Parallel Deterministic and Stochastic Global Minimization of Functions with Very Many Minima

The optimization of three problems with high dimensionality and many local minima are investigated under five different optimization algorithms: DIRECT, simulated annealing, Spall’s SPSA algorithm, the KNITRO package, and QNSTOP, a new algorithm developed at Indiana University

Automatic construction of boundary parametrizations for geometric multigrid solvers

We present an algorithm that constructs parametrizations of boundary and interface surfaces automatically. Starting with high-resolution triangulated surfaces describing the computational domains, we iteratively simplify the surfaces yielding a coarse approximation of the boundaries with the same topological type. While simplifying we construct a function that is defined on the coarse surface and whose image is the original surface. This function allows access to the correct shape and surface normals of the original surface as well as to any kind of data defined on it. Such information can be used by geometric multigrid solvers doing adaptive mesh refinement. Our algorithm runs stable on all types of input surfaces, including those that describe domains consisting of several materials. We have used our method with success in different fields and we discuss examples from structural mechanics and biomechanics

Conformal Tracking For Virtual Environments

A virtual environment is a set of surroundings that appears to exist to a user through sensory stimuli provided by a computer. By virtual environment, we mean to include environments supporting the full range from VR to pure reality. A necessity for virtual environments is knowledge of the location of objects in the environment. This is referred to as the tracking problem, which points to the need for accurate and precise tracking in virtual environments. Marker-based tracking is a technique which employs fiduciary marks to determine the pose of a tracked object. A collection of markers arranged in a rigid configuration is called a tracking probe. The performance of marker-based tracking systems depends upon the fidelity of the pose estimates provided by tracking probes. The realization that tracking performance is linked to probe performance necessitates investigation into the design of tracking probes for proponents of marker-based tracking. The challenges involved with probe design include prediction of the accuracy and precision of a tracking probe, the creation of arbitrarily-shaped tracking probes, and the assessment of the newly created probes. To address these issues, we present a pioneer framework for designing conformal tracking probes. Conformal in this work means to adapt to the shape of the tracked objects and to the environmental constraints. As part of the framework, the accuracy in position and orientation of a given probe may be predicted given the system noise. The framework is a methodology for designing tracking probes based upon performance goals and environmental constraints. After presenting the conformal tracking framework, the elements used for completing the steps of the framework are discussed. We start with the application of optimization methods for determining the probe geometry. Two overall methods for mapping markers on tracking probes are presented, the Intermediary Algorithm and the Viewpoints Algorithm. Next, we examine the method used for pose estimation and present a mathematical model of error propagation used for predicting probe performance in pose estimation. The model uses a first-order error propagation, perturbing the simulated marker locations with Gaussian noise. The marker locations with error are then traced through the pose estimation process and the effects of the noise are analyzed. Moreover, the effects of changing the probe size or the number of markers are discussed. Finally, the conformal tracking framework is validated experimentally. The assessment methods are divided into simulation and post-fabrication methods. Under simulation, we discuss testing of the performance of each probe design. Then, post-fabrication assessment is performed, including accuracy measurements in orientation and position. The framework is validated with four tracking probes. The first probe is a six-marker planar probe. The predicted accuracy of the probe was 0.06 deg and the measured accuracy was 0.083 plus/minus 0.015 deg. The second probe was a pair of concentric, planar tracking probes mounted together. The smaller probe had a predicted accuracy of 0.206 deg and a measured accuracy of 0.282 plus/minus 0.03 deg. The larger probe had a predicted accuracy of 0.039 deg and a measured accuracy of 0.017 plus/minus 0.02 deg. The third tracking probe was a semi-spherical head tracking probe. The predicted accuracy in orientation and position was 0.54 plus/minus 0.24 deg and 0.24 plus/minus 0.1 mm, respectively. The experimental accuracy in orientation and position was 0.60 plus/minus 0.03 deg and 0.225 plus/minus 0.05 mm, respectively. The last probe was an integrated, head-mounted display probe, created using the conformal design process. The predicted accuracy of this probe was 0.032 plus/minus 0.02 degrees in orientation and 0.14 plus/minus 0.08 mm in position. The measured accuracy of the probe was 0.028 plus/minus 0.01 degrees in orientation and 0.11 plus/minus 0.01 mm in position. These results constitute an order of magnitude improvement over current marker-based tracking probes in orientation, indicating the benefits of a conformal tracking approach. Also, this result translates to a predicted positional overlay error of a virtual object presented at 1m of less than 0.5 mm, which is well above reported overlay performance in virtual environments

