# Shape and measurements

An optimal shape operator to produce an accurate detection and localization of a given shape.

The response of the local image of an object to the operator having geometric configuration is

If we assume that the image is corrupted by noise , then the observation is given by

where is the noise response. Since we sample the observations over the course of time, we denote the observation process by

# The Zakai equation and the branching particle method

## The Zakai equation

The state vector representing the geometric parameters of an object is governed by the equation

Here is a Brownian motion, and models the state noise structure.

The tracking problem is solved if we can compute the state updates, given information from the observations. We are interested in estimating some statistic of the states, of the form

given the observation history . Zakai et al. have shown that the unnormalized conditional density satisfies a partial differential equation, usually called the Zakai equation:

where

## The branching particle algorithm

The unnormalized optimal filter which is the solution to the problem is given by

We construct a sequence of branching particle systems as in [#!Crisan-branching-98!#], which can be proved to approach the solution : .

Let be a sequence of branching particle systems on , the standard measure space on the state space.

Initial condition

0. is the empirical measure of particles of mass , i.e., , where , for every .

Evolution in the interval ,

1. At time , the process consists of the occupation measure of particles of mass ( denotes the number of particles alive at time ).

2. During the interval, the particles move independently with the same law as the signal . Let , be the trajectory of a generic particle during this interval.

3. At , each particle branches into particles with a mechanism depending on its trajectory in the interval. The mean number of offspring for a particle given the -field of events up to time is

so that the variance of is minimal, consistent with the number of offspring being an integer. More specifically, we determine the number of offsprings by

# Time update of the state

In the discrete-time process:

where is the displacement vector which involves the estimation of velocity and acceleration. can be further refined to ensure the maximum observation likelihood:

This seemingly trivial addition of a prediction adjustment is found to achieve a great degree of stability.

The time update step yields the prior estimate of the state and the covariance matrix:

Here and denote the posterior estimates after the measurement update (the application of the Kalman gain), which is equivalent to the observation and branching steps in the proposed algorithm. The a priori and a posteriori error covariance matrices are formally defined as

These matrices are estimated by bootstrapping the particles and the prior/posterior state estimates into the above expressions. We use the error covariance estimated from the particles at time for the diffusion at time :

## Camera model and filter construction

Given , the hypothetical geometric parameters of the head and feature (simply denoted by ), we compute the inverse projection on the ellipsoid, to construct the shape operator. We first compute the inverse rotation and translation to get . Suppose the feature curve on the ellipsoid is the intersection (with the ellipsoid) of the circle centered at which is on the ellipsoid. Let be any point in the image. The inverse projection of is the line defined by the projection equation. The point on the ellipsoid is computed by solving the projection equation combined with the quadratic equation . This solution exists and is unique, since we seek the solution on the visible side of the ellipsoid. The point on the reference ellipsoid is computed using the inverse projection.

If we define the mapping from to by we can construct the shape filter as

<

## Experiments on real data

An example, where the person repeatedly moves his head left and right, and the rotation of the head is naturally coupled with the translation.

The contribution of the maximum observation likelihood prediction adjustment and the adaptive perturbation is verified.

A small translation of the head in the vertical direction can be confused with a nodding' motion. The following figure depicts the ambiguity present in the same sequence by plotting the projections of particles onto the plane. Initial distribution shows the correlation between and . As more information is provided (), the particles show multi-modal concentrations. The concentration is dispersed when the motion is rapid, and shrinks when the head motion is close to one of the two extreme' points. The parameters eventually settle into a dominant configuration ( and )

The following shows another example in which local feature motion is tracked in addition to global object motion; the motions of the irises and upper eyelids are more carefully tracked, so that squinting and gaze are recognized.

3D Object Tracking

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The translation was initiated by Hankyu Moon on 2001-02-20

Hankyu Moon 2001-02-20