Sparse image reconstruction on the sphere: implications of a new sampling theorem

We study the impact of sampling theorems on the fidelity of sparse image reconstruction on the sphere. We discuss how a reduction in the number of samples required to represent all information content of a band-limited signal acts to improve the fidelity of sparse image reconstruction, through both the dimensionality and sparsity of signals. To demonstrate this result we consider a simple inpainting problem on the sphere and consider images sparse in the magnitude of their gradient. We develop a framework for total variation (TV) inpainting on the sphere, including fast methods to render the inpainting problem computationally feasible at high-resolution. Recently a new sampling theorem on the sphere was developed, reducing the required number of samples by a factor of two for equiangular sampling schemes. Through numerical simulations we verify the enhanced fidelity of sparse image reconstruction due to the more efficient sampling of the sphere provided by the new sampling theorem.Comment: 11 pages, 5 figure

Implications for compressed sensing of a new sampling theorem on the sphere

A sampling theorem on the sphere has been developed recently, requiring half as many samples as alternative equiangular sampling theorems on the sphere. A reduction by a factor of two in the number of samples required to represent a band-limited signal on the sphere exactly has important implications for compressed sensing, both in terms of the dimensionality and sparsity of signals. We illustrate the impact of this property with an inpainting problem on the sphere, where we show the superior reconstruction performance when adopting the new sampling theorem compared to the alternative.Comment: 1 page, 2 figures, Signal Processing with Adaptive Sparse Structured Representations (SPARS) 201

Sympathetic activity and early mobilization in patients in intensive and intermediate care with severe brain injuries: a preliminary prospective randomized study.

Patients who experience severe brain injuries are at risk of secondary brain damage, because of delayed vasospasm and edema. Traditionally, many of these patients are kept on prolonged bed rest in order to maintain adequate cerebral blood flow, especially in the case of subarachnoid hemorrhage. On the other hand, prolonged bed rest carries important morbidity. There may be a clinical benefit in early mobilization and our hypothesis is that early gradual mobilization is safe in these patients. The aim of this study was to observe and quantify the changes in sympathetic activity, mainly related to stress, and blood pressure in gradual postural changes by the verticalization robot (Erigo®) and after training by a lower body ergometer (MOTOmed-letto®), after prolonged bed rest of minimum 7 days. Thirty patients with severe neurological injuries were randomized into 3 groups with different protocols of mobilization: Standard, MOTOmed-letto® or Erigo® protocol. We measured plasma catecholamines, metanephrines and blood pressure before, during and after mobilization. Blood pressure does not show any significant difference between the 3 groups. The analysis of the catecholamines suggests a significant increase in catecholamine production during Standard mobilization with physiotherapists and with MOTOmed-letto® and no changes with Erigo®. This preliminary prospective randomized study shows that the mobilization of patients with severe brain injuries by means of Erigo® does not increase the production of catecholamines. It means that Erigo® is a well-tolerated method of mobilization and can be considered a safe system of early mobilization of these patients. Further studies are required to validate our conclusions. The study was registered in the ISRCTN registry with the trial registration number ISRCTN56402432 . Date of registration: 08.03.2016. Retrospectively registered

Outcome Prediction of Consciousness Disorders in the Acute Stage Based on a Complementary Motor Behavioural Tool.

Attaining an accurate diagnosis in the acute phase for severely brain-damaged patients presenting Disorders of Consciousness (DOC) is crucial for prognostic validity; such a diagnosis determines further medical management, in terms of therapeutic choices and end-of-life decisions. However, DOC evaluation based on validated scales, such as the Revised Coma Recovery Scale (CRS-R), can lead to an underestimation of consciousness and to frequent misdiagnoses particularly in cases of cognitive motor dissociation due to other aetiologies. The purpose of this study is to determine the clinical signs that lead to a more accurate consciousness assessment allowing more reliable outcome prediction. From the Unit of Acute Neurorehabilitation (University Hospital, Lausanne, Switzerland) between 2011 and 2014, we enrolled 33 DOC patients with a DOC diagnosis according to the CRS-R that had been established within 28 days of brain damage. The first CRS-R assessment established the initial diagnosis of Unresponsive Wakefulness Syndrome (UWS) in 20 patients and a Minimally Consciousness State (MCS) in the remaining13 patients. We clinically evaluated the patients over time using the CRS-R scale and concurrently from the beginning with complementary clinical items of a new observational Motor Behaviour Tool (MBT). Primary endpoint was outcome at unit discharge distinguishing two main classes of patients (DOC patients having emerged from DOC and those remaining in DOC) and 6 subclasses detailing the outcome of UWS and MCS patients, respectively. Based on CRS-R and MBT scores assessed separately and jointly, statistical testing was performed in the acute phase using a non-parametric Mann-Whitney U test; longitudinal CRS-R data were modelled with a Generalized Linear Model. Fifty-five per cent of the UWS patients and 77% of the MCS patients had emerged from DOC. First, statistical prediction of the first CRS-R scores did not permit outcome differentiation between classes; longitudinal regression modelling of the CRS-R data identified distinct outcome evolution, but not earlier than 19 days. Second, the MBT yielded a significant outcome predictability in the acute phase (p<0.02, sensitivity>0.81). Third, a statistical comparison of the CRS-R subscales weighted by MBT became significantly predictive for DOC outcome (p<0.02). The association of MBT and CRS-R scoring improves significantly the evaluation of consciousness and the predictability of outcome in the acute phase. Subtle motor behaviour assessment provides accurate insight into the amount and the content of consciousness even in the case of cognitive motor dissociation

