Spatio-Temporal Scale Selection in Video Data

Abstract

This work presents a theory and methodology for simultaneous detection of local spatial and temporal scales in video data. The underlying idea is that if we process video data by spatio-temporal receptive fields at multiple spatial and temporal scales, we would like to generate hypotheses about the spatial extent and the temporal duration of the underlying spatio-temporal image structures that gave rise to the feature responses. For two types of spatio-temporal scale-space representations, (i) a non-causal Gaussian spatio-temporal scale space for offline analysis of pre-recorded video sequences and (ii) a time-causal and time-recursive spatio-temporal scale space for online analysis of real-time video streams, we express sufficient conditions for spatio-temporal feature detectors in terms of spatio-temporal receptive fields to deliver scale-covariant and scale-invariant feature responses. We present an in-depth theoretical analysis of the scale selection properties of eight types of spatio-temporal interest point detectors in terms of either: (i)–(ii) the spatial Laplacian applied to the first- and second-order temporal derivatives, (iii)–(iv) the determinant of the spatial Hessian applied to the first- and second-order temporal derivatives, (v) the determinant of the spatio-temporal Hessian matrix, (vi) the spatio-temporal Laplacian and (vii)–(viii) the first- and second-order temporal derivatives of the determinant of the spatial Hessian matrix. It is shown that seven of these spatio-temporal feature detectors allow for provable scale covariance and scale invariance. Then, we describe a time-causal and time-recursive algorithm for detecting sparse spatio-temporal interest points from video streams and show that it leads to intuitively reasonable results. An experimental quantification of the accuracy of the spatio-temporal scale estimates and the amount of temporal delay obtained from these spatio-temporal interest point detectors is given, showing that: (i) the spatial and temporal scale selection properties predicted by the continuous theory are well preserved in the discrete implementation and (ii) the spatial Laplacian or the determinant of the spatial Hessian applied to the first- and second-order temporal derivatives leads to much shorter temporal delays in a time-causal implementation compared to the determinant of the spatio-temporal Hessian or the first- and second-order temporal derivatives of the determinant of the spatial Hessian matrix.