We derive a pair of algorithms, one optimal and the other approximate, for recognizing three-dimensional objects from a collection of points chosen from their surface according to some probabilistic mechanism. The measurements are assumed to be noisy, and the measured location of a given point is translated according to a noise probability distribution. Distributions governing surface point selection and measurement noise can take a variety of forms depending upon the particular measurement scenario. At one extreme, each measurement is assumed to yield values restricted to a one-dimensional ray, a special case commonly adopted in the literature. At the other extreme, measured points are chosen uniformly from the object's surface, and the noise distribution is spherically symmetric, a worst-case scenario that involves no prior information about the measurements. We apply these two algorithms to shape recognition problems involving simple geometrical objects, and examine their relative behavior using a combination of analytical derivation and Monte Carlo simulation. We show that the approximate algorithm can be far simpler to compute, and its performance is competitive with the optimal algorithm when noise levels are relatively low. We show the existence of a critical noise level, beyond which the approximate algorithm exhibits catastrophic failure.