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**View-Invariant Analysis of Cyclic Motion**

S. M. Seitz and C. R. Dyer,
Int. J. Computer Vision, **25(3)**, 1997, 231-251.

### Abstract

This paper presents a general framework for image-based analysis of 3D repeating motions that addresses two limitations in the state of the art. First, the assumption that a motion be perfectly even from one cycle to the next is relaxed. Real repeating motions tend not to be perfectly even, i.e., the length of a cycle varies through time because of physically important changes in the scene. A generalization of {\em period} is defined for repeating motions that makes this temporal variation explicit. This representation, called the period trace, is compact and purely temporal, describing the evolution of an object or scene without reference to spatial quantities such as position or velocity. Second, the requirement that the observer be stationary is removed. Observer motion complicates image analysis because an object that undergoes a 3D repeating motion will generally not produce a repeating sequence of images. Using principles of affine invariance, we derive necessary and sufficient conditions for an image sequence to be the projection of a 3D repeating motion, accounting for changes in viewpoint and other camera parameters. Unlike previous work in visual invariance, however, our approach is applicable to objects and scenes whose motion is highly non-rigid. Experiments on real image sequences demonstrate how the approach may be used to detect several types of purely temporal motion features, relating to motion trends and irregularities. Applications to athletic and medical motion analysis are discussed.