Shift & 2D Rotation Invariant Sparse Coding for Multivariate Signals

Abstract

International audienceClassical dictionary learning algorithms (DLA) allow unicomponent signals to be processed. Due to our interest in two-dimensional (2D) motion signals, we wanted to mix the two components to provide rotation invariance. So, multicomponent frameworks are examined here. In contrast to the well-known multichannel framework, a multivariate framework is first introduced as a tool to easily solve our problem and to preserve the data structure. Within this multivariate framework, we then present sparse coding methods: multivariate orthogonal matching pursuit (M-OMP), which provides sparse approximation for multivariate signals, and multivariate DLA (M-DLA), which empirically learns the characteristic patterns (or features) that are associated to a multivariate signals set, and combines shift-invariance and online learning. Once the multivariate dictionary is learned, any signal of this considered set can be approximated sparsely. This multivariate framework is introduced to simply present the 2D rotation invariant (2DRI) case. By studying 2D motions that are acquired in bivariate real signals, we want the decompositions to be independent of the orientation of the movement execution in the 2D space. The methods are thus specified for the 2DRI case to be robust to any rotation: 2DRI-OMP and 2DRI-DLA. Shift and rotation invariant cases induce a compact learned dictionary and provide robust decomposition. As validation, our methods are applied to 2D handwritten data to extract the elementary features of this signals set, and to provide rotation invariant decomposition

Decomposition and dictionary learning for 3D trajectories

“…In signal processing, a multicomponent temporal signal is described in [7] as the sum of the multicomponent patterns. Considering here the particular case of tricomponent data, a 3D trajectory of N samples is viewed as the sum of K 3D trajectories.…”

“…The advantage of model (2) is to deal with 3D trajectory patterns φ k ∈ R 3×N where the three components can be different, contrary to model (1), which has the same pattern on the three components. The differences between model (1), known as the multichannel framework, and model (2), known as the multivariate framework, are detailed in [7]. In order to be clear, our study deals with tricomponent signals, and not tridimensional ones.…”

“…So, a manual trade-off (between K and the residual error) is used on an exhaustive search on all possibilities to determinate the optimal vectors [6]. To solve this problem, multichannel sparse approximations [30], [7] could be used.…”

“…From the beginning of Section 3, explanations were given for univariate signals. However, they are extended to trivariate signals by multivariate OMP (M-OMP) [7].…”

“…(9) is retrieved, with trivariate signals, and this case is solved by M-OMP. Note that 2DRI-OMP [7] simply tackles Eq. (10) in the 2D case using complexes.…”