Quantitative comparison of reconstruction methods for intra-voxel fiber recovery from diffusion MRI.

Validation is arguably the bottleneck in the diffusion magnetic resonance imaging (MRI) community. This paper evaluates and compares 20 algorithms for recovering the local intra-voxel fiber structure from diffusion MRI data and is based on the results of the "HARDI reconstruction challenge" organized in the context of the "ISBI 2012" conference. Evaluated methods encompass a mixture of classical techniques well known in the literature such as diffusion tensor, Q-Ball and diffusion spectrum imaging, algorithms inspired by the recent theory of compressed sensing and also brand new approaches proposed for the first time at this contest. To quantitatively compare the methods under controlled conditions, two datasets with known ground-truth were synthetically generated and two main criteria were used to evaluate the quality of the reconstructions in every voxel: correct assessment of the number of fiber populations and angular accuracy in their orientation. This comparative study investigates the behavior of every algorithm with varying experimental conditions and highlights strengths and weaknesses of each approach. This information can be useful not only for enhancing current algorithms and develop the next generation of reconstruction methods, but also to assist physicians in the choice of the most adequate technique for their studies.

Accelerated MR imaging with spread spectrum encoding

We advocate the use of a novel compressed sensing technique for accelerating the magnetic resonance image acquisition process, coined spread spectrum MR imaging or simply s2MRI. The method resides in pre-modulating the signal of interest by a linear chirp, resulting from the application of quadratic phase profiles, before random k-space under-sampling with uniform average density. The effectiveness of the procedure is theoretically underpinned by the optimization of the coherence between the sparsity and sensing bases. The application of the technique for single coil acquisitions is thoroughly studied by means of numerical simulations as well as phantom and in vivo experiments on a 7T scanner. The corresponding results suggest a favorable comparison with state-of-the-art variable density k-space under-sampling approaches.

Spread spectrum magnetic resonance imaging.

We propose a novel compressed sensing technique to accelerate the magnetic resonance imaging (MRI) acquisition process. The method, coined spread spectrum MRI or simply s(2)MRI, consists of premodulating the signal of interest by a linear chirp before random k-space under-sampling, and then reconstructing the signal with nonlinear algorithms that promote sparsity. The effectiveness of the procedure is theoretically underpinned by the optimization of the coherence between the sparsity and sensing bases. The proposed technique is thoroughly studied by means of numerical simulations, as well as phantom and in vivo experiments on a 7T scanner. Our results suggest that s(2)MRI performs better than state-of-the-art variable density k-space under-sampling approaches.

Transport properties across the La2/3Ca1/3MnO3/SrTiO3 heterointerface

The transport properties across La2/3Ca1/3MnO3/SrTiO3 (LCMO/STO) heterostructures with different thicknesses of the STO insulating barrier have been studied by using atomic force microscopy measurements in the current sensing (CS) mode. To avoid intrinsic problems of the CS method we have developed a nanostructured contact geometry of Au dots. The conduction process across the LCMO/STO interface exhibits the typical features of a tunneling process.

Cardiovascular magnetic resonance imaging techniques for motion-suppressed multi-dimensional coronary artery imaging with high spatial and temporal resolution

