Accelerated edge-preserving image restoration without boundary artifacts

Fast X-ray diffraction (XRD) tomography for enhanced identification of materials

X-ray computed tomography (CT) is a commercially established modality for imaging large objects like passenger luggage. CT can provide the density and the effective atomic number, which is not always sufficient to identify threats like explosives and narcotics, since they can have a similar composition to benign plastics, glass, or light metals. In these cases, X-ray diffraction (XRD) may be better suited to distinguish the threats. Unfortunately, the diffracted photon flux is typically much weaker than the transmitted one. Measurement of quality XRD data is therefore slower compared to CT, which is an economic challenge for potential customers like airports. In this article we numerically analyze a novel low-cost scanner design which captures CT and XRD signals simultaneously, and uses the least possible collimation to maximize the flux. To simulate a realistic instrument, we propose a forward model that includes the resolution-limiting effects of the polychromatic spectrum, the detector, and all the finite-size geometric factors. We then show how to reconstruct XRD patterns from a large phantom with multiple diffracting objects. We include a reasonable amount of photon counting noise (Poisson statistics), as well as measurement bias (incoherent scattering). Our XRD reconstruction adds material-specific information, albeit at a low resolution, to the already existing CT image, thus improving threat detection. Our theoretical model is implemented in GPU (Graphics Processing Unit) accelerated software which can be used to further optimize scanner designs for applications in security, healthcare, and manufacturing quality control.

Accelerating MR Parameter Mapping Using Nonlinear Compressive Manifold Learning and Regularized Pre-Imaging

In this study, we presented a novel method to reconstruct the MR parametric maps from highly undersampled k-space data. Specifically, we utilized a nonlinear model to sparsely represent the unknown MR parameter-weighted images in high-dimensional feature space. Each image at a specific time point is assumed to belong to a low-dimensional manifold which is learned from training images created based on the parametric model. The final reconstruction is carried out by venturing the sparse representation of the images in the feature space back to the input space, using the pre-imaging technique. Particularly, among an infinite number of solutions that satisfy the data consistency, the one that is closest to the manifold is selected as the desired solution. The underlying optimization problem is solved using kernel trick, sparse coding, and split Bregman iteration algorithm. In addition, both spatial and temporal regularizations were utilized to further improve the reconstruction quality. The proposed method was validated on both phantom and in vivo human brain T2 mapping data. Results showed the proposed method was superior to the conventional linear model-based reconstruction methods, in terms of artifact removal and quantitative estimate accuracy. The proposed method could be potentially beneficial for quantitative MR applications.

3D MRI of the Hand and Wrist: Technical Considerations and Clinical Applications

In the last few years, major developments have been observed in the field of magnetic resonance imaging (MRI). Advances in both scanner hardware and software technologies have witnessed great leaps, enhancing the diagnostic quality and, therefore, the value of MRI. In musculoskeletal radiology, three-dimensional (3D) MRI has become an integral component of the diagnostic pathway at our institutions. This technique is particularly relevant in patients with hand and wrist symptoms, due to the intricate nature of the anatomical structures and the wide range of differential diagnoses for most presentations. We review the benefits of 3D MRI of the hand and wrist, commonly used pulse sequences, clinical applications, limitations, and future directions. We offer guidance for enhancing the image quality and tips for image interpretation of 3D MRI of the hand and wrist.

Multidimensional correlation spectroscopic imaging of exponential decays: From theoretical principles to in vivo human applications

Multiexponential modeling of relaxation or diffusion MR signal decays is a popular approach for estimating and spatially mapping different microstructural tissue compartments. While this approach can be quite powerful, it is also limited by the fact that one-dimensional multiexponential modeling is an ill-posed inverse problem with substantial ambiguities. In this article, we present an overview of a recent multidimensional correlation spectroscopic imaging approach to this problem. This approach helps to alleviate ill-posedness by making advantageous use of multidimensional contrast encoding (e.g., 2D diffusion-relaxation encoding or 2D relaxation-relaxation encoding) combined with a regularized spatial-spectral estimation procedure. Theoretical calculations, simulations, and experimental results are used to illustrate the benefits of this approach relative to classical methods. In addition, we demonstrate an initial proof-of-principle application of this kind of approach to in vivo human MRI experiments.

