For certain sensing matrices, the Approximate Message Passing (AMP) algorithm efficiently reconstructs undersampled signals. However, in Magnetic Resonance Imaging (MRI), where Fourier coefficients of a natural image are sampled with variable density, AMP encounters convergence problems. In response we present an algorithm based on Orthogonal AMP constructed specifically for variable density partial Fourier sensing matrices. For the first time in this setting a state evolution has been observed. A practical advantage of state evolution is that Stein's Unbiased Risk Estimate (SURE) can be effectively implemented, yielding an algorithm with no free parameters. We empirically evaluate the effectiveness of the parameter-free algorithm on simulated data and find that it converges over 5x faster and to a lower mean-squared error solution than Fast Iterative Shrinkage-Thresholding (FISTA).

We propose to learn a low-dimensional probabilistic deformation model from data which can be used for the registration and the analysis of deformations. The latent variable model maps similar deformations close to each other in an encoding space. It enables to compare deformations, to generate normal or pathological deformations for any new image, or to transport deformations from one image pair to any other image. Our unsupervised method is based on the variational inference. In particular, we use a conditional variational autoencoder network and constrain transformations to be symmetric and diffeomorphic by applying a differentiable exponentiation layer with a symmetric loss function. We also present a formulation that includes spatial regularization such as the diffusion-based filters. In addition, our framework provides multi-scale velocity field estimations. We evaluated our method on 3-D intra-subject registration using 334 cardiac cine-MRIs. On this dataset, our method showed the state-of-the-art performance with a mean DICE score of 81.2% and a mean Hausdorff distance of 7.3 mm using 32 latent dimensions compared to three state-of-the-art methods while also demonstrating more regular deformation fields. The average time per registration was 0.32 s. Besides, we visualized the learned latent space and showed that the encoded deformations can be used to transport deformations and to cluster diseases with a classification accuracy of 83% after applying a linear projection.

Denoising is an indispensable step in processing low-dose X-ray fluoroscopic images that requires development of specialized high-quality algorithms able to operate in near real-time. We address this problem with an efficient deep learning approach based on the process-centric view of traditional iterative thresholding methods. We develop a novel trainable patch-based multiscale framework for sparse image representation. In a computationally efficient way, it allows us to accurately reconstruct important image features on multiple levels of decomposition with patch dictionaries of reduced size and complexity. The flexibility of the chosen machine learning approach allows us to tailor the learned basis for preserving important structural information in the image and noticeably minimize the amount of artifacts. Our denoising results obtained with real clinical data demonstrate significant quality improvement and are computed much faster in comparison with the BM3D algorithm.

In this work, we study the links between the recovery properties of sparse signals for Orthogonal Matching Pursuit (OMP) and the whole General MP class over nested supports. We show that the optimality of those algorithms is not locally nested: there is a dictionary and supports I and J with J included in I such that OMP will recover all signals of support I, but not all signals of support J. We also show that the optimality of OMP is globally nested: if OMP can recover all s-sparse signals, then it can recover all s'-sparse signals with s' smaller than s. We also provide a tighter version of Donoho and Elad's spark theorem, which allows us to complete Tropp's proof that sparse representation algorithms can only be optimal for all s-sparse signals if s is strictly lower than half the spark of the dictionary.

It is shown that Theorem 10 (Non-Nestedness of ERC) in [Plumbley, IEEE Trans. Inf. Theory, vol. 53, pp. 3188-3195, Sep. 2007] neglects the derivations of the exact recovery conditions (ERCs) of constrained l1-minimization (BP) and orthogonal matching pursuit. This means that it does not reflect the recovery properties of these algorithms. Furthermore, an ERC of BP more general than that in [Tropp, IEEE Trans. Inf. Theory, vol. 50, pp. 2231-2242, Oct. 2004] is shown.

This work considers the problem of learning an incoherent dictionary that is both adapted to a set of training data and incoherent so that existing sparse approximation algorithms can recover the sparsest representation. A new decorrelation method is presented that computes a fixed coherence dictionary close to a given dictionary. That step iterates pairwise decorrelations of atoms in the dictionary. Dictionary learning is then performed by adding this decorrelation method as an intermediate step in the K-SVD learning algorithm. The proposed algorithm INK-SVD is tested on musical data and compared to another existing decorrelation method. INK-SVD can compute a dictionary that approximates the training data as well as K-SVD while decreasing the coherence from 0.6 to 0.2.

We propose a new method for ventricular cancellation and atrial modelling in the ECG of patients suffering from atrial fibrillation. Our method is based on dictionary learning. It extends both the average beat subtraction and the sparse source separation approaches. Experiments on synthetic data show that this method can almost completely suppress the ventricular activity, but it generates some artifacts. Contrary to other ventricular cancellations methods, our approach also learns a model for the atrial activity.

We propose a variant of orthogonal matching pursuit (OMP), called LoCOMP, for scalable sparse signal approximation. The algorithm is designed for shift-invariant signal dictionaries with localized atoms, such as time-frequency dictionaries, and achieves approximation performance comparable to OMP at a computational cost similar to matching pursuit. Numerical experiments with a large audio signal show that, compared to OMP and gradient pursuit, the proposed algorithm runs in over 500 less time while leaving the approximation error almost unchanged.

This paper introduces p-thresholding, an algorithm to compute simultaneous sparse approximations of multichannel signals over redundant dictionaries. We work out both worst case and average case recovery analyses of this algorithm and show that the latter results in much weaker conditions on the dictionary. Numerical simulations confirm our theoretical findings and show that p-thresholding is an interesting low complexity alternative to simultaneous greedy or convex relaxation algorithms for processing sparse multichannel signals with balanced coefficients.

Real-world phenomena involve complex interactions between multiple signal modalities. As a consequence, humans are used to integrate at each instant perceptions from all their senses in order to enrich their understanding of the surrounding world. This paradigm can be also extremely useful in many signal processing and computer vision problems involving mutually related signals. The simultaneous processing of multimodal data can, in fact, reveal information that is otherwise hidden when considering the signals independently. However, in natural multimodal signals, the statistical dependencies between modalities are in general not obvious. Learning fundamental multimodal patterns could offer deep insight into the structure of such signals. In this paper, we present a novel model of multimodal signals based on their sparse decomposition over a dictionary of multimodal structures. An algorithm for iteratively learning multimodal generating functions that can be shifted at all positions in the signal is proposed, as well. The learning is defined in such a way that it can be accomplished by iteratively solving a generalized eigenvector problem, which makes the algorithm fast, flexible, and free of user-defined parameters. The proposed algorithm is applied to audiovisual sequences and it is able to discover underlying structures in the data. The detection of such audio-video patterns in audiovisual clips allows to effectively localize the sound source on the video in presence of substantial acoustic and visual distractors, outperforming state-of-the-art audiovisual localization algorithms.