L'imagerie par résonance magnétique (IRM) peut fournir aux cardiologues des informations diagnostiques importantes sur l'état de la maladie de l'artère coronarienne dans les patients. Le défi majeur pour l'IRM cardiaque est de gérer toutes les sources de mouvement qui peuvent affecter la qualité des images en réduisant l'information diagnostique. Cette thèse a donc comme but de développer des nouvelles techniques d'acquisitions des images IRM, en changeant les techniques de compensation du mouvement, pour en augmenter l'efficacité, la flexibilité, la robustesse et pour obtenir plus d'information sur le tissu et plus d'information temporelle. Les techniques proposées favorisent donc l'avancement de l'imagerie des coronaires dans une direction plus maniable et multi-usage qui peut facilement être transférée dans l'environnement clinique. La première partie de la thèse s'est concentrée sur l'étude du mouvement des artères coronariennes sur des patients en utilisant la techniques d'imagerie standard (rayons x), pour mesurer la précision avec laquelle les artères coronariennes retournent dans la même position battement après battement (repositionnement des coronaires). Nous avons découvert qu'il y a des intervalles dans le cycle cardiaque, tôt dans la systole et à moitié de la diastole, où le repositionnement des coronaires est au minimum. En réponse nous avons développé une nouvelle séquence d'acquisition (T2-post) capable d'acquérir les données aussi tôt dans la systole. Cette séquence a été testée sur des volontaires sains et on a pu constater que la qualité de visualisation des artère coronariennes est égale à celle obtenue avec les techniques standard. De plus, le rapport signal sur bruit fourni par la séquence d'acquisition proposée est supérieur à celui obtenu avec les techniques d'imagerie standard. La deuxième partie de la thèse a exploré un paradigme d'acquisition des images cardiaques complètement nouveau pour l'imagerie du coeur entier. La technique proposée dans ce travail acquiert les données sans arrêt (free-running) au lieu d'être synchronisée avec le mouvement cardiaque. De cette façon, l'efficacité de la séquence d'acquisition est augmentée de manière significative et les images produites représentent le coeur entier dans toutes les phases cardiaques (quatre dimensions, 4D). Par ailleurs, l'auto-navigation de la respiration permet d'effectuer cette acquisition en respiration libre. Cette technologie rend possible de visualiser et évaluer l'anatomie du coeur et de ses vaisseaux ainsi que la fonction cardiaque en quatre dimensions et avec une très haute résolution spatiale et temporelle, sans la nécessité d'injecter un moyen de contraste. Le pas essentiel qui a permis le développement de cette technique est l'utilisation d'une trajectoire d'acquisition radiale 3D basée sur l'angle d'or. Avec cette trajectoire, il est possible d'acquérir continûment les données d'espace k, puis de réordonner les données et choisir les paramètres temporel des images 4D a posteriori. L'acquisition 4D a été aussi couplée avec un algorithme de reconstructions itératif (compressed sensing) qui permet d'augmenter la résolution temporelle tout en augmentant la qualité des images. Grâce aux images 4D, il est possible maintenant de visualiser les artères coronariennes entières dans chaque phase du cycle cardiaque et, avec les mêmes données, de visualiser et mesurer la fonction cardiaque. La qualité des artères coronariennes dans les images 4D est la même que dans les images obtenues avec une acquisition 3D standard, acquise en diastole Par ailleurs, les valeurs de fonction cardiaque mesurées au moyen des images 4D concorde avec les valeurs obtenues avec les images 2D standard. Finalement, dans la dernière partie de la thèse une technique d'acquisition a temps d'écho ultra-court (UTE) a été développée pour la visualisation in vivo des calcifications des artères coronariennes. Des études récentes ont démontré que les acquisitions UTE permettent de visualiser les calcifications dans des plaques athérosclérotiques ex vivo. Cepandent le mouvement du coeur a entravé jusqu'à maintenant l'utilisation des techniques UTE in vivo. Pour résoudre ce problème nous avons développé une séquence d'acquisition UTE avec trajectoire radiale 3D et l'avons testée sur des volontaires. La technique proposée utilise une auto-navigation 3D pour corriger le mouvement respiratoire et est synchronisée avec l'ECG. Trois échos sont acquis pour extraire le signal de la calcification avec des composants au T2 très court tout en permettant de séparer le signal de la graisse depuis le signal de l'eau. Les résultats sont encore préliminaires mais on peut affirmer que la technique développé peut potentiellement montrer les calcifications des artères coronariennes in vivo. En conclusion, ce travail de thèse présente trois nouvelles techniques pour l'IRM du coeur entier capables d'améliorer la visualisation et la caractérisation de la maladie athérosclérotique des coronaires. Ces techniques fournissent des informations anatomiques et fonctionnelles en quatre dimensions et des informations sur la composition du tissu auparavant indisponibles. CORONARY artery magnetic resonance imaging (MRI) has the potential to provide the cardiologist with relevant diagnostic information relative to coronary artery disease of patients. The major challenge of cardiac MRI, though, is dealing with all sources of motions that can corrupt the images affecting the diagnostic information provided. The current thesis, thus, focused on the development of new MRI techniques that change the standard approach to cardiac motion compensation in order to increase the efficiency of cardioavscular MRI, to provide more flexibility and robustness, new temporal information and new tissue information. The proposed approaches help in advancing coronary magnetic resonance angiography (MRA) in the direction of an easy-to-use and multipurpose tool that can be translated to the clinical environment. The first part of the thesis focused on the study of coronary artery motion through gold standard imaging techniques (x-ray angiography) in patients, in order to measure the precision with which the coronary arteries assume the same position beat after beat (coronary artery repositioning). We learned that intervals with minimal coronary artery repositioning occur in peak systole and in mid diastole and we responded with a new pulse sequence (T2~post) that is able to provide peak-systolic imaging. Such a sequence was tested in healthy volunteers and, from the image quality comparison, we learned that the proposed approach provides coronary artery visualization and contrast-to-noise ratio (CNR) comparable with the standard acquisition approach, but with increased signal-to-noise ratio (SNR). The second part of the thesis explored a completely new paradigm for whole- heart cardiovascular MRI. The proposed techniques acquires the data continuously (free-running), instead of being triggered, thus increasing the efficiency of the acquisition and providing four dimensional images of the whole heart, while respiratory self navigation allows for the scan to be performed in free breathing. This enabling technology allows for anatomical and functional evaluation in four dimensions, with high spatial and temporal resolution and without the need for contrast agent injection. The enabling step is the use of a golden-angle based 3D radial trajectory, which allows for a continuous sampling of the k-space and a retrospective selection of the timing parameters of the reconstructed dataset. The free-running 4D acquisition was then combined with a compressed sensing reconstruction algorithm that further increases the temporal resolution of the 4D dataset, while at the same time increasing the overall image quality by removing undersampling artifacts. The obtained 4D images provide visualization of the whole coronary artery tree in each phases of the cardiac cycle and, at the same time, allow for the assessment of the cardiac function with a single free- breathing scan. The quality of the coronary arteries provided by the frames of the free-running 4D acquisition is in line with the one obtained with the standard ECG-triggered one, and the cardiac function evaluation matched the one measured with gold-standard stack of 2D cine approaches. Finally, the last part of the thesis focused on the development of ultrashort echo time (UTE) acquisition scheme for in vivo detection of calcification in the coronary arteries. Recent studies showed that UTE imaging allows for the coronary artery plaque calcification ex vivo, since it is able to detect the short T2 components of the calcification. The heart motion, though, prevented this technique from being applied in vivo. An ECG-triggered self-navigated 3D radial triple- echo UTE acquisition has then been developed and tested in healthy volunteers. The proposed sequence combines a 3D self-navigation approach with a 3D radial UTE acquisition enabling data collection during free breathing. Three echoes are simultaneously acquired to extract the short T2 components of the calcification while a water and fat separation technique allows for proper visualization of the coronary arteries. Even though the results are still preliminary, the proposed sequence showed great potential for the in vivo visualization of coronary artery calcification. In conclusion, the thesis presents three novel MRI approaches aimed at improved characterization and assessment of atherosclerotic coronary artery disease. These approaches provide new anatomical and functional information in four dimensions, and support tissue characterization for coronary artery plaques.

Radiotherapy Treatment Assessment using DCE-MRI

Abstract

The goal of modern radiotherapy is to precisely deliver a prescribed radiation dose to delineated target volumes that contain a significant amount of tumor cells while sparing the surrounding healthy tissues/organs. Precise delineation of treatment and avoidance volumes is the key for the precision radiation therapy. In recent years, considerable clinical and research efforts have been devoted to integrate MRI into radiotherapy workflow motivated by the superior soft tissue contrast and functional imaging possibility. Dynamic contrast-enhanced MRI (DCE-MRI) is a noninvasive technique that measures properties of tissue microvasculature. Its sensitivity to radiation-induced vascular pharmacokinetic (PK) changes has been preliminary demonstrated. In spite of its great potential, two major challenges have limited DCE-MRI’s clinical application in radiotherapy assessment: the technical limitations of accurate DCE-MRI imaging implementation and the need of novel DCE-MRI data analysis methods for richer functional heterogeneity information.

This study aims at improving current DCE-MRI techniques and developing new DCE-MRI analysis methods for particular radiotherapy assessment. Thus, the study is naturally divided into two parts. The first part focuses on DCE-MRI temporal resolution as one of the key DCE-MRI technical factors, and some improvements regarding DCE-MRI temporal resolution are proposed; the second part explores the potential value of image heterogeneity analysis and multiple PK model combination for therapeutic response assessment, and several novel DCE-MRI data analysis methods are developed.

