Abstract: The ${\ell _1}/{\ell _2}$ ratio regularization function has shown good performance for retrieving sparse signals in a number of recent works, in the context of blind deconvolution. Indeed, it benefits from a scale invariance property much desirable in the blind context. However, the ${\ell _1}/{\ell _2}$ function raises some difficulties when solving the nonconvex and nonsmooth minimization problems resulting from the use of such a penalty term in current restoration methods. In this paper, we propose a new penalty based on a smooth approximation to the ${\ell _1}/{\ell _2}$ function. In addition, we develop a proximal-based algorithm to solve variational problems involving this function and we derive theoretical convergence results. We demonstrate the effectiveness of our method through a comparison with a recent alternating optimization strategy dealing with the exact ${\ell _1}/{\ell _2}$ term, on an application to seismic data blind deconvolution.

Abstract: Scanning radar is of notable importance for ground surveillance, terrain mapping and disaster rescue. However, the angular resolution of a scanning radar image is poor compared to the achievable range resolution. This paper presents a deconvolution algorithm for angular super-resolution in scanning radar based on Bayesian theory, which states that the angular super-resolution can be realized by solving the corresponding deconvolution problem with the maximum a posteriori (MAP) criterion. The algorithm considers that the noise is composed of two mutually independent parts, i.e., a Gaussian signal-independent component and a Poisson signal-dependent component. In addition, the Laplace distribution is used to represent the prior information about the targets under the assumption that the radar image of interest can be represented by the dominant scatters in the scene. Experimental results demonstrate that the proposed deconvolution algorithm has higher precision for angular super-resolution compared with the conventional algorithms, such as the Tikhonov regularization algorithm, the Wiener filter and the Richardson–Lucy algorithm.

Abstract: Blind deconvolution (BD), the resolution of a signal and a filter given their convolution, arises in many applications. Without further constraints, BD is ill-posed. In practice, subspace or sparsity constraints have been imposed to reduce the search space, and have shown some empirical success. However, existing theoretical analysis on uniqueness in BD is rather limited. As an effort to address the still mysterious question, we derive sufficient conditions under which two vectors can be uniquely identified from their circular convolution, subject to subspace or sparsity constraints. These sufficient conditions provide the first algebraic sample complexities for BD. We first derive a sufficient condition that applies to almost all bases or frames. For blind deconvolution of vectors in $\mathbb{C}^n$, with two subspace constraints of dimensions $m_1$ and $m_2$, the required sample complexity is $n\geq m_1m_2$. Then we impose a sub-band structure on one basis, and derive a sufficient condition that involves a relaxed sample complexity $n\geq m_1+m_2-1$, which we show to be optimal. We present the extensions of these results to BD with sparsity constraints or mixed constraints, with the sparsity level replacing the subspace dimension. The cost for the unknown support in this case is an extra factor of 2 in the sample complexity.

Abstract: The localization of sound sources with delay-and-sum (DAS) beamforming is limited by a poor spatial resolution—particularly at low frequencies. Various methods based on deconvolution are examined to improve the resolution of the beamforming map, which can be modeled by a convolution of the unknown acoustic source distribution and the beamformer's response to a point source, i.e., point-spread function. A significant limitation of deconvolution is, however, an additional computational effort compared to beamforming. In this paper, computationally efficient deconvolution algorithms are examined with computer simulations and experimental data. Specifically, the deconvolution problem is solved with a fast gradient projection method called Fast Iterative Shrikage-Thresholding Algorithm (FISTA), and compared with a Fourier-based non-negative least squares algorithm. The results indicate that FISTA tends to provide an improved spatial resolution and is up to 30% faster and more robust to noise. In the spirit of reproducible research, the source code is available online.

Abstract: Subsampled blind deconvolution is the recovery of two unknown signals from samples of their convolution. To overcome the ill-posedness of this problem, solutions based on priors tailored to specific application have been developed in practical applications. In particular, sparsity models have provided promising priors. However, in spite of empirical success of these methods in many applications, existing analyses are rather limited in two main ways: by disparity between the theoretical assumptions on the signal and/or measurement model versus practical setups; or by failure to provide a performance guarantee for parameter values within the optimal regime defined by the information theoretic limits. In particular, it has been shown that a naive sparsity model is not a strong enough prior for identifiability in the blind deconvolution problem. Instead, in addition to sparsity, we adopt a conic constraint, which enforces spectral flatness of the signals. Under this prior, we provide an iterative algorithm that achieves guaranteed performance in blind deconvolution at near optimal sample complexity. Numerical results show the empirical performance of the iterative algorithm agrees with the performance guarantee.

