Optical flow cannot be computed locally, since only one independent measurement is available from the image sequence at a point, while the flow velocity has two components. A second constraint is needed. A method for finding the optical flow pattern is presented which assumes that the apparent velocity of the brightness pattern varies smoothly almost everywhere in the image. An iterative implementation is shown which successfully computes the optical flow for a number of synthetic image sequences. The algorithm is robust in that it can handle image sequences that are quantized rather coarsely in space and time. It is also insensitive to quantization of brightness levels and additive noise. Examples are included where the assumption of smoothness is violated at singular points or along lines in the image.

Here we search fi)r the curve which has the smallest integral of the square of curvature, while passing through two given points with given orientation. This is the true shape of a spline used in lofting. In computer-aided design, curves have been sought which maximize "smoothness". The curve discussed here is the one arising in this way from a commonly used measure of smoothness. The human visual system may use such a curve when it constructs a subjective contour.

A new active contour model for finding and mapping the outer cortex in brain images is developed. A cross-section of the brain cortex is modeled as a ribbon, and a constant speed mapping of its spine is sought. A variational formulation, an associated force balance condition, and a numerical approach are proposed to achieve this goal. The primary difference between this formulation and that of snakes is in the specification of the external force acting on the active contour. A study of the uniqueness and fidelity of solutions is made through convexity and frequency domain analyses, and a criterion for selection of the regularization coefficient is developed. Examples demonstrating the performance of this method on simulated and real data are provided.

In this paper we identify sources of error in global illumination algorithms and derive bounds for each distinct category. Errors arise from three sources: inaccuracies in the boundary data, discretization, and computation. Boundary data consist of surface geometry, reflectance functions, and emission functions, all of which may be perturbed by errors in measurement or simulation, or by simplifications made for computational efficiency. Discretization error is introduced by replacing the continuous radiative transfer equation with a finite-dimensional linear system, usually by means of boundaryelements and a corresponding projection method. Finally, computational errors perturb the finite-dimensional linear system through imprecise form factors, inner products, visibility, etc., as well as by halting iterative solvers after a finite number of steps. Using the error taxonomy introduced in the paper we examine existing global illumination algorithms and suggest new avenues of research. ...

Abstract-This paper is a review of the theory-of laser beams and resonators. It is meant to be tutorial in nature and useful in scope. No attempt is made to be exhaustive in the treatment. Rather, emphasis is placed on formulations and derivations which lead to basic understanding and on results which bear practical significance. 1.

These notes introduce the basic concepts of integral equations and their application in global illumination. Much of the discussion is expressed in the language of linear operators to simplify the notation and to emphasize the algebraic properties of the integral equations. We start by reviewing some facts about linear operators and examining some of the operators that occur in global illumination. Six general methods of solving operator and integral equations are then discussed: the Neumann series, successive approximations, the Nystrom method, collocation, least squares, and the Galerkin method. Finally, we look at some of the steps involved in applying these techniques in the context of global illumination. 1 Introduction The transfer of energy by radiation has a character fundamentally different from the processes of conduction and convection. One reason for this difference is that the radiant energy passing through a point in space cannot be completely described by a single scala...

In this paper, we study triply periodic surfaces with minimal surface area under a constraint in the volume fraction of the regions (phases) that the surface separates. Using a variational level set method formulation, we present a theoretical characterization of and a numerical algorithm for computing these surfaces. We use our theoretical and computational formulation to study the optimality of the Schwartz P, Schwartz D, and Schoen G surfaces when the volume fractions of the two phases are equal and explore the properties of optimal structures when the volume fractions of the two phases not equal. Due to the computational cost of the fully, threedimensional shape optimization problem, we implement our numerical simulations using a parallel level set method software package. 1.