Efficient estimation for a subclass of shape invariant models

In this paper, we observe a fixed number of unknown $2\pi$-periodic functions differing from each other by both phases and amplitude. This semiparametric model appears in literature under the name "shape invariant model." While the common shape is unknown, we introduce an asymptotically efficient estimator of the finite-dimensional parameter (phases and amplitude) using the profile likelihood and the Fourier basis. Moreover, this estimation method leads to a consistent and asymptotically linear estimator for the common shape.Comment: Published in at http://dx.doi.org/10.1214/07-AOS566 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Estimation of Translation, Rotation, and Scaling between Noisy Images Using the Fourier–Mellin Transform

In this paper we focus on extended Euclidean registration of a set of noisy images. We provide an appropriate statistical model for this kind of registration problems, and a new criterion based on Fourier-type transforms is proposed to estimate the translation, rotation and scaling parameters to align a set of images. This criterion is a two step procedure which does not require the use of a reference template onto which aligning all the images. Our approach is based on M-estimation and we prove the consistency of the resulting estimators. A small scale simulation study and real examples are used to illustrate the numerical performances of our procedure

A statistical analysis of particle trajectories in living cells

Recent advances in molecular biology and fluorescence microscopy imaging have made possible the inference of the dynamics of single molecules in living cells. Such inference allows to determine the organization and function of the cell. The trajectories of particles in the cells, computed with tracking algorithms, can be modelled with diffusion processes. Three types of diffusion are considered : (i) free diffusion; (ii) subdiffusion or (iii) superdiffusion. The Mean Square Displacement (MSD) is generally used to determine the different types of dynamics of the particles in living cells (Qian, Sheetz and Elson 1991). We propose here a non-parametric three-decision test as an alternative to the MSD method. The rejection of the null hypothesis -- free diffusion -- is accompanied by claims of the direction of the alternative (subdiffusion or a superdiffusion). We study the asymptotic behaviour of the test statistic under the null hypothesis, and under parametric alternatives which are currently considered in the biophysics literature, (Monnier et al,2012) for example. In addition, we adapt the procedure of Benjamini and Hochberg (2000) to fit with the three-decision test setting, in order to apply the test procedure to a collection of independent trajectories. The performance of our procedure is much better than the MSD method as confirmed by Monte Carlo experiments. The method is demonstrated on real data sets corresponding to protein dynamics observed in fluorescence microscopy.Comment: Revised introduction. A clearer and shorter description of the model (section 2

Semiparametric estimation of shifts on compact Lie groups for image registration

In this paper we focus on estimating the deformations that may exist between similar images in the presence of additive noise when a reference template is unknown. The deformations aremodeled as parameters lying in a finite dimensional compact Lie group. A generalmatching criterion based on the Fourier transformand itswell known shift property on compact Lie groups is introduced. M-estimation and semiparametric theory are then used to study the consistency and asymptotic normality of the resulting estimators. As Lie groups are typically nonlinear spaces, our tools rely on statistical estimation for parameters lying in a manifold and take into account the geometrical aspects of the problem. Some simulations are used to illustrate the usefulness of our approach and applications to various areas in image processing are discussed

Inférence statistique par des transformées de Fourier pour des modèles de régression semi-paramétriques

version du 1 novembre 2007The shape invariant model consist of the observation of a fixed number of regression functions which differ only by a parametric warping operator. This type of models finds applications in the problems of alignment of continuous signals (images 2D, circadian rhythms,. . .) or discrete (electroencephalogram,. . .). For various warping groups, we propose M-estimators for the parameters characterizing the warping operators associated with the regression functions. These estimators minimize or maximize criteria which are defined with the synchronized average of the Fourier transforms of the data. Moreover, for one of the studied models, we prove the semi-parametric efficiency of the proposed estimator, and we build a test of adequacy of the shape invariant model from one of the criteria.Dans cette thèse, nous étudions des modèles semi-paramétriques dits de forme invariante. Ces modèles consistent en l'observation d'un nombre fixés de fonctions de régression identiques à un opérateur de déformation paramétriques près. Ce type de modèles trouve des applications dans les problèmes d'alignement de signaux continus (images 2D, rythmes biologiques, ...) ou discrets (electroencéphalogramme, ...). Pour différents groupes de déformations, nous proposons des M-estimateurs pour les paramètres caractérisant les opérateurs associés aux fonctions de régression. Ces estimateurs minimisent ou maximisent des fonctions de contraste, construites à partir de la moyenne synchronisée des transformées de Fourier des données. De plus, pour l'un des modèles étudiés, nous prouvons l'efficacité semi-paramétrique de cet estimateur ainsi défini, et nous proposons un test d'adéquation du modèle de forme invariante construit à partir d'une des fonctions de contraste

