A Geometric Approach to Particle Filtering-Based Visual Tracking Junghyun Kwon
Frank C. Park
School of Electrical Engineering and Computer Science Seoul National University, Korea
[email protected]
School of Mechanical and Aerospace Engineering Seoul National University, Korea
[email protected]
Abstract - We develop a geometric particle filtering algorithm that explicitly takes into account the geometry of the affine group for the visual tracking problem. The geometric issues arising in a particle filtering on the affine group, i.e., the state equation and sample mean formula, are resolved in a coordinate-invariant way. The superiority of our geometric particle filtering-based visual tracker to the local coordinate representation-based ones is demonstrated via comparative experiments. Keywords - Visual tracking, Particle filtering, Lie group, Affine group 1. I NTRODUCTION A visual tracking can be defined as successive localization of the object of interest on video image frames delivered continuously. Sequential filtering-based approaches based on the state space model are popularly used for the visual tracking because the tracking result of the current frame in general heavily depends on those of the previous frames [1], [2]. Among various filtering methods, particle filtering algorithms recently have gained much interest due to their simplicity and flexibility with rapidly increasing computing power after the pioneering work of [3]. The particle filtering algorithms, also known as sequential Monte Carlo methods, offer a practical approximation to the optimal Bayesian solution to the filtering problem via Monte Carlo simulation [4], [5], [6]. After the condensation algorithm [3] has shown the applicability of the particle filtering to the visual tracking problem, numerous particle filtering-based visual trackers were proposed; most efforts were made to guarantee the robust measurement likelihood calculation even under difficult conditions [7], [8], [9], [10], [11], [12] while some tried to improve the particle filtering algorithm itself [13], [14], [15]. When the particle filtering is applied to the visual tracking problem, what should be resolved first is how to define the state for the filtering. If we represent the tracked object as an image region surrounding the object, e.g., circle or rectangle, it is desirable to estimate not only the 2-D translation but the shape deformation of the object image region between adjacent image frames for a better performance of the simultaneous
tracking and recognition [8], [16] which can be effectively adopted for the video surveillance and human-robot interaction applications. For this case, the state can be a 2-D affine geometric transformation using matrix multiplication, which can be understood as a matrix Lie group. There exists much literature explicitly introducing the affine transformation into the state to resolve the visual tracking problem via the particle filtering-based approach [8], [10], [12]. What is common among them is a set of local coordinates is used to represent the geometric transformation in a vector form to make the particle filtering readily feasible. However, it is unclear what the implications of ignoring the intrinsic geometry of the underlying space will be, particularly as one approaches the extremes of the operating regime. Moreover, the tracking results will depend on the choice of local coordinates. There exist several previous works dealing with the particle filtering problem on Lie groups, e.g., [17], [18], [19], especially for special orthogonal group (SO(n)) and special Euclidean group (SE(n)). Recently, in [20], we have proposed a particle filtering algorithm on the Euclidean group for simultaneous state and covariance parameter estimation, by a geometric coordinate-invariant generalization of Liu and West’s method [21] to a space that is the Cartesian product of the Euclidean group and symmetric positive definite matrices. Our paper’s main contribution is to build a geometrically well-defined particle filtering framework where the state is the affine group itself, which thus can be readily applied to the visual tracking problem. We resolve the following geometric issues arising in a particle filtering on the affine group in a coordinate-invariant way: (i) the state particle propagation on the affine group, and (ii) the sample mean calculation of the particles of the affine group. Furthermore, we demonstrate the superiority of our geometric particle filtering-based visual tracker to the local coordinate representation-based ones via comparative experiments. The paper is organized as follows. In Section II, the geometric machinery required to formulate the particle filtering on the affine group is discussed. The comparison of our geometric approach with the local coordinate representationbased approach is also given in Section II. The experimental results demonstrating the superiority of the proposed visual tracker to the local coordinate representation-based ones are
presented in Section III, while Section IV concludes with a summary. 2. PARTICLE F ILTERING ON THE A FFINE G ROUP 2.1. State Representation The affine transformation of a point p ∈ where U , D, 0 λ2 and V are results of a SVD of G. In [12] and [10], the state x is defined as x = (θ, φ, λ1 , α, tx , ty )> ∈ . For both cases, the state equation can be represented via a general stochastic differential equation as dx = f (x, t)dt + F (x, t)dw,
(2)
where f :