Local Activation Time Estimation in Fractionated Electrograms of Cardiac Mappings

In this study, we propose a novel approach for estimation of local activation times (LATs) in fractionated electrograms. Using an electrophysiological tissue model, we first formulate the electrogram array as a convolution of transmembrane currents with a distance kernel. These currents are more local activities and less affected by the heterogeneity in the tissue compared to electrograms. We then deconvolve the distance kernel with the electrograms to reconstruct the transmembrane current. To stabilize the solution of this ill-posed deconvolution, we use spatio-temporal total variation as a regularization. This helps to preserve sharp spatial and temporal deflections in the currents that are of higher importance in LAT estimation. Finally, the maximum negative slope of the reconstructed transmembrane currents are used to estimate the LATs. Instrumental comparison to two reference approaches shows that the proposed approach performs better in estimating the LATs in fractionated electrograms.

Measure and model a 3-D space-variant PSF for fluorescence microscopy image deblurring

Conventional deconvolution methods assume that the microscopy system is spatially invariant, introducing considerable errors. We developed a method to more precisely estimate space-variant point-spread functions from sparse measurements. To this end, a space-variant version of deblurring algorithm was developed and combined with a total-variation regularization. Validation with both simulation and real data showed that our PSF model is more accurate than the piecewise-invariant model and the blending model. Comparing with the orthogonal basis decomposition based PSF model, our proposed model also performed with a considerable improvement. We also evaluated the proposed deblurring algorithm. Our new deblurring algorithm showed a significantly better signal-to-noise ratio and higher image quality than those of the conventional space-invariant algorithm.

Convolutional Dictionary Learning: Acceleration and Convergence

Convolutional dictionary learning (CDL or sparsifying CDL) has many applications in image processing and computer vision. There has been growing interest in developing efficient algorithms for CDL, mostly relying on the augmented Lagrangian (AL) method or the variant alternating direction method of multipliers (ADMM). When their parameters are properly tuned, AL methods have shown fast convergence in CDL. However, the parameter tuning process is not trivial due to its data dependence and, in practice, the convergence of AL methods depends on the AL parameters for nonconvex CDL problems. To moderate these problems, this paper proposes a new practically feasible and convergent Block Proximal Gradient method using a Majorizer (BPG-M) for CDL. The BPG-M-based CDL is investigated with different block updating schemes and majorization matrix designs, and further accelerated by incorporating some momentum coefficient formulas and restarting techniques. All of the methods investigated incorporate a boundary artifacts removal (or, more generally, sampling) operator in the learning model. Numerical experiments show that, without needing any parameter tuning process, the proposed BPG-M approach converges more stably to desirable solutions of lower objective values than the existing state-of-the-art ADMM algorithm and its memory-efficient variant do. Compared with the ADMM approaches, the BPG-M method using a multi-block updating scheme is particularly useful in single-threaded CDL algorithm handling large data sets, due to its lower memory requirement and no polynomial computational complexity. Image denoising experiments show that, for relatively strong additive white Gaussian noise, the filters learned by BPG-M-based CDL outperform those trained by the ADMM approach.

Blurred image restoration using knife-edge function and optimal window Wiener filtering

Motion blur in images is usually modeled as the convolution of a point spread function (PSF) and the original image represented as pixel intensities. The knife-edge function can be used to model various types of motion-blurs, and hence it allows for the construction of a PSF and accurate estimation of the degradation function without knowledge of the specific degradation model. This paper addresses the problem of image restoration using a knife-edge function and optimal window Wiener filtering. In the proposed method, we first calculate the motion-blur parameters and construct the optimal window. Then, we use the detected knife-edge function to obtain the system degradation function. Finally, we perform Wiener filtering to obtain the restored image. Experiments show that the restored image has improved resolution and contrast parameters with clear details and no discernible ringing effects.

