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

Overviews of Optimization Techniques for Geometric Estimation

We summarize techniques for optimal geometric estimation from noisy observations for computer vision applications. We first discuss the interpretation of optimality and point out that geometric estimation is different from the standard statistical estimation. We also describe our noise modeling and a theoretical accuracy limit called the KCR lower bound. Then, we formulate estimation techniques based on minimization of a given cost function: least squares (LS), maximum likelihood (ML), which includes reprojection error minimization as a special case, and Sampson error minimization. We describe bundle adjustment and the FNS scheme for numerically solving them and the hyperaccurate correction that improves the accuracy of ML. Next, we formulate estimation techniques not based on minimization of any cost function: iterative reweight, renormalization, and hyper-renormalization. Finally, we show numerical examples to demonstrate that hyper-renormalization has higher accuracy than ML, which has widely been regarded as the most accurate method of all. We conclude that hyper-renormalization is robust to noise and currently is the best method

Advanced Biostatistical Methods for Curved and Censored Biomedical Data

This research was dedicated to analyze two types of biomedical data: curved data lying on a manifold and censored survival data from clinical trials. The main part of the research aims at developing a general regression framework for the analysis of a manifold-valued response in a Riemannian symmetric space (RSS) and its association with Euclidean covariates of interest, such as age. Such data arises frequently in medical imaging, computational biology, and computer vision, among many others. We developed an intrinsic regression model solely based on an intrinsic conditional moment assumption, avoiding specifying any parametric distribution on RSS. We proposed various link functions from the Euclidean space of covariates to the RSS of responses. We constructed parameter estimates and test statistics, and determined their asymptotic distributions and geometric invariant properties. Simulation studies were used to evaluate the finite sample properties of our method. We applied our model to investigate the association between covariates, including gender, age, and diagnosis, and the shape of the Corpus Callosum contours from the Alzheimer's Disease Neuroimaging Initiative dataset, in both cross-sectional and longitudinal cases. In oncology clinical trials, progression-free survival (PFS) has been a key endpoint to support licensing approval, and it is recommended to have the investigator's tumor assessments verified by an independent review committee blinded to study treatments, especially in open-label studies. Agreement between these evaluations may vary for subjects with short or long PFS, while there exist no such statistical quantities that can completely account for this temporal pattern of agreements. We proposed a new method to assess temporal agreement between two time-to-event endpoints, assuming they have a positive probability of being identical. Overall scores of agreement over a period of time are also proposed. We used maximum likelihood estimation to infer the proposed agreement measures using empirical data, accounting for different censoring mechanisms including reader's censoring (event from one reader dependently censored by event from the other reader). The proposed method is demonstrated to perform well in small-sample via extensive simulation studies and is illustrated through a head and neck cancer trial.Doctor of Philosoph

Untangling hotel industry’s inefficiency: An SFA approach applied to a renowned Portuguese hotel chain

The present paper explores the technical efficiency of four hotels from Teixeira Duarte Group - a renowned Portuguese hotel chain. An efficiency ranking is established from these four hotel units located in Portugal using Stochastic Frontier Analysis. This methodology allows to discriminate between measurement error and systematic inefficiencies in the estimation process enabling to investigate the main inefficiency causes. Several suggestions concerning efficiency improvement are undertaken for each hotel studied.info:eu-repo/semantics/publishedVersio

The Disk Structure of Late Type Galaxies: Determining the Black Hole Mass Function of Low Surface Brightness Galaxies Through Logarithmic Spiral Arm Pitch Angle Measurement

This dissertation pertains to the geometric structure of late type (spiral) galaxies, specifically on the relation between the logarithmic spiral pitch angle of the galactic spiral arms with other properties of the galaxy, such as central Supermassive Black Hole (SMBH) mass. Our work continues a study of the Black Hole Mass Function (BHMF) in local galaxies by recording the pitch angles of spiral galaxies with lower surface brightness than were previously included. We also conduct a case study on the structure of an interestingly shaped galaxy, UGC 4599. Previous studies on the topic of spiral arm pitch angles have measured the pitch angle of galaxies using a variety of image analysis techniques. Here the effectiveness of two of these techniques are assessed under different galaxy conditions and their errors and failure modes are probed as the measurement characteristics of simulated galaxies are manipulated and degraded. This is done for the purpose of recognizing and accounting for the limits of techniques for measuring the pitch angles of galaxies as they increase in redshift or decrease in surface brightness or angular resolution (pixel size). As a result, imaging based relations in galaxy structure may be applied to extend measurements from the local universe to greater distances as long as image degradation with distance is accounted for. In exploring populations of galaxies, errors in distribution studies might result from gaps in selection; galaxies with too little apparent structure or too faint a surface brightness to be recognized as spirals and included in the study. Errors might also result from inaccuracy or failure on the measurement side, where low resolution, low surface brightness galaxies produce pitch angle measurements characterized by higher failure rates and higher associated errors for successful measurements. We work to employ new techniques to minimize these errors as well as understand and account for their effects on the distributions being measured

Proceedings of the Second International Workshop on Mathematical Foundations of Computational Anatomy (MFCA'08) - Geometrical and Statistical Methods for Modelling Biological Shape Variability

