Learning Speed Invariant Gait Template via Thin Plate Spline Kernel Manifold Fitting

Abstract

Similar to the solutions of general gait recognition approaches, there are two ways to handle the cross-speed gait recognition issues. The first approach is model-based approach which is to model the walking action using static or dynamic body parameters [6]. The second class of crossspeed gait recognition approach is model free approach, also known as feature or appearance-based approach. It focuses on directly extracting holistic gait features from gait sequences [4, 5]. Our proposed method, Speed Invariant Gait Template (SIGT), belongs to the second category. According to some recent studies, the walking action is considered as residing on a manifold which is topologically equivalent to a unit circle [1]. Thus, we can format the gait feature extraction issue as a gait manifold fitting issue. To address the cross-speed gait recognition issue, Thin Plate Spline (TPS) kernel based RBF interpolation is used to fit the gait manifold, since TPS has a desirable property [7] that it separates the mapping coefficients into an affine component and a non-affine component. And a natural assumption, that the affine component depicts the manifold deformation caused by walking and the non-affine component preserves the static features irrelevant to motion, can be given. Consequently, the non-affine component is a gait feature which is robust to the walking speed variation and we name this new feature Speed Invariant Gait Template (SIGT) (see Figure 1). We show in this paper how to use our proposed TPS-based manifold model to extract SIGTs from the full cyclic gait silhouette sequences. Firstly, The implicit function-based representation is used to represent each gait silhouette, since this representation is robust to the noise and silhouette fragmentation. Each pixel x of a gait silhouette is represented as 0 when it is on the contour, as dc(x) when it is inside the contour and as−dc(x) when it is outside the contour, where dc(x) denotes the distance to the closest point on the contour. Next, TPS Kernel based RBF interpolation is used to learn the mapping between the manifold embedding space and the represented gait space. Let Y = {yi ∈ Rl , i = 1, · · · ,N} be a gait sequence in the gait space and X = {xi ∈ R2, i = 1, · · · ,N} be the corresponding points in the embedding space. T = {t j ∈ R2, i = 1, · · · ,Nt} denotes Nt equally spaced centers in the embedding space. We can solve for multiple TPS kernel interpolants f k : R2→ R where k is the kth element (dimension) of the gait vector (the vectorizated represented gait image) in the gait space and f k is a RBF interpolant. We minimize a regularized risk criteria to learn nonlinear mapping from the embedding space to each individual dimension in the gait space that satisfies yi = f (xi). From the representer theorem [3], such a function admits a representation of the form of linear combination of basis functions around arbitrary points (centers). Therefore, to the kth dimension of the input, the form of function f k(x) is as follows: