This article presents an algorithm for computing the exact aspect graph of an opaque solid bounded by a smooth algebraic surface. Orthographic projection is assumed. The algorithm is based on a catalog of visual events available from singularity theory. It uses curve tracing, cell decomposition, homotopy continuation, and ray tracing to construct the regions of the view sphere delineated by visual-event curves. The algorithm has been fully implemented, and examples are presented.

Computing exact aspect graphs of curved objects: algebraic surfaces

Contributors. Preface. Introduction. Polynomial Continuation and its Relationship to the Symbolic Reduction of Polynomial Systems. Elimination Methods: an Introduction. On the Solutions of a Set of Polynomial Equations. Quantifier Elimation for Conjunction of Linear Constraints via a Convex Hull Algoritm. Elimation Theory and Computer Vision: Recognition and Positioning of Curved 3D Objects from Range, Intensity, or Contours. 2D and 3D Object Recognition and Posiitioning with Algebraic Invariants and Covariants. Applications of Invariant Theory in Computer Vision. Distance Metrics for Comparing Shapes in the Plane. A Mathematical Framework for Combinational/Structural analysis of Linear Dynamical Systems by Means of Matroids. Symbolic Methods for the Simulation of Planar Mechanical Systems in Design. Basic Requirements for the Automatic Generation of FORTRAN code. Symbolic and Parallel Adaptive Methods for Partial Differential Equations. An Interactive Symbolic-Numeric Interface to Paralles ELLPACK for Building General PDE Solvers. Symbolic / Numeric Techniques in Modeling and Simulation. Symbolic and Numeric Computation: the Example of IRENA. Author Index. Index.

/pdf/symbolic-and-numerical-computation-for-artificial-44kq02s5k9.pdf

Symbolic and Numerical Computation for Artificial Intelligence

Currently the aspect graph is computed from the theoretical standpoint of perfect resolution in object shape, the viewpoint and the projected image. This means that the aspect graph may include details that an observer could never see in practice. Introducing the notion of scale into the aspect graph framework provides a mechanism for selecting a level of detail that is "large enough" to merit explicit representation. This effectively allows control over the number of nodes retained in the aspect graph. This paper introduces the concept of the scale space aspect graph, defines three different interpretations of the scale dimension, and presents a detailed example for a simple class of objects, with scale defined in terms of the spatial extent of features in the image. >

The scale space aspect graph

This thesis addresses the problem of synthesizing images of real scenes under threedimensional transformations in viewpoint and appearance. Solving this problem enables interactive viewing of remote scenes on a computer, in which a user can move a virtual camera through the environment and virtually paint or sculpt objects in the scene. It is demonstrated that a variety of three-dimensional scene transformations can be rendered on a video display device by applying simple transformations to a set of basis images of the scene. The virtue of these transformations is that they operate directly on images and recover only the scene information that is required in order to accomplish the desired e ect. Consequently, they are applicable in situations where accurate three-dimensional models are di cult or impossible to obtain. A central topic is the problem of view synthesis, i.e., rendering images of a real scene from di erent camera viewpoints by processing a set of basis images. Towards this end, two algorithms are described that warp and resample pixels in a set of basis images to produce new images that are physically-valid, i.e., they correspond to what a real camera would see from the speci ed viewpoints. Techniques for synthesizing other types of transformations, e.g., non-rigid shape and color transformations, are also discussed. The techniques are found to perform well on a wide variety of real and synthetic images. A basic question is uniqueness, i.e., for which views is the appearance of the scene uniquely determined from the information present in the basis views. An important contribution is a uniqueness result for the no-occlusion case, which proves that all views on the line segment between the two camera centers are uniquely determined from two uncalibrated views of a scene. Importantly, neither dense pixel correspondence nor camera information is needed. From this result, a view morphing algorithm is derived that produces high quality viewpoint and shape transformations from two uncalibrated images.

Image-based transformation of viewpoint and scene appearance

A new approach is presented for explicitly relating image observables to models of curved three-dimensional objects. This relationship is used for object recognition and positioning. Object models consist of collections of parametric surface patches. The image observables considered are raw range data, surface normal and Gaussian curvature, raw image intensity and intensity gradient, raw image contours, and contour orientation and curvature. Elimination theory provides a method for constructing an implicit equation that relates these observables to the three-dimensional position and orientation of object models. Determining the unknown pose parameters is reduced to a fitting problem between the implicit equation and the observed data points. By considering translation-independent observables such as surface normal and curvature, this process is further decomposed into first determining orientation and then determining translation. Applications to object recognition are described, and an implementation is presented.

On using CAD models to compute the pose of curved 3D objects

A viewer-centered approach is presented to modeling the geometry of the visible occluding contour of solid 3-D shape. The rim appearance representation models the exact appearance of the occluding contour formed by the edges of a polyhedron. An algorithm is presented for constructing the rim appearance representation. Bounds on space and time are given, and implementation results show that the rim appearance representation is significantly smaller than both the aspect graph and the asp representation. >

Modeling the rim appearance

The paper addresses the problem of how to constrain the set of viewpoints from which a 3D model will project to a given a set of occluding contour features. The approach is new in that it relies on a precomputed occluding contour representation for polyhedral models that makes T-junctions, contour terminals and the change in contour topology explicit. Potential model-image correspondences of occluding contour features can be formulated as regions in viewpoint space defined by constraint boundaries since features are only visible from a restricted set of viewpoints. The implementation results demonstrate that T-junction features can efficiently constrain the space of viewpoints to produce a small, bounded viewpoint set that accounts for the occluding contour features. >

http://ieeexplore.ieee.org/iel2/339/3939/00148757.pdf

W.B. Seales

Papers

Modeling the rim appearance

Constrained viewpoint from occluding contour