The least-squares error for structure from infinitesimal motion

We analyze the least-squares error for structure from motion with a single infinitesimal motion ("structure from optical flow"). We present asymptotic approximations to the noiseless error over two, complementary regions of motion estimates: roughly forward and non-forward translations. Our approximations are powerful tools for understanding the error. Experiments show that they capture its detailed behavior over the entire range of motions. We illustrate the use of our approximations by deriving new properties of the least-squares error. We generalize the earlier results of Jepson/Heeger/Maybank on the bas-relief ambiguity and of Oliensis on the reflected minimum. We explain the error's complexity and its multiple local minima for roughly forward translation estimates (epipoles within the field of view) and identify the factors that make this complexity likely. For planar scenes, we clarify the effects of the two-fold ambiguity, show the existence of a new, double bas-relief ambiguity, and analyze the error's local minima. For nonplanar scenes, we derive simplified error approximations for reasonable assumptions on the image and scene. For example, we show that the error tends to have a simpler form when many points are tracked. We show experimentally that our analysis for zero image noise gives a good model of the error for large noise. We show theoretically and experimentally that the error for projective structure from motion is simpler but flatter than the error for calibrated images.