Abstract: We present a method to extract a video sequence from a single motion-blurred image. Motion-blurred images are the result of an averaging process, where instant frames are accumulated over time during the exposure of the sensor. Unfortunately, reversing this process is nontrivial. Firstly, averaging destroys the temporal ordering of the frames. Secondly, the recovery of a single frame is a blind deconvolution task, which is highly ill-posed. We present a deep learning scheme that gradually reconstructs a temporal ordering by sequentially extracting pairs of frames. Our main contribution is to introduce loss functions invariant to the temporal order. This lets a neural network choose during training what frame to output among the possible combinations. We also address the ill-posedness of deblurring by designing a network with a large receptive field and implemented via resampling to achieve a higher computational efficiency. Our proposed method can successfully retrieve sharp image sequences from a single motion blurred image and can generalize well on synthetic and real datasets captured with different cameras.

Abstract: We propose a non-iterative image deconvolution algorithm for data corrupted by Poisson or mixed Poisson-Gaussian noise. Many applications involve such a problem, ranging from astronomical to biological imaging. We parameterize the deconvolution process as a linear combination of elementary functions, termed as linear expansion of thresholds. This parameterization is then optimized by minimizing a robust estimate of the true mean squared error, the Poisson unbiased risk estimate. Each elementary function consists of a Wiener filtering followed by a pointwise thresholding of undecimated Haar wavelet coefficients. In contrast to existing approaches, the proposed algorithm merely amounts to solving a linear system of equations, which has a fast and exact solution. Simulation experiments over different types of convolution kernels and various noise levels indicate that the proposed method outperforms the state-of-the-art techniques, in terms of both restoration quality and computational complexity. Finally, we present some results on real confocal fluorescence microscopy images and demonstrate the potential applicability of the proposed method for improving the quality of these images.

Abstract: We consider simultaneous blind deconvolution of $r$ source signals from their noisy superposition, a problem also referred to blind demixing and deconvolution . This signal processing problem occurs in the context of the Internet of Things where a massive number of sensors sporadically communicate only short messages over unknown channels. We show that robust recovery of message and channel vectors can be achieved via convex optimization when random linear encoding using i.i.d. complex Gaussian matrices is used at the devices and the number of required measurements at the receiver scales with the degrees of freedom of the overall estimation problem. Since the scaling is linear in $r$ our result significantly improves over recent works.

Abstract: Hyperspectral imaging is a cutting-edge type of remote sensing used for mapping vegetation properties, rock minerals and other materials. A major drawback of hyperspectral imaging devices is their intrinsic low spatial resolution. In this paper, we propose a method for increasing the spatial resolution of a hyperspectral image by fusing it with an image of higher spatial resolution that was obtained with a different imaging modality. This is accomplished by solving a variational problem in which the regularization functional is the directional total variation. To accommodate for possible mis-registrations between the two images, we consider a non-convex blind super-resolution problem where both a fused image and the corresponding convolution kernel are estimated. Using this approach, our model can realign the given images if needed. Our experimental results indicate that the non-convexity is negligible in practice and that reliable solutions can be computed using a variety of different optimization algorithms. Numerical results on real remote sensing data from plant sciences and urban monitoring show the potential of the proposed method and suggests that it is robust with respect to the regularization parameters, mis-registration and the shape of the kernel.

Abstract: Most existing non-blind restoration methods are based on the assumption that a precise degradation model is known. As the degradation process can only be partially known or inaccurately modeled, images may not be well restored. Rain streak removal and image deconvolution with inaccurate blur kernels are two representative examples of such tasks. For rain streak removal, although an input image can be decomposed into a scene layer and a rain streak layer, there exists no explicit formulation for modeling rain streaks and the composition with scene layer. For blind deconvolution, as estimation error of blur kernel is usually introduced, the subsequent non-blind deconvolution process does not restore the latent image well. In this paper, we propose a principled algorithm within the maximum a posterior framework to tackle image restoration with a partially known or inaccurate degradation model. Specifically, the residual caused by a partially known or inaccurate degradation model is spatially dependent and complexly distributed. With a training set of degraded and ground-truth image pairs, we parameterize and learn the fidelity term for a degradation model in a task-driven manner. Furthermore, the regularization term can also be learned along with the fidelity term, thereby forming a simultaneous fidelity and regularization learning model. Extensive experimental results demonstrate the effectiveness of the proposed model for image deconvolution with inaccurate blur kernels, deconvolution with multiple degradations and rain streak removal.

