Magnetic Resonance Imaging (MRI) is considered today the golden-standard modality for soft tissues. The long acquisition times, however, make it more prone to motion artifacts as well as contribute to the relative high costs of this examination. Over the years, multiple studies concentrated on designing reduced measurement schemes and image reconstruction schemes for MRI, however these problems have been so far addressed separately. On the other hand, recent works in optical computational imaging have demonstrated growing success of simultaneous learning-based design of the acquisition and reconstruction schemes manifesting significant improvement in the reconstruction quality with a constrained time budget. Inspired by these successes, in this work, we propose to learn accelerated MR acquisition schemes (in the form of Cartesian trajectories) jointly with the image reconstruction operator. To this end, we propose an algorithm for training the combined acquisition-reconstruction pipeline end-to-end in a differentiable way. We demonstrate the significance of using the learned Cartesian trajectories at different speed up rates. Code available at https://github.com/tomer196/fastMRI-Cartesian.

Multidimensional scaling (MDS) is a dimensionality reduction tool used for information analysis, data visualization and manifold learning. Most MDS procedures embed data points in low-dimensional euclidean (flat) domains, such that distances between the points are as close as possible to given inter-point dissimilarities. We present an efficient solver for classical scaling, a specific MDS model, by extrapolating the information provided by distances measured from a subset of the points to the remainder. The computational and space complexities of the new MDS methods are thereby reduced from quadratic to quasi-linear in the number of data points. Incorporating both local and global information about the data allows us to construct a low-rank approximation of the inter-geodesic distances between the data points. As a by-product, the proposed method allows for efficient computation of geodesic distances.

REgularization by Denoising (RED) is an attractive framework for solving inverse problems by incorporating state-of-the-art denoising algorithms as the priors. A drawback of this approach is the high computational complexity of denoisers, which dominate the computation time. In this paper, we apply a general framework called weighted proximal methods (WPMs) to solve RED efficiently. We first show that two recently introduced RED solvers (using the fixed point and accelerated proximal gradient methods) are particular cases of WPMs. Then we show by numerical experiments that slightly more sophisticated variants of WPM can lead to reduced run times for RED by requiring a significantly smaller number of calls to the denoiser.

Compressed sensing (CS) is a signal processing framework for efficiently reconstructing a signal from a small number of measurements, obtained by linear projections of the signal. Block-based CS is a lightweight CS approach that is mostly suitable for processing very high-dimensional images and videos: it operates on local patches, employs a low-complexity reconstruction operator and requires significantly less memory to store the sensing matrix. In this paper we present a deep learning approach for block-based CS, in which a fully-connected network performs both the block-based linear sensing and non-linear reconstruction stages. During the training phase, the sensing matrix and the non-linear reconstruction operator are jointly optimized, and the proposed approach out-performs state-of-the-art both in terms of reconstruction quality and computation time. For example, at a 25% sensing rate the average PSNR advantage is 0.77dB and computation time is over 200-times faster.

We present SEBOOST, a technique for boosting the performance of existing stochastic optimization methods. SEBOOST applies a secondary optimization process in the subspace spanned by the last steps and descent directions. The method was inspired by the SESOP optimization method for large-scale problems, and has been adapted for the stochastic learning framework. It can be applied on top of any existing optimization method with no need to tweak the internal algorithm. We show that the method is able to boost the performance of different algorithms, and make them more robust to changes in their hyper-parameters. As the boosting steps of SEBOOST are applied between large sets of descent steps, the additional subspace optimization hardly increases the overall computational burden. We introduce two hyper-parameters that control the balance between the baseline method and the secondary optimization process. The method was evaluated on several deep learning tasks, demonstrating promising results.

Recent work in image processing suggests that operating on (overlapping) patches in an image may lead to state-of-the-art results. This has been demonstrated for a variety of problems including denoising, inpainting, deblurring, and super-resolution. The work reported in [I. Ram, I. Cohen, and M. Elad, IEEE Trans. Image Process., 23 (2014), pp. 2779--2792] and [I. Ram, M. Elad, and I. Cohen, IEEE Trans. Image Process., 22 (2013), pp. 2764--2774] takes an extra step forward by showing that ordering these patches to form an approximate shortest path can be leveraged for better processing. The core idea is to apply a simple filter on the resulting 1D smoothed signal obtained after the patch-permutation. This idea has been also explored in combination with a wavelet pyramid, leading eventually to a sophisticated and highly effective regularizer for inverse problems in imaging. In this work we further study the patch-permutation concept and harness it to propose a new simple yet effective regularization for image restoration problems. Our approach builds on the classic maximum a posteriori probability (MAP), with a penalty function consisting of a regular log-likelihood term and a novel permutation-based regularization term. Using a plain 1D Laplacian, the proposed regularization forces robust smoothness ($L1$) on the permuted pixels. Since the permutation originates from patch ordering, we propose to accumulate the smoothness terms over all of the patches' pixels. Furthermore, we take into account the found distances between adjacent patches in the ordering by weighting the Laplacian outcome. We demonstrate the proposed scheme on the following diverse set of problems: (i) severe Poisson image denoising, (ii) Gaussian image denoising, (iii) image deblurring, and (iv) single image super-resolution. In all of these cases, we use recent methods that handle these problems as initialization to our scheme. This is followed by an L-BFGS optimization of the above-described penalty function, leading to state-of-the-art results, especially for highly ill-posed cases.

Multidimensional-scaling (MDS) is an information analysis tool. It involves the evaluation of distances between data points, which is a quadratic space-time problem. Then, MDS procedures find an embedding of the points in a low dimensional Euclidean (flat) domain, optimizing for the similarity of inter-points distances. We present an efficient solver for Classical Scaling (a specific MDS model) by extending the distances measured from a subset of the points to the rest, while exploiting the smoothness property of the distance functions. The smoothness is measured by the L2 norm of the Laplace-Beltrami operator applied to the unknown distance function. The Laplace Beltrami reflects the local differential relations between points, and can be computed in linear time. Classical-scaling is thereby reformulated into a quasi-linear space-time complexities procedure.

We propose a supervised machine learning approach for boosting existing signal and image recovery methods and demonstrate its efficacy on example of image reconstruction in computed tomography. Our technique is based on a local nonlinear fusion of several image estimates, all obtained by applying a chosen reconstruction algorithm with different values of its control parameters. Usually such output images have different bias/variance trade-off. The fusion of the images is performed by feed-forward neural network trained on a set of known examples. Numerical experiments show an improvement in reconstruction quality relatively to existing direct and iterative reconstruction methods.

According to the uniformization theorem, any surface can be conformally mapped into a domain of a constant Gaussian curvature. The conformal factor indicates the local scaling introduced by such a mapping. This process could be used to compute geometric quantities in a simplified flat domain with zero Gaussian curvature. For example, the computation of geodesic distances on a curved surface can be mapped into solving an eikonal equation in a plane weighted by the conformal factor. Solving an eikonal equation on the weighted plane can then be done by regular sampling of the domain using, for example, the celebrated fast marching method (FMM). The connection between the conformal factor on the plane and the surface geometry can be justified analytically. Still, in order to construct consistent numerical solvers that exploit this relation, one needs to prove that the conformal factor is bounded. We provide theoretical bounds of the conformal factor and introduce optimization formulations that control its behavior. It is demonstrated that without such restrictions the numerical results are unboundedly inaccurate. Putting all ingredients in the right order, we introduce a method for computing geodesic distances on a two-dimensional manifold by using the FMM on a weighted flat domain. It is also shown how a metric on a curved domain can be reconstructed by reformulating the nonflat metric restoration problem into a weighted flat domain, again, with bounded weights for consistent results.

We propose a sinogram restoration method which consists of a patch-wise non-linear processing, based on a sparsity prior in terms of a learned dictionary. An off-line learning process uses a statistical model of the sinogram noise and minimizes an error measure in the image domain over the training set. The error measure is designed to preserve low-contrast edges for visibility of soft tissues. Our numerical study shows that the algorithm improves on the performance of the standard Filtered Back-Projection algorithm and effectively allows to halve the radiation dose for the same image quality.

In this paper, we address the problem of fully automated decomposition of hyperspectral images for transmission light microscopy. The hyperspectral images are decomposed into spectrally homogeneous compounds. The resulting compounds are described by their spectral characteristics and optical density. We present the multiplicative physical model of image formation in transmission light microscopy, justify reduction of a hyperspectral image decomposition problem to a blind source separation problem, and provide method for hyperspectral restoration of separated compounds. In our approach, dimensionality reduction using principal component analysis (PCA) is followed by a blind source separation (BSS) algorithm. The BSS method is based on sparsifying transformation of observed images and relative Newton optimization procedure. The presented method was verified on hyperspectral images of biological tissues. The method was compared to the existing approach based on nonnegative matrix factorization. Experiments showed that the presented method is faster and better separates the biological compounds from imaging artifacts. The results obtained in this work may be used for improving automatic microscope hardware calibration and computer-aided diagnostics.

