Michael Kass

Snakes: Active Contour Models

A snake is an energy-minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges. Snakes are active contour models: they lock onto nearby edges, localizing them accurately. Scale-space continuation can be used to enlarge the capture region surrounding a feature. Snakes provide a unified account of a number of visual problems, including detection of edges, lines, and subjective contours; motion tracking; and stereo matching. We have used snakes successfully for interactive interpretation, in which user-imposed constraint forces guide the snake near features of interest.

Constraints on deformable models: recovering 3D shape and nongrid motion

Abstract Inferring the 3D structures of nonrigidly moving objects from images is a difficult yet basic problem in computational vision. Our approach makes use of dynamic, elastically deformable object models that offer the geometric flexibility to satisfy a diversity of real-world visual constraints. We specialize these models to include intrinsic forces inducing a preference for axisymmetry. Image-based constraints are applied as extrinsic forces that mold the symmetry-seeking model into shapes consistent with image data. We describe an extrinsic force that applies constraints derived from profiles of monocularly viewed objects. We generalize this constraint force to incorporate profile information from multiple views and use it to exploit binocular image data. For time-varying images, the force becomes dynamic and the model is able to infer not only depth, but nonrigid motion as well. We demonstrate the recovery of 3D shape and nonrigid motion from natural imagery.

Analyzing oriented patterns

Abstract Oriented patterns, such as those produced by propagation, accretion, or deformation, are common in nature and therefore an important class for visual analysis. Our approach to understanding such patterns is to decompose them into two parts: the flow field, describing the direction of anisotropy; and the residual pattern obtained by describing the image in a coordinate system built from the flow field. We develop a method for the local estimation of anisotropy and a method for combining the estimates to construct a flow coordinate system. Several examples of the use of these methods are presented. These include the use of the flow coordinates to provide preferred directions for edge detection, detection of anomalies, fitting simple models to the straightened pattern, and detecting singularities in the flow field.

Subdivision surfaces in character animation

The creation of believable and endearing characters in computer graphics presents a number of technical challenges, including the modeling, animation and rendering of complex shapes such as heads, hands, and clothing. Traditionally, these shapes have been modeled with NURBS surfaces despite the severe topological restrictions that NURBS impose. In order to move beyond these restrictions, we have recently introduced subdivision surfaces into our production environment. Subdivision surfaces are not new, but their use in high-end CG production has been limited. Here we describe a series of developments that were required in order for subdivision surfaces to meet the demands of high-end production. First, we devised a practical technique for constructing provably smooth variable-radius fillets and blends. Second, we developed methods for using subdivision surfaces in clothing simulation including a new algorithm for efficient collision detection. Third, we developed a method for constructing smooth scalar fields on subdivision surfaces, thereby enabling the use of a wider class of programmable shaders. These developments, which were used extensively in our recently completed short film Geri’s game, have become a highly valued feature of our production environment. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling; I.3.3 [Computer Graphics]: Picture/Image Generation.

Hierarchical Z-buffer visibility

An ideal visibility algorithm should a) quickly reject most of the hidden geometry in a model and b) exploit the spatial and perhaps temporal coherence of the images being generated. Ray casting with spatial subdivision does well on criterion (a), but poorly on criterion (b). Traditional Z-buffer scan conversion does well on criterion (b), but poorly on criterion (a). Here we present a hierarchical Z-buffer scan-conversion algorithm that does well on both criteria. The method uses two hierarchical data structures, an object-space octree and an image-space Z pyramid, to accelerate scan conversion. The two hierarchical data structures make it possible to reject hidden geometry very rapidly while rendering visible geometry with the speed of scan conversion. For animation, the algorithm is also able to exploit temporal coherence. The method is well suited to models with high depth complexity, achieving orders of magnitude acceleration in some cases compared to ordinary Z-buffer scan conversion. CR

Rapid, stable fluid dynamics for computer graphics

We present a new method for animating water based on a simple, rapid and stable solution of a set of partial differential equations resulting from an approximation to the shallow water equations. The approximation gives rise to a version of the wave equation on a height-field where the wave velocity is proportional to the square root of the depth of the water. The resulting wave equation is then solved with an alternating-direction implicit method on a uniform finite-difference grid. The computational work required for an iteration consists mainly of solving a simple tridiagonal linear system for each row and column of the height field. A single iteration per frame suffices in most cases for convincing animation.Like previous computer-graphics models of wave motion, the new method can generate the effects of wave refraction with depth. Unlike previous models, it also handles wave reflections, net transport of water and boundary conditions with changing topology. As a consequence, the model is suitable for animating phenomena such as flowing rivers, raindrops hitting surfaces and waves in a fish tank as well as the classic phenomenon of waves lapping on a beach. The height-field representation prevents it from easily simulating phenomena such as breaking waves, except perhaps in combination with particle-based fluid models. The water is rendered using a form of caustic shading which simulates the refraction of illuminating rays at the water surface. A wetness map is also used to compute the wetting and drying of sand as the water passes over it.

Reaction-diffusion textures

We present a method for texture synthesis based on the simulation of a process of local nonlinear interaction, called reaction-diffusion, which has been proposed as a model of biological pattern formation. We extend traditional reaction-diffusion systems by allowing anisotropic and spatially non-uniform diffusion, as well as multiple competing directions of diffusion. We adapt reaction-diffusion system to the needs of computer graphics by presenting a method to synthesize patterns which compensate for the effects of non-uniform surface parameterization. Finally, we develop efficient algorithms for simulating reaction-diffusion systems and display a collection of resulting textures using standard texture- and displacement-mapping techniques.

Symmetry-Seeking Models and 3D Object Reconstruction

We propose models of 3D shape which may be viewed as deformable bodies composed of simulated elastic material. In contrast to traditional, purely geometric models of shape, deformable models are active—their shapes change in response to externally applied forces. We develop a deformable model for 3D shape which has a preference for axial symmetry. Symmetry is represented even though the model does not belong to a parametric shape family such as (generalized) cylinders. Rather, a symmetry-seeking property is designed into internal forces that constrain the deformations of the model. We develop a framework for 3D object reconstruction based on symmetry-seeking models. Instances of these models are formed from monocular image data through the action of external forces derived from the data. The forces proposed in this paper deform the model in space so that the shape of its projection into the image plane is consistent with the 2D silhouette of an object of interest. The effectiveness of our approach is demonstrated using natural images.

CONDOR: constraint-based dataflow

CONDOR is an interactive constraint-based dataflow programming environment which is particularly suited to problems that arise in computer graphics. In addition to traditional dataflow functions, CONDOR dataflow elements can perform efficient derivative evaluation and interval arithmetic. As a result, CONDOR is able to support constraints, dynamics, surface trimming, collision testing, and a variety of other computations which are difficult or impossible to implement in traditional dataflow systems. CONDOR includes a graphical interface in which mathematical functions are represented as boxes with vector or scalar inputs and outputs. The functions can be composed by interactively connecting together their inputs and outputs, CONDOR performance is sufficiently fast to make it suitable for creating shaders, parametric surfaces, and complex constrained models.

Error-bounded antialiased rendering of complex environments

In previous work, we presented an algorithm to accelerate z-buffer rendering of enormously complex scenes. Here, we extend the approach to antialiased rendering with an algorithm that guarantees that each pixel of the output image is within a user-specified error tolerance of the filtered underlying continuous image. As before, we use an object-space octree to cull hidden geometry rapidly. However, instead of using an image-space depth pyramid to test visibility of collections of pixel samples, we use a quadtree data structure to test visibility throughout image-space regions. When regions are too complex, we use quadtree subdivision to simplify the geometry as in Warnocks algorithm. Subdivison stops when the algorithm can either analytically filter the required region or bound the convolution integral appropriately with interval methods. To the best of our knowledge, this is the first algorithm to antialias with guaranteed accuracy scenes consisting of hundreds of millions of polygons.