Optical techniques for 3D surface reconstruction in computer-assisted laparoscopic surgery

One of the main challenges for computer-assisted surgery (CAS) is to determine the intra-opera- tive morphology and motion of soft-tissues. This information is prerequisite to the registration of multi-modal patient-specific data for enhancing the surgeon’s navigation capabilites by observ- ing beyond exposed tissue surfaces and for providing intelligent control of robotic-assisted in- struments. In minimally invasive surgery (MIS), optical techniques are an increasingly attractive approach for in vivo 3D reconstruction of the soft-tissue surface geometry. This paper reviews the state-of-the-art methods for optical intra-operative 3D reconstruction in laparoscopic surgery and discusses the technical challenges and future perspectives towards clinical translation. With the recent paradigm shift of surgical practice towards MIS and new developments in 3D opti- cal imaging, this is a timely discussion about technologies that could facilitate complex CAS procedures in dynamic and deformable anatomical regions

kPCA-Based Parametric Solutions Within the PGD Framework

Parametric solutions make possible fast and reliable real-time simulations which, in turn allow real time optimization, simulation-based control and uncertainty propagation. This opens unprecedented possibilities for robust and efficient design and real-time decision making. The construction of such parametric solutions was addressed in our former works in the context of models whose parameters were easily identified and known in advance. In this work we address more complex scenarios in which the parameters do not appear explicitly in the model—complex microstructures, for instance. In these circumstances the parametric model solution requires combining a technique to find the relevant model parameters and a solution procedure able to cope with high-dimensional models, avoiding the well-known curse of dimensionality. In this work, kPCA (kernel Principal Component Analysis) is used for extracting the hidden model parameters, whereas the PGD (Proper Generalized Decomposition) is used for calculating the resulting parametric solution

Classical and all-floating FETI methods for the simulation of arterial tissues

High-resolution and anatomically realistic computer models of biological soft tissues play a significant role in the understanding of the function of cardiovascular components in health and disease. However, the computational effort to handle fine grids to resolve the geometries as well as sophisticated tissue models is very challenging. One possibility to derive a strongly scalable parallel solution algorithm is to consider finite element tearing and interconnecting (FETI) methods. In this study we propose and investigate the application of FETI methods to simulate the elastic behavior of biological soft tissues. As one particular example we choose the artery which is - as most other biological tissues - characterized by anisotropic and nonlinear material properties. We compare two specific approaches of FETI methods, classical and all-floating, and investigate the numerical behavior of different preconditioning techniques. In comparison to classical FETI, the all-floating approach has not only advantages concerning the implementation but in many cases also concerning the convergence of the global iterative solution method. This behavior is illustrated with numerical examples. We present results of linear elastic simulations to show convergence rates, as expected from the theory, and results from the more sophisticated nonlinear case where we apply a well-known anisotropic model to the realistic geometry of an artery. Although the FETI methods have a great applicability on artery simulations we will also discuss some limitations concerning the dependence on material parameters.Comment: 29 page

MEMS Capacitive Strain Sensing Elements for Integrated Total Knee Arthroplasty Prosthesis Monitoring

