Teleological computer graphics modeling

Summary form only give. Teleological modeling, a developing approach for creating abstractions and mathematical representations of physically realistic time-dependent objects, is described. In this approach, geometric constraint-properties, mechanical properties of objects, the parameters representing an object, and the control of the object are incorporated into a single conceptual framework. A teleological model incorporates time-dependent goals of behavior of purpose as the primary abstraction and representation of what the object is. A teleological implementation takes a geometrically incomplete specification of the motion, position, and shape of an object, and produces a geometrically complete description of the object's shape and behavior as a function of time. Teleological modeling techniques may be suitable for consideration in computer vision algorithms by extending the current notions about how to make mathematical representations of objects. Teleological descriptions can produce compact representations for many of the physically derivable quantities controlling the shapes, combining-operations, and constraints which govern the formation and motion of objects

Improving the chi-squared approximation for bivariate normal tolerance regions

Let X be a two-dimensional random variable distributed according to N2(mu,Sigma) and let bar-X and S be the respective sample mean and covariance matrix calculated from N observations of X. Given a containment probability beta and a level of confidence gamma, we seek a number c, depending only on N, beta, and gamma such that the ellipsoid R = (x: (x - bar-X)'S(exp -1) (x - bar-X) less than or = c) is a tolerance region of content beta and level gamma; i.e., R has probability gamma of containing at least 100 beta percent of the distribution of X. Various approximations for c exist in the literature, but one of the simplest to compute -- a multiple of the ratio of certain chi-squared percentage points -- is badly biased for small N. For the bivariate normal case, most of the bias can be removed by simple adjustment using a factor A which depends on beta and gamma. This paper provides values of A for various beta and gamma so that the simple approximation for c can be made viable for any reasonable sample size. The methodology provides an illustrative example of how a combination of Monte-Carlo simulation and simple regression modelling can be used to improve an existing approximation