Physics in the year 2006 is tightly constrained by experiment, observation, and mathematical consistency. The Standard Model provides a remarkably precise de- scription of particle physics, and general relativity is quite successful in describing gravitational phenomena. At the same time, it is clear that a more fundamental theory is needed for several distinct reasons. Here we consider a new approach, which begins with the unusually ambitious point of view that a truly fundamental theory should aspire to explaining the origins of Lorentz invariance, gravity, gauge fields and their symmetry, supersymmetry, fermionic fields, bosonic fields, quantum mechanics and spacetime. The present dissertation is organized so that it starts with the most conventional ideas for extending the Standard Model and ends with a microscopic statistical picture, which is actually the logical starting point of the theory, but which is also the most remote excursion from conventional physics. One motivation for the present work is the fact that a Euclidean path integral in quantum physics is equivalent to a partition function in statistical physics. This suggests that the most fundamental description of nature may be statistical. This dissertation may be regarded as an attempt to see how far one can go with this premise in explaining the observed phenomena, starting with the simplest statistical picture imaginable. It may be that nature is richer than the model assumed here, but the present results are quite suggestive, because, with a set of assumptions that are not unreasonable, one recovers the phenomena listed above. At the end, the present theory leads back to conventional physics, except that Lorentz invariance and supersymmetry are violated at extremely high energy. To be more specific, one obtains local Lorentz invariance (at low energy compared to the Planck scale), an SO(N) unified gauge theory (with N = 10 as the simplest possibility), supersymmetry of Standard Model fermions and their sfermion partners, and other familiar features of standard physics. Like other attempts at superunification, the present theory involves higher dimensions and topological defects.

Human actions are the essence of a human life and a natural product of the human mind. Analysis of human activities by a machine has attracted the attention of many researchers. This analysis is very important in a variety of domains including surveillance, video retrieval, human-computer interaction, athlete performance investigation, etc. This dissertation makes three major contributions to automatic analysis of human actions. First, we conjecture that the relationship between body joints of two actors in the same posture can be described by a 3D rigid transformation. This transformation simultaneously captures different poses and various sizes and proportions. As a consequence of this conjecture, we show that there exists a fundamental matrix between the imaged positions of the body joints of two actors, if they are in the same posture. Second, we propose a novel projection model for cameras moving at a constant velocity in 3D space, \emph cameras, and derive the Galilean fundamental matrix and apply it to human action recognition. Third, we propose a novel use for the invariant ratio of areas under an affine transformation and utilizing the epipolar geometry between two cameras for 2D model-based tracking of human body joints. In the first part of the thesis, we propose an approach to match human actions using semantic correspondences between human bodies. These correspondences are used to provide geometric constraints between multiple anatomical landmarks ( e.g. hands, shoulders, and feet) to match actions observed from different viewpoints and performed at different rates by actors of differing anthropometric proportions. The fact that the human body has approximate anthropometric proportion allows for innovative use of the machinery of epipolar geometry to provide constraints for analyzing actions performed by people of different anthropometric sizes, while ensuring that changes in viewpoint do not affect matching. A novel measure in terms of rank of matrix constructed only from image measurements of the locations of anatomical landmarks is proposed to ensure that similar actions are accurately recognized. Finally, we describe how dynamic time warping can be used in conjunction with the proposed measure to match actions in the presence of nonlinear time warps. We demonstrate the versatility of our algorithm in a number of challenging sequences and applications including action synchronization , odd one out, following the leader, analyzing periodicity etc. Next, we extend the conventional model of image projection to video captured by a camera moving at constant velocity. We term such moving camera Galilean camera. To that end, we derive the spacetime projection and develop the corresponding epipolar geometry between two Galilean cameras. Both perspective imaging and linear pushbroom imaging form specializations of the proposed model and we show how six different ``fundamental" matrices including the classic fundamental matrix, the Linear Pushbroom (LP) fundamental matrix, and a fundamental matrix relating Epipolar Plane Images (EPIs) are related and can be directly recovered from a Galilean fundamental matrix. We provide linear algorithms for estimating the parameters of the the mapping between videos in the case of planar scenes. For applying fundamental matrix between Galilean cameras to human action recognition, we propose a measure that has two important properties. First property makes it possible to recognize similar actions, if their execution rates are linearly related. Second property allows recognizing actions in video captured by Galilean cameras. Thus, the proposed algorithm guarantees that actions can be correctly matched despite changes in view, execution rate, anthropometric proportions of the actor, and even if the camera moves with constant velocity. Finally, we also propose a novel 2D model based approach for tracking human body parts during articulated motion. The human body is modeled as a 2D stick figure of thirteen body joints and an action is considered as a sequence of these stick figures. Given the locations of these joints in every frame of a model video and the first frame of a test video, the joint locations are automatically estimated throughout the test video using two geometric constraints. First, invariance of the ratio of areas under an affine transformation is used for initial estimation of the joint locations in the test video. Second, the epipolar geometry between the two cameras is used to refine these estimates. Using these estimated joint locations, the tracking algorithm determines the exact location of each landmark in the test video using the foreground silhouettes. The novelty of the proposed approach lies in the geometric formulation of human action models, the combination of the two geometric constraints for body joints prediction, and the handling of deviations in anthropometry of individuals, viewpoints, execution rate, and style of performing action. The proposed approach does not require extensive training and can easily adapt to a wide variety of articulated actions.