Manifold embedding for curve registration

We focus on the problem of finding a good representative of a sample of random curves warped from a common pattern f. We first prove that such a problem can be moved onto a manifold framework. Then, we propose an estimation of the common pattern f based on an approximated geodesic distance on a suitable manifold. We then compare the proposed method to more classical methods

Manifold Learning for Natural Image Sets, Doctoral Dissertation August 2006

The field of manifold learning provides powerful tools for parameterizing high-dimensional data points with a small number of parameters when this data lies on or near some manifold. Images can be thought of as points in some high-dimensional image space where each coordinate represents the intensity value of a single pixel. These manifold learning techniques have been successfully applied to simple image sets, such as handwriting data and a statue in a tightly controlled environment. However, they fail in the case of natural image sets, even those that only vary due to a single degree of freedom, such as a person walking or a heart beating. Parameterizing data sets such as these will allow for additional constraints on traditional computer vision problems such as segmentation and tracking. This dissertation explores the reasons why classical manifold learning algorithms fail on natural image sets and proposes new algorithms for parameterizing this type of data

Template estimation for samples of curves and functional calibration estimation via the method of maximum entropy on the mean

L'une des principales difficultés de l'analyse des données fonctionnelles consiste à extraire un motif commun qui synthétise l'information contenue par toutes les fonctions de l'échantillon. Le Chapitre 2 examine le problème d'identification d'une fonction qui représente le motif commun en supposant que les données appartiennent à une variété ou en sont suffisamment proches, d'une variété non linéaire de basse dimension intrinsèque munie d'une structure géométrique inconnue et incluse dans un espace de grande dimension. Sous cette hypothèse, un approximation de la distance géodésique est proposé basé sur une version modifiée de l'algorithme Isomap. Cette approximation est utilisée pour calculer la fonction médiane empirique de Fréchet correspondante. Cela fournit un estimateur intrinsèque robuste de la forme commune. Le Chapitre 3 étudie les propriétés asymptotiques de la méthode de normalisation quantile développée par Bolstad, et al. (2003) qui est devenue l'une des méthodes les plus populaires pour aligner des courbes de densité en analyse de données de microarrays en bioinformatique. Les propriétés sont démontrées considérant la méthode comme un cas particulier de la procédure de la moyenne structurelle pour l'alignement des courbes proposée par Dupuy, Loubes and Maza (2011). Toutefois, la méthode échoue dans certains cas. Ainsi, nous proposons une nouvelle méthode, pour faire face à ce problème. Cette méthode utilise l'algorithme développée dans le Chapitre 2. Dans le Chapitre 4, nous étendons le problème d'estimation de calage pour la moyenne d'une population finie de la variable de sondage dans un cadre de données fonctionnelles. Nous considérons le problème de l'estimation des poids de sondage fonctionnel à travers le principe du maximum d'entropie sur la moyenne -MEM-. En particulier, l'estimation par calage est considérée comme un problème inverse linéaire de dimension infinie suivant la structure de l'approche du MEM. Nous donnons un résultat précis d'estimation des poids de calage fonctionnels pour deux types de mesures aléatoires a priori: la measure Gaussienne centrée et la measure de Poisson généralisée.One of the main difficulties in functional data analysis is the extraction of a meaningful common pattern that summarizes the information conveyed by all functions in the sample. The problem of finding a meaningful template function that represents this pattern is considered in Chapter 2 assuming that the functional data lie on an intrinsically low-dimensional smooth manifold with an unknown underlying geometric structure embedding in a high-dimensional space. Under this setting, an approximation of the geodesic distance is developed based on a robust version of the Isomap algorithm. This approximation is used to compute the corresponding empirical Fréchet median function, which provides a robust intrinsic estimator of the template. The Chapter 3 investigates the asymptotic properties of the quantile normalization method by Bolstad, et al. (2003) which is one of the most popular methods to align density curves in microarray data analysis. The properties are proved by considering the method as a particular case of the structural mean curve alignment procedure by Dupuy, Loubes and Maza (2011). However, the method fails in some case of mixtures, and a new methodology to cope with this issue is proposed via the algorithm developed in Chapter 2. Finally, the problem of calibration estimation for the finite population mean of a survey variable under a functional data framework is studied in Chapter 4. The functional calibration sampling weights of the estimator are obtained by matching the calibration estimation problem with the maximum entropy on the mean -MEM- principle. In particular, the calibration estimation is viewed as an infinite-dimensional linear inverse problem following the structure of the MEM approach. A precise theoretical setting is given and the estimation of functional calibration weights assuming, as prior measures, the centered Gaussian and compound Poisson random measures is carried out

