Bruce M. Bennett

The computation of structure from fixed-axis motion: Rigid structures

We show that three distinct orthographic views of three points in a rigid configuration are compatibel with at most 64 interpretations of the three-dimensional structure and motion of the points. If, in addition, one assumes that the three points spin about a fixed axis over the three views, then with probability one there is a unique three-dimensional interpretation (plus a reflection). Moreover the probability of false targets is zero. In the special case that the axis of rotation is parallel to the image plane three views of the three points are sufficient to obtain at most two interpretations (plus reflections)-unless one assumes the angular velocity about the axis is constant, in which case three views of two points are sufficient to determine a unique interpretation. Closed form solutions are obtained for each of these cases. The systems of equations studied here are in each case overconstraining (i.e. there are more independent equations than unknowns) and are amenable to solution by nonlinear programming. These two properties make possible the construction of noise insensitive algorithms for computer vision systems. Our uniqueness proofs employ the Principle of upper semicontinuity, a principle which underlies a general mathematical framework for the analysis of solutions to overconstraining systems of equations.

Minimum Points and Views for the Recovery of Three-Dimensional Structure

Mathematical analyses of motion perception have established minimum combinations of points and distinct views that are sufficient to recover three-dimensional (3D) structure from two-dimensional (2D) images, using such regularities as rigid motion, fixed axis of rotation, and constant angular velocity. To determine whether human subjects could recover 3D information at these theoretical levels, we presented subjects with pairs of displays and asked them to determine whether they represented the same or different 3D structures. Number of points was varied between two and five; number of views was varied between two and six; and the motion was fixed axis with constant angular velocity, fixed axis with variable velocity, or variable axis with variable velocity. Accuracy increased with views, decreased with points, and was greater with fixed-axis motion. Subjects performed above chance levels even when motion was eliminated, indicating that they exploited regularities in addition to those in the theoretical analyses.

Description of solid shape and its inference from occluding contours

We explore a method of representing solid shape that is useful for visual recognition. We assume that complex shapes are constructed from convex, compact shapes and that construction involves three operations: solid union (to form humps), solid subtraction (to leave dents), and smoothing (to remove discontinuities). The boundaries between shapes joined through these operations are contours of extrema of a principal curvature. Complex objects can be decomposed along these boundaries into convex shapes, the so-called parts. We suggest that this decomposition into parts forms the basis for a shape memory. We show that the part boundaries of an object can be inferred from its occluding contours, at least up to a number of ambiguities.

Inferring the relative three-dimensional positions of two moving points.

We show that four orthographic projections of two rigidly linked points are compatible with at most four interpretations of the relative three-dimensional positions of the points if the points rotate about a fixed axis--even when the points as a system undergo arbitrary rigid translations. A fifth view (projection) yields a unique interpretation and makes zero the probability that randomly chosen image points will receive a three-dimensional interpretation. Assuming that the points rotate at a constant angular velocity, instead of adding a fifth view, also yields a unique interpretation and makes zero the probability that randomly chosen image points will receive a three-dimensional interpretation.

Dual Quaternion Synthesis of Constrained Robots

This paper presents a synthesis methodology for robots that have less than six degrees of freedom, termed constrained robots. The goal is to determine the physical parameters of the chain that fit its workspace to a given set of spatial positions. Our formulation uses the dual quaternion form of the kinematics equations of the constrained robot. Here we develop the theory and formulate the synthesis equations for the spatial RPR robot. Their solution ensures that the three dimensional workspace of this robot contains a given set of four spatial positions.

Inferring 3D structure from three points in rigid motion

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.

Inferring 3D Structure from Image Motion: The Constraint of Poinsot Motion*

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.

PERCEPTION AND COMPUTATION

In this chapter we indicate how the class of observers properly contains the class of Turing machines. We discuss the simulation of observers by Turing machines.

Inferring structure from motion: a homotopy algorithm

Theoretical investigations of the inference of three-dimensional structure from image motion often result in systems of coupled nonlinear equations which must be solved to infer the third dimension. If closed-form solutions cannot be obtained, then various search procedures, such as simulated annealing, are often used. Here, the authors discuss a relatively novel approach to solving coupled nonlinear systems of equations, an approach based on the so-called homotopy principle. This approach is discussed in the context of developing an algorithm for inferring structure from motion using an assumption of rigid fixed-axis motion. This approach is also discussed in the more general context of observer theory, a mathematical framework for the field of perception.<<ETX>>

CHAPTER FOUR – SEMANTICS

We have a definition of observer, but not of the observed. A theory of perception cannot be complete without some account of the objects of perception. Parsimony suggests that we not postulate a new ontological category for these objects. We therefore explore the possibility that the objects of perception are themselves observers. We develop this proposal in the context of an investigation of the meaning and truth conditions of conclusion measures. To this end, we introduce a “primitive semantics” and an “extended semantics” for the representations appearing in the definition of observer.