Inferring 3D structure from three points in rigid motion

Abstract

Abstract. We prove the following: Given four (or more) orthographic views of three points then (a) the views almost surely have no rigid interpretation but (b) if they do then they almost surely have at most thirty-two rigid interpretations. Part (a) means that the measure of "false targets", viz., the measure of nonrigid motions that project to views having rigid interpretations, is zero. Part (b) means that rigid interpretations, when they exist, are not unique. Uniqueness of interpretation can be obtained if a point is added, but not if views are added. Our proof relies on an upper semicontinuity theorem for proper mappings of complex algebraic varieties. We note some psychophysical motivations of the theory.

Non-rigid structure from locally-rigid motion

“…Here we consider the problem of finding three point rigid models that exactly explain a set of image trajectories. The problem of counting such rigid interpretations in F = 3 or F = 4 frames has been considered previously [7,24]. In contrast, we demonstrate how a simple algorithm allows for the recovery of solutions in F ≥ 3 frames when they exist.…”

“…For F = 3 frames, we have shown how to recover up to two possible link length solutions yielding up to 2 • 2 3 = 16 possible rigid interpretations. In contrast, the work [7] shows that there are at most 32 different rigid interpretations for F = 4 frames, and the work [24] shows that there are up to two distinct link length solutions that yield up to 2 • 2 3 = 32 different rigid interpretations for F = 3 frames. We suggest that these works are likely counting, for each length solution, the 2 F depth flip ambiguities that result from each of the two triangles that can be specified using γ ± ∈ T ± γ (see Section 3.1.1).…”

“…In this chapter, we consider what can be said about 3 rigid scene points that have been orthographically imaged from F viewpoints as illustrated in Figure 3.1. Previous work has focused on the exact case, where it is assumed that no noise has entered the imaging process [7,24]. That work has aimed to identify, given a set of image projections, whether such a rigid configuration exists to explain those projections and if so, how many such configurations exist.…”

“…One might also wonder whether one can decide whether three points belong to a rigid configuration by examining the residual errors provided by 3-SFM. It was demonstrated by Bennett and Hoffman [7] that in 4 generic viewpoints, the set of rigid models that can exactly explain a set of observations is measure zero. If we wish to allow some level of noise or deviance from rigidity, we can use 3-SFM to assign a non-rigidity score based on RMS reprojection error.…”

“…This paradigm, when formalized as a problem to be solved computationally, is called the structure from motion (SFM) problem. When the scene is assumed to be static (or rigid), a rich theory detailing the required minimal configurations of points and views for scene reconstruction [7,51] is available, as well as efficient reconstruction algorithms [51]. When the scene, however, is allowed to deform arbitrarily, the problem becomes highly ambiguous and new constraints need to be added to regularize the problem.…”