An Efficient Recursive Factorization Method for Determining Structure from Motion

A recursive method is presented for recovering 3D object shape and camera motion under orthography from an extended sequence of video images. This may be viewed as a natural extension of both the original [9] and the sequential [7] factorization methods. A critical aspect of these factorization approaches is the estimation of the so-called shape space [7], and they may in part be characterized by the manner in which this subspace is computed. If P points are tracked through F frames, the recursive leastsquares method proposed in this paper updates the shape space with complexity O(P ) per frame. In contrast, the sequential factorization method updates the shape space with complexity O(P 2) per frame. The original factorization method is intended to be used in batch mode using points tracked across all available frames. It effectively computes the shape space with complexity O(FP 2) after F frames. Unlike other methods, the recursive approach does not require the estimation or updating of a large measurement or covariance matrix. Experiments with real and synthetic image sequences confirm the recursive method’s low computational complexity and good performance, and indicate that it is well suited to real-time applications.