Sufficient Image Structure for 3-D Motion and Shape Estimation

We derive suucient 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 rst order spatio-temporal derivative or of the image is given and that the image intensity is continuously diierentiable everywhere or that image contours are continuously diierentiable. This means that only the component of the image motion eld orthogonal to iso-intensity contours, the so called normal ow, can be measured. By deening a tangent line at a point as the line orthogonal to the gradient or normal the suuciency condition on image structure can be stated as: If each point (x; y) in the innnitely 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 rst order spatio temporal derivatives, except for speciic combinations of surface texture and depth. The exceptions are speciic 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 ow eld lines are aligned with the contours of the image. 1 Normal ow Image motion with components (u; v) can be shown to be related to rst order spatio temporal image intensity gradients as: @f @x u(x; y) + @f @y v(x; y) + @f @t = 0 (1) This equation is in general used as the starting point for trying to recover the image ow eld (u; v). Since it is one equation and two unknowns for each image point (x; y) , there can never be a unique ow eld solution. In order to obtain a unique solution, This equation is therefore often supplemented with additional