Inferring 3D structure from image motion: The constraint of Poinsot motion

Abstract

Abstract. Monocular observers perceive as three-dimensional (3D) many displays that depict three points rotating rigidly in space but rotating about an axis that is itself tumbling. No theory of structure from motion currently available can account for this ability. We propose a formal theory for this ability based on the constraint of Poinsot motion, i.e., rigid motion with constant angular momentum. In particular, we prove that three (or more) views of three (or more) points are sufficient to decide if the motion of the points conserves angular momentum and, if it does, to compute a unique 3D interpretation. Our proof relies on an upper semicontinuity theorem for finite morphisms of algebraic varieties. We discuss some psychophysical implications of the theory.

Inferring 3D structure from three points in rigid motion

“…It has been shown elsewhere [3] that the six equations of (4) corresponding to the first three views (i.e., for which 1 < i < 2) have, for generic parameters, precisely sixty four complex affine solutions for the six Zm,i,(m = 1,2; i = 1,2,3).…”

“…To prove part (a) of Theorem 1, i.e., to prove that S has Lebesgue measure zero in Y, we use the following key fact. Let Yc = C 16 be the complexification of Y, and Ec the complexification of E. It is shown elsewhere [3] that varieties Ec which arise as complex solution spaces of systems of equations of the type of (2) have the following property: their projective completion in the zm,+ variables has no more points than Ec itself. Thus (t) Ec is a family of projective varieties parametrized by Yc.…”

“…Because of the property (t) above, the function f is upper sernicontinuous in the Zariski topology [3]. In the Zariski topology the closed sets are closed algebraic varieties, i.e., solution sets to systems of polynomial equations.…”

“…Constraints examined in the literature include rigid motion [1,4,7,8,12,14,17], rigid fixedaxis motion [10,19], nonrigid fixed-axis motion [2], planar rigid motion [10], rigid motion about a vertical axis [15], certain bending motions [12], and rigid motion that conserves angular momentum [3]. This latter constraint led to the following theorem: Given three distinct orthographic projections of three points, (a) the projections are almost surely incompatible with any three-dimensional interpretation in which the points move rigidly and conserve angular momentum, but (b) if the projections are compatible with such an interpretation then, generically, they are compatible with at most two such interpretations.…”