed. The network is initialized randomly without the need to train it on any dataset first. In tandem with variational optimization, this approach not only introduces smoothness in the spatial estimate of the emissions but also reduces the number of unknowns by enforcing a prior covariance structure in the source term. The strengths of this method are demonstrated on the case of 137Cs emissions during the Chernobyl wildfires in 2020.

The Residual ISI for Which the Convolutional Noise Probability Density Function Associated with the Blind Adaptive Deconvolution Problem Turns Approximately Gaussian

In a blind adaptive deconvolution problem, the convolutional noise observed at the output of the deconvolution process, in addition to the required source signal, is-according to the literature-assumed to be a Gaussian process when the deconvolution process (the blind adaptive equalizer) is deep in its convergence state. Namely, when the convolutional noise sequence or, equivalently, the residual inter-symbol interference (ISI) is considered small. Up to now, no closed-form approximated expression is given for the residual ISI, where the Gaussian model can be used to describe the convolutional noise probability density function (pdf). In this paper, we use the Maximum Entropy density technique, Lagrange's Integral method, and quasi-moment truncation technique to obtain an approximated closed-form equation for the residual ISI where the Gaussian model can be used to approximately describe the convolutional noise pdf. We will show, based on this approximated closed-form equation for the residual ISI, that the Gaussian model can be used to approximately describe the convolutional noise pdf just before the equalizer has converged, even at a residual ISI level where the "eye diagram" is still very closed, namely, where the residual ISI can not be considered as small.

Heavy-Tailed Self-Similarity Modeling for Single Image Super Resolution

Self-similarity is a prominent characteristic of natural images that can play a major role when it comes to their denoising, restoration or compression. In this paper, we propose a novel probabilistic model that is based on the concept of image patch similarity and applied to the problem of Single Image Super Resolution. Based on this model, we derive a Variational Bayes algorithm, which super resolves low-resolution images, where the assumed distribution for the quantified similarity between two image patches is heavy-tailed. Moreover, we prove mathematically that the proposed algorithm is both an extended and superior version of the probabilistic Non-Local Means (NLM). Its prime advantage remains though, which is that it requires no training. A comparison of the proposed approach with state-of-the-art methods, using various quantitative metrics shows that it is almost on par, for images depicting rural themes and in terms of the Structural Similarity Index (SSIM) with the best performing methods that rely on trained deep learning models. On the other hand, it is clearly inferior to them, for urban themed images and in terms of all metrics, especially for the Mean-Squared-Error (MSE). In addition, qualitative evaluation of the proposed approach is performed using the Perceptual Index metric, which has been introduced to better mimic the human perception of the image quality. This evaluation favors our approach when compared to the best performing method that requires no training, even if they perform equally in qualitative terms, reinforcing the argument that MSE is not always an accurate metric for image quality.

A New Efficient Expression for the Conditional Expectation of the Blind Adaptive Deconvolution Problem Valid for the Entire Range ofSignal-to-Noise Ratio

In the literature, we can find several blind adaptive deconvolution algorithms based on closed-form approximated expressions for the conditional expectation (the expectation of the source input given the equalized or deconvolutional output), involving the maximum entropy density approximation technique. The main drawback of these algorithms is the heavy computational burden involved in calculating the expression for the conditional expectation. In addition, none of these techniques are applicable for signal-to-noise ratios lower than 7 dB. In this paper, I propose a new closed-form approximated expression for the conditional expectation based on a previously obtained expression where the equalized output probability density function is calculated via the approximated input probability density function which itself is approximated with the maximum entropy density approximation technique. This newly proposed expression has a reduced computational burden compared with the previously obtained expressions for the conditional expectation based on the maximum entropy approximation technique. The simulation results indicate that the newly proposed algorithm with the newly proposed Lagrange multipliers is suitable for signal-to-noise ratio values down to 0 dB and has an improved equalization performance from the residual inter-symbol-interference point of view compared to the previously obtained algorithms based on the conditional expectation obtained via the maximum entropy technique.