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.

Efficient, Convergent SENSE MRI Reconstruction for Nonperiodic Boundary Conditions via Tridiagonal Solvers

Undersampling is an effective method for reducing scan acquisition time for MRI. Strategies for accelerated MRI such as parallel MRI and Compressed Sensing MRI present challenging image reconstruction problems with non-differentiable cost functions and computationally demanding operations. Variable splitting (VS) can simplify implementation of difficult image reconstruction problems, such as the combination of parallel MRI and Compressed Sensing, CS-SENSE-MRI. Combined with augmented Lagrangian (AL) and alternating minimization strategies, variable splitting can yield iterative minimization algorithms with simpler auxiliary variable updates. However, arbitrary variable splitting schemes are not guaranteed to converge. Many variable splitting strategies are combined with periodic boundary conditions. The resultant circulant Hessians enable 𝒪(n log n) computation but may compromise image accuracy at the spatial boundaries. We propose two methods for CS-SENSE-MRI that use regularization with non-periodic boundary conditions to prevent wrap-around artifacts. Each algorithm computes one of the resulting variable updates efficiently in 𝒪(n) time using a parallelizable tridiagonal solver. AL-tridiag is a VS method designed to enable efficient computation for non-periodic boundary conditions. Another proposed algorithm, ADMM-tridiag, uses a similar VS scheme but also ensures convergence to a minimizer of the proposed cost function using the Alternating Direction Method of Multipliers (ADMM). AL-tridiag and ADMM-tridiag show speeds competitive with previous VS CS-SENSE-MRI reconstruction algorithm AL-P2. We also apply the tridiagonal VS approach to a simple image inpainting problem.

A Framework for Fast Image Deconvolution With Incomplete Observations

In image deconvolution problems, the diagonalization of the underlying operators by means of the fast Fourier transform (FFT) usually yields very large speedups. When there are incomplete observations (e.g., in the case of unknown boundaries), standard deconvolution techniques normally involve non-diagonalizable operators, resulting in rather slow methods or, otherwise, use inexact convolution models, resulting in the occurrence of artifacts in the enhanced images. In this paper, we propose a new deconvolution framework for images with incomplete observations that allows us to work with diagonalized convolution operators, and therefore is very fast. We iteratively alternate the estimation of the unknown pixels and of the deconvolved image, using, e.g., an FFT-based deconvolution method. This framework is an efficient, high-quality alternative to existing methods of dealing with the image boundaries, such as edge tapering. It can be used with any fast deconvolution method. We give an example in which a state-of-the-art method that assumes periodic boundary conditions is extended, using this framework, to unknown boundary conditions. Furthermore, we propose a specific implementation of this framework, based on the alternating direction method of multipliers (ADMM). We provide a proof of convergence for the resulting algorithm, which can be seen as a "partial" ADMM, in which not all variables are dualized. We report experimental comparisons with other primal-dual methods, where the proposed one performed at the level of the state of the art. Four different kinds of applications were tested in the experiments: deconvolution, deconvolution with inpainting, superresolution, and demosaicing, all with unknown boundaries.

Generalized recovery algorithm for 3D super-resolution microscopy using rotating point spread functions

Super-resolution microscopy with phase masks is a promising technique for 3D imaging and tracking. Due to the complexity of the resultant point spread functions, generalized recovery algorithms are still missing. We introduce a 3D super-resolution recovery algorithm that works for a variety of phase masks generating 3D point spread functions. A fast deconvolution process generates initial guesses, which are further refined by least squares fitting. Overfitting is suppressed using a machine learning determined threshold. Preliminary results on experimental data show that our algorithm can be used to super-localize 3D adsorption events within a porous polymer film and is useful for evaluating potential phase masks. Finally, we demonstrate that parallel computation on graphics processing units can reduce the processing time required for 3D recovery. Simulations reveal that, through desktop parallelization, the ultimate limit of real-time processing is possible. Our program is the first open source recovery program for generalized 3D recovery using rotating point spread functions.