Abstract: An FTIR spectrometer often suffers from common problems of band overlap and Poisson noises. In this paper, we show that the issue of infrared (IR) spectrum degradation can be considered as a maximum a posterior (MAP) problem and solved by minimized a cost function that includes a likelihood term and two prior terms. In the MAP framework, the likelihood probability density function (PDF) is constructed based on the observed Poisson noise model. A fitted distribution of curvelet transform coefficient is used as spectral prior PDF, and the instrument response function (IRF) prior is described based on a Gauss-Markov function. Moreover, the split Bregman iteration method is employed to solve the resulting minimization problem, which highly reduces the computational load. As a result, the Poisson noises are perfectly removed, while the spectral structure information is well preserved. The novelty of the proposed method lies in its ability to estimate the IRF and latent spectrum in a joint framework, thus eliminating the degradation effects to a large extent. The reconstructed IR spectrum is more convenient for extracting the spectral feature and interpreting the unknown chemical or biological materials.

Abstract: We introduce a family of novel approaches to single-image blind deconvolution, i.e., the problem of recovering a sharp image and a blur kernel from a single blurry input. This problem is highly ill-posed, because infinite (image, blur) pairs produce the same blurry image. Most research effort has been devoted to the design of priors for natural images and blur kernels, which can drastically prune the set of possible solutions. Unfortunately, these priors are usually not sufficient to favor the sharp solution. In this paper we address this issue by looking at a much less studied aspect: the relative scale ambiguity between the sharp image and the blur. Most prior work eliminates this ambiguity by fixing the \(L^1\) norm of the blur kernel. In principle, however, this choice is arbitrary. We show that a careful design of the blur normalization yields a blind deconvolution formulation with remarkable accuracy and robustness to noise. Specifically, we show that using the Frobenius norm to fix the scale ambiguity enables convex image priors, such as the total variation, to achieve state-of-the-art results on both synthetic and real datasets.

Abstract: In this paper, we propose a Riemannian steepest descent method for solving a blind deconvolution problem, which is to recover two unknown signals from their circular convolution. We assume that the...

Abstract: We consider the multichannel blind deconvolution problem where we observe the output of multiple channels that are all excited with the same unknown input. From these observations, we wish to estimate the impulse responses of each of the channels. We show that this problem is well-posed if the channels follow a bilinear model where the ensemble of channel responses is modeled as lying in a low-dimensional subspace but with each channel modulated by an independent gain. Under this model, we show how the channel estimates can be found by minimizing a quadratic function over a non-convex set. We analyze two methods for solving this non-convex program, and provide performance guarantees for each. The first is a method of alternating eigenvectors that breaks the program down into a series of eigenvalue problems. The second is a truncated power iteration, which can roughly be interpreted as a method for finding the largest eigenvector of a symmetric matrix with the additional constraint that it adheres to our bilinear model. As with most non-convex optimization algorithms, the performance of both of these algorithms is highly dependent on having a good starting point. We show how such a starting point can be constructed from the channel measurements. Our performance guarantees are non-asymptotic, and provide a sufficient condition on the number of samples observed per channel in order to guarantee channel estimates of certain accuracy. Our analysis uses a model with a “generic” subspace that is drawn at random, and we show the performance bounds hold with high probability. Mathematically, the key estimates are derived by quantifying how well the eigenvectors of certain random matrices approximate the eigenvectors of their mean. We also present a series of numerical results demonstrating that the empirical performance is consistent with the presented theory.

Abstract: The quality of images of the Sun obtained from the ground are severely limited by the perturbing effect of the turbulent Earth's atmosphere. The post-facto correction of the images to compensate for the presence of the atmosphere require the combination of high-order adaptive optics techniques, fast measurements to freeze the turbulent atmosphere and very time consuming blind deconvolution algorithms. Under mild seeing conditions, blind deconvolution algorithms can produce images of astonishing quality. They can be very competitive with those obtained from space, with the huge advantage of the flexibility of the instrumentation thanks to the direct access to the telescope. In this contribution we leverage deep learning techniques to significantly accelerate the blind deconvolution process and produce corrected images at a peak rate of ~100 images per second. We present two different architectures that produce excellent image corrections with noise suppression while maintaining the photometric properties of the images. As a consequence, polarimetric signals can be obtained with standard polarimetric modulation without any significant artifact. With the expected improvements in computer hardware and algorithms, we anticipate that on-site real-time correction of solar images will be possible in the near future.

