Finding Edges and Lines in Images

The problem of detecting intensity changes in images is canonical in vision. Edge detection operators are typically designed to optimally estimate first or second derivative over some (usually small) support. Other criteria such as output signal to noise ratio or bandwidth have also been argued for. This thesis is an attempt to formulate a set of edge detection criteria that capture as directly as possible the desirable properties of an edge operator. Variational techniques are used to find a solution over the space of all linear shift invariant operators. The first criterion is that the detector have low probability of error i.e. failing to mark edges or falsely marking non-edges. The second is that the marked points should be as close as possible to the centre of the true edge. The third criterion is that there should be low probability of more than one response to a single edge. The technique is used to find optimal operators for step edges and for extended impulse profiles (ridges or valleys in two dimensions). The extension of the one dimensional operators to two dimentions is then discussed. The result is a set of operators of varying width, length and orientation. The problem of combining these outputs into a single description is discussed, and a set of heuristics for the integration are given.

Analysis of Differential and Matching Methods for Optical Flow

Several algorithms for optical flow are studied theoretically and experimentally. Differential and matching methods are examined; these two methods have differing domains of application- differential methods are best when displacements in the image are small (<2 pixels) while matching methods work well for moderate displacements but do not handle sub-pixel motions. Both types of optical flow algorithm can use either local or global constraints, such as spatial smoothness. Local matching and differential techniques and global differential techniques will be examined. Most algorithms for optical flow utilize weak assumptions on the local variation of the flow and on the variation of image brightness. Strengthening these assumptions improves the flow computation. The computational consequence of this is a need for larger spatial and temporal support. Global differential approaches can be extended to local (patchwise) differential methods and local differential methods using higher derivatives. Using larger support is valid when constraint on the local shape of the flow are satisfied. We show that a simple constraint on the local shape of the optical flow, that there is slow spatial variation in the image plane, is often satisfied. We show how local differential methods imply the constraints for related methods using higher derivatives. Experiments show the behavior of these optical flow methods on velocity fields which so not obey the assumptions. Implementation of these methods highlights the importance of numerical differentiation. Numerical approximation of derivatives require care, in two respects: first, it is important that the temporal and spatial derivatives be matched, because of the significant scale differences in space and time, and, second, the derivative estimates improve with larger support.

Maxwell's Demon, Rectifiers, and the Second Law: Computer Simulation of Smoluchowski's Trapdoor

We have simulated numerically an automated Maxwell's demon inspired by Smoluchowski's ideas of 1912. Two gas chambers of equal area are connected via an opening that is covered by a trapdoor. The trapdoor can open to the left but not to the right, and is intended to rectify naturally occurring variations in density between the two chambers. Our results confirm that though the trapdoor behaves as a rectifier when large density differences are imposed by external means, it can not extract useful work from the thermal motion of the molecules when left on its own.

Proceedings of the Second PHANToM User's Group Workshop

On October 19-22, 1997 the Second PHANToM Users Group Workshop was held at the MIT Endicott House in Dedham, Massachusetts. Designed as a forum for sharing results and insights, the workshop was attended by more than 60 participants from 7 countries. These proceedings report on workshop presentations in diverse areas including rigid and compliant rendering, tool kits, development environments, techniques for scientific data visualization, multi-modal issues and a programming tutorial.