Real time feature-based facial tracking using Lie algebras

We have developed a novel human facial tracking system that operates in real time at a video frame rate without needing any special hardware. The approach is based on the use of Lie algebra, and uses three-dimensional feature points on the targeted human face. It is assumed that the roughly estimated facial model (relative coordinates of the three-dimensional feature points) is known. First, the initial feature positions of the face are determined using a model fitting technique. Then, the tracking is operated by the following sequence: (1) capture the new video frame and render feature points to the image plane; (2) search for new positions of the feature points on the image plane; (3) get the Euclidean matrix from the moving vector and the three-dimensional information for the points; and (4) rotate and translate the feature points by using the Euclidean matrix, and render the new points on the image plane. The key algorithm of this tracker is to estimate the Euclidean matrix by using a least square technique based on Lie algebra. The resulting tracker performed very well on the task of tracking a human face.

Mathematical relationships between representations of structure in linear interconnected dynamical systems

A dynamical system can exhibit structure on multiple levels. Different system representations can capture different elements of a dynamical system's structure. We consider LTI input-output dynamical systems and present four representations of structure: complete computational structure, subsystem structure, signal structure, and input output sparsity structure. We then explore some of the mathematical relationships that relate these different representations of structure. In particular, we show that signal and subsystem structure are fundamentally different ways of representing system structure. A signal structure does not always specify a unique subsystem structure nor does subsystem structure always specify a unique signal structure. We illustrate these concepts with a numerical example. © 2011 AACC American Automatic Control Council.

Flexible representations of dynamics are used in object manipulation.

To manipulate an object skillfully, the brain must learn its dynamics, specifying the mapping between applied force and motion. A fundamental issue in sensorimotor control is whether such dynamics are represented in an extrinsic frame of reference tied to the object or an intrinsic frame of reference linked to the arm. Although previous studies have suggested that objects are represented in arm-centered coordinates [1-6], all of these studies have used objects with unusual and complex dynamics. Thus, it is not known how objects with natural dynamics are represented. Here we show that objects with simple (or familiar) dynamics and those with complex (or unfamiliar) dynamics are represented in object- and arm-centered coordinates, respectively. We also show that objects with simple dynamics are represented with an intermediate coordinate frame when vision of the object is removed. These results indicate that object dynamics can be flexibly represented in different coordinate frames by the brain. We suggest that with experience, the representation of the dynamics of a manipulated object may shift from a coordinate frame tied to the arm toward one that is linked to the object. The additional complexity required to represent dynamics in object-centered coordinates would be economical for familiar objects because such a representation allows object use regardless of the orientation of the object in hand.