Registration techniques for computer assisted orthopaedic surgery

The registration of 3D preoperative medical data to patients is a key task in developing computer assisted surgery systems. In computer assisted surgery, the patient in the operation theatre must be aligned with the coordinate system in which the preoperative data has been acquired, so that the planned surgery based on the preoperative data can be carried out under the guidance of the computer assisted surgery system.The aim of this research is to investigate registration algorithms for developing computer assisted bone surgery systems. We start with reference mark registration. New interpretations are given to the development of well knowm algorithms based on singular value decomposition, polar decomposition techniques and the unit quaternion representation of the rotation matrix. In addition, a new algorithm is developed based on the estimate of the rotation axis. For non-land mark registration, we first develop iterative closest line segment and iterative closest triangle patch registrations, similar to the well known iterative closest point registration, when the preoperative data are dense enough. We then move to the situation where the preoperative data are not dense enough. Implicit fitting is considered to interpolate the gaps between the data . A new ellipsoid fitting algorithm and a new constructive implicit fitting strategy are developed. Finally, a region to region matching procedure is proposed based on our novel constructive implicit fitting technique. Experiments demonstrate that the new algorithm is very stable and very efficient

Energy preserving model order reduction of the nonlinear Schr\"odinger equation

An energy preserving reduced order model is developed for two dimensional nonlinear Schr\"odinger equation (NLSE) with plane wave solutions and with an external potential. The NLSE is discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of Hamiltonian ordinary differential equations are integrated in time by the energy preserving average vector field (AVF) method. The mass and energy preserving reduced order model (ROM) is constructed by proper orthogonal decomposition (POD) Galerkin projection. The nonlinearities are computed for the ROM efficiently by discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and mass are shown for the full order model (FOM) and for the ROM which ensures the long term stability of the solutions. Numerical simulations illustrate the preservation of the energy and mass in the reduced order model for the two dimensional NLSE with and without the external potential. The POD-DMD makes a remarkable improvement in computational speed-up over the POD-DEIM. Both methods approximate accurately the FOM, whereas POD-DEIM is more accurate than the POD-DMD

Statistical and numerical methods for diffusion processes with multiple scales

In this thesis we address the problem of data-driven coarse-graining, i.e. the process of inferring simplified models, which describe the evolution of the essential characteristics of a complex system, from available data (e.g. experimental observation or simulation data). Specifically, we consider the case where the coarse-grained model can be formulated as a stochastic differential equation. The main part of this work is concerned with data-driven coarse-graining when the underlying complex system is characterised by processes occurring across two widely separated time scales. It is known that in this setting commonly used statistical techniques fail to obtain reasonable estimators for parameters in the coarse-grained model, due to the multiscale structure of the data. To enable reliable data-driven coarse-graining techniques for diffusion processes with multiple time scales, we develop a novel estimation procedure which decisively relies on combining techniques from mathematical statistics and numerical analysis. We demonstrate, both rigorously and by means of extensive simulations, that this methodology yields accurate approximations of coarse-grained SDE models. In the final part of this work, we then discuss a systematic framework to analyse and predict complex systems using observations. Specifically, we use data-driven techniques to identify simple, yet adequate, coarse-grained models, which in turn allow to study statistical properties that cannot be investigated directly from the time series. The value of this generic framework is exemplified through two seemingly unrelated data sets of real world phenomena.Open Acces

Une méthode de région de confiance avec ensemble actif pour l'optimisation non linéaire sans dérivées avec contraintes de bornes appliquée à des problèmes aérodynamiques bruités