On Motion Parameterizations in Image Sequences from Fixed Viewpoints

This dissertation addresses the problem of parameterizing object motion within a set of images taken with a stationary camera. We develop data-driven methods across all image scales: characterizing motion observed at the scale of individual pixels, along extended structures such as roads, and whole image deformations such as lungs deforming over time. The primary contributions include: a) fundamental studies of the relationship between spatio-temporal image derivatives accumulated at a pixel, and the object motions at that pixel,: b) data driven approaches to parameterize breath motion and reconstruct lung CT data volumes, and: c) defining and offering initial results for a new class of Partially Unsupervised Manifold Learning: PUML) problems, which often arise in medical imagery. Specifically, we create energy functions for measuring how consistent a given velocity vector is with observed spatio-temporal image derivatives. These energy functions are used to fit parametric snake models to roads using velocity constraints. We create an automatic data-driven technique for finding the breath phase of lung CT scans which is able to replace external belt measurements currently in use clinically. This approach is extended to automatically create a full deformation model of a CT lung volume during breathing or heart MRI during breathing and heartbeat. Additionally, motivated by real use cases, we address a scenario in which a dataset is collected along with meta-data which describes some, but not all, aspects of the dataset. We create an embedding which displays the remaining variability in a dataset after accounting for variability related to the meta-data

Constrained manifold learning for the characterization of pathological deviations from normality

International audienceThis paper describes a technique to (1) learn the representation of a pathological motion pattern from a given population, and (2) compare individuals to this population. Our hypothesis is that this pattern can be modeled as a deviation from normal motion by means of non-linear embedding techniques. Each subject is represented by a 2D map of local motion abnormalities, obtained from a statistical atlas of myocardial motion built from a healthy population. The algorithm estimates a manifold from a set of patients with varying degrees of the same disease, and compares individuals to the training population using a mapping to the manifold and a distance to normality along the manifold. The approach extends recent manifold learning techniques by constraining the manifold to pass by a physiologically meaningful origin representing a normal motion pattern. Interpolation techniques using locally adjustable kernel improve the accuracy of the method. The technique is applied in the context of cardiac resynchronization therapy (CRT), focusing on a specific motion pattern of intra-ventricular dyssynchrony called septal flash (SF). We estimate the manifold from 50 CRT candidates with SF and test it on 37 CRT candidates and 21 healthy volunteers. Experiments highlight the relevance of nonlinear techniques to model a pathological pattern from the training set and compare new individuals to this pattern

Manifold Modeling of the Beating Heart Motion

Modeling the heart motion has important applications for diagnosis and intervention. We present a new method for modeling the deformation of the myocardium in the cardiac cycle. Our approach is based on manifold learning to build a representation of shape variation across time. We experiment with various manifold types to identify the best manifold method, and with real patient data extracted from cine MRIs. We obtain a representation, common to all subjects, that can discriminate cardiac cycle phases and heart function types

Manifold modeling of the beating heart motion

We present a new method for modeling the deformation of the myocardium in the cardiac cycle. Our approach is based on manifold learning to build a representation of shape variation across time. We experiment with various manifold types to identify the best manifold method, and with real patient data extracted from cine MRIs. We obtain a representation, common to all subjects, that can discriminate cardiac cycle phases and heart function types