Cue Integration Using Affine Arithmetic and Gaussians

Thumbnail Image
Penn collection
Technical Reports (CIS)
Degree type
statistical cue integration
deformable model tracking
affine arithmetic
visual motion
Grant number
Copyright date
Related resources
Goldenstein, Siome
Vogler, Christian
Metaxas, Dimitris

In this paper we describe how the connections between affine forms, zonotopes, and Gaussian distributions help us devise an automated cue integration technique for tracking deformable models. This integration technique is based on the confidence estimates of each cue. We use affine forms to bound these confidences. Affine forms represent bounded intervals, with a well-defined set of arithmetic operations. They are constructed from the sum of several independent components. An n-dimensional affine form describes a complex convex polytope, called a zonotope. Because these components lie in bounded intervals, Lindeberg's theorem, a modified version of the central limit theorem,can be used to justify a Gaussian approximation of the affine form. We present a new expectation-based algorithm to find the best Gaussian approximation of an affine form. Both the new and the previous algorithm run in O(n2m) time, where n is the dimension of the affine form, and m is the number of independent components. The constants in the running time of new algorithm, however, are much smaller, and as a result it runs 40 times faster than the previous one for equal inputs. We show that using the Berry-Esseen theorem it is possible to calculate an upper bound for the error in the Gaussian approximation. Using affine forms and the conversion algorithm, we create a method for automatically integrating cues in the tracking process of a deformable model. The tracking process is described as a dynamical system, in which we model the force contribution of each cue as an affine form. We integrate their Gaussian approximations using a Kalman filter as a maximum likelihood estimator. This method not only provides an integrated result that is dependent on the quality of each on of the cues, but also provides a measure of confidence in the final result. We evaluate our new estimation algorithm in experiments, and we demonstrate our deformable model-based face tracking system as an application of this algorithm.

Date Range for Data Collection (Start Date)
Date Range for Data Collection (End Date)
Digital Object Identifier
Series name and number
Publication date
Volume number
Issue number
Publisher DOI
Journal Issue
University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-02-06.
University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-02-06.
Recommended citation