Sufficient Image Structure for 3-D Motion and Shape Estimation

Abstract

We derive sufficient conditions on image structure that permits determination of 3-D motion parameters and depth from motion relative a rigid surface in front of the camera. We assume that only the first order spatio-temporal derivative or of the image is given and that the image intensity is continuously differentiable everywhere or that image contours are continuously differentiable. This means that only the component of the image motion field orthogonal to iso-intensity contours, the so called normal flow, can be measured. By defining a tangent line at a point as the line orthogonal to the gradient or normal the sufficiency condition on image structure can be stated as: If each point ( x, y ) in the infinitely extended image plane is the intersection of at least 6 tangent lines, it is possible to compute unique 3-D motion and positive depth from first order spatio temporal derivatives, except for specific combinations of surface texture and depth. The exceptions are specific texture patterns for any surface, for which the problem is inherently ambiguous, e.g. the so called “barber-pole”. These patterns have the property that there exist a relative motion to the surface such that the image flow field lines are aligned with the contours of the image.