Optimal instantaneous rigid motion estimation insensitive to local minima

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Abstract

A novel method is introduced for optimal estimation of rigid camera motion from instantaneous velocity measurements. The error surface associated with this problem is highly complex and existing algorithms suffer heavily from local minima. Repeated minimization with different random initializations and selection of the minimum-cost solution are a common (albeit ad hoc) procedure to increase the likelihood of finding the global minimum. We instead show that the optimal estimation problem can be transformed into one of arbitrary complexity, which allows for a gradual regularization of the error function. A simple reweighting scheme is presented that smoothly increases the problem complexity at each iteration. We show that the resulting method retains all the desirable properties of optimal algorithms, such as unbiasedness and minimal variance of the parameter estimates, but is substantially more robust to local minima. This robustness comes at the expense of a slightly increased computational complexity.

Introduction

The instantaneous velocity or optic flow field encountered by a moving observer contains an enormous amount of information related to the three dimensional (3D) structure of the environment and to the presence and motion of independently moving objects. Knowledge of the egomotion or self-motion of the observer is a necessary prerequisite to obtain this valuable information. Since small observer motions can have large effects on the optic flow field, it is advisable to extract the egomotion parameters from the optic flow field itself. This, however, is non-trivial and an active topic of research.

The field has matured a lot over the years and a number of ‘optimal’ algorithms (unbiased and minimal variance of the estimates) have appeared [1], [2]. The error function of the optimal problem formulation is however highly nonlinear and contains a large number of local minima [3], [4], which renders these algorithms unreliable and hard to use in practical applications. The earlier approaches [5], [6], [7], [8], which operate on a linearization of the problem, are no valid alternative. Compared to optimal algorithms, they are extremely sensitive to noise [1], [2], [9] and the estimates they provide are unsuitable, even as initializations for the optimal methods.

As an alternative to the time-consuming process of repeatedly minimizing with different, random initializations and selection of the minimum-cost solution, we propose to regularize the error function. We reformulate the problem in such a way that the complexity of the error function (the likelihood that algorithms end up in local minima) is controlled by a single parameter. We propose a reweighting scheme that gradually increases the problem complexity during the minimization, until the optimal problem formulation is obtained. We demonstrate, both in simulation and on real data, that the proposed method retains the accuracy of optimal algorithms, but is much less sensitive to local minima. On the extensive set of data investigated, these improvements come at the cost of less than a doubling in computation time compared to previous optimal algorithms.

Section snippets

Problem statement

Under a static environment assumption, the motion of all points in space, relative to a coordinate system centered in the nodal point of the observer’s eye, is determined by the translational velocity, t = (tx, ty, tz)T, and rotational velocity, ω = (ωx, ωy, ωz)T, of the moving observer. The 3D velocity, v = (vx, vy, vz)T, of a point in space, x = (x, y, z)T, is then [10]v=-t-ω×x.Under perspective projection and assuming, without loss of generality, a focal length equal to unity, these 3D motion vectors are

Previous algorithms

A wide variety of egomotion-estimation methods have been proposed in the past. An important distinction can be made between the earlier approaches, which suffer from biased and/or widely varying estimates, and the more recent optimal algorithms.

Proposed method

As mentioned in the introduction, the optimal algorithms suffer heavily from local minima. These minima are due to singularities in the unweighted error function that arise from the normalization of the bilinear constraints (Eq. (6)) by ∥A(x)t∥. As a consequence, a singularity exists for each feature where t  (x, y, 1)T. Under certain conditions, which are not uncommon in real-world optic flow fields, these singularities interact and influence larger regions of heading space [3], [4]. Optimal

Experiments

In this section, the proposed method is extensively compared to some of the algorithms discussed in Section 3. First, in Section 5.1, the algorithms are compared in terms of accuracy of the parameter estimates. This evaluation involves synthetic data only and is applied to both optimal and non-optimal algorithms. Next, in Section 5.2, the proposed method’s superior robustness to local minima as compared to other optimal algorithms is demonstrated. For this purpose, a synthetic problem is

Discussion

We have presented a novel method that reduces the sensitivity to local minima of optimal egomotion-estimation algorithms by gradually increasing the problem complexity during the optimization. We have demonstrated that the local minima encountered by these algorithms are related to the feature (or feature cluster) locations and, as such, their values can be arbitrary and unrelated to the true solution. This makes these algorithms hard to use in practical applications.

As a remedy, it has been

Acknowledgments

Thanks to Dr. Tong Zhang and Dr. Tina Y. Tian and coworkers for providing the source code of some of the egomotion algorithms used in this paper. Thanks also to Dr. Temujin Gautama for helpful suggestions on the manuscript. Karl Pauwels and Marc M. Van Hulle are supported by the Belgian Fund for Scientific Research—Flanders (G.0248.03, G.0234.04), the Flemish Regional Ministry of Education (Belgium) (GOA 2000/11), the Belgian Science Policy (IUAP P5/04), and the European Commission

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