Abstract: We consider the problem of demixing a sequence of source signals from the sum of noisy bilinear measurements. It is a generalized mathematical model for blind demixing with blind deconvolution, which is prevalent across the areas of dictionary learning, image processing, and communications. However, state-of-the-art convex methods for blind demixing via semidefinite programming are computationally infeasible for large-scale problems. Although the existing nonconvex algorithms are able to address the scaling issue, they normally require proper regularization to establish optimality guarantees. The additional regularization yields tedious algorithmic parameters and pessimistic convergence rates with conservative step sizes. To address the limitations of exiting methods, we thus develop a provable nonconvex demixing procedure via Wirtinger flow, much like vanilla gradient descent, to harness the benefits of regularization-free fast convergence rate with aggressive step size and computational optimality guarantees. This is achieved by exploiting the benign geometry of the blind demixing problem, thereby revealing that Wirtinger flow enforces the regularization-free iterates in the region of strong convexity and qualified level of smoothness, where the step size can be chosen aggressively.

Abstract: Multichannel blind deconvolution is the problem of recovering an unknown signal $f$ and multiple unknown channels $x_i$ from convolutional measurements $y_i=x_i \circledast f$ ($i=1,2,\dots,N$). We consider the case where the $x_i$'s are sparse, and convolution with $f$ is invertible. Our nonconvex optimization formulation solves for a filter $h$ on the unit sphere that produces sparse output $y_i\circledast h$. Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of $f$ up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of $f$ and $x_i$ using a simple manifold gradient descent algorithm with random initialization. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods.

Abstract: Most non-blind deconvolution methods are developed under the error-free kernel assumption, and are not robust to inaccurate blur kernel. Unfortunately, despite the great progress in blind deconvolution, estimation error remains inevitable during blur kernel estimation. Consequently, severe artifacts such as ringing effects and distortions are likely to be introduced in the non-blind deconvolution stage. In this paper, we tackle this issue by suggesting: 1) a partial map in the Fourier domain for modeling kernel estimation error, and 2) a partial deconvolution model for robust deblurring with inaccurate blur kernel. The partial map is constructed by detecting the reliable Fourier entries of estimated blur kernel. And partial deconvolution is applied to wavelet-based and learning-based models to suppress the adverse effect of kernel estimation error. Furthermore, an E-M algorithm is developed for estimating the partial map and recovering the latent sharp image alternatively. Experimental results show that our partial deconvolution model is effective in relieving artifacts caused by inaccurate blur kernel, and can achieve favorable deblurring quality on synthetic and real blurry images.

Abstract: This chapter introduces basic concepts, criteria, and algorithms for Blind signal separation (BSS) and blind deconvolution and explores relationships between the BSS and blind deconvolution tasks. The chapter considers open issues and challenges within these related fields. BSS is sometimes used interchangeably with independent component analysis (ICA), technically, BSS and ICA are different tasks. BSS is most appropriate in situations where a linear mixture model is plausible. BSS offers the potential of extracting coherent and identifiable signal features that can be more easily tied to specific bodily functions or ailments. Two formulations of BSS task have been extensively explored: those that use spatial independence and non-Gaussianity, and those that use spatial decorrelation and temporal correlation. Density matching BSS methods rely heavily on concepts in information theory, a half-century-young field with applications in numerous fields including communications, economics, neuro-science, and physics. The chapter outlines spatio-temporal extensions of BSS and blind deconvolution, namely, the multichannel blind deconvolution and convolutive BSS tasks, respectively.

Abstract: Non-blind image deconvolution is an ill-posed problem. The presence of noise and band-limited blur kernels makes the solution of this problem non-unique. Existing deconvolution techniques produce a residual between the sharp image and the estimation that is highly correlated with the sharp image, the kernel, and the noise. In most cases, different restoration models must be constructed for different blur kernels and different levels of noise, resulting in low computational efficiency or highly redundant model parameters. Here we aim to develop a single model that handles different types of kernels and different levels of noise: general non-blind deconvolution. Specifically, we propose a very deep convolutional neural network that predicts the residual between a pre-deconvolved image and the sharp image rather than the sharp image. The residual learning strategy makes it easier to train a single model for different kernels and different levels of noise, encouraging high effectiveness and efficiency. Quantitative evaluations demonstrate the practical applicability of the proposed model for different blur kernels. The model also shows the state-of-the-art performance on synthesized blurry images.

