Information Processing by Neuron Populations in the Central Nervous System: Mathematical Structure of Data and Operations

In the intricate architecture of the mammalian central nervous system, neurons form populations. Axonal bundles communicate between these clusters using spike trains as their medium. However, these neuron populations' precise encoding and operations have yet to be discovered. In our analysis, the starting point is a state-of-the-art mechanistic model of a generic neuron endowed with plasticity. From this simple framework emerges a profound mathematical construct: The representation and manipulation of information can be precisely characterized by an algebra of finite convex cones. Furthermore, these neuron populations are not merely passive transmitters. They act as operators within this algebraic structure, mirroring the functionality of a low-level programming language. When these populations interconnect, they embody succinct yet potent algebraic expressions. These networks allow them to implement many operations, such as specialization, generalization, novelty detection, dimensionality reduction, inverse modeling, prediction, and associative memory. In broader terms, this work illuminates the potential of matrix embeddings in advancing our understanding in fields like cognitive science and AI. These embeddings enhance the capacity for concept processing and hierarchical description over their vector counterparts.Comment: 34 pages, 12 figure

Evolved Topology Generalized Multi-layer Perceptron (GMLP) for Anatomical Joint Constraint Modelling

The accurate simulation of anatomical joint models is becoming increasingly important for both medical diagnosis and realistic animation applications. Quaternion algebra has been increasingly applied to model rotations providing a compact representation while avoiding singularities. We propose the use of Artificial Neural Networks to accurately simulate joint constraints based on recorded data. This paper describes the application of Genetic Algorithm approaches to neural network training in order to model corrective piece-wise linear / discontinuous functions required to maintain valid joint configurations. The results show that artificial Neural Networks are capable of modeling constraints on the rotation of and around a virtual limb

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)

Geometric Inference in Bayesian Hierarchical Models with Applications to Topic Modeling

Unstructured data is available in abundance with the rapidly growing size of digital information. Labeling such data is expensive and impractical, making unsupervised learning an increasingly important field. Big data collections often have rich latent structure that statistical modeler is challenged to uncover. Bayesian hierarchical modeling is a particularly suitable approach for complex latent patterns. Graphical model formalism has been prominent in developing various procedures for inference in Bayesian models, however the corresponding computational limits often fall behind the demands of the modern data sizes. In this thesis we develop new approaches for scalable approximate Bayesian inference. In particular, our approaches are driven by the analysis of latent geometric structures induced by the models. Our specific contributions include the following. We develop full geometric recipe of the Latent Dirichlet Allocation topic model. Next, we study several approaches for exploiting the latent geometry to first arrive at a fast weighted clustering procedure augmented with geometric corrections for topic inference, and then a nonparametric approach based on the analysis of the concentration of mass and angular geometry of the topic simplex, a convex polytope constructed by taking the convex hull of vertices representing the latent topics. Estimates produced by our methods are shown to be statistically consistent under some conditions. Finally, we develop a series of models for temporal dynamics of the latent geometric structures where inference can be performed in online and distributed fashion. All our algorithms are evaluated with extensive experiments on simulated and real datasets, culminating at a method several orders of magnitude faster than existing state-of-the-art topic modeling approaches, as demonstrated by experiments working with several million documents in a dozen minutes.PHDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/146051/1/moonfolk_1.pd

Towards Visual Localization, Mapping and Moving Objects Tracking by a Mobile Robot: a Geometric and Probabilistic Approach

Dans cette thèse, nous résolvons le problème de reconstruire simultanément une représentation de la géométrie du monde, de la trajectoire de l'observateur, et de la trajectoire des objets mobiles, à l'aide de la vision. Nous divisons le problème en trois étapes : D'abord, nous donnons une solution au problème de la cartographie et localisation simultanées pour la vision monoculaire qui fonctionne dans les situations les moins bien conditionnées géométriquement. Ensuite, nous incorporons l'observabilité 3D instantanée en dupliquant le matériel de vision avec traitement monoculaire. Ceci élimine les inconvénients inhérents aux systèmes stéréo classiques. Nous ajoutons enfin la détection et suivi des objets mobiles proches en nous servant de cette observabilité 3D. Nous choisissons une représentation éparse et ponctuelle du monde et ses objets. La charge calculatoire des algorithmes de perception est allégée en focalisant activement l'attention aux régions de l'image avec plus d'intérêt. ABSTRACT : In this thesis we give new means for a machine to understand complex and dynamic visual scenes in real time. In particular, we solve the problem of simultaneously reconstructing a certain representation of the world's geometry, the observer's trajectory, and the moving objects' structures and trajectories, with the aid of vision exteroceptive sensors. We proceeded by dividing the problem into three main steps: First, we give a solution to the Simultaneous Localization And Mapping problem (SLAM) for monocular vision that is able to adequately perform in the most ill-conditioned situations: those where the observer approaches the scene in straight line. Second, we incorporate full 3D instantaneous observability by duplicating vision hardware with monocular algorithms. This permits us to avoid some of the inherent drawbacks of classic stereo systems, notably their limited range of 3D observability and the necessity of frequent mechanical calibration. Third, we add detection and tracking of moving objects by making use of this full 3D observability, whose necessity we judge almost inevitable. We choose a sparse, punctual representation of both the world and the moving objects in order to alleviate the computational payload of the image processing algorithms, which are required to extract the necessary geometrical information out of the images. This alleviation is additionally supported by active feature detection and search mechanisms which focus the attention to those image regions with the highest interest. This focusing is achieved by an extensive exploitation of the current knowledge available on the system (all the mapped information), something that we finally highlight to be the ultimate key to success