Abstract: Microphone arrays and beamforming have become a standard method to localize aeroacoustic sources. Deconvolution techniques have been developed to improve spatial resolution of beamforming maps. The deconvolution approach for the mapping of acoustic sources (DAMAS) is a standard deconvolution technique, which has been enhanced via a sparsity approach called sparsity constrained deconvolution approach for the mapping of acoustic sources (SC-DAMAS). In this paper, the DAMAS inverse problem is solved using the orthogonal matching pursuit (OMP) and compared with beamforming and SC-DAMAS. The resulting noise source maps show that OMP-DAMAS is an efficient source localization technique in the case of uncorrelated or correlated acoustic sources. Moreover, the computation time is clearly reduced as compared to SC-DAMAS.

Abstract: In this work we devise two novel algorithms for blind deconvolution based on a family of logarithmic image priors. In contrast to recent approaches, we consider a minimalistic formulation of the blind deconvolution problem where there are only two energy terms: a least-squares term for the data fidelity and an image prior based on a lower-bounded logarithm of the norm of the image gradients. We show that this energy formulation is sufficient to achieve the state of the art in blind deconvolution with a good margin over previous methods. Much of the performance is due to the chosen prior. On the one hand, this prior is very effective in favoring sparsity of the image gradients. On the other hand, this prior is non convex. Therefore, solutions that can deal effectively with local minima of the energy become necessary. We devise two iterative minimization algorithms that at each iteration solve convex problems: one obtained via the primal-dual approach and one via majorization-minimization. While the former is computationally efficient, the latter achieves state-of-the-art performance on a public dataset.

Abstract: Identifying concentrations of components from an observed mixture is a fundamental problem in signal processing. It has diverse applications in fields ranging from hyperspectral imaging to denoising biomedical sensors. This paper focuses on in-silico deconvolution of signals associated with complex tissues into their constitutive cell-type specific components, along with a quantitative characterization of the cell-types. Deconvolving mixed tissues/cell-types is useful in the removal of contaminants (e.g., surrounding cells) from tumor biopsies, as well as in monitoring changes in the cell population in response to treatment or infection. In these contexts, the observed signal from the mixture of cell-types is assumed to be a linear combination of the expression levels of genes in constitutive cell-types. The goal is to use known signals corresponding to individual cell-types along with a model of the mixing process to cast the deconvolution problem as a suitable optimization problem. In this paper, we present a survey of models, methods, and assumptions underlying deconvolution techniques. We investigate the choice of the different loss functions for evaluating estimation error, constraints on solutions, preprocessing and data filtering, feature selection, and regularization to enhance the quality of solutions, along with the impact of these choices on the performance of regression-based methods for deconvolution. We assess different combinations of these factors and use detailed statistical measures to evaluate their effectiveness. We identify shortcomings of current methods and avenues for further investigation. For many of the identified shortcomings, such as normalization issues and data filtering, we provide new solutions. We summarize our findings in a prescriptive step-by-step process, which can be applied to a wide range of deconvolution problems.

Abstract: Bilinear inverse problems (BIPs), the resolution of two vectors given their image under a bilinear mapping, arise in many applications. Without further constraints, BIPs are usually ill-posed. In practice, properties of natural signals are exploited to solve BIPs. For example, subspace constraints or sparsity constraints are imposed to reduce the search space. These approaches have shown some success in practice. However, there are few results on uniqueness in BIPs. For most BIPs, the fundamental question of under what condition the problem admits a unique solution, is yet to be answered. For example, blind gain and phase calibration (BGPC) is a structured bilinear inverse problem, which arises in many applications, including inverse rendering in computational relighting (albedo estimation with unknown lighting), blind phase and gain calibration in sensor array processing, and multichannel blind deconvolution (MBD). It is interesting to study the uniqueness of such problems. In this paper, we define identifiability of a BIP up to a group of transformations. We derive necessary and sufficient conditions for such identifiability, i.e., the conditions under which the solutions can be uniquely determined up to the transformation group. Applying these results to BGPC, we derive sufficient conditions for unique recovery under several scenarios, including subspace, joint sparsity, and sparsity models. For BGPC with joint sparsity or sparsity constraints, we develop a procedure to compute the relevant transformation groups. We also give necessary conditions in the form of tight lower bounds on sample complexities, and demonstrate the tightness of these bounds by numerical experiments. The results for BGPC not only demonstrate the application of the proposed general framework for identifiability analysis, but are also of interest in their own right.

Abstract: Blind image deconvolution involves two key objectives: 1) latent image and 2) blur estimation For latent image estimation, we propose a fast deconvolution algorithm, which uses an image prior of nondimensional Gaussianity measure to enforce sparsity and an undetermined boundary condition methodology to reduce boundary artifacts For blur estimation, a linear inverse problem with normalization and nonnegative constraints must be solved However, the normalization constraint is ignored in many blind image deblurring methods, mainly because it makes the problem less tractable In this paper, we show that the normalization constraint can be very naturally incorporated into the estimation process by using a Dirichlet distribution to approximate the posterior distribution of the blur Making use of variational Dirichlet approximation, we provide a blur posterior approximation that considers the uncertainty of the estimate and removes noise in the estimated kernel Experiments with synthetic and real data demonstrate that the proposed method is very competitive to the state-of-the-art blind image restoration methods

Abstract: We propose a novel camera calibration method for defocused images using a smartphone under the assumption that the defocus blur is modeled as a convolution of a sharp image with a Gaussian point spread function (PSF). In contrast to existing calibration approaches which require well-focused images, the proposed method achieves accurate camera calibration with severely defocused images. This robustness to defocus is due to the proposed set of unidirectional binary patterns, which simplifies 2D Gaussian deconvolution to a 1D Gaussian deconvolution problem with multiple observations. By capturing the set of patterns consecutively displayed on a smartphone, we formulate the feature extraction as a deconvolution problem to estimate feature point locations in sub-pixel accuracy and the blur kernel in each location. We also compensate the error in camera parameters due to refraction of the glass panel of the display device. We evaluate the performance of the proposed method on synthetic and real data. Even under severe defocus, our method shows accurate camera calibration result.

Abstract: We derive near optimal performance guarantees for subsampled blind deconvolution. Blind deconvolution is an ill-posed bilinear inverse problem and additional subsampling makes the problem even more challenging. Sparsity and spectral flatness priors on unknown signals are introduced to overcome these difficulties. While being crucial for deriving desired near optimal performance guarantees, unlike the sparsity prior with a nice union-of-subspaces structure, the spectral flatness prior corresponds to a nonconvex cone structure, which is not preserved by elementary set operations. This prohibits the operator arising in subsampled blind deconvolution from satisfying the standard restricted isometry property (RIP) at near optimal sample complexity, which motivated us to study other RIP-like properties. Combined with the performance guarantees derived using these RIP-like properties in a companion paper, we show that subsampled blind deconvolution is provably solved at near optimal sample complexity by a practical algorithm.

Abstract: Various applications in signal processing and machine learning give rise to highly structured spectral optimization problems characterized by low-rank solutions. Two important examples that motivate this work are optimization problems from phase retrieval and from blind deconvolution, which are designed to yield rank-1 solutions. An algorithm is described that is based on solving a certain constrained eigenvalue optimization problem that corresponds to the gauge dual which, unlike the more typical Lagrange dual, has an especially simple constraint. The dominant cost at each iteration is the computation of rightmost eigenpairs of a Hermitian operator. A range of numerical examples illustrate the scalability of the approach.

Abstract: A common problem of imaging 3-D objects into image plane is superposition of the projected structures. In dynamic imaging, projection overlaps of organs and tissues complicate extraction of signals specific to individual structures with different dynamics. The problem manifests itself also in dynamic tomography as tissue mixtures are present in voxels. Separation of signals specific to dynamic structures belongs to the category of blind source separation. It is an underdetermined problem with many possible solutions. Existing separation methods select the solution that best matches their additional assumptions on the source model. We propose a novel blind source separation method based on probabilistic model of dynamic image sequences assuming each source dynamics as convolution of an input function and a source specific kernel (modeling organ impulse response or retention function). These assumptions are formalized as a Bayesian model with hierarchical prior and solved by the Variational Bayes method. The proposed prior distribution assigns higher probability to sparse source images and sparse convolution kernels. We show that the results of separation are relevant to selected tasks of dynamic renal scintigraphy. Accuracy of tissue separation with simulated and clinical data provided by the proposed method outperformed accuracy of previously developed methods measured by the mean square and mean absolute errors of estimation of simulated sources and the sources separated by an expert physician. MATLAB implementation of the algorithm is available for download.