Abstract: In this paper, a novel method to enhance Frequency Modulated Continuous Wave (FMCW) THz imaging resolution beyond its diffraction limit is proposed. Our method comprises two stages. Firstly, we reconstruct the signal in depth-direction using a sinc-envelope, yielding a significant improvement in depth estimation and signal parameter extraction. The resulting high precision depth estimate is used to deduce an accurate reflection intensity THz image. This image is fed in the second stage of our method to a 2D blind deconvolution procedure, adopted to enhance the lateral THz image resolution beyond the diffraction limit. Experimental data acquired with a FMCW system operating at 577 GHz with a bandwidth of 126 GHz shows that the proposed method enhances the lateral resolution by a factor of 2.29 to 346.2um with respect to the diffraction limit. The depth accuracy is 91um. Interestingly, the lateral resolution enhancement achieved with this blind deconvolution concept leads to better results in comparison to conventional gaussian deconvolution. Experimental data on a PCB resolution target is presented, in order to quantify the resolution enhancement and to compare the performance with established image enhancement approaches. The presented technique allows exposure of the interwoven fibre reinforced embedded structures of the PCB test sample.

Abstract: The mechanical structure or transmission path between fault source and sensor location always distorts the impulsive signatures of machine faults. It is thus an important task to estimate the desired impulsive feature and the influence of transmission path simultaneously from the noisy observation signals. Therefore, a convolutional sparse learning model (ConvSLM) is proposed to perform impulsive feature detection. The ConvSLM directly models the modulation process of the transmission path and is completely different from the indirect inverse filter design scheme as popular deconvolution techniques adopted. Meanwhile, to overcome the inherent drawbacks of the popular Kurtosis maximization strategy, the sparse structure of the impulsive feature is integrated into the objective function of the ConvSLM. Different from the recently developed two-stage solver, a new iterative algorithm with only one stage is also developed under a multiple-block nonconvex alternating direction method of multiplier framework to cope with the nonconvexity and nonsmoothness of the sparsity-regularized objective function, which not only reduces the algorithmic complexity but also has a convergence guarantee. Numerical experiments on synthetic data and test results corroborate the efficacy of the advocated approach. Compared with the state-of-the-art blind deconvolution techniques, the ConvSLM’s superiority is sufficiently verified through its application on the impulsive detection of the wind turbine gearbox gear.

Abstract: For more than half a century, seismic deconvolution has been of a great interest in reflection seismics. Its goal is to remove the effect of the seismic wavelet, i.e., the waveform produced by the seismic source, from the data. Its importance to the industry is that it increases the resolution of the seismic image, yielding a more interpretable seismic section. From a theoretical standpoint, what is interesting is that neither the wavelet nor the characteristics of the subsurface are known, which makes the problem particularly challenging. Over the years, several seismic deconvolution approaches have been proposed, based on different assumptions about the seismic wavelet and the reflectivity series.

Abstract: Given the noise-corrupted seismic recordings, blind deconvolution simultaneously solves for the reflectivity series and the wavelet Blind deconvolution can be formulated as a fully perturb

Abstract: The quality of images of the Sun obtained from the ground are severely limited by the perturbing effect of the Earth’s turbulent atmosphere. The post-facto correction of the images to compensate for the presence of the atmosphere require the combination of high-order adaptive optics techniques, fast measurements to freeze the turbulent atmosphere, and very time-consuming blind deconvolution algorithms. Under mild seeing conditions, blind deconvolution algorithms can produce images of astonishing quality. They can be very competitive with those obtained from space, with the huge advantage of the flexibility of the instrumentation thanks to the direct access to the telescope. In this contribution we make use of deep learning techniques to significantly accelerate the blind deconvolution process and produce corrected images at a peak rate of ∼100 images per second. We present two different architectures that produce excellent image corrections with noise suppression while maintaining the photometric properties of the images. As a consequence, polarimetric signals can be obtained with standard polarimetric modulation without any significant artifact. With the expected improvements in computer hardware and algorithms, we anticipate that on-site real-time correction of solar images will be possible in the near future.

Abstract: Raman spectroscopy often suffers from the problems of band overlap and random noise. In this work, we develop a nonlocal low-rank regularization (NLR) approach toward exploiting structured sparsity and explore its applications in Raman spectral deconvolution. Motivated by the observation that the rank of a ground-truth spectrum matrix is lower than that of the observed spectrum, a Raman spectral deconvolution model is formulated in our method to regularize the rank of the observed spectrum by total variation regularization. Then, an effective optimization algorithm is described to solve this model, which alternates between the instrument broadening function and latent spectrum until convergence. In addition to conceptual simplicity, the proposed method has achieved highly competent objective performance compared to several state-of-the-art methods in Raman spectrum deconvolution tasks. The restored Raman spectra are more suitable for extracting spectral features and recognizing the unknown materials or targets.

