A Variational Principle for Model-based Morphing

Given a multidimensional data set and a model of its density, we consider how to define the optimal interpolation between two points. This is done by assigning a cost to each path through space, based on two competing goals-one to interpolate through regions of high density, the other to minimize arc length. From this path functional, we derive the Euler-Lagrange equations for extremal motionj given two points, the desired interpolation is found by solving aboundary value problem. We show that this interpolation can be done efficiently, in high dimensions, for Gaussian, Dirichlet, and mixture models. 1 Introduction The problem of nonlinear interpolation arises frequently in image, speech, and signal processing. Consider the following two examples: (i) given two profiles of the same face, connect them by a smooth animation of intermediate poses[l]j (ii) given a telephone signal masked by intermittent noise, fill in the missing speech. Both these examples may be viewed as instances of the same abstract problem. In qualitative terms, we can state the problem as follows[2]: given a multidimensional data set, and two points from this set, find a smooth adjoining path that is consistent with available models of the data. We will refer to this as the problem of model-based morphing. In this paper, we examine this problem it arises from statistical models of multidimensional data.Specifically, our focus is on models that have been derived from Current address: AT&T Labs, 600 Mountain Ave 2D-439, Murray Hill, NJ 07974 268 LK.