Algebraic Functions For Recognition

In the general case, a trilinear relationship between three perspective views is shown to exist. The trilinearity result is shown to be of much practical use in visual recognition by alignment --- yielding a direct method that cuts through the computations of camera transformation, scene structure and epipolar geometry. The proof of the central result may be of further interest as it demonstrates certain regularities across homographies of the plane and introduces new view invariants. Experiments on simulated and real image data were conducted, including a comparative analysis with epipolar intersection and the linear combination methods, with results indicating a greater degree of robustness in practice and a higher level of performance in re-projection tasks.

Image Based Rendering Using Algebraic Techniques

This paper presents an image-based rendering system using algebraic relations between different views of an object. The system uses pictures of an object taken from known positions. Given three such images it can generate "virtual'' ones as the object would look from any position near the ones that the two input images were taken from. The extrapolation from the example images can be up to about 60 degrees of rotation. The system is based on the trilinear constraints that bind any three view so fan object. As a side result, we propose two new methods for camera calibration. We developed and used one of them. We implemented the system and tested it on real images of objects and faces. We also show experimentally that even when only two images taken from unknown positions are given, the system can be used to render the object from other view points as long as we have a good estimate of the internal parameters of the camera used and we are able to find good correspondence between the example images. In addition, we present the relation between these algebraic constraints and a factorization method for shape and motion estimation. As a result we propose a method for motion estimation in the special case of orthographic projection.

Geometric and Algebraic Aspects of 3D Affine and Projective Structures from Perspective 2D Views

We investigate the differences --- conceptually and algorithmically --- between affine and projective frameworks for