In this paper we revisit the sparse multiple measurement vector (MMV) problem, where the aim is to recover a set of jointly sparse multichannel vectors from incomplete measurements. This problem has received increasing interest as an extension of single channel sparse recovery, which lies at the heart of the emerging field of compressed sensing. However, MMV approximation has origins in the field of array signal processing as we discuss in this paper.Inspired by these links, we introduce a new family of MMV algorithms based on the well-know MUSIC method in array processing. We particularly highlight the role of the rank of the unknown signal matrix X in determining the difficulty of the recovery problem. We begin by deriving necessary and sufficient conditions for the uniqueness of the sparse MMV solution, which indicates that the larger the rank of X the less sparse X needs to be to ensure uniqueness. We also show that as the rank of X increases, the computational effort required to solve the MMV problem through a combinatorial search is reduced.In the second part of the paper we consider practical suboptimal algorithms for MMV recovery. We examine the rank awareness of popular methods such as Simultaneous Orthogonal Matching Pursuit (SOMP) and mixed norm minimization techniques and show them to be rank blind in terms of worst case analysis. We then consider a family of greedy algorithms that are rank aware. The simplest such method is a discrete version of the MUSIC algorithm popular in array signal processing. This approach is guaranteed to recover the sparse vectors in the full rank MMV setting under mild conditions. We then extend this idea to develop a rank aware pursuit algorithm that naturally reduces to Order Recursive Matching Pursuit (ORMP) in the single measurement case. This approach also provides guaranteed recovery in the full rank setting. Numerical simulations demonstrate that the rank aware techniques are significantly better than existing methods in dealing with multiple measurements.

No abstract

Abstract. Inspired by the recently proposed Magnetic Resonance Fingerprinting (MRF) technique, we develop a principled compressed sensing framework for quantitative MRI. The three key components are: a random pulse excitation sequence following the MRF technique; a random EPI subsampling strategy and an iterative projection algorithm that imposes consistency with the Bloch equations. We show that theoretically, as long as the excitation sequence possesses an appropriate form of persistent excitation, we are able to accurately recover the proton density, T1, T2 and off-resonance maps simultaneously from a limited number of samples. These results are further supported through extensive simulations using a brain phantom.Key words. Compressed sensing, MRI, Bloch equations, manifolds, Johnston-Linderstrauss embedding 1. Introduction. Inspired by the recently proposed procedure of Magnetic Resonance Fingerprinting (MRF), which gives a new technique for quantitative MRI, we investigate this idea from a compressed sensing perspective. While MRF itself, was inspired by the recent growth of compressed sensing (CS) techniques in MRI [29], the exact link to CS was not made explicit, and the paper does not consider a full CS formulation. Indeed the role of sparsity, random excitation and sampling are not clarified. The goal of this current paper is to make the links with CS explicit, shed light on the appropriate acquisition and reconstruction procedures and hence to develop a full compressed sensing strategy for quantitative MRI.In particular, we identify separate roles for the pulse excitation and the subsampling of k-space. We identify the Bloch response manifold as the appropriate low dimensional signal model on which the CS acquisition is performed, and interpret the "model-based" dictionary of [29] as a natural discretization of this response manifold We also discuss what is necessary in order to have an appropriate CS-type acquisition scheme.Having identified the underlying signal model we next turn to the reconstruction process. In [29] this was performed through pattern matching using a matched filter based on the model-based dictionary. However, this does not offer the opportunity for exact reconstruction, even if the signal is hypothesised to be 1-sparse in this dictionary due to the undersampling of k-space. This suggests that we should look to a model based CS framework that directly supports such manifold models [4]. Recent algorithmic work in this direction has been presented by Iwen and Maggioni [23], however, their approach is not practical in the present context as the computational cost of their scheme grows exponentially with the dimension of the manifold. Instead, we leverage recent results from [11] and develop a recovery algorithm based on the Projected Landweber Algorithm (PLA). This method also has the appealing interpretation of an iterated refinement of the original MRF scheme.The remainder of the paper is set out as follows. We begin by giving a brief overview of MRI acquisition. Then we discuss ...

This review aims to summarize the current knowledge of molecular pathways and their clinical relevance in melanoma. Metastatic melanoma was a grim diagnosis, but in recent years tremendous advances have been made in treatments. Chemotherapy provided little benefit in these patients, but development of targeted and new immune approaches made radical changes in prognosis. This would not have happened without remarkable advances in understanding the biology of disease and tremendous progress in the genomic (and other “omics”) scale analyses of tumors. The big problems facing the field are no longer focused exclusively on the development of new treatment modalities, though this is a very busy area of clinical research. The focus shifted now to understanding and overcoming resistance to targeted therapies, and understanding the underlying causes of the heterogeneous responses to immune therapy.

Takens' Embedding Theorem forms the basis of virtually all approaches to the analysis of time series generated by nonlinear deterministic dynamical systems. It typically allows us to reconstruct an unknown dynamical system which gave rise to a given observed scalar time series simply by constructing a new state space out of successive values of the time series. This provides the theoretical foundation for many popular techniques, including those for the measurement of fractal dimensions and Liapunov exponents, for the prediction of future behaviour, for noise reduction and signal separation, and most recently for control and targeting. Current versions of Takens' Theorem assume that the underlying system is autonomous (and noise free). Unfortunately this is not the case for many real systems. In a previous paper, one of us showed how to extend Takens' Theorem to deterministically forced systems. Here, we use similar techniques to prove a number of delay embedding theorems for arbitrarily and stochastically forced systems. As a special case, we obtain embedding results for Iterated Functions Systems, and we also briefly consider noisy observations.