Semiparametric estimation of rigid transformations on compact Lie groups

International audienceWe study a simple model for the estimation of rigid transformations between noisy images. The transformations are supposed to belong to a compact Lie group, and a new matching criteria based on the Fourier transform is proposed. Consistency and asymptotic normality of the resulting estimators are studied. Some simulations are used to illustrate the methodology, and we describe potential applications of this approach to various image registration problems

A Statistical Analysis of Particle Trajectories in Living Cells

Recent advances in molecular biology and fluorescence microscopy imaging have made possible the inference of the dynamics of single molecules in living cells. Such inference allows to determine the organization and function of the cell. The trajectories of particles in the cells, computed with tracking algorithms, can be modelled with diffusion processes. Three types of diffusion are considered : (i) free diffusion ; (ii) subdiffusion or (iii) superdiffusion. The Mean Square Displacement (MSD) is generally used to determine the different types of dynamics of the particles in living cells (Qian, Sheetz and Elson, 1991). We propose here a non-parametric three-decision test as an alternative to the MSD method. The rejection of the null hypothesis – free diffusion – is accompanied by claims of the direction of the alternative (subdiffusion or a superdiffusion). We study the asymp-totic behaviour of the test statistic under the null hypothesis, and under parametric alternatives. In addition, we adapt the procedure of Benjamini and Hochberg (2000) to fit with the three-decision test setting, in order to apply the test procedure to a collection of independent trajectories. The performance of our procedure is much better than the MSD method as confirmed by Monte Carlo experiments. The method is demonstrated on real data sets corresponding to protein dynamics observed in fluorescence microscopy

Test statistique pour détecter les diffusions non browniennes

National audienceLa modélisation de la dynamique des particules intracellulaires permet de répondre à des problèmes biologiques. Dans ce travail, nous supposons que le mouvement de ces particules est décrit à l'aide de processus stochastiques particuliers: les processus de diffusion. Nous développons ici un test statistique permettant de classer les trajectoires observées en trois groupes: la diffusion confinée, dirigée et libre (ou mouvement Brownien). Cette méthode est une alternative à la l'analyse du déplacement carré moyen (Mean Square Displacement) utilisé dans la littérature biophysique. Notre procédure est évaluée sur des simulations et des cas réels

An overview of diffusion models for intracellular dynamics analysis

We present an overview of diffusion models commonly used for quantifying the dynamics of intracellular particles (e.g., biomolecules) inside living cells. It is established that inference on the modes of mobility of molecules is central in cell biology since it reflects interactions between structures and determines functions of biomolecules in the cell. In that context, Brownian motion is a key component in short distance transportation (e.g., connectivity for signal transduction). Another dynamical process that have been heavily studied in the past decade is the motor-mediated transport (e.g., dynein, kinesin, myosin) of molecules. Primarily supported by actin filament and microtubule network, it ensures spatial organization and temporal synchronization in the intracellular mechanisms and structures. Nevertheless, the complexity of internal structures and molecular processes in the living cell influence the molecular dynamics and prevent the systematic application of pure Brownian or directed motion modeling. On the one hand, cytoskeleton density will hinder the free displacement of the particle, a phenomenon called subdiffusion. On the other hand, the cytoskeleton elasticity combined with thermal bending can contribute a phenomenon called superdiffusion. This paper discusses the basics of diffusion modes observed in cells, by introducing the essential properties of these processes. Applications of diffusion models include protein trafficking and transport, and membrane diffusion