Blind Deconvolution Problem Research Articles

Image denoising and model-independent parameterization for IVIM MRI

Objective. To improve intravoxel incoherent motion imaging (IVIM) magnetic resonance Imaging quality using a new image denoising technique and model-independent parameterization of the signal versus b-value curve. Approach. IVIM images were acquired for 13 head-and-neck patients prior to radiotherapy. Post-radiotherapy scans were also acquired for five of these patients. Images were denoised prior to parameter fitting using neural blind deconvolution, a method of solving the ill-posed mathematical problem of blind deconvolution using neural networks. The signal decay curve was then quantified in terms of several area under the curve (AUC) parameters. Improvements in image quality were assessed using blind image quality metrics, total variation (TV), and the correlations between parameter changes in parotid glands with radiotherapy dose levels. The validity of blur kernel predictions was assessed by the testing the method's ability to recover artificial ‘pseudokernels’. AUC parameters were compared with monoexponential, biexponential, and triexponential model parameters in terms of their correlations with dose, contrast-to-noise (CNR) around parotid glands, and relative importance via principal component analysis. Main results. Image denoising improved blind image quality metrics, smoothed the signal versus b-value curve, and strengthened correlations between IVIM parameters and dose levels. Image TV was reduced and parameter CNRs generally increased following denoising. AUC parameters were more correlated with dose and had higher relative importance than exponential model parameters. Significance. IVIM parameters have high variability in the literature and perfusion-related parameters are difficult to interpret. Describing the signal versus b-value curve with model-independent parameters like the AUC and preprocessing images with denoising techniques could potentially benefit IVIM image parameterization in terms of reproducibility and functional utility.

Blind multi-Poissonian image deconvolution with sparse log-step gradient prior.

Blind image deconvolution plays a very important role in the fields such as astronomical observation and fluorescence microscopy imaging, in which the noise follows Poisson distribution. However, due to the ill-posedness, it is a very challenging task to reach a satisfactory result from a single blurred image especially when the power of the Poisson noise is at a high level. Therefore, in this paper, we try to achieve high-quality restoration results with multi-blurred images which are contaminated by Poisson noise. Firstly, we design a novel sparse log-step gradient prior which adopts a mixture of logarithm and step functions to regularize the image gradients and combine it with the Poisson distribution to formulate the blind multi-image deconvolution problem. Secondly, we incorporate the methods of variable splitting and Lagrange multiplier to convert the original problem into sub-problems, then we alternately solve them to achieve the estimation of all the blur kernels of corresponding blurred images. Besides, we also design a non-blind multi-image deconvolution algorithm which is based on the log-step gradient prior to reach the final restored image. Experimental results on both synthetic and real-world blurred images show that the proposed prior is very capable of suppressing negative artifacts caused by ill-posedness. The algorithm can achieve restored image of very high quality which is competitive with some state-of-the-art methods.

Neural blind deconvolution with Poisson data

Blind Deconvolution problem is a challenging task in several scientific imaging domains, such as Microscopy, Medicine and Astronomy. The Point Spread Function inducing the blur effect on the acquired image can be solely approximately known, or just a mathematical model may be available. Blind deconvolution aims to reconstruct the image when only the recorded data is available. In the last years, among the standard variational approaches, Deep Learning techniques have gained interest thanks to their impressive performances. The Deep Image Prior framework has been employed for solving this task, giving rise to the so-called neural blind deconvolution (NBD), where the unknown blur and image are estimated via two different neural networks. In this paper, we consider microscopy images, where the predominant noise is of Poisson type, hence signal-dependent: this leads to consider the generalized Kullback–Leibler as loss function and to couple it with regularization terms on both the blur operator and on the image. Furthermore, we propose to modify the standard NBD formulation problem, by including for the blur kernel an upper bound which depends on the optical instrument. A numerical solution is obtained by an alternating Proximal Gradient Descent-Ascent procedure, which results in the Double Deep Image Prior for Poisson noise algorithm. We evaluate the proposed strategy on both synthetic and real-world images, achieving promising results and proving that the correct choice of the loss and regularization functions strongly depends on the application at hand.

