Consistent Dynamic Mode Decomposition

We propose a new method for computing Dynamic Mode Decomposition (DMD) evolution matrices, which we use to analyze dynamical systems. Unlike the majority of existing methods, our approach is based on a variational formulation consisting of data alignment penalty terms and constitutive orthogonality constraints. Our method does not make any assumptions on the structure of the data or their size, and thus it is applicable to a wide range of problems including non-linear scenarios or extremely small observation sets. In addition, our technique is robust to noise that is independent of the dynamics and it does not require input data to be sequential. Our key idea is to introduce a regularization term for the forward and backward dynamics. The obtained minimization problem is solved efficiently using the Alternating Method of Multipliers (ADMM) which requires two Sylvester equation solves per iteration. Our numerical scheme converges empirically and is similar to a provably convergent ADMM scheme. We compare our approach to various state-of-the-art methods on several benchmark dynamical systems

Hybrid-State Free Precession in Nuclear Magnetic Resonance

The dynamics of large spin-1/2 ensembles in the presence of a varying magnetic field are commonly described by the Bloch equation. Most magnetic field variations result in unintuitive spin dynamics, which are sensitive to small deviations in the driving field. Although simplistic field variations can produce robust dynamics, the captured information content is impoverished. Here, we identify adiabaticity conditions that span a rich experiment design space with tractable dynamics. These adiabaticity conditions trap the spin dynamics in a one-dimensional subspace. Namely, the dynamics is captured by the absolute value of the magnetization, which is in a transient state, while its direction adiabatically follows the steady state. We define the hybrid state as the co-existence of these two states and identify the polar angle as the effective driving force of the spin dynamics. As an example, we optimize this drive for robust and efficient quantification of spin relaxation times and utilize it for magnetic resonance imaging of the human brain

Text mining with constrained tensor decomposition

International audienceText mining, as a special case of data mining, refers to the estimation of knowledge or parameters necessary for certain purposes, such as unsupervised clustering by observing various documents. In this context, the topic of a document can be seen as a hidden variable, and words are multi-view variables related to each other by a topic. The main goal in this paper is to estimate the probability of topics, and conditional probability of words given topics. To this end, we use non negative Canonical Polyadic (CP) decomposition of a third order moment tensor of observed words. Our computer simulations show that the proposed algorithm has better performance compared to a previously proposed algorithm, which utilizes the Robust tensor power method after whitening by second order moment. Moreover, as our cost function includes the non negativity constraint on estimated probabilities, we never obtain negative values in our estimated probabilities , whereas it is often the case with the power method combined with deflation. In addition, our algorithm is capable of handling over-complete cases, where the number of hidden variables is larger than that of multi-view variables, contrary to deflation-based techniques. Further, the method proposed therein supports a larger over-completeness compared to modified versions of the tensor power method, which has been customized to handle over-complete case

Total variation regularization for manifold-valued data

We consider total variation minimization for manifold valued data. We propose a cyclic proximal point algorithm and a parallel proximal point algorithm to minimize TV functionals with $\ell^p$-type data terms in the manifold case. These algorithms are based on iterative geodesic averaging which makes them easily applicable to a large class of data manifolds. As an application, we consider denoising images which take their values in a manifold. We apply our algorithms to diffusion tensor images, interferometric SAR images as well as sphere and cylinder valued images. For the class of Cartan-Hadamard manifolds (which includes the data space in diffusion tensor imaging) we show the convergence of the proposed TV minimizing algorithms to a global minimizer

Robust Orthogonal Complement Principal Component Analysis

Recently, the robustification of principal component analysis has attracted lots of attention from statisticians, engineers and computer scientists. In this work we study the type of outliers that are not necessarily apparent in the original observation space but can seriously affect the principal subspace estimation. Based on a mathematical formulation of such transformed outliers, a novel robust orthogonal complement principal component analysis (ROC-PCA) is proposed. The framework combines the popular sparsity-enforcing and low rank regularization techniques to deal with row-wise outliers as well as element-wise outliers. A non-asymptotic oracle inequality guarantees the accuracy and high breakdown performance of ROC-PCA in finite samples. To tackle the computational challenges, an efficient algorithm is developed on the basis of Stiefel manifold optimization and iterative thresholding. Furthermore, a batch variant is proposed to significantly reduce the cost in ultra high dimensions. The paper also points out a pitfall of a common practice of SVD reduction in robust PCA. Experiments show the effectiveness and efficiency of ROC-PCA in both synthetic and real data

Steady-state anatomical and quantitative magnetic resonance imaging of the heart using RF-frequencymodulated techniques