We consider an underdetermined linear system of equations Ax = b with non-negative entries in A and b, and seek a non-negative solution x. We generalize known equivalence results for the basis pursuit, for an arbitrary matrix A, and an arbitrary monotone element-wise concave penally replacing the lscr1-norm in the objective function. This result is then used to show that if there exists a sufficiently sparse solution to Ax = b, x > 0, it is necessarily unique.

An underdetermined linear system of equationsAx=bwith nonnegativity constraintxges 0 is considered. It is shown that for matricesAwith a row-span intersecting the positive orthant, if this problem admits a sufficiently sparse solution, it is necessarily unique. The bound on the required sparsity depends on a coherence property of the matrixA. This coherence measure can be improved by applying a conditioning stage onA, thereby strengthening the claimed result. The obtained uniqueness theorem relies on an extended theoretical analysis of the lscr0- lscr1equivalence developed here as well, considering a matrixAwith arbitrary column norms, and an arbitrary monotone element-wise concave penalty replacing the lscr1-norm objective function. Finally, from a numerical point of view, a greedy algorithm-a variant of the matching pursuit-is presented, such that it is guaranteed to find this sparse solution. It is further shown how this algorithm can benefit from well-designed conditioning ofA.

Shrinkage is a well known and appealing denoising technique. The use of shrinkage is known to be optimal for Gaussian white noise, provided that the sparsity on the signal’s representation is enforced using a unitary transform. Still, shrinkage is also practiced successfully with nonunitary, and even redundant representations. In this paper we shed some light on this behavior. We show that simple shrinkage could be interpreted as the first iteration of an algorithm that solves the basis pursuit denoising (BPDN) problem. Thus, this work leads to a novel iterative shrinkage algorithm that can be considered as an effective pursuit method. We demonstrate this algorithm, both on synthetic data, and for the image denoising problem, where we learn the image prior parameters directly from the given image. The results in both cases are superior to several popular alternatives. 1

We pose the problem of tissue classification in MRI as a blind source separation (BSS) problem and solve it by means of sparse component analysis (SCA). Assuming that most MR images can be sparsely represented, we consider their optimal sparse representation. Sparse components define a physically-meaningful feature space for classification. We demonstrate our approach on simulated and real multi-contrast MRI data. The proposed framework is general in that it is applicable to other modalities of medical imaging as well, whenever the linear mixing model is applicable.

The relative Newton algorithm, previously proposed for quasi-maximum likelihood blind source separation and blind deconvolution of one-dimensional signals is generalized for blind deconvolution of images. Smooth approximation of the absolute value is used as the nonlinear term for sparse sources. In addition, we propose a method of sparsification, which allows blind deconvolution of arbitrary sources, and show how to find optimal sparsifying transformations by supervised learning.

We present an efficient Newton-like algorithm for quasimaximum likelihood (QML) blind deconvolution of images. This algorithm exploits the sparse structure of the Hessian. An optimal distribution-shaping approach by means of sparsification allows one to use simple and convenient sparsity prior for processing of a wide range of natural images. Simulation results demonstrate the efficiency of the proposed method.

We address the problem of blind source separation (BSS) of superimposed images and, in particular, consider the recovery of a scene recorded through a semireflective medium (e.g. glass windshield) from its mixture with a virtual reflected image. We extend the sparse ICA (SPICA) approach to BSS and apply it to the separation of the desired image from the superimposed images, without having any a priori knowledge about its structure and/or statistics. Advances in the SPICA approach are discussed. Simulations and experimental results illustrate the efficiency of the proposed approach, and of its specific implementation in a simple algorithm of a low computational cost. The approach and the algorithm are generic in that they can be adapted and applied to a wide range of BSS problems involving one-dimensional signals or images.

The presented relative Newton method for quasi-maximum likelihood blind source separation significantly outperforms the natural gradient descent in batch mode. The structure of the corresponding Hessian matrix allows its fast inversion without assembling. Experiments with sparsely representable signals demonstrate super-efficient separation. More experiments with natural images are presented elsewhere (http://ie.technion.ac.il//spl sim/mcib/).

In this paper, we consider the problem of recovery of a scene recorded through a semireflective medium from its mixture with a virtual reflected image using the blind source separation (BSS) framework. We extend the sparse ICA (SPICA) approach and apply it to the separation of the desired image from the superimposed images, without having any a priory knowledge about its structure and/or statistics. Advances in the SPICA approach are discussed. Simulations and experimental results illustrate the efficiency of the proposed approach, and of its specific implementation in a simple algorithm of a low computational cost. The approach and the algorithm are generic and can be adapted and applied to a wide range of BSS problems involving one-dimensional signals or images.