Abstract: Multichannel blind deconvolution is the problem of recovering an unknown signal $f$ and multiple unknown channels $x_i$ from their circular convolution $y_i=x_i \circledast f$ ($i=1,2,\dots,N$). We consider the case where the $x_i$'s are sparse, and convolution with $f$ is invertible. Our nonconvex optimization formulation solves for a filter $h$ on the unit sphere that produces sparse output $y_i\circledast h$. Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of $f$ up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of $f$ and $x_i$ using a simple manifold gradient descent (MGD) algorithm. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods.

Abstract: This paper generalizes the classical joint problem of signal demixing and blind deconvolution to the realm of graphs We investigate a setup where a single observation formed by the sum of multiple graph signals is available The main assumption is that each individual signal is generated by an originally sparse input diffused through the graph via the application of a graph filter In this context, we address the related problems of: 1) separating the individual graph signals, 2) identifying the unknown input supports, and 3) estimating the coefficients of the diffusing graph filters We first consider the case where each signal – prior to mixing – is diffused in a different graph We then particularize the results for the more challenging case where all the signals are diffused in the same graph The corresponding demixing and blind graph-signal deconvolution problems are formulated, convex relaxations are presented, and recovery conditions are discussed Numerical experiments in both the single and multiple graph cases show the capabilities of demixing in synthetic and biology-inspired graphs

Abstract: Objective: Common biological measurements are in the form of noisy convolutions of signals of interest with possibly unknown and transient blurring kernels. Examples include EEG and calcium imaging data. Thus, signal deconvolution of these measurements is crucial in understanding the underlying biological processes. The objective of this paper is to develop fast and stable solutions for signal deconvolution from noisy, blurred, and undersampled data, where the signals are in the form of discrete events distributed in time and space. Methods: We introduce compressible state-space models as a framework to model and estimate such discrete events. These state-space models admit abrupt changes in the states and have a convergent transition matrix, and are coupled with compressive linear measurements. We consider a dynamic compressive sensing optimization problem and develop a fast solution, using two nested expectation maximization algorithms, to jointly estimate the states as well as their transition matrices. Under suitable sparsity assumptions on the dynamics, we prove optimal stability guarantees for the recovery of the states and present a method for the identification of the underlying discrete events with precise confidence bounds. Results: We present simulation studies as well as application to calcium deconvolution and sleep spindle detection, which verify our theoretical results and show significant improvement over existing techniques. Conclusion: Our results show that by explicitly modeling the dynamics of the underlying signals, it is possible to construct signal deconvolution solutions that are scalable, statistically robust, and achieve high temporal resolution. Significance: Our proposed methodology provides a framework for modeling and deconvolution of noisy, blurred, and undersampled measurements in a fast and stable fashion, with potential application to a wide range of biological data.

Abstract: The radar autofocus problem arises in situations where radar measurements are acquired of a scene using antennas that suffer from position ambiguity. Current techniques model the antenna ambiguity as a global phase error affecting the received radar measurement at every antenna. However, the phase error signal model is only valid in the far field regime where the position error can be approximated by a one dimensional shift in the down-range direction. We propose in this paper an alternate formulation where the antenna position error is modeled using a two-dimensional shift operator in the image-domain. The radar autofocus problem then becomes a multichannel two-dimensional blind deconvolution problem where the static radar image is convolved with a two dimensional shift kernel for each antenna measurement. We develop an alternating minimization framework that leverages the sparsity and piece-wise smoothness of the radar scene, as well as the one-sparse property of the two dimensional shift kernels.

Abstract: This paper concerns solving the sparse deconvolution and demixing problem using l1,2-minimization. We show that under a certain structured random model, robust and stable recovery is possible. The results extend results of Ling and Strohmer (Inverse Probl. 31, 115002 2015), and in particular theoretically explain certain experimental findings from that paper. Our results do not only apply to the deconvolution and demixing problem, but to recovery of column-sparse matrices in general.

Abstract: Blurring image caused by a number of factors such as de focus, motion, and limited sensor resolution. Most of existing blind deconvolution research concentrates at recovering a single blurring kernel for the entire image. We proposed adaptive blind- non reference image quality assessment method for estimation the blur function (i.e. point spread function PSF) from the image acquired under low-lighting conditions and defocus images using Bayesian Blind Deconvolution. It is based on predicting a sharp version of a blurry inter image and uses the two images to solve a PSF. The estimation down by trial and error experimentation, until an acceptable restored image quality is obtained. Assessments the qualities of images have done through the applications of a set of quality metrics. Our method is fast and produces accurate results.