I. Improvement of DCE-MRI temporal resolution. First, the feasibility of improving DCE-MRI temporal resolution via image undersampling was studied. Specifically, a novel MR image iterative reconstruction algorithm was studied for DCE-MRI reconstruction. This algorithm was built on the recently developed compress sensing (CS) theory. By utilizing a limited k-space acquisition with shorter imaging time, images can be reconstructed in an iterative fashion under the regularization of a newly proposed total generalized variation (TGV) penalty term. In the retrospective study of brain radiosurgery patient DCE-MRI scans under IRB-approval, the clinically obtained image data was selected as reference data, and the simulated accelerated k-space acquisition was generated via undersampling the reference image full k-space with designed sampling grids. Two undersampling strategies were proposed: 1) a radial multi-ray grid with a special angular distribution was adopted to sample each slice of the full k-space; 2) a Cartesian random sampling grid series with spatiotemporal constraints from adjacent frames was adopted to sample the dynamic k-space series at a slice location. Two sets of PK parameters’ maps were generated from the undersampled data and from the fully-sampled data, respectively. Multiple quantitative measurements and statistical studies were performed to evaluate the accuracy of PK maps generated from the undersampled data in reference to the PK maps generated from the fully-sampled data. Results showed that at a simulated acceleration factor of four, PK maps could be faithfully calculated from the DCE images that were reconstructed using undersampled data, and no statistically significant differences were found between the regional PK mean values from undersampled and fully-sampled data sets. DCE-MRI acceleration using the investigated image reconstruction method has been suggested as feasible and promising.

Second, for high temporal resolution DCE-MRI, a new PK model fitting method was developed to solve PK parameters for better calculation accuracy and efficiency. This method is based on a derivative-based deformation of the commonly used Tofts PK model, which is presented as an integrative expression. This method also includes an advanced Kolmogorov-Zurbenko (KZ) filter to remove the potential noise effect in data and solve the PK parameter as a linear problem in matrix format. In the computer simulation study, PK parameters representing typical intracranial values were selected as references to simulated DCE-MRI data for different temporal resolution and different data noise level. Results showed that at both high temporal resolutions (<1s) and clinically feasible temporal resolution (~5s), this new method was able to calculate PK parameters more accurate than the current calculation methods at clinically relevant noise levels; at high temporal resolutions, the calculation efficiency of this new method was superior to current methods in an order of 102. In a retrospective of clinical brain DCE-MRI scans, the PK maps derived from the proposed method were comparable with the results from current methods. Based on these results, it can be concluded that this new method can be used for accurate and efficient PK model fitting for high temporal resolution DCE-MRI.

II. Development of DCE-MRI analysis methods for therapeutic response assessment. This part aims at methodology developments in two approaches. The first one is to develop model-free analysis method for DCE-MRI functional heterogeneity evaluation. This approach is inspired by the rationale that radiotherapy-induced functional change could be heterogeneous across the treatment area. The first effort was spent on a translational investigation of classic fractal dimension theory for DCE-MRI therapeutic response assessment. In a small-animal anti-angiogenesis drug therapy experiment, the randomly assigned treatment/control groups received multiple fraction treatments with one pre-treatment and multiple post-treatment high spatiotemporal DCE-MRI scans. In the post-treatment scan two weeks after the start, the investigated Rényi dimensions of the classic PK rate constant map demonstrated significant differences between the treatment and the control groups; when Rényi dimensions were adopted for treatment/control group classification, the achieved accuracy was higher than the accuracy from using conventional PK parameter statistics. Following this pilot work, two novel texture analysis methods were proposed. First, a new technique called Gray Level Local Power Matrix (GLLPM) was developed. It intends to solve the lack of temporal information and poor calculation efficiency of the commonly used Gray Level Co-Occurrence Matrix (GLCOM) techniques. In the same small animal experiment, the dynamic curves of Haralick texture features derived from the GLLPM had an overall better performance than the corresponding curves derived from current GLCOM techniques in treatment/control separation and classification. The second developed method is dynamic Fractal Signature Dissimilarity (FSD) analysis. Inspired by the classic fractal dimension theory, this method measures the dynamics of tumor heterogeneity during the contrast agent uptake in a quantitative fashion on DCE images. In the small animal experiment mentioned before, the selected parameters from dynamic FSD analysis showed significant differences between treatment/control groups as early as after 1 treatment fraction; in contrast, metrics from conventional PK analysis showed significant differences only after 3 treatment fractions. When using dynamic FSD parameters, the treatment/control group classification after 1st treatment fraction was improved than using conventional PK statistics. These results suggest the promising application of this novel method for capturing early therapeutic response.