Variational Dirichlet Blur Kernel Estimation

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.

High-accuracy total variation with application to compressed video sensing

Numerous total variation (TV) regularizers, engaged in image restoration problem, encode the gradients by means of simple [-1, 1] finite-impulse-response (FIR) filter. Despite its low computational processing, this filter severely distorts signal's high-frequency components pertinent to edge/ discontinuous information and cause several deficiency issues known as texture and geometric loss. This paper addresses this problem by proposing an alternative model to the TV regularization problem via high-order accuracy differential FIR filters to preserve rapid transitions in signal recovery. A numerical encoding scheme is designed to extend the TV model into multidimensional representation (tensorial decomposition). We adopt this design to regulate the spatial and temporal redundancy in compressed video sensing problem to jointly recover frames from undersampled measurements. We then seek the solution via alternating direction methods of multipliers and find a unique solution to quadratic minimization step with capability of handling different boundary conditions. The resulting algorithm uses much lower sampling rate and highly outperforms alternative state-of-the-art methods. This is evaluated both in terms of restoration accuracy and visual quality of the recovered frames.

Disease prediction based on functional connectomes using a scalable and spatially-informed support vector machine

Substantial evidence indicates that major psychiatric disorders are associated with distributed neural dysconnectivity, leading to a strong interest in using neuroimaging methods to accurately predict disorder status. In this work, we are specifically interested in a multivariate approach that uses features derived from whole-brain resting state functional connectomes. However, functional connectomes reside in a high dimensional space, which complicates model interpretation and introduces numerous statistical and computational challenges. Traditional feature selection techniques are used to reduce data dimensionality, but are blind to the spatial structure of the connectomes. We propose a regularization framework where the 6-D structure of the functional connectome (defined by pairs of points in 3-D space) is explicitly taken into account via the fused Lasso or the GraphNet regularizer. Our method only restricts the loss function to be convex and margin-based, allowing non-differentiable loss functions such as the hinge-loss to be used. Using the fused Lasso or GraphNet regularizer with the hinge-loss leads to a structured sparse support vector machine (SVM) with embedded feature selection. We introduce a novel efficient optimization algorithm based on the augmented Lagrangian and the classical alternating direction method, which can solve both fused Lasso and GraphNet regularized SVM with very little modification. We also demonstrate that the inner subproblems of the algorithm can be solved efficiently in analytic form by coupling the variable splitting strategy with a data augmentation scheme. Experiments on simulated data and resting state scans from a large schizophrenia dataset show that our proposed approach can identify predictive regions that are spatially contiguous in the 6-D "connectome space," offering an additional layer of interpretability that could provide new insights about various disease processes.

Deconvolving images with unknown boundaries using the alternating direction method of multipliers

The alternating direction method of multipliers (ADMM) has recently sparked interest as a flexible and efficient optimization tool for inverse problems, namely, image deconvolution and reconstruction under non-smooth convex regularization. ADMM achieves state-of-the-art speed by adopting a divide and conquer strategy, wherein a hard problem is split into simpler, efficiently solvable sub-problems (e.g., using fast Fourier or wavelet transforms, or simple proximity operators). In deconvolution, one of these sub-problems involves a matrix inversion (i.e., solving a linear system), which can be done efficiently (in the discrete Fourier domain) if the observation operator is circulant, i.e., under periodic boundary conditions. This paper extends ADMM-based image deconvolution to the more realistic scenario of unknown boundary, where the observation operator is modeled as the composition of a convolution (with arbitrary boundary conditions) with a spatial mask that keeps only pixels that do not depend on the unknown boundary. The proposed approach also handles, at no extra cost, problems that combine the recovery of missing pixels (i.e., inpainting) with deconvolution. We show that the resulting algorithms inherit the convergence guarantees of ADMM and illustrate its performance on non-periodic deblurring (with and without inpainting of interior pixels) under total-variation and frame-based regularization.