Blurry Facial-Image Deconvolution via Model-Guided Deep Neural Network Inspired From Edge Regularization

The blind facial-image deconvolution problem is ill-posed so that the estimation of latent image and blur kernel is hard to obtain without additional priors or heuristics. Although data-driven methods with deep neural networks have achieved excellent performance in this issue, most existing methods are designed like black boxes without transparency and interpretability. Thus, a model-guided explainable approach is proposed to address the above issues on blurry-facial images’ recovery, named the model-guided deep neural network (MG-DNN). To break the barrier between the network design and interpretability, we optimize blind image deconvolution based on salient edge regularization (BID-SER) and use it to guide the proposed MG-DNN. First, basic units for processing image and kernel features are designed with convolutional operators. Then, unfolding the optimization of BID-SER, the multiblocks concatenation is designed by several kernel networks and image networks with basic units. Finally, comprehensive experiments evaluated by several metrics demonstrate that the proposed MG-DNN performs favorably with existing state-of-the-art methods while processing on three different face data sets.

Unrolled Compressed Blind-Deconvolution

The problem of sparse multichannel blind deconvolution (S-MBD) arises frequently in many engineering applications such as radar/sonar/ultrasound imaging. To reduce its computational and implementation cost, we propose a compression method that enables blind recovery from much fewer measurements with respect to the full received signal in time. The proposed compression measures the signal through a filter followed by a subsampling, allowing for a significant reduction in implementation cost. We derive theoretical guarantees for the identifiability and recovery of a sparse filter from compressed measurements. Our results allow for the design of a wide class of compression filters. We, then, propose a data-driven unrolled learning framework to learn the compression filter and solve the S-MBD problem. The encoder is a recurrent inference network that maps compressed measurements into an estimate of sparse filters. We demonstrate that our unrolled learning method is more robust to choices of source shapes and has better recovery performance compared to optimization-based methods. Finally, in data-limited applications (fewshot learning), we highlight the superior generalization capability of unrolled learning compared to conventional deep learning.

$L_1$-Norm Regularization for Short-and-Sparse Blind Deconvolution: Point Source Separability and Region Selection

Blind deconvolution is about estimating both the convolution kernel and the latent signal from their convolution. Many blind deconvolution problems have a short-and-sparse (SaS) structure; i.e., the signal (or its gradient) is sparse and the kernel size is much smaller than the signal size. While $\ell_1$-norm relating regularizations have been widely used for solving SaS blind deconvolution problems, the so-called region/edge selection technique brings great empirical improvement to such $\ell_1$-norm relating regularizations in image deblurring. The essence of region/edge selection is during an alternative iterative scheme of SaS blind deconvolution: one estimates the kernel on an estimate of the latent image with well-separated image edges instead of the one with the least fitting error. In this paper, we first examine the validity and soundness of $\ell_1$-norm relating regularization in the setting of 1D SaS blind deconvolution. The analysis reveals the importance of the separation of nonzero signal entries toward the soundness of such a regularization. The studies laid out the foundation of region selection technique; i.e., during the iteration, an estimate of the latent image with well-separated edges is a better candidate for estimating the kernel than the one with the least fitting error. Based on the studies conducted in this paper, an alternating iterative scheme with region selection model is developed for SaS blind deconvolution, which is then applied to blind motion deblurring. The experiments show its effectiveness over many existing $\ell_1$-norm relating approaches.

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.

Blind inverse problems with isolated spikes

Abstract Assume that an unknown integral operator living in some known subspace is observed indirectly, by evaluating its action on a discrete measure containing a few isolated Dirac masses at an unknown location. Is this information enough to recover the impulse response location and the operator with a sub-pixel accuracy? We study this question and bring to light key geometrical quantities for exact and stable recovery. We also propose an in-depth study of the presence of additive white Gaussian noise. We illustrate the well-foundedness of this theory on the challenging optical imaging problem of blind deconvolution and blind deblurring with non-stationary operators.

