Optimal and Suboptimal Structure From Motion

\Structure From Motion" (SFM) refers to the problem of estimating three-dimensional information about the environment from the motion of its two-dimensional projection onto a surface (for instance the retina). Noise plays an important role in this problem, but it has been addressed only marginally in more than twenty years of research. We present an analysis of SFM from the point of view of noise. This analysis results in algorithms that are provably convergent and provably optimal with respect to a chosen norm. In particular, we cast SFM as a nonlinear optimization problem and deene a bilinear projection iteration that converges to xed points of a certain cost-function. We then show that such xed points are \fundamental", i.e. are intrinsic to the problem of SFM and not an artifact introduced by our algorithms. We classify and interpret geometrically local extrema, and we argue that they correspond to phenomena observed in visual psychophysics. Finally, we show under what conditions it it is possible, given convergence to a local extremum, to \jump" to the valley containing the optimum.