Graduate

published_or_final_version / abstract / toc / Electrical and Electronic Engineering / Master / Master of Philosophy

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In a recent manuscript (Stratification and domination in graphs. Discrete Math. 272 (2003), 171-185) a new mathematical framework for studying domination is presented. It is shown that the domination number and many domination related parameters can be interpreted as restricted 2-stratifications or 2-colorings. This framework places the domination number in a new perspective and suggests many other parameters of a graph which are related in some way to the domination number. In this thesis, we continue this study of domination and stratification in graphs. Let F be a 2-stratified graph with one fixed blue vertex v specified. We say that F is rooted at the blue vertex v. An F-coloring of a graph G is a red-blue coloring of the vertices of G such that every blue vertex v of G belongs to a copy of F (not necessarily induced in G) rooted at v. The F-domination number yF(GQ of G is the minimum number of red vertices of G in an F-coloring of G. Chapter 1 is an introduction to the chapters that follow. In Chapter 2, we investigate the X-domination number of prisms when X is a 2-stratified 4-cycle rooted at a blue vertex where a prism is the cartesian product Cn x K2, n > 3, of a cycle Cn and a K2. In Chapter 3 we investigate the F-domination number when (i) F is a 2-stratified path P3 on three vertices rooted at a blue vertex which is an end-vertex of the F3 and is adjacent to a blue vertex and with the remaining vertex colored red. In particular, we show that for a tree of diameter at least three this parameter is at most two-thirds its order and we characterize the trees attaining this bound. (ii) We also investigate the F-domination number when F is a 2-stratified K3 rooted at a blue vertex and with exactly one red vertex. We show that if G is a connected graph of order n in which every edge is in a triangle, then for n sufficiently large this parameter is at most (n — /n)/2 and this bound is sharp. In Chapter 4, we further investigate the F-domination number when F is a 2- stratified path P3 on three vertices rooted at a blue vertex which is an end-vertex of the P3 and is adjacent to a blue vertex with the remaining vertex colored red. We show that for a connected graph of order n with minimum degree at least two this parameter is bounded above by (n —1)/2 with the exception of five graphs (one each of orders four, five and six and two of order eight). For n > 9, we characterize those graphs that achieve the upper bound of (n — l)/2. In Chapter 5, we define an f-coloring of a graph to be a red-blue coloring of the vertices such that every blue vertex is adjacent to a blue vertex and to a red vertex, with the red vertex itself adjacent to some other red vertex. The f-domination number yz{G) of a graph G is the minimum number of red vertices of G in an f-coloring of G. Let G be a connected graph of order n > 4 with minimum degree at least 2. We prove that (i) if G has maximum degree A where A 4 with maximum degree A where A 5 with maximum degree A where 3 / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2006.

Current hybrid coders such as H.261/263/264 or MPEG-l/-2 cannot always offer high quality-to-compression ratios for video transfer over the (low-bandwidth) wireless channels typical of handheld devices (such as smartphones and PDAs). Often these devices are utilised in videophone and teleconferencing scenarios, where the subjects of inte:est in the scene are peoples faces. In these cases, an alternative coding scheme known as Model-Based Video Coding (MBVC) can be employed. MBVC systems for face scenes utilise geometrically and photorealistically accurate computer graphic models to represent head !md shoulder views of people in a scene. High compression ratios are achieved at the encoder by extracting and transmitting only the parameters which represent the explicit shape and motion changes occurring on the face in the scene. With some a priori knowledge (such as the MPEG-4 standard for facial animation parameters), the transmitted parameters can be used at the decoder to accurately animate the graphical model and a synthesised version of the scene (originally appearing at the encoder) can be output. Primary components for facial re-animation at the decoder are a set of local and global motion parameters extracted from the video sequence appearing at the encoder. Local motion describes the changes in facial expression occurring on the face. Global motion describes the three-dimensional motion· of the entire head as a rigid object. Extraction of this three-dimensional global motion is often called head tracking. This thesis focuses on the tracking of rigid head pose in a monocular video sequence. The system framework utilises the recursive Structure from Motion (SfM) method of Azarbayejani and Pentland. Integral to the SfM solution are a large number of manually selected two-dimensional feature points, which are tracked throughout the sequence using an efficient image registration technique. The trajectories of the feature points are simultaneously processed by an extended Kalman filter (EKF) to stably recover camera geometry and the rigid three-dimensional structure and pose of the head. To improve estimation accuracy and stability, adaptive estimation is harnessed within the Kalman filter by dynamically varying the noise associated with each of the feature measurements. A closed loop approach is used to constrain feature tracking in each frame. The Kalman filter's estimate of motion and structure of the face are used to predict the trajectory of the features, thereby constraining the search space for the next frame in the video sequence. Further robustness in feature tracking is achieved through the integration of a linear appearance basis to accommodate variations in illumination or changes in aspect on the face. Synthetic experiments are performed for both the SfM and the feature tracking algorithm. The accuracy of the SfM solution is evaluated against synthetic ground truth. Further experimentation demonstrates the stability of the framework to significant noise corruption on arriving measurement data. The accuracy of obtained pixel measurements in the feature tracking algorithm is also evaluated against known ground truth. Additional experiments confirm feature tracking stability despite significant changes in target appearance. Experiments with real video sequences illustrate robustness of the complete head tracker to partial occlusions on the face. The SfM solution (including two-dimensional tracking) runs near real time at 12 Hz. The limits of Pitch, Yaw and Roll (rotational) recovery are 45°,45° and 90° respectively. Large translational recovery (especially depth) is also demonstrated. The estimated motion trajectories are validated against (publically available) ground truth motion captured using a commercial magnetic orientation tracking system. Rigid reanimation of an overlayed wire frame face model is further used as a visually subjective analysis technique. These combined results serve to confirm the suitability of the proposed head tracker as the global (rigid) motion estimator in an MBVC system. / Thesis (M.Sc.Eng.)-University of KwaZulu-Natal, Durban, 2005.

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