A spatio-temporal Gaussian-Conical wavelet with high aperture   selectivity for motion and speed analysis
  The construction of a spatio-temporal wavelet and its tuning to speed was first realized in the 90s on the Morlet wavelet by M. Duval-Destin \cite{Duval-Destin91a,Duval-Destin92}. This enabled to demonstrate the capacities of the speed-tuned Morlet for psychovisual analysis. This construction was also used very efficiently in a powerful aerial target tracking algorithm by Mujica et al.\cite{Mujica99,Mujica2000}. In the last decade, this tool was proposed as an elegant and efficient alternative framework to the Optical Flow (OF), the Block Matching (BM) or the phase difference, for the study of motion estimation in image sequences. Nevertheless, the aperture selectivity of the 2D+T Morlet wavelet presents some difficulties. Here we propose to replace the 2D Morlet wavelet by a Gaussian-Conical (GC) wavelet for the spatial part of the spatio-temporal wavelet, since the GC wavelet has a better aperture selectivity and allows a very simple adjustment of the aperture. Therefore we build a new, highly directional, speed-tuned wavelet called Gaussian-Conical-Morlet (GCM) wavelet. Like the speed-tuned 2D+T Morlet, the new wavelet presents very good characteristics in motion estimation and tracking, namely long temporal dependence, robustness to noise and to occlusions, and supersedes the OF (Optical Flow) and BM (Block Matching) techniques. However, for aperture selectivity, directional speed-capture and spectral recognition and tracking, GCM easily outperforms Morlet. This paper describes the GCM construction, utilization and aperture performances. 
Patrice Brault and Jean-Pierre Antoine
math-ph math.MP