The second approach of developing novel DCE-MRI methods is to combine PK information from multiple PK models. Currently, the classic Tofts model or its alternative version has been widely adopted for DCE-MRI analysis as a gold-standard approach for therapeutic response assessment. Previously, a shutter-speed (SS) model was proposed to incorporate transcytolemmal water exchange effect into contrast agent concentration quantification. In spite of richer biological assumption, its application in therapeutic response assessment is limited. It might be intriguing to combine the information from the SS model and from the classic Tofts model to explore potential new biological information for treatment assessment. The feasibility of this idea was investigated in the same small animal experiment. The SS model was compared against the Tofts model for therapeutic response assessment using PK parameter regional mean value comparison. Based on the modeled transcytolemmal water exchange rate, a biological subvolume was proposed and was automatically identified using histogram analysis. Within the biological subvolume, the PK rate constant derived from the SS model were proved to be superior to the one from Tofts model in treatment/control separation and classification. Furthermore, novel biomarkers were designed to integrate PK rate constants from these two models. When being evaluated in the biological subvolume, this biomarker was able to reflect significant treatment/control difference in both post-treatment evaluation. These results confirm the potential value of SS model as well as its combination with Tofts model for therapeutic response assessment.

In summary, this study addressed two problems of DCE-MRI application in radiotherapy assessment. In the first part, a method of accelerating DCE-MRI acquisition for better temporal resolution was investigated, and a novel PK model fitting algorithm was proposed for high temporal resolution DCE-MRI. In the second part, two model-free texture analysis methods and a multiple-model analysis method were developed for DCE-MRI therapeutic response assessment. The presented works could benefit the future DCE-MRI routine clinical application in radiotherapy assessment.

Convex Optimization Algorithms and Statistical Bounds for Learning Structured Models

Thesis (Ph.D.)--University of Washington, 2016-08

Likelihood-Based Inference for Partially Observed Multi-Type Markov Branching Processes

Thesis (Ph.D.)--University of Washington, 2016-08

Tensor Completion for Multidimensional Inverse Problems with Applications to Magnetic Resonance Relaxometry

This thesis deals with tensor completion for the solution of multidimensional inverse problems. We study the problem of reconstructing an approximately low rank tensor from a small number of noisy linear measurements. New recovery guarantees, numerical algorithms, non-uniform sampling strategies, and parameter selection algorithms are developed. We derive a fixed point continuation algorithm for tensor completion and prove its convergence. A restricted isometry property (RIP) based tensor recovery guarantee is proved. Probabilistic recovery guarantees are obtained for sub-Gaussian measurement operators and for measurements obtained by non-uniform sampling from a Parseval tight frame. We show how tensor completion can be used to solve multidimensional inverse problems arising in NMR relaxometry. Algorithms are developed for regularization parameter selection, including accelerated k-fold cross-validation and generalized cross-validation. These methods are validated on experimental and simulated data. We also derive condition number estimates for nonnegative least squares problems. Tensor recovery promises to significantly accelerate N-dimensional NMR relaxometry and related experiments, enabling previously impractical experiments. Our methods could also be applied to other inverse problems arising in machine learning, image processing, signal processing, computer vision, and other fields.

Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems

We present a detailed analysis of the application of a multi-scale Hierarchical Reconstruction method for solving a family of ill-posed linear inverse problems. When the observations on the unknown quantity of interest and the observation operators are known, these inverse problems are concerned with the recovery of the unknown from its observations. Although the observation operators we consider are linear, they are inevitably ill-posed in various ways. We recall in this context the classical Tikhonov regularization method with a stabilizing function which targets the specific ill-posedness from the observation operators and preserves desired features of the unknown. Having studied the mechanism of the Tikhonov regularization, we propose a multi-scale generalization to the Tikhonov regularization method, so-called the Hierarchical Reconstruction (HR) method. First introduction of the HR method can be traced back to the Hierarchical Decomposition method in Image Processing. The HR method successively extracts information from the previous hierarchical residual to the current hierarchical term at a finer hierarchical scale. As the sum of all the hierarchical terms, the hierarchical sum from the HR method provides an reasonable approximate solution to the unknown, when the observation matrix satisfies certain conditions with specific stabilizing functions. When compared to the Tikhonov regularization method on solving the same inverse problems, the HR method is shown to be able to decrease the total number of iterations, reduce the approximation error, and offer self control of the approximation distance between the hierarchical sum and the unknown, thanks to using a ladder of finitely many hierarchical scales. We report numerical experiments supporting our claims on these advantages the HR method has over the Tikhonov regularization method.

Sparse Signal Representation in Digital and Biological Systems

Theories of sparse signal representation, wherein a signal is decomposed as the sum of a small number of constituent elements, play increasing roles in both mathematical signal processing and neuroscience. This happens despite the differences between signal models in the two domains. After reviewing preliminary material on sparse signal models, I use work on compressed sensing for the electron tomography of biological structures as a target for exploring the efficacy of sparse signal reconstruction in a challenging application domain. My research in this area addresses a topic of keen interest to the biological microscopy community, and has resulted in the development of tomographic reconstruction software which is competitive with the state of the art in its field. Moving from the linear signal domain into the nonlinear dynamics of neural encoding, I explain the sparse coding hypothesis in neuroscience and its relationship with olfaction in locusts. I implement a numerical ODE model of the activity of neural populations responsible for sparse odor coding in locusts as part of a project involving offset spiking in the Kenyon cells. I also explain the validation procedures we have devised to help assess the model's similarity to the biology. The thesis concludes with the development of a new, simplified model of locust olfactory network activity, which seeks with some success to explain statistical properties of the sparse coding processes carried out in the network.

Intelligent Sensor Systems for Structural Health Monitoring Applications

The convergence between the recent developments in sensing technologies, data science, signal processing and advanced modelling has fostered a new paradigm to the Structural Health Monitoring (SHM) of engineered structures, which is the one based on intelligent sensors, i.e., embedded devices capable of stream processing data and/or performing structural inference in a self-contained and near-sensor manner. To efficiently exploit these intelligent sensor units for full-scale structural assessment, a joint effort is required to deal with instrumental aspects related to signal acquisition, conditioning and digitalization, and those pertaining to data management, data analytics and information sharing. In this framework, the main goal of this Thesis is to tackle the multi-faceted nature of the monitoring process, via a full-scale optimization of the hardware and software resources involved by the {SHM} system. The pursuit of this objective has required the investigation of both: i) transversal aspects common to multiple application domains at different abstraction levels (such as knowledge distillation, networking solutions, microsystem {HW} architectures), and ii) the specificities of the monitoring methodologies (vibrations, guided waves, acoustic emission monitoring). The key tools adopted in the proposed monitoring frameworks belong to the embedded signal processing field: namely, graph signal processing, compressed sensing, ARMA System Identification, digital data communication and TinyML.

Problemas de programação matemática com restrições lineares de equilíbrio

Neste trabalho é desenvolvido um algoritmo enumerativo paramétrico de optimização global para a resolução de Problemas de Programação Matemática com Restrições de Equilíbrio ou de Complementaridade (MPEC). A comparação com outras técnicas globais da literatura é efectuada para um leque variado de problemas, de modo a poder avaliar a eficiência do processo proposto. A utilização de algoritmos de MPEC para a resolução de alguns problemas de optimização global é o outro grande objectivo desta tese. Nesse sentido são introduzidas novas formula¸c˜oes de programas bilineares e lineares complementares como MPECs. São ainda analisadas e discutidas formulaçõess MPEC para o problema de programação linear inteira 0-1, para a determinação do Conjunto Independente Máximo de um Grafo (MIS) e para a estimação do Número de Condição de uma Matriz. Para o problema MIS é desenvolvido um algoritmo de ramificação e limitação, baseado na decomposição de uma função quadrática numa diferença de duas funçõess convexas (DC). Finalmente é introduzida uma técnica MPEC local para a estimação do número de condição com a norma l1 e é estabelecido para matrizes de Minkowski que o número de condição nessa norma pode ser estimado com apenas um sistema de equações lineares. Em todos os desenvolvimentos houve uma grande preocupação em testar as novas formulações e algoritmos com problemas conhecidos da literatura, de modo a aferir da qualidade e interesse dessas propostas.

Estimação paramétrica e não-paramétrica em modelos de markov ocultos

In this work we study the Hidden Markov Models with finite as well as general state space. In the finite case, the forward and backward algorithms are considered and the probability of a given observed sequence is computed. Next, we use the EM algorithm to estimate the model parameters. In the general case, the kernel estimators are used and to built a sequence of estimators that converge in L1-norm to the density function of the observable process