Spatially adaptive blind deconvolution methods for optical coherence tomography

Optical coherence tomography (OCT) is a powerful noninvasive imaging technique for detecting microvascular abnormalities. Following optical imaging principles, an OCT image will be blurred in the out-of-focus domain. Digital deconvolution is a commonly used method for image deblurring. However, the accuracy of traditional digital deconvolution methods, e.g., the Richardson-Lucy method, depends on the prior knowledge of the point spread function (PSF), which varies with the imaging depth and is difficult to determine. In this paper, a spatially adaptive blind deconvolution framework is proposed for recovering clear OCT images from blurred images without a known PSF. First, a depth-dependent PSF is derived from the Gaussian beam model. Second, the blind deconvolution problem is formalized as a regularized energy minimization problem using the least squares method. Third, the clear image and imaging depth are simultaneously recovered from blurry images using an alternating optimization method. To improve the computational efficiency of the proposed method, an accelerated alternating optimization method is proposed based on the convolution theorem and Fourier transform. The proposed method is numerically implemented with various regularization terms, including total variation, Tikhonov, and l1 norm terms. The proposed method is used to deblur synthetic and experimental OCT images. The influence of the regularization term on the deblurring performance is discussed. The results show that the proposed method can accurately deblur OCT images. The proposed acceleration method can significantly improve the computational efficiency of blind demodulation methods.

Blind Image Deblurring Using Laplacian of Gaussian (LoG) Based Image Prior

Blind image deconvolution, a technique for obtaining restored image as well as the blur kernel from an inexact image. This research uses spatial characteristics to tackle the problem of blind image deconvolution. To work, the proposed method does not necessitate prior information about the blur kernel. Many applications, such as remote sensing, astronomy, and medical X-ray imaging, necessitate blind image deconvolution algorithms. This study used the maximum a posteriori (MAP) paradigm to create a new blind deblurring approach for removing blur from images. In beginning, we employed a Laplacian of Gaussian (LoG)-based image before regularising the gradients of an image. In the second phase, we used an operator known as the Iterative Shrinkage Thresholding Algorithm (ISTA) to cope with the non-convex challenge that develops during the entire deblurring procedure. Finally, we compared our method to several well-known methods in terms of quantitative and qualitative qualities, and we were able to determine which strategy was the most effective. Our findings show that the strategy we propose outperforms the others by a large margin.

Bayesian Sparse Blind Deconvolution Using MCMC Methods Based on Normal-Inverse-Gamma Prior

Bayesian estimation methods for sparse blind deconvolution problems conventionally employ Bernoulli-Gaussian (BG) prior for modeling sparse sequences and utilize Markov Chain Monte Carlo (MCMC) methods for the estimation of unknowns. However, the discrete nature of the BG model creates computational bottlenecks, preventing efficient exploration of the probability space even with the recently proposed enhanced sampler schemes. To address this issue, we propose an alternative MCMC method by modeling the sparse sequences using the Normal-Inverse-Gamma (NIG) prior. We derive effective Gibbs samplers for this prior and illustrate that the computational burden associated with the BG model can be eliminated by transferring the problem into a completely continuous-valued framework. In addition to sparsity, we also incorporate time and frequency domain constraints on the convolving sequences. We demonstrate the effectiveness of the proposed methods via extensive simulations and characterize computational gains relative to the existing methods that utilize BG modeling.

Improved Approach for the Maximum Entropy Deconvolution Problem.

The probability density function (pdf) valid for the Gaussian case is often applied for describing the convolutional noise pdf in the blind adaptive deconvolution problem, although it is known that it can be applied only at the latter stages of the deconvolution process, where the convolutional noise pdf tends to be approximately Gaussian. Recently, the deconvolutional noise pdf was approximated with the Edgeworth Expansion and with the Maximum Entropy density function for the 16 Quadrature Amplitude Modulation (QAM) input but no equalization performance improvement was seen for the hard channel case with the equalization algorithm based on the Maximum Entropy density function approach for the convolutional noise pdf compared with the original Maximum Entropy algorithm, while for the Edgeworth Expansion approximation technique, additional predefined parameters were needed in the algorithm. In this paper, the Generalized Gaussian density (GGD) function and the Edgeworth Expansion are applied for approximating the convolutional noise pdf for the 16 QAM input case, with no need for additional predefined parameters in the obtained equalization method. Simulation results indicate that improved equalization performance is obtained from the convergence time point of view of approximately 15,000 symbols for the hard channel case with our new proposed equalization method based on the new model for the convolutional noise pdf compared to the original Maximum Entropy algorithm. By convergence time, we mean the number of symbols required to reach a residual inter-symbol-interference (ISI) for which reliable decisions can be made on the equalized output sequence.

Redrawing-resampling rejection controlled sequential importance sampling

Monte Carlo computation has been widely applied in the field of dynamic systems. This paper focuses on the general framework in the implementation of sequential importance sampling by combining redrawing, resampling and rejection control simultaneously. The proposed algorithm is named as Redrawing Resampling Rejection Controlled Sequential Importance Sampling (RR-RC-SIS). It can reduce sampling computation and meanwhile maintain the diversity of random samples. Theoretical basis is given to prove that RR-RC-SIS has advantages in comparison with Rejection Controlled Sequential Importance Sampling. It also has practical value as illustrated in numerical simulation on blind deconvolution problem in digital communications.

Learning to do multiframe wavefront sensing unsupervised: Applications to blind deconvolution

Context. Observations from ground-based telescopes are severely perturbed by the presence of the Earth’s atmosphere. The use of adaptive optics techniques has allowed us to partly overcome this limitation. However, image-selection or post-facto image-reconstruction methods applied to bursts of short-exposure images are routinely needed to reach the diffraction limit. Deep learning has recently been proposed as an efficient way to accelerate these image reconstructions. Currently, these deep neural networks are trained with supervision, meaning that either standard deconvolution algorithms need to be applied a priori or complex simulations of the solar magneto-convection need to be carried out to generate the training sets. Aims. Our aim here is to propose a general unsupervised training scheme that allows multiframe blind deconvolution deep learning systems to be trained with observations only. The approach can be applied for the correction of point-like as well as extended objects. Methods. Leveraging the linear image formation theory and a probabilistic approach to the blind deconvolution problem produces a physically motivated loss function. Optimization of this loss function allows end-to-end training of a machine learning model composed of three neural networks. Results. As examples, we apply this procedure to the deconvolution of stellar data from the FastCam instrument and to solar extended data from the Swedish Solar Telescope. The analysis demonstrates that the proposed neural model can be successfully trained without supervision using observations only. It provides estimations of the instantaneous wavefronts, from which a corrected image can be found using standard deconvolution techniques. The network model is roughly three orders of magnitude faster than applying standard deconvolution based on optimization and shows potential to be used on real-time at the telescope.

An ADMM-LAP method for total variation myopic deconvolution of adaptive optics retinal images

Adaptive optics corrected flood imaging of the retina is a popular technique for studying the retinal structure and function in the living eye. However, the raw retinal images are usually of poor contrast and the interpretation of such images requires image deconvolution. Different from standard deconvolution problems where the point spread function (PSF) is completely known, the PSF in these retinal imaging problems is only partially known which leads to the more complicated myopic (mildly blind) deconvolution problem. In this paper, we propose an efficient numerical scheme for solving this myopic deconvolution problem with total variational (TV) regularization. First, we apply the alternating direction method of multipliers (ADMM) to tackle the TV regularizer. Specifically, we reformulate the TV problem as an equivalent equality constrained problem where the objective function is separable, and then minimize the augmented Lagrangian function by alternating between two (separated) blocks of unknowns to obtain the solution. Due to the structure of the retinal images, the subproblems with respect to the fidelity term appearing within each ADMM iteration are tightly coupled and a variation of the linearize and project method is designed to solve these subproblems efficiently. The proposed method is called the ADMM-LAP method. Theoretically, we establish the subsequence convergence of the ADMM-LAP method to a stationary point. Both the theoretical complexity analysis and numerical results are provided to demonstrate the efficiency of the ADMM-LAP method.

A class of multi-modulus blind deconvolution algorithms using hyperbolic and Givens rotations for MIMO systems

This paper targets the blind deconvolution problem for multiple-input multiple-output communication systems, using small and moderate constellation’s size signals, i.e. PSK and QAM. We introduce four different blind deconvolution algorithms based on four different techniques. These algorithms come as a natural extension of the successful work done by Shah et al in 2018 for blind source separation (BSS). The first two methods are considered as two-step based methods, where the first one performs the BSS for the spatio-temporal system followed by a pairing and sorting phase. While the second is accomplished by performing a cascaded linear equalization, using one of the existing subspace-methods, followed by the BSS routine. The third method is based on the minimization of a hybrid cost function, and the last one is a deflation-based method. These solutions summarize the main possible paths that can be followed to extend any of the existing instantaneous de-mixing algorithms. Experimental results are provided to compare and highlight the unique characteristics of each of the four different methods.

Composite optimization for robust rank one bilinear sensing

Abstract We consider the task of recovering a pair of vectors from a set of rank one bilinear measurements, possibly corrupted by noise. Most notably, the problem of robust blind deconvolution can be modeled in this way. We consider a natural nonsmooth formulation of the rank one bilinear sensing problem and show that its moduli of weak convexity, sharpness and Lipschitz continuity are all dimension independent, under favorable statistical assumptions. This phenomenon persists even when up to half of the measurements are corrupted by noise. Consequently, standard algorithms, such as the subgradient and prox-linear methods, converge at a rapid dimension-independent rate when initialized within a constant relative error of the solution. We complete the paper with a new initialization strategy, complementing the local search algorithms. The initialization procedure is both provably efficient and robust to outlying measurements. Numerical experiments, on both simulated and real data, illustrate the developed theory and methods.

Phase retrieval of low-rank matrices by anchored regression

Abstract We study the low-rank phase retrieval problem, where our goal is to recover a $d_1\times d_2$ low-rank matrix from a series of phaseless linear measurements. This is a fourth-order inverse problem, as we are trying to recover factors of a matrix that have been observed, indirectly, through some quadratic measurements. We propose a solution to this problem using the recently introduced technique of anchored regression. This approach uses two different types of convex relaxations: we replace the quadratic equality constraints for the phaseless measurements by a search over a polytope and enforce the rank constraint through nuclear norm regularization. The result is a convex program in the space of $d_1 \times d_2$ matrices. We analyze two specific scenarios. In the first, the target matrix is rank-$1$, and the observations are structured to correspond to a phaseless blind deconvolution. In the second, the target matrix has general rank, and we observe the magnitudes of the inner products against a series of independent Gaussian random matrices. In each of these problems, we show that anchored regression returns an accurate estimate from a near-optimal number of measurements given that we have access to an anchor matrix of sufficient quality. We also show how to create such an anchor in the phaseless blind deconvolution problem from an optimal number of measurements and present a partial result in this direction for the general rank problem.

Secant variable projection method for solving nonnegative separable least squares problems

The variable projection method is a classical and efficient method for solving separable nonlinear least squares (SNLLS) problems. However, it is hard to handle the constrained SNLLS problems since the explicit form of the Jacobian matrix is required in each iteration. In this paper, we propose a secant variable projection (SVP) method, which employs a rank-one update to estimate the Jacobian matrices. The main advantages of our method are efficiency and ease of applicability to constrained SNLLS problems. The local convergence of our SVP method is also analyzed. Finally, some data fitting and image processing problems are solved to compare the performance of our proposed method with the classical variable projection method. Numerical results illustrate the efficiency and stability of our proposed SVP method in solving the SNLLS problems arising from the blind deconvolution problems.

Exact Recovery of Multichannel Sparse Blind Deconvolution via Gradient Descent

We study the multichannel sparse blind deconvolution (MCS-BD) problem, whose task is to simultaneously recover a kernel $a$ and multiple sparse inputs $\{x_i\}_{i=1}^p$ from their circulant convolu...