Harmonic Wavelet Transform and Image Approximation

In 2006, Saito and Remy proposed a new transform called the Laplace Local Sine Transform (LLST) in image processing as follows. Let f be a twice continuously differentiable function on a domain Ω. First we approximate f by a harmonic function u such that the residual component v=f−u vanishes on the boundary of Ω. Next, we do the odd extension for v, and then do the periodic extension, i.e. we obtain a periodic odd function v *. Finally, we expand v * into Fourier sine series. In this paper, we propose to expand v * into a periodic wavelet series with respect to biorthonormal periodic wavelet bases with the symmetric filter banks. We call this the Harmonic Wavelet Transform (HWT). HWT has an advantage over both the LLST and the conventional wavelet transforms. On the one hand, it removes the boundary mismatches as LLST does. On the other hand, the HWT coefficients reflect the local smoothness of f in the interior of Ω. So the HWT algorithm approximates data more efficiently than LLST, periodic wavelet transform, folded wavelet transform, and wavelets on interval. We demonstrate the superiority of HWT over the other transforms using several standard images

Decoding of Emotional Information in Voice-Sensitive Cortices

The ability to correctly interpret emotional signals from others is crucial for successful social interaction. Previous neuroimaging studies showed that voice-sensitive auditory areas [1-3] activate to a broad spectrum of vocally expressed emotions more than to neutral speech melody (prosody). However, this enhanced response occurs irrespective of the specific emotion category, making it impossible to distinguish different vocal emotions with conventional analyses [4-8]. Here, we presented pseudowords spoken in five prosodic categories (anger, sadness, neutral, relief, joy) during event-related functional magnetic resonance imaging (fMRI), then employed multivariate pattern analysis [9, 10] to discriminate between these categories on the basis of the spatial response pattern within the auditory cortex. Our results demonstrate successful decoding of vocal emotions from fMRI responses in bilateral voice-sensitive areas, which could not be obtained by using averaged response amplitudes only. Pairwise comparisons showed that each category could be classified against all other alternatives, indicating for each emotion a specific spatial signature that generalized across speakers. These results demonstrate for the first time that emotional information is represented by distinct spatial patterns that can be decoded from brain activity in modality-specific cortical areas

Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering

Discontinuous Galerkin (DG) methods are a popular class of numerical techniques to solve partial differential equations due to their higher order of accuracy. However, the inter-element discontinuity of a DG solution hinders its utility in various applications, including visualization and feature extraction. This shortcoming can be alleviated by postprocessing of DG solutions to increase the inter-element smoothness. A class of postprocessing techniques proposed to increase the inter-element smoothness is SIAC filtering. In addition to increasing the inter-element continuity, SIAC filtering also raises the convergence rate from order k+1k+1 to order 2k+12k+1 . Since the introduction of SIAC filtering for univariate hyperbolic equations by Cockburn et al. (Math Comput 72(242):577–606, 2003), many generalizations of SIAC filtering have been proposed. Recently, the idea of dimensionality reduction through rotation has been the focus of studies in which a univariate SIAC kernel has been used to postprocess a two-dimensional DG solution (Docampo-Sánchez et al. in Multi-dimensional filtering: reducing the dimension through rotation, 2016. arXiv preprint arXiv:1610.02317). However, the scope of theoretical development of multidimensional SIAC filters has never gone beyond the usage of tensor product multidimensional B-splines or the reduction of the filter dimension. In this paper, we define a new SIAC filter called hexagonal SIAC (HSIAC) that uses a nonseparable class of two-dimensional spline functions called hex splines. In addition to relaxing the separability assumption, the proposed HSIAC filter provides more symmetry to its tensor-product counterpart. We prove that the superconvergence property holds for a specific class of structured triangular meshes using HSIAC filtering and provide numerical results to demonstrate and validate our theoretical results

Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem