We present a unified derivation of covariant time derivatives, which transform as tensors under a time-dependent coordinate change. Such derivatives are essential for formulating physical laws in a frame-independent manner. Three specific derivatives are described: convective, corotational, and directional. The covariance is made explicit by working in arbitrary time-dependent coordinates, inst...

The points P0, P1 and P2 are the vertices of the triangle. P is a point at which interpolation will be performed. For purposes of this document, point P0 has cartesian coordinates (x0,y0,w0), color coordinates (r0,g0,b0) and texture coordinates (u0,v0). Points P1 and P2 are similarly defined. The cartesian coordinates are defined in perspective space, i.e., they have been multiplied by a perspe...

Some general remarks are made about the quantum theory of scalar fields and the definition of momentum in curved space. Special emphasis is given to field theory in anti-de Sitter space, as it represents a maximally symmetric space-time of constant curvature which could arise in the local description of matter interactions in the small regions of space-time. Transform space rules for evaluating...

Over the past decade, computer scientists and psychologists have made great efforts to collect and analyze facial dynamics data that exhibit different expressions and emotions. Such data is commonly captured as videos and are transformed into feature-based time-series prior to any analysis. However, the analytical tasks, such as expression classification, have been hindered by the lack of under...