L'optimisation sans dérivées (OSD) a connu un regain d'intérêt ces dernières années, principalement motivée par le besoin croissant de résoudre les problèmes d'optimisation définis par des fonctions dont les valeurs sont calculées par simulation (par exemple, la conception technique, la restauration d'images médicales ou de nappes phréatiques). Ces dernières années, un certain nombre de méthodes d'optimisation sans dérivée ont été développées et en particulier des méthodes fondées sur un modèle de région de confiance se sont avérées obtenir de bons résultats. Dans cette thèse, nous présentons un nouvel algorithme de région de confiance, basé sur l'interpolation, qui se montre efficace et globalement convergent (en ce sens que sa convergence vers un point stationnaire est garantie depuis tout point de départ arbitraire). Le nouvel algorithme repose sur la technique d'auto-correction de la géométrie proposé par Scheinberg and Toint (2010). Dans leur théorie, ils ont fait avancer la compréhension du rôle de la géométrie dans les méthodes d'OSD à base de modèles. Dans notre travail, nous avons pu améliorer considérablement l'efficacité de leur méthode, tout en maintenant ses bonnes propriétés de convergence. De plus, nous examinons l'influence de différents types de modèles d'interpolation sur les performances du nouvel algorithme. Nous avons en outre étendu cette méthode pour prendre en compte les contraintes de borne par l'application d'une stratégie d'activation. Considérer une méthode avec ensemble actif pour l'optimisation basée sur des modèles d'interpolation donne la possibilité d'économiser une quantité importante d'évaluations de fonctions. Il permet de maintenir les ensembles d'interpolation plus petits tout en poursuivant l'optimisation dans des sous-espaces de dimension inférieure. L'algorithme résultant montre un comportement numérique très compétitif. Nous présentons des résultats sur un ensemble de problèmes-tests issu de la collection CUTEr et comparons notre méthode à des algorithmes de référence appartenant à différentes classes de méthodes d'OSD. Pour réaliser des expériences numériques qui intègrent le bruit, nous créons un ensemble de cas-tests bruités en ajoutant des perturbations à l'ensemble des problèmes sans bruit. Le choix des problèmes bruités a été guidé par le désir d'imiter les problèmes d'optimisation basés sur la simulation. Enfin, nous présentons des résultats sur une application réelle d'un problème de conception de forme d'une aile fourni par Airbus. ABSTRACT : Derivative-free optimization (DFO) has enjoyed renewed interest over the past years, mostly motivated by the ever growing need to solve optimization problems defined by functions whose values are computed by simulation (e.g. engineering design, medical image restoration or groundwater supply). In the last few years, a number of derivative-free optimization methods have been developed and especially model-based trust-region methods have been shown to perform well. In this thesis, we present a new interpolation-based trust-region algorithm which shows to be efficient and globally convergent (in the sense that its convergence is guaranteed to a stationary point from arbitrary starting points). The new algorithm relies on the technique of self-correcting geometry proposed by Scheinberg and Toint [128] in 2009. In their theory, they advanced the understanding of the role of geometry in model-based DFO methods, in our work, we improve the efficiency of their method while maintaining its good theoretical convergence properties. We further examine the influence of different types of interpolation models on the performance of the new algorithm. Furthermore, we extended this method to handle bound constraints by applying an activeset strategy. Considering an active-set method in bound-constrained model-based optimization creates the opportunity of saving a substantial amount of function evaluations. It allows to maintain smaller interpolation sets while proceeding optimization in lower dimensional subspaces. The resulting algorithm is shown to be numerically highly competitive. We present results on a test set of smooth problems from the CUTEr collection and compare to well-known state-of-the-art packages from different classes of DFO methods. To report numerical experiments incorporating noise, we create a test set of noisy problems by adding perturbations to the set of smooth problems. The choice of noisy problems was guided by a desire to mimic simulation-based optimization problems. Finally, we will present results on a real-life application of a wing-shape design problem provided by Airbus. optimisation sans dérivées, région de confiance, contraintes de borne, fonctions bruitées

Hemodynamic Deconvolution Demystified: Sparsity-Driven Regularization at Work

Deconvolution of the hemodynamic response is an important step to access short timescales of brain activity recorded by functional magnetic resonance imaging (fMRI). Albeit conventional deconvolution algorithms have been around for a long time (e.g., Wiener deconvolution), recent state-of-the-art methods based on sparsity-pursuing regularization are attracting increasing interest to investigate brain dynamics and connectivity with fMRI. This technical note revisits the main concepts underlying two main methods, Paradigm Free Mapping and Total Activation, in the most accessible way. Despite their apparent differences in the formulation, these methods are theoretically equivalent as they represent the synthesis and analysis sides of the same problem, respectively. We demonstrate this equivalence in practice with their best-available implementations using both simulations, with different signal-to-noise ratios, and experimental fMRI data acquired during a motor task and resting-state. We evaluate the parameter settings that lead to equivalent results, and showcase the potential of these algorithms compared to other common approaches. This note is useful for practitioners interested in gaining a better understanding of state-of-the-art hemodynamic deconvolution, and aims to answer questions that practitioners often have regarding the differences between the two methods.Comment: 18 pages, 6 figures, submitted to Apertur

A Bayesian approach to single-particle electron cryo-tomography in RELION-4.0

We present a new approach for macromolecular structure determination from multiple particles in electron cryo-tomography (cryo-ET) data sets. Whereas existing subtomogram averaging approaches are based on 3D data models, we propose to optimise a regularised likelihood target that approximates a function of the 2D experimental images. In addition, analogous to Bayesian polishing and contrast transfer function (CTF) refinement in single-particle analysis, we describe the approaches that exploit the increased signal-to-noise ratio in the averaged structure to optimise tilt-series alignments, beam-induced motions of the particles throughout the tilt-series acquisition, defoci of the individual particles, as well as higher-order optical aberrations of the microscope. Implementation of our approaches in the open-source software package RELION aims to facilitate their general use, particularly for those researchers who are already familiar with its single-particle analysis tools. We illustrate for three applications that our approaches allow structure determination from cryo-ET data to resolutions sufficient for de novo atomic modelling.This work was funded by the UK Research and Innovation (UKRI) Medical Research Council (MC_UP_A025_1013 to SHWS; and MC_UP_1201/16 to JAGB), the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (ERC-CoG-2014, grant 648432, MEMBRANEFUSION to JAGB and ERC StG-2019, grant 852915 CRYTOCOP to GZ); the Swiss National Science Foundation (grant 205321_179041/1 to DC-D), the Max Planck Society (to JAGB) and the UKRI Biotechnology and Biological Sciences Research Council (grant BB/T002670/1 to GZ). TAMB is a recipient of a Sir Henry Dale Fellowship, jointly funded by the Wellcome Trust and the Royal Society (202231/Z/16/Z). JZ was partially funded by the European Union’s Horizon 2020 research and innovation program (ERC-ADG-2015, grant 692726, GlobalBioIm to Michael Unser)