It was previously shown that sparse representations can improve and simplify the estimation of an unknown mixing matrix of a set of images and thereby improve the quality of separation of source images. Here we propose a multiscale approach to the problem of blind separation of images from a set of their mixtures. We take advantage of the properties of multiscale transforms such as wavelet packets and decompose signals and images according to sets of local features. The resulting partial representations on a tree of data structure depict various degrees of sparsity. We show how the separation error is affected by the sparsity of the decomposition coefficients, and by the misfit between the prior, formulated in accordance with the probabilistic model of the coefficients distribution, and the actual distribution of the coefficients. Our error estimator, based on the Taylor expansion of the quasi log-likelihood function, is used in selection of the best subsets of coefficients, utilized in turn for further separation. The performance of the proposed method is assessed by separation of noise-free and noisy data. Experiments with simulated and real signals and images demonstrate significant improvement of separation quality over previously reported results.

We show an iterative reconstruction framework for diffraction ultrasound tomography. The use of broad-band illumination allows significant reduction of the number of projections compared to straight ray tomography. The proposed algorithm makes use of forward nonuniform fast Fourier transform (NUFFT) for iterative Fourier inversion. Incorporation of total variation regularization allows the reduction of noise and Gibbs phenomena while preserving the edges. The complexity of the NUFFT-based reconstruction is comparable to the frequency-domain interpolation (gridding) algorithm, whereas the reconstruction accuracy (in sense of the L2 and the L(infinity) norm) is better.

A classical technique for reconstruction of emission tomography (ET) images from measured projections is based on the maximum likelihood (ML) estimation, achieved with the expectation maximization (EM) algorithm. We incorporate the wavelet transform (WT) and total variation (TV) based penalties into the ML framework, and compare performance of the EM algorithm and the previously proposed conjugate barrier (CB) algorithm. Using the WT- and TV-based penalties allows one to embed regularization procedures into the iterative process. In the case of the WT-based penalty, we impose a subset of wavelet coefficients with a desired resolution on the objective function. It appears that the CB algorithm outperforms substantially the EM algorithm in penalized reconstruction. The properties of the optimization algorithms along with WTand TV-based regularization are demonstrated on image reconstructions of a synthetic brain phantom, and the quality of reconstruction is compared with standard methods.

The blind source separation problem is concerned with extraction of the underlying source signals from a set of their linear mixtures, where the mixing matrix is unknown. It was discovered recently, that exploiting the sparsity of sources in their representation according to some signal dictionary, dramatically improves the quality of separation. It is especially useful in image processing problems, wherein signals possess strong spatial sparsity. We use multiscale transforms, such as wavelet or wavelet packets, to decompose signals into sets of local features with various degrees of sparsity. We use this intrinsic property for selecting the best (most sparse) subsets of features for further separation. Experiments with 1D signals and images demonstrate significant improvement of separation quality.

We consider a problem of blind source separation from a set of instan(cid:173) taneous linear mixtures, where the mixing matrix is unknown. It was discovered recently, that exploiting the sparsity of sources in an appro(cid:173) priate representation according to some signal dictionary, dramatically improves the quality of separation. In this work we use the property of multi scale transforms, such as wavelet or wavelet packets, to decompose signals into sets of local features with various degrees of sparsity. We use this intrinsic property for selecting the best (most sparse) subsets of features for further separation. The performance of the algorithm is ver(cid:173) ified on noise-free and noisy data. Experiments with simulated signals, musical sounds and images demonstrate significant improvement of sep(cid:173) aration quality over previously reported results.

The blind source separation problem is to extract the underlying source signals from a set of linear mixtures, where the mixing matrix is unknown. This situation is common in acoustics, radio, medical signal and image processing, hyperspectral imaging, and other areas. We suggest a two-stage separation process: a priori selection of a possibly overcomplete signal dictionary (for instance, a wavelet frame or a learned dictionary) in which the sources are assumed to be sparsely representable, followed by unmixing the sources by exploiting the their sparse representability. We consider the general case of more sources than mixtures, but also derive a more efficient algorithm in the case of a nonovercomplete dictionary and an equal numbers of sources and mixtures. Experiments with artificial signals and musical sounds demonstrate significantly better separation than other known techniques.

In this paper,we present coincidence-list-ordered sets expectation-maximization (COSEM), an algorithm for iterative image reconstruction directly from list-mode coincidence acquisition data. The COSEM algorithm is based on the ordered sets EM algorithm for binned data but has several extensions that makes it suitable for rotating two planar detector tomographs. We develop the COSEM algorithm and extend it to include analytic calculation of detection probability, noise reducing iterative filtering schemes, and on-the-fly attenuation correction methods. We present an adaptation of COSEM to the Varicam\VG camera and show results from clinical and phantom studies.