International audienceThe goal of computational anatomy is to analyze and to statistically model the anatomy of organs in different subjects. Computational anatomic methods are generally based on the extraction of anatomical features or manifolds which are then statistically analyzed, often through a non-linear registration. There are nowadays a growing number of methods that can faithfully deal with the underlying biomechanical behavior of intra-subject deformations. However, it is more difficult to relate the anatomies of different subjects. In the absence of any justified physical model, diffeomorphisms provide a general mathematical framework that enforce topological consistency. Working with such infinite dimensional space raises some deep computational and mathematical problems, in particular for doing statistics. Likewise, modeling the variability of surfaces leads to rely on shape spaces that are much more complex than for curves. To cope with these, different methodological and computational frameworks have been proposed (e.g. smooth left-invariant metrics, focus on well-behaved subspaces of diffeomorphisms, modeling surfaces using courants, etc.) The goal of the Mathematical Foundations of Computational Anatomy (MFCA) workshop is to foster the interactions between the mathematical community around shapes and the MICCAI community around computational anatomy applications. It targets more particularly researchers investigating the combination of statistical and geometrical aspects in the modeling of the variability of biological shapes. The workshop aims at being a forum for the exchange of the theoretical ideas and a source of inspiration for new methodological developments in computational anatomy. A special emphasis is put on theoretical developments, applications and results being welcomed as illustrations. Following the very successful first edition of this workshop in 2006 (see http://www.inria.fr/sophia/asclepios/events/MFCA06/), the second edition was held in New-York on September 6, in conjunction with MICCAI 2008. Contributions were solicited in Riemannian and group theoretical methods, Geometric measurements of the anatomy, Advanced statistics on deformations and shapes, Metrics for computational anatomy, Statistics of surfaces. 34 submissions were received, among which 9 were accepted to MICCAI and had to be withdrawn from the workshop. Each of the remaining 25 paper was reviewed by three members of the program committee. To guaranty a high level program, 16 papers only were selected

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

{3D} Morphable Face Models -- Past, Present and Future

In this paper, we provide a detailed survey of 3D Morphable Face Models over the 20 years since they were first proposed. The challenges in building and applying these models, namely capture, modeling, image formation, and image analysis, are still active research topics, and we review the state-of-the-art in each of these areas. We also look ahead, identifying unsolved challenges, proposing directions for future research and highlighting the broad range of current and future applications

Theory and Methods for Large Spatial Data

Correlated Gaussian processes are of central importance to the study of time series, spatial statistics, computer experiments, and many machine learning models. Large spatially or temporally indexed datasets bring with them a host of computational and mathematical challenges. Parameter estimation of these processes often relies on maximum likelihood, which for Gaussian processes involves manipulations of the covariance matrix including solving systems of equations and determinant calculations. The score function, on the other hand, avoids direct calculation of the determinant, but still requires solving a large number of linear equations. We propose an equivalent kernel approximation to the score function of a stationary Gaussian process. A nugget effect is required for the approximation. We suggest two approximations, and for large sample sizes, our proposals are fast, accurate, and compare well against existing approaches.We then present a method for simulating time series of high frequency wind data calibrated by real data. The method provides and fits a parametric model for local wind directions by embedding them into the angular projection of a bivariate normal. Incorporating a temporal autocorrelation structure in that normal induces a continuous angular correlation over time in the simulated wind directions. The final joint model for speed and direction can be decomposed into the simulation of a single multivariate normal and a series of transformations thereof, allowing for fast and easy repeated generations of long time series. This is compared to a state of the art approach for simulating angular time series of swapping between discrete regimes of wind direction, a method that does not fully translate to high frequency data

Proceedings of the Third International Workshop on Mathematical Foundations of Computational Anatomy - Geometrical and Statistical Methods for Modelling Biological Shape Variability

International audienceComputational anatomy is an emerging discipline at the interface of geometry, statistics and image analysis which aims at modeling and analyzing the biological shape of tissues and organs. The goal is to estimate representative organ anatomies across diseases, populations, species or ages, to model the organ development across time (growth or aging), to establish their variability, and to correlate this variability information with other functional, genetic or structural information. The Mathematical Foundations of Computational Anatomy (MFCA) workshop aims at fostering the interactions between the mathematical community around shapes and the MICCAI community in view of computational anatomy applications. It targets more particularly researchers investigating the combination of statistical and geometrical aspects in the modeling of the variability of biological shapes. The workshop is a forum for the exchange of the theoretical ideas and aims at being a source of inspiration for new methodological developments in computational anatomy. A special emphasis is put on theoretical developments, applications and results being welcomed as illustrations. Following the successful rst edition of this workshop in 20061 and second edition in New-York in 20082, the third edition was held in Toronto on September 22 20113. Contributions were solicited in Riemannian and group theoretical methods, geometric measurements of the anatomy, advanced statistics on deformations and shapes, metrics for computational anatomy, statistics of surfaces, modeling of growth and longitudinal shape changes. 22 submissions were reviewed by three members of the program committee. To guaranty a high level program, 11 papers only were selected for oral presentation in 4 sessions. Two of these sessions regroups classical themes of the workshop: statistics on manifolds and diff eomorphisms for surface or longitudinal registration. One session gathers papers exploring new mathematical structures beyond Riemannian geometry while the last oral session deals with the emerging theme of statistics on graphs and trees. Finally, a poster session of 5 papers addresses more application oriented works on computational anatomy