Abstract: The time (vertical) resolution enhancement of ground-penetrating radar (GPR) data by deconvolution is a long-standing problem due to the mixed-phase characteristics of the source wavelet. Several approaches have been proposed, which take the mixed-phase nature of the GPR source wavelet into account. However, most of these schemes are usually laborious and/or computationally intensive and have not yet found widespread use. Here, we propose a simple and fast approach to GPR deconvolution that requires only a minimal user input. First, a trace-by-trace minimum-phase (spiking) deconvolution is applied to remove the minimum-phase part of the mixed-phase GPR wavelet. Then, a global phase rotation is applied to maximize the sparseness (kurtosis) of the minimum-phase deconvolved data to correct for phase distortions that remain after the minimum-phase deconvolution. Applications of this scheme to synthetic and field data demonstrate that a significant improvement in image quality can be achieved, leading to deconvolved data that are a closer representation of the underlying reflectivity structure than the input or minimum-phase deconvolved data. Synthetic-data tests indicate that, because of the temporal and spatial correlation inherent in the GPR data due to the frequency- and wavenumber-bandlimited nature of the GPR source wavelet and the reflectivity structure, a significant number of samples are required for a reliable sparseness (kurtosis) estimate and stable phase rotation. This observation calls into question the blithe application of kurtosis-based methods within short time windows such as that for time-variant deconvolution.

Abstract: Blind deconvolution is the recovery of two unknown signals from their convolution. To overcome the ill-posedness of this problem, solutions based on priors tailored to specific application have been developed in practical applications. In particular, sparsity models have provided promising priors. In spite of empirical success in many applications, existing analyses are rather limited in two main ways: by disparity between theoretical assumptions on the signal and/or measurement model versus practical setups; or by failure to demonstrate success for parameter values within the optimal regime defined by the information theoretic limits. In particular, it has been shown that a naive sparsity model is not strong enough as a prior for identifiability in blind deconvolution problem. In addition to sparsity, we adopt a conic constraint by Ahmed et al., which enforces flat spectra in the Fourier domain. Under this prior, we provide an iterative algorithm that achieves guaranteed performance in blind deconvolution with number of measurements proportional (up to a logarithmic factor) to the sparsity level of the signal.

Abstract: In this paper we analyze the blind deconvolution of an image and an unknown blur in a coded imaging system. The measurements consist of subsampled convolution of an unknown blurring kernel with multiple random binary modulations (coded masks) of the image. To perform the deconvolution, we consider a standard lifting of the image and the blurring kernel that transforms the measurements into a set of linear equations of the matrix formed by their outer product. Any rank-one solution to this system of equations provides a valid pair of an image and a blur. We first express the necessary and sufficient conditions for the uniqueness of a rank-one solution under some additional assumptions (uniform subsampling and no limit on the number of coded masks). These conditions are a special case of a previously established result regarding identifiability in the matrix completion problem. We also characterize a low-dimensional subspace model for the blur kernel that is sufficient to guarantee identifiability, includin...

Abstract: In this paper, we extend our recent registration method designed specifically for registering blurred images. The original method works for unknown blurs, assuming the blurring point-spread function (PSF) exhibits an $N$ -fold rotational symmetry. Here, we also generalize the theory to the case of dihedrally symmetric blurs, which are produced by the PSFs having both rotational and axial symmetries. Such kind of blurs are often found in unfocused images acquired by digital cameras, as in out-of-focus shots the PSF typically mimics the shape of the shutter aperture. This makes our registration algorithm particularly well-suited in applications where blurred image registration must be used as a preprocess step of an image fusion algorithm, and where common registration methods fail, due to the amount of blur. We demonstrate that the proposed method leads to an improvement of the registration performance, and we show its applicability to real images by providing successful examples of blurred image registration followed by depth-of-field extension and multichannel blind deconvolution.

Abstract: Time reversal techniques are used in ocean acoustics, medical imaging, seismology, and non-destructive evaluation to backpropagate recorded signals to the source of origin. We demonstrate experimentally a technique which improves the temporal focus achieved at the source location by utilizing deconvolution. One experiment consists of propagating a signal from a transducer within a concrete block to a single receiver on the surface, and then applying time reversal or deconvolution to focus the energy back at the source location. Another two experiments are run to study the robust nature of deconvolution by investigating the effect of changing the stabilization constant used in the deconvolution and the impact multiple sources have upon deconvolutions’ focusing abilities. The results show that we are able to generate an improved temporal focus at the source transducer using deconvolution while maintaining the robust nature of time reversal. Additionally, deconvolution’s costs are negligible due to it being a preprocessing step to the recorded data. The technique can be applied for detailed investigation of the source mechanisms (e.g. cracks) but also for monitoring purposes.

Abstract: We investigate efficient algorithmic realisations for robust deconvolution of grey-value images with known space-invariant point-spread function, with emphasis on 1D motion blur scenarios. The goal is to make deconvolution suitable as preprocessing step in automated image processing environments with tight time constraints. Candidate deconvolution methods are selected for their restoration quality, robustness and efficiency. Evaluation of restoration quality and robustness on synthetic and real-world test images leads us to focus on a combination of Wiener filtering with few iterations of robust and regularised Richardson–Lucy deconvolution. We discuss algorithmic optimisations for specific scenarios. In the case of uniform linear motion blur in coordinate direction, it is possible to achieve real-time performance (less than 50 ms) in single-threaded CPU computation on images of $$256\times 256$$ pixels. For more general space-invariant blur settings, still favourable computation times are obtained. Exemplary parallel implementations demonstrate that the proposed method also achieves real-time performance for general 1D motion blurs in a multi-threaded CPU setting and for general 2D blurs on a GPU.

Abstract: Fluorescence lifetime microscopy imaging (FLIM) is an optic technique that allows a quantitative characterization of the fluorescent components of a sample. However, for an accurate interpretation of FLIM, an initial processing step is required to deconvolve the instrument response of the system from the measured fluorescence decays. In this paper, we present a novel strategy for the deconvolution of FLIM data based on a library of exponentials. Our approach searches for the scaling coefficients of the library by non-negative least squares approximations plus Thikonov/l(2) or l(1) regularization terms. The parameters of the library are given by the lower and upper bounds in the characteristic lifetimes of the exponential functions and the size of the library, where we observe that this last variable is not a limiting factor in the resulting fitting accuracy. We compare our proposal to nonlinear least squares and global non-linear least squares estimations with a multi-exponential model, and also to constrained Laguerre-base expansions, where we visualize an advantage of our proposal based on Thikonov/l(2) regularization in terms of estimation accuracy, computational time, and tuning strategy. Our validation strategy considers synthetic datasets subject to both shot and Gaussian noise and samples with different lifetime maps, and experimental FLIM data of ex-vivo atherosclerotic plaques and human breast cancer cells.

Abstract: The recorded spectra often suffer noise and band overlapping, and deconvolution methods are always used for spectral recovery. However, during the process of spectral recovery, the details cannot always be preserved. To solve this problem, two regularization terms are introduced and proposed. First, the conditions on the regularization term are analyzed for smoothing noise and preserving detail, and according to these conditions, φHL regularization is introduced into the spectral deconvolution model. In view of the deficiency of φHL under noisy condition, adaptive φHL regularization (φAHL) is proposed. Then semi-blind deconvolution methods based on φHL regularization (SBD-HL) and based on adaptive φHL regularization (SBD-AHL) are proposed, respectively. The simulation experimental results indicate that the proposed SBD-HL and SBD-AHL methods have better recovery, and SBD-AHL is superior to SBD-HL, especially in the noisy case.

Abstract: The main hindrance in easy detection of bearing faults from vibration data is that the signal is noise ridden, and only an efficient method for noise reduction will effectively bring out the fault characteristics. This paper proposes a novel method for such noise reduction using Lucy–Richardson deconvolution, which is an iterative technique for deblurring images. Its application in signal processing more specially in bearing fault diagnosis is being studied in this paper. The characteristics of this deconvolution with different shapes of point spread function and their effectiveness are also shown.