Measuring the in vivo load state of Total Knee Arthroplasty (TKA) components is required to understand the structural environment and wear characteristics of the devices. The ability to acquire this information gives tremendous insight into the mechanics of the joint replacement prosthesis. Data corresponding to normal loads, in-plane loads, shear loads, load center, contact area, and the rate of loading is needed to fully understand the kinematics and kinetics of the orthopedic implant. In this research, a novel sensing system has been developed which is capable of fully characterizing three-dimensional strain and stress at a single location. Capacitance-based sensors were chosen to avoid the power loss and drift characteristics typical of resistive elements due to resistive heating effects. A design and optimization methodology has been developed by combining conformal mapping electrostatic analysis techniques with methods from micromechanics of composite materials. Results of the design and optimization technique are used to understand the behavior of the sensing system. Simulation of these systems was performed using multiphysics finite element analysis, and novel methods for fabricating the sensors were adapted from techniques for fabricating microelectromechanical systems (MEMS) using biocompatible materials. An array of six sensors was fabricated with a critical dimension of 2.25 micrometers. This array consisted of a parallel plate capacitor for measuring normal strain, two differential elements for sensing shear strain normal to the plane of the array, and three interdigitated transducer (IDT) elements for characterizing strain in the plane of the sensor. The normal strain sensor exhibited a sensitivity of 1.54×10-3 picofarads per megapascal, and the shear sensor had a sensitivity of 4.77×10-5 picofarads per megapascal. Testing results showed that all sensors had linear response to loading and insignificant drift. Multiaxial testing results illustrated the ability of the differential sensors to determine loading direction. A multiaxial, MEMS sensor array has been developed for use in orthopedic, load-measuring conditions. This system has been optimized for use in soft materials such as ultra-high molecular weight polyethylene (UHMWPE). In the future, arrays of sensors will be embedded in orthopedic components to determine the total state of stress at local positions within the component

Tracking and modelling motion for biomechanical analysis

This thesis focuses on the problem of determining appropriate skeletal configurations for which a virtual animated character moves to desired positions as smoothly, rapidly, and as accurately as possible. During the last decades, several methods and techniques, sophisticated or heuristic, have been presented to produce smooth and natural solutions to the Inverse Kinematics (IK) problem. However, many of the currently available methods suffer from high computational cost and production of unrealistic poses. In this study, a novel heuristic method, called Forward And Backward Reaching Inverse Kinematics (FABRIK), is proposed, which returns visually natural poses in real-time, equally comparable with highly sophisticated approaches. It is capable of supporting constraints for most of the known joint types and it can be extended to solve problems with multiple end effectors, multiple targets and closed loops. FABRIK was compared against the most popular IK approaches and evaluated in terms of its robustness and performance limitations. This thesis also includes a robust methodology for marker prediction under multiple marker occlusion for extended time periods, in order to drive real-time centre of rotation (CoR) estimations. Inferred information from neighbouring markers has been utilised, assuming that the inter-marker distances remain constant over time. This is the first time where the useful information about the missing markers positions which are partially visible to a single camera is deployed. Experiments demonstrate that the proposed methodology can effectively track the occluded markers with high accuracy, even if the occlusion persists for extended periods of time, recovering in real-time good estimates of the true joint positions. In addition, the predicted positions of the joints were further improved by employing FABRIK to relocate their positions and ensure a fixed bone length over time. Our methodology is tested against some of the most popular methods for marker prediction and the results confirm that our approach outperforms these methods in estimating both marker and CoR positions. Finally, an efficient model for real-time hand tracking and reconstruction that requires a minimum number of available markers, one on each finger, is presented. The proposed hand model is highly constrained with joint rotational and orientational constraints, restricting the fingers and palm movements to an appropriate feasible set. FABRIK is then incorporated to estimate the remaining joint positions and to fit them to the hand model. Physiological constraints, such as inertia, abduction, flexion etc, are also incorporated to correct the final hand posture. A mesh deformation algorithm is then applied to visualise the movements of the underlying hand skeleton for comparison with the true hand poses. The mathematical framework used for describing and implementing the techniques discussed within this thesis is Conformal Geometric Algebra (CGA)