Elsevier

Signal Processing

Volume 82, Issue 9, September 2002, Pages 1205-1214
Signal Processing

A HMM approach to the estimation of random trajectories on manifolds

https://doi.org/10.1016/S0165-1684(02)00242-6Get rights and content

Abstract

Dynamic image analysis requires the estimation of time-varying model parameters (e.g., shape coefficients). This can be seen as states of a dynamic model which are restricted to a subset of Euclidean space. This paper describes an algorithm for the estimation of the state evolution on manifolds exploiting three sources of information: the manifold geometry, the motion model and the sensor model. Examples are provided to illustrate the performance of this method in situations where classic procedures cannot perform well.

Introduction

Dynamic image analysis requires the estimation of time-varying model parameters (e.g., shape coefficients). This can be seen as states of a dynamic model which are restricted to a subset of Euclidean space. Typical examples are the estimation of objects motions (e.g., cars) or the evolution of objects shapes (e.g., mouth contour, heart cavities) from video sequences. Hereafter, state is to be understood in this way. A common factor present in most estimation problems is the fact that unknown variables exist in high-dimensional spaces but they cannot take arbitrary values. Instead, they are usually restricted to smooth subsets (e.g., surfaces) (Fig. 1). These subsets often have a complex structure and must be estimated from the observation data. The following examples illustrate this point. Consider the problem of visual tracking of cars in a lane. A simple prototype situation is considered below in this paper. Since the lane geometry will enforce the trajectory, this is a valuable information for reducing the computational load of the search algorithm and furthermore enhances the robustness of the position estimation. Classical algorithms i.e., which do not restrict the state variables to a manifold, are more prone to yield instabilities in the tracking error. This drawback is even more serious when the observations of the car position are drawn from omnidirectional sensors which measure only the distance to the target. An example is given below. Where these examples refer to the actual position of the moving object constrained to a manifold, one can also think of its motion parameters also being constrained. For instance, let the car velocity be modeled as the lowpass filtering of white noise with a given bandwidth (BW). If BW slowly varies in time, being constrained to some subset of values, the methods to be considered in this paper may also be applied with advantage.

In static problems (e.g., in image reconstruction and object recognition) attempts have been made for incorporating known restrictions in the estimation process [6], [11], [13]. Two types of constraints are usually considered: hard constraints which lead to the use of constrained optimization methods (e.g., POCS [1], [7]) or soft constraints based on the use of regularization techniques or prior information [12]. In [6] three-dimensional object views are represented as a linear combination of eigen images multiplied by appropriate coefficients. Although the number of coefficients is very high, the number of degrees of freedom is much smaller i.e., when the view changes the coefficients describe a trajectory on a low dimension manifold. Advantage is taken of this fact for object recognition.

In dynamic scene analysis, state constraints play an even more important role for two main reasons: (i) they significantly improve the trajectory estimates (the improvement being often dramatic, e.g., a nonobservable system may become observable if appropriate restrictions are used) and (ii) they allow one to formulate the estimation problem in a lower dimension subspace (the dimension of the working subspace depends on the manifold dimension and not on the data dimension). Both effects are instrumental for achieving good results and should be considered in the design of trajectory estimation algorithms [3], [2]. In [3] geometric restrictions are used for lip tracking. Although lips are represented by 40 control points belonging to a space of dimension 80, by exploiting the constraints in the control point movement, estimation has only to be performed in a space of dimension 5. Other examples are provided in the analysis of Human gestures. In [2] gestures are described by a one-dimensional manifold denoted as principal curve.

In this paper, by considering a general framework for trajectory estimation on manifolds, a specific algorithm to solve this problem under a general hypothesis is proposed. By relying on discrete approximation and hidden Markov model (HMM) techniques, an algorithm for trajectory estimation on manifolds is derived. Illustrative examples are presented.

Section snippets

Estimation framework

The problem is stated as follows: Let x be an unknown trajectory defined in a manifold M⊂Rn. The trajectory x is to be retrieved from a sequence of nonlinear and noisy observations y. It will be assumed that x is a realization of a stochastic process defined on the manifold and y consists of nonlinear and noisy observations of x values; these processes are characterized by a motion model p(xt/xt−1) and by a sensor model p(yt/xt) that have to be estimated from the data.

The overall solution to

The discrete manifold analysis (DMA) algorithm

This section describes an algorithm for the estimation of trajectories on manifolds. This algorithm, denoted as DMA, provides a solution for the three problems described before: manifold learning; motion/sensor model learning and trajectory retrieval.

Experimental results

Hereafter, the DMA algorithm is evaluated with both synthetic and real data. Three examples will be described to illustrate the concepts presented in the paper.

Conclusion

This paper exploits geometric restrictions for solving estimation problems in dynamic scene analysis. Although the primary motivation stems from image processing problems, the methods described may also be used in relation to control systems where parameters slowly move on a manifold due to changes in plant operating condition. An algorithm is proposed to estimate unknown state trajectories in manifolds. This algorithm denoted as discrete manifold analysis (DMA) allows to use the available

Acknowledgements

The authors thank the anonymous reviewers for the valuable comments which helped to improve the paper.

References (13)

  • J. Biemond et al.

    Iterative methods for image deblurring

    Proc. IEEE

    (1990)
  • A.F. Bobick, A.D. Wilson, A state-based approach to the representation and recognition of gesture, Trans. Pattern Anal....
  • C. Bregler, S. Omohundro, Nonlinear manifold learning for visual speech recognition, Internat. Conf. Comput. Vision...
  • J. Clark

    An introduction to stochastic differential equations on manifolds, in: Geometric Methods on Systems Theory

    (1973)
  • R. Marino et al.

    Nonlinear Control Design

    (1995)
  • H. Murase et al.

    Visual learning and recognition of 3-D objects from appearance

    Internat. J. Comput. Vision

    (1995)
There are more references available in the full text version of this article.

Cited by (3)

This work was partially supported by the POSI Program of the 3rd EC Framework and by PRAXIS XXI under project EEI/12050/1998 (Tracking of Moving Objects).

View full text