Lipschitz-constrained neural networks have several advantages over unconstrained ones and can be applied to a variety of problems, making them a topic of attention in the deep learning community. Unfortunately, it has been shown both theoretically and empirically that they perform poorly when equipped with ReLU activation functions. By contrast, neural networks with learnable 1-Lipschitz linear splines are known to be more expressive. In this paper, we show that such networks correspond to global optima of a constrained functional optimization problem that consists of the training of a neural network composed of 1-Lipschitz linear layers and 1-Lipschitz freeform activation functions with second-order total-variation regularization. Further, we propose an efficient method to train these neural networks. Our numerical experiments show that our trained networks compare favorably with existing 1-Lipschitz neural architectures.

Thanks to its capability of acquiring full-view frames at multiple kilohertz, ultrafast ultrasound imaging unlocked the analysis of rapidly changing physical phenomena in the human body, with pioneering applications such as ultrasensitive flow imaging in the cardiovascular system or shear-wave elastography. The accuracy achievable with these motion estimation techniques is strongly contingent upon two contradictory requirements: a high quality of consecutive frames and a high frame rate. Indeed, the image quality can usually be improved by increasing the number of steered ultrafast acquisitions, but at the expense of a reduced frame rate and possible motion artifacts. To achieve accurate motion estimation at uncompromised frame rates and immune to motion artifacts, the proposed approach relies on single ultrafast acquisitions to reconstruct high-quality frames and on only two consecutive frames to obtain 2-D displacement estimates. To this end, we deployed a convolutional neural network-based image reconstruction method combined with a speckle tracking algorithm based on cross-correlation. Numerical and in vivo experiments, conducted in the context of plane-wave imaging, demonstrate that the proposed approach is capable of estimating displacements in regions where the presence of side lobe and grating lobe artifacts prevents any displacement estimation with a state-of-the-art technique that relies on conventional delay-and-sum beamforming. The proposed approach may therefore unlock the full potential of ultrafast ultrasound, in applications such as ultrasensitive cardiovascular motion and flow analysis or shear-wave elastography.

We investigate the benefits of known partial support for the recovery of joint-sparse signals and demonstrate that it is advantageous in terms of recovery performance for both rank-blind and rank-aware algorithms. We suggest extensions of several joint-sparse recovery algorithms, e.g., simultaneous normalized iterative hard thresholding, subspace greedy methods and subspace-augmented multiple signal classification techniques. We describe a direct application of the proposed methods for compressive multiplexing of ultrasound (US) signals. The technique exploits the compressive multiplexer architecture for signal compression and relies on joint-sparsity of US signals in the frequency domain for signal reconstruction. We validate the proposed algorithms on numerical experiments and show their superiority against state-of-the-art approaches in rank-defective cases. We also demonstrate that the techniques lead to a significant increase of the image quality on in vivo carotid images compared to reconstruction without partially known support. The supporting code is available on https://github.com/AdriBesson/spl2018_joint_sparse.

This paper considers the problem of reconstructing raw signals from random projections in the context of time-of-flight imaging with an array of sensors. It presents a new signal model, coined as multi-channel pulse-stream model, which exploits pulse-stream models and accounts for additional structure induced by inter-sensor dependencies. We propose a sampling theorem and a reconstruction algorithm, based on ℓ -minimization, for signals belonging to such a model. We demonstrate the benefits of the proposed approach by means of numerical simulations and on a real non-destructive-evaluation application where the peak-signal-to-noise-ratio is increased by 3 dB compared to standard compressed-sensing strategies.

We propose to map the fast iterative soft thresholding algorithm to a deep neural network (DNN), with a sparsity prior in a concatenation of wavelet bases, in the context of compressive imaging. We exploit the DNN architecture to learn the optimal weight matrix of the corresponding reweighted ℓ1-minimization problem. We later use the learned weight matrix for the image reconstruction process, which is recast as a simple ℓ1-minimization problem. The approach, denoted as learned extended FISTA, shows promising results in terms of image quality, compared to state-of-the art algorithms, and significantly reduces the reconstruction time required to solve the reweighted ℓ1-minimization problem.