Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems

Stochastic physical problems governed by nonlinear conservation laws are challenging due to solution discontinuities in stochastic and physical space. In this paper, we present a level set method to track discontinuities in stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed function that vanishes at discontinuities, the iso-zero of the level set problem coincide with the discontinuities of the conservation law. The level set problem is solved on a sequence of successively finer grids in stochastic space. The method is adaptive in the sense that costly evaluations of the conservation law of interest are only performed in the vicinity of the discontinuities during the refinement stage. In regions of stochastic space where the solution is smooth, a surrogate method replaces expensive evaluations of the conservation law. The proposed method is tested in conjunction with different sets of localized orthogonal basis functions on simplex elements, as well as frames based on piecewise polynomials conforming to the level set function. The performance of the proposed method is compared to existing adaptive multi-element generalized polynomial chaos methods

Graph Laplacian for Image Anomaly Detection

Reed-Xiaoli detector (RXD) is recognized as the benchmark algorithm for image anomaly detection; however, it presents known limitations, namely the dependence over the image following a multivariate Gaussian model, the estimation and inversion of a high-dimensional covariance matrix, and the inability to effectively include spatial awareness in its evaluation. In this work, a novel graph-based solution to the image anomaly detection problem is proposed; leveraging the graph Fourier transform, we are able to overcome some of RXD's limitations while reducing computational cost at the same time. Tests over both hyperspectral and medical images, using both synthetic and real anomalies, prove the proposed technique is able to obtain significant gains over performance by other algorithms in the state of the art.Comment: Published in Machine Vision and Applications (Springer

Multi-resolution analysis for ENO schemes

Given an function, u(x), which is represented by its cell-averages in cells which are formed by some unstructured grid, we show how to decompose the function into various scales of variation. This is done by considering a set of nested grids in which the given grid is the finest, and identifying in each locality the coarsest grid in the set from which u(x) can be recovered to a prescribed accuracy. This multi-resolution analysis was applied to essentially non-oscillatory (ENO) schemes in order to advance the solution by one time-step. This is accomplished by decomposing the numerical solution at the beginning of each time-step into levels of resolution, and performing the computation in each locality at the appropriate coarser grid. An efficient algorithm for implementing this program in the 1-D case is presented; this algorithm can be extended to the multi-dimensional case with Cartesian grids

Signal Decomposition Methods for Reducing Drawbacks of the DWT

Besides many advantages of wavelet transform, it has several drawbacks, e.g. ringing, shift variance, aliasing and lack of directionality. Some of them can be eliminated by using wavelet packet transform, stationary wavelet transform, complex wavelet transform, adaptive directional lifting-based wavelet transform, or adaptive wavelet filter banks that use either L2 or L1 norm. This paper contains an overview of these methods

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

Graph Priors, Optimal Transport, and Deep Learning in Biomedical Discovery

Recent advances in biomedical data collection allows the collection of massive datasets measuring thousands of features in thousands to millions of individual cells. This data has the potential to advance our understanding of biological mechanisms at a previously impossible resolution. However, there are few methods to understand data of this scale and type. While neural networks have made tremendous progress on supervised learning problems, there is still much work to be done in making them useful for discovery in data with more difficult to represent supervision. The flexibility and expressiveness of neural networks is sometimes a hindrance in these less supervised domains, as is the case when extracting knowledge from biomedical data. One type of prior knowledge that is more common in biological data comes in the form of geometric constraints. In this thesis, we aim to leverage this geometric knowledge to create scalable and interpretable models to understand this data. Encoding geometric priors into neural network and graph models allows us to characterize the models’ solutions as they relate to the fields of graph signal processing and optimal transport. These links allow us to understand and interpret this datatype. We divide this work into three sections. The first borrows concepts from graph signal processing to construct more interpretable and performant neural networks by constraining and structuring the architecture. The second borrows from the theory of optimal transport to perform anomaly detection and trajectory inference efficiently and with theoretical guarantees. The third examines how to compare distributions over an underlying manifold, which can be used to understand how different perturbations or conditions relate. For this we design an efficient approximation of optimal transport based on diffusion over a joint cell graph. Together, these works utilize our prior understanding of the data geometry to create more useful models of the data. We apply these methods to molecular graphs, images, single-cell sequencing, and health record data

Echocardiography

The book "Echocardiography - New Techniques" brings worldwide contributions from highly acclaimed clinical and imaging science investigators, and representatives from academic medical centers. Each chapter is designed and written to be accessible to those with a basic knowledge of echocardiography. Additionally, the chapters are meant to be stimulating and educational to the experts and investigators in the field of echocardiography. This book is aimed primarily at cardiology fellows on their basic echocardiography rotation, fellows in general internal medicine, radiology and emergency medicine, and experts in the arena of echocardiography. Over the last few decades, the rate of technological advancements has developed dramatically, resulting in new techniques and improved echocardiographic imaging. The authors of this book focused on presenting the most advanced techniques useful in today's research and in daily clinical practice. These advanced techniques are utilized in the detection of different cardiac pathologies in patients, in contributing to their clinical decision, as well as follow-up and outcome predictions. In addition to the advanced techniques covered, this book expounds upon several special pathologies with respect to the functions of echocardiography

Enhancing spatial resolution of remotely sensed data for mapping freshwater environments

Freshwater environments are important for ecosystem services and biodiversity. These environments are subject to many natural and anthropogenic changes, which influence their quality; therefore, regular monitoring is required for their effective management. High biotic heterogeneity, elongated land/water interaction zones, and logistic difficulties with access make field based monitoring on a large scale expensive, inconsistent and often impractical. Remote sensing (RS) is an established mapping tool that overcomes these barriers. However, complex and heterogeneous vegetation and spectral variability due to water make freshwater environments challenging to map using remote sensing technology. Satellite images available for New Zealand were reviewed, in terms of cost, and spectral and spatial resolution. Particularly promising image data sets for freshwater mapping include the QuickBird and SPOT-5. However, for mapping freshwater environments a combination of images is required to obtain high spatial, spectral, radiometric, and temporal resolution. Data fusion (DF) is a framework of data processing tools and algorithms that combines images to improve spectral and spatial qualities. A range of DF techniques were reviewed and tested for performance using panchromatic and multispectral QB images of a semi-aquatic environment, on the southern shores of Lake Taupo, New Zealand. In order to discuss the mechanics of different DF techniques a classification consisting of three groups was used - (i) spatially-centric (ii) spectrally-centric and (iii) hybrid. Subtract resolution merge (SRM) is a hybrid technique and this research demonstrated that for a semi aquatic QuickBird image it out performed Brovey transformation (BT), principal component substitution (PCS), local mean and variance matching (LMVM), and optimised high pass filter addition (OHPFA). However some limitations were identified with SRM, which included the requirement for predetermined band weights, and the over-representation of the spatial edges in the NIR bands due to their high spectral variance. This research developed three modifications to the SRM technique that addressed these limitations. These were tested on QuickBird (QB), SPOT-5, and Vexcel aerial digital images, as well as a scanned coloured aerial photograph. A visual qualitative assessment and a range of spectral and spatial quantitative metrics were used to evaluate these modifications. These included spectral correlation and root mean squared error (RMSE), Sobel filter based spatial edges RMSE, and unsupervised classification. The first modification addressed the issue of predetermined spectral weights and explored two alternative regression methods (Least Absolute Deviation, and Ordinary Least Squares) to derive image-specific band weights for use in SRM. Both methods were found equally effective; however, OLS was preferred as it was more efficient in processing band weights compared to LAD. The second modification used a pixel block averaging function on high resolution panchromatic images to derive spatial edges for data fusion. This eliminated the need for spectral band weights, minimised spectral infidelity, and enabled the fusion of multi-platform data. The third modification addressed the issue of over-represented spatial edges by introducing a sophisticated contrast and luminance index to develop a new normalising function. This improved the spatial representation of the NIR band, which is particularly important for mapping vegetation. A combination of the second and third modification of SRM was effective in simultaneously minimising the overall spectral infidelity and undesired spatial errors for the NIR band of the fused image. This new method has been labelled Contrast and Luminance Normalised (CLN) data fusion, and has been demonstrated to make a significant contribution in fusing multi-platform, multi-sensor, multi-resolution, and multi-temporal data. This contributes to improvements in the classification and monitoring of fresh water environments using remote sensing