Cardiovascular disease (CVD) is the leading cause of death in the United States and Europe and generates healthcare costs of hundreds of billions of dollars annually. Conventional methods of diagnosing CVD are often invasive and carry risks for the patient. For example, the gold standard for diagnosing coronary artery disease, a major class of CVD, is x-ray coronary angiography, which has the disadvantages of being invasive, being expensive, using ionizing radiation, and having a ris k of complications. Conversely, coronary MR angiography (MRA) does not use ionizing radiation, can effectively visualize tissues without the need for exogenous contrast agents, and benefits from an adaptable temporal resolution. However, the acquisition time of cardiac MRI is far longer than the temporal scales of cardiac and respiratory motion, necessitating some method of compensating for this motion. The free-running framework is a novel development in our lab, benefitting from advances over the past three decades, that attempts to address disadvantages of previous cardiac MRI approaches: it provides fully self-gated 5D cardiac MRI with a simplified workflow, improved ease-of-use, reduced operator dependence, and automatic patient-specific motion detection. Free-running imaging increases the amount of information available to the clinician and is flexible enough to be translated to different app lications within cardiac MRI. Moreover, the self-gating of the free-running framework decoupled the acquisition from the motion compensation and thereby opened up cardiac MRI to the wider class of steady-state-based techniques utilizing balanced steady-state free precession (bSSFP) sequences, which have the benefits of practical simplicity and high signal-to-noise ratio. The focus of this thesis was therefore on the application of steady- state techniques to cardiac MRI. The first part addressed the long acquisition time of the current free-running framework and focused on anatomical coronary imaging. The published protocol of the free- running framework used an interrupted bSSFP acquisition where CHESS fat saturation modules were inserted to provide blood-fat contrast, as they suppress the signal of fat tissue surrounding the coronary arteries, and were followed by ramp-up pulses to reduce artefacts arising from the return to steady-state. This interrupted acquisition, however, suffered from an interrupted steady-state, reduced time efficiency, and higher specific absorption rate (SAR). Using novel lipid-insensitive binomial off-resonant RF excitation (LIBRE) pulses developed in our lab, the first project showed that LIBRE pulses incorporated into an uninterrupted free-running bSSFP sequence could be successfully used for 5D cardiac MRI at 1.5T. The free-running LIBRE approach reduced the acquisition time and SAR relative to the previous interrupted approach while maintaining image quality and vessel conspicuity. Furthermore, this had been the first successful use of a fat-suppressing RF excitation pulse in an uninterrupted bSSFP sequence for cardiac imaging, demonstrating that uninterrupted bSSFP can be used for cardiac MRI and addressing the problem of clinical sequence availability. Inspired by the feasibility of uninterrupted bSSFP for cardiac MRI, the second part investigated the potential of PLANET, a novel 3D multiparametric mapping technique, for free-running 5D myocardial mapping. PLANET utilizes a phase-cycled bSSFP acquisition and a direct ellipse-fitting algorithm to calculate T1 and T2 relaxation times, which suggested that it could be readily integrated into the free-running framework without interrupting the steady-state. After initially calibrating the acquisition, the possibility of accelerating the static PLANET acquisition was explored prior to applying it to the moving heart. It was shown that PLANET accuracy and precision could be maintained with two-fold acceleration with a 3D Cartesian spiral trajectory, suggesting that PLANET for myocardial mapping with the free-running 5D radial acquisition is feasible. Further work should investigate optimizing the reconstruction scheme, improving the coil sensitivity estimate, and examining the use of the radial trajectory with a view to implementing free-running 5D myocardial T1 and T2 mapping. This thesis presents two approaches utilizing RF-frequency-modulated steady-state techniques for cardiac MRI. The first approach involved the novel application of an uninterrupted bSSFP acquisition with off-resonant RF excitation for anatomical coronary imaging. The second approach investigated the use of phase-cycled bSSFP for free-running 5D myocardial T1 and T2 mapping. Both methods addressed the challenge of clinical availability of sequences in cardiac MRI, by showing that a common and simple sequence like bSSFP can be used for acquisition while the steps of motion compensation and reconstruction can be handled offline, and thus have the potential to improve adoption of cardiac MRI. -- Les maladies cardiovasculaires (MCV) représentent la principale cause de décès aux États-Unis et en Europe et génèrent des coûts de santé de plusieurs centaines de milliards de dollars par an. Les méthodes conventionnelles de diagnostic des MCV sont souvent invasives et comportent des risques pour le patient. Par exemple, la méthode de référence pour le diagnostic de la maladie coronarienne, une catégorie majeure de MCV, est la coronarographie par rayons X qui a comme inconvénients son caractère invasif, son coût, l’utilisation de rayonnements ionisants et le risque de complications. A l’inverse, l'angiographie coronarienne par résonance magnétique (ARM) n'utilise pas de rayonnements ionisants, permet de visualiser efficacement les tissus sans avoir recours à des agents de contraste exogènes et bénéficie d'une résolution temporelle ajustable. Cependant, le temps d'acquisition en IRM cardiaque est bien plus long que les échelles temporelles des mouvements cardiaques et respiratoires en jeu, ce qui rend la compensation de ces mouvements indispensable. Le cadre dit de « free -running » est un nouveau développement de notre laboratoire qui bénéficie des progrès réalisés au cours des trois dernières décennies et tente de remédier aux inconvénients des approches précédentes pour l'IRM cardiaque : il fournit une IRM cardiaque en cinq dimensions (5D) complètement « self-gated » , c’est-à-dire capable de détecter les mouvements cardiaques et respiratoires, forte d’une implémentation simplifiée, d’une plus grande facilité d'utilisation, d’une dépendance réduite vis-à-vis de l'opérateur et d’une détection automatique des mouvements spécifiques du patient. L'imagerie « free- running » augmente la quantité d'informations à disposition du clinicien et est suffisamment flexible pour être appliquée à différents domaines de l'IRM cardiaque. De plus, le « self-gating » du cadre « free-running » a découplé l'acquisition de la compensation de mouvement et a ainsi ouvert l'IRM cardiaque à la classe plus large des techniques basées sur l'état stationnaire utilisant des séquences de précession libre équilibrée en état stationnaire (bSSFP), qui se distinguent par leur simplicité d’utilisation et leur rapport signal sur bruit élevé. Le thème de cette thèse est donc l'application des techniques basées sur l'état stationnaire à l'IRM cardiaque. La première partie porte sur le long temps d'acquisition de l'actuel cadre « free-running» et se concentre sur l'imagerie anatomique coronaire. Le protocole publié utilise une acquisition bSSFP interrompue où des modules de saturation de graisse (CHESS) sont insérés de façon à fournir un contraste sang-graisse puisqu’ils suppriment le signal du tissu graisseux entourant les artères coronaires, et sont suivis par des impulsions en rampe pour réduire les artefacts résultant du retour à l'état stable. Cette acquisition interrompue souffre cependant d'un état d'équilibre interrompu, d'une efficacité temporelle réduite et d'un débit d'absorption spécifique (DAS) plus élevé. En utilisant les nouvelles impulsions d'excitation radiofréquence (RF) binomiales hors -résonance insensibles aux lipides (LIBRE) développées dans notre laboratoi re, ce premier projet montre que les impulsions LIBRE incorporées dans une séquence bSSFP ininterrompue et « free-running » peuvent être utilisées avec succès pour l'IRM cardiaque 5D à 1,5 T. L'approche « free-running LIBRE » permet de réduire le temps d'acquisition et le DAS par rapport à l'approche interrompue précédente, tout en maintenant la perceptibilité des artères coronariennes. En outre, il s'agit de la première utilisation réussie d'une impulsion d'excitation RF supprimant la graisse dans une séquence bSSFP ininterrompue pour l'imagerie cardiaque, ce qui démontre le potentiel d’utilisation de la séquence bSSFP ininterrompue pour l'IRM cardiaque et résout le problème de la disponibilité de la séquence en clinique. Inspirée par la faisabilité d’utilisation de la séquence bSSFP ininterrompue pour l'IRM cardiaque, la deuxième partie étudie le potentiel de PLANET, une nouvelle technique de cartographie 3D multiparamétrique, pour la cartographie 5D du myocarde via l’imagerie « free-running ». PLANET utilise une acquisition bSSFP à cycle de phase et un algorithme d'ajustement d'ellipse direct pour calculer les temps de relaxation T1 et T2, ce qui suggère que cette méthode pourrait être facilement intégrée au cadre « free - running » sans interruption de l’état d'équilibre. Après calibration de l'acquisition, nous explorons la possibilité d'accélérer l'acquisition statique de PLANET pour l'appliquer au cœur. Nous démontrons que l'exactitude et la précision de PLANET peuvent être maintenues pour une accélération double avec une trajectoire 3D cartésienne en spirale, ce qui suggère que PLANET est réalisable pour la cartographie du myocarde avec une acquisition radiale 5D « free-running ». D'autres travaux devraient porter sur l'optimisation du schéma de reconstruction, l'amélioration de l'estimation de la sensibilité de l’antenne et l'examen de l'utilisation de la trajectoire radiale en vue de la mise en œuvre de la cartographie 5D « free-running » T1 et T2 du myocarde. Cette thèse présente deux approches utilisant des techniques de modulation de fréquence radio en état stationnaire pour l'IRM cardiaque. La première approche implique l'application nouvelle d'une acquisition bSSFP ininterrompue avec une excitation RF hors résonance pour l'imagerie anatomique coronaire. La seconde approche porte sur l'utilisation d’une séquence bSSFP à cycle de phase pour la cartographie 5D T1 et T2 du myocarde. Ces deux méthodes permettent de répondre au défi posé par la disponibilité des séquences en IRM cardiaque en montrant qu'une séquence commune et simple comme la bSSFP peut être utilisée pour l'acquisition, tandis que les étapes de compensation du mouvement et de reconstruction peuvent être traitées hors ligne. Ainsi, ces méthodes ont le potentiel de favoriser l'adoption de l'IRM cardiaque