Blind Deconvolution With Model Discrepancies

Blind deconvolution is a strongly ill-posed problem comprising of simultaneous blur and image estimation. Recent advances in prior modeling and/or inference methodology led to methods that started to perform reasonably well in real cases. However, as we show here, they tend to fail if the convolution model is violated even in a small part of the image. Methods based on variational Bayesian inference play a prominent role. In this paper, we use this inference in combination with the same prior for noise, image, and blur that belongs to the family of independent non-identical Gaussian distributions, known as the automatic relevance determination prior. We identify several important properties of this prior useful in blind deconvolution, namely, enforcing non-negativity of the blur kernel, favoring sharp images over blurred ones, and most importantly, handling non-Gaussian noise, which, as we demonstrate, is common in real scenarios. The presented method handles discrepancies in the convolution model, and thus extends applicability of blind deconvolution to real scenarios, such as photos blurred by camera motion and incorrect focus.

A Regularization Approach to Blind Deblurring and Denoising of QR Barcodes

QR bar codes are prototypical images for which part of the image is a priori known (required patterns). Open source bar code readers, such as ZBar, are readily available. We exploit both these facts to provide and assess purely regularization-based methods for blind deblurring of QR bar codes in the presence of noise.

Multi-Observation Blind Deconvolution with an Adaptive Sparse Prior

This paper describes a robust algorithm for estimating a single latent sharp image given multiple blurry and/or noisy observations. The underlying multi-image blind deconvolution problem is solved by linking all of the observations together via a Bayesian-inspired penalty function, which couples the unknown latent image along with a separate blur kernel and noise variance associated with each observation, all of which are estimated jointly from the data. This coupled penalty function enjoys a number of desirable properties, including a mechanism whereby the relative-concavity or sparsity is adapted as a function of the intrinsic quality of each corrupted observation. In this way, higher quality observations may automatically contribute more to the final estimate than heavily degraded ones, while troublesome local minima can largely be avoided. The resulting algorithm, which requires no essential tuning parameters, can recover a sharp image from a set of observations containing potentially both blurry and noisy examples, without knowing a priori the degradation type of each observation. Experimental results on both synthetic and real-world test images clearly demonstrate the efficacy of the proposed method.

Estimation method of point spread function based on Kalman filter for accurately evaluating real optical properties of photonic crystal fibers

The evaluation accuracy of real optical properties of photonic crystal fibers (PCFs) is determined by the accurate extraction of air hole edges from microscope images of cross sections of practical PCFs. A novel estimation method of point spread function (PSF) based on Kalman filter is presented to rebuild the micrograph image of the PCF cross-section and thus evaluate real optical properties for practical PCFs. Through tests on both artificially degraded images and microscope images of cross sections of practical PCFs, we prove that the proposed method can achieve more accurate PSF estimation and lower PSF variance than the traditional Bayesian estimation method, and thus also reduce the defocus effect. With this method, we rebuild the microscope images of two kinds of commercial PCFs produced by Crystal Fiber and analyze the real optical properties of these PCFs. Numerical results are in accord with the product parameters.

Semi-blind sparse image reconstruction with application to MRFM

We propose a solution to the image deconvolution problem where the convolution kernel or point spread function (PSF) is assumed to be only partially known. Small perturbations generated from the model are exploited to produce a few principal components explaining the PSF uncertainty in a high-dimensional space. Unlike recent developments on blind deconvolution of natural images, we assume the image is sparse in the pixel basis, a natural sparsity arising in magnetic resonance force microscopy (MRFM). Our approach adopts a Bayesian Metropolis-within-Gibbs sampling framework. The performance of our Bayesian semi-blind algorithm for sparse images is superior to previously proposed semi-blind algorithms such as the alternating minimization algorithm and blind algorithms developed for natural images. We illustrate our myopic algorithm on real MRFM tobacco virus data.

Framelet-based blind motion deblurring from a single image

How to recover a clear image from a single motion-blurred image has long been a challenging open problem in digital imaging. In this paper, we focus on how to recover a motion-blurred image due to camera shake. A regularization-based approach is proposed to remove motion blurring from the image by regularizing the sparsity of both the original image and the motion-blur kernel under tight wavelet frame systems. Furthermore, an adapted version of the split Bregman method is proposed to efficiently solve the resulting minimization problem. The experiments on both synthesized images and real images show that our algorithm can effectively remove complex motion blurring from natural images without requiring any prior information of the motion-blur kernel.

Blind deconvolution using generalized cross-validation approach to regularization parameter estimation

In this paper, we propose and present an algorithm for total variation (TV)-based blind deconvolution. Both the unknown image and blur can be estimated within an alternating minimization framework. With the generalized cross-validation (GCV) method, the regularization parameters associated with the unknown image and blur can be updated in alternating minimization steps. Experimental results confirm that the performance of the proposed algorithm is better than variational Bayesian blind deconvolution algorithms with Student's-t priors or a total variation prior.

Joint NDT image restoration and segmentation using Gauss-Markov-Potts prior models and variational Bayesian computation

In this paper, we propose a method to simultaneously restore and to segment piecewise homogeneous images degraded by a known point spread function (PSF) and additive noise. For this purpose, we propose a family of nonhomogeneous Gauss-Markov fields with Potts region labels model for images to be used in a Bayesian estimation framework. The joint posterior law of all the unknowns (the unknown image, its segmentation (hidden variable) and all the hyperparameters) is approximated by a separable probability law via the variational Bayes technique. This approximation gives the possibility to obtain practically implemented joint restoration and segmentation algorithm. We will present some preliminary results and comparison with a MCMC Gibbs sampling based algorithm. We may note that the prior models proposed in this work are particularly appropriate for the images of the scenes or objects that are composed of a finite set of homogeneous materials. This is the case of many images obtained in nondestructive testing (NDT) applications.

Adaptive Langevin sampler for separation of t-distribution modelled astrophysical maps

We propose to model the image differentials of astrophysical source maps by Student's t-distribution and to use them in the Bayesian source separation method as priors. We introduce an efficient Markov Chain Monte Carlo (MCMC) sampling scheme to unmix the astrophysical sources and describe the derivation details. In this scheme, we use the Langevin stochastic equation for transitions, which enables parallel drawing of random samples from the posterior, and reduces the computation time significantly (by two orders of magnitude). In addition, Student's t-distribution parameters are updated throughout the iterations. The results on astrophysical source separation are assessed with two performance criteria defined in the pixel and the frequency domains.

A fast algorithm for robust mixtures in the presence of measurement errors

In experimental and observational sciences, detecting atypical, peculiar data from large sets of measurements has the potential of highlighting candidates of interesting new types of objects that deserve more detailed domain-specific followup study. However, measurement data is nearly never free of measurement errors. These errors can generate false outliers that are not truly interesting. Although many approaches exist for finding outliers, they have no means to tell to what extent the peculiarity is not simply due to measurement errors. To address this issue, we have developed a model-based approach to infer genuine outliers from multivariate data sets when measurement error information is available. This is based on a probabilistic mixture of hierarchical density models, in which parameter estimation is made feasible by a tree-structured variational expectation-maximization algorithm. Here, we further develop an algorithmic enhancement to address the scalability of this approach, in order to make it applicable to large data sets, via a K-dimensional-tree based partitioning of the variational posterior assignments. This creates a non-trivial tradeoff between a more detailed noise model to enhance the detection accuracy, and the coarsened posterior representation to obtain computational speedup. Hence, we conduct extensive experimental validation to study the accuracy/speed tradeoffs achievable in a variety of data conditions. We find that, at low-to-moderate error levels, a speedup factor that is at least linear in the number of data points can be achieved without significantly sacrificing the detection accuracy. The benefits of including measurement error information into the modeling is evident in all situations, and the gain roughly recovers the loss incurred by the speedup procedure in large error conditions. We analyze and discuss in detail the characteristics of our algorithm based on results obtained on appropriately designed synthetic data experiments, and we also demonstrate its working in a real application example.

Fast splitting algorithm for multiframe total variation blind video deconvolution

We consider the recovery of degraded videos without complete knowledge about the degradation. A spatially shift-invariant but temporally shift-varying video formation model is used. This leads to a simple multiframe degradation model that relates each original video frame with multiple observed frames and point spread functions (PSFs). We propose a variational method that simultaneously reconstructs each video frame and the associated PSFs from the corresponding observed frames. Total variation (TV) regularization is used on both the video frames and the PSFs to further reduce the ill-posedness and to better preserve edges. In order to make TV minimization practical for video sequences, we propose an efficient splitting method that generalizes some recent fast single-image TV minimization methods to the multiframe case. Both synthetic and real videos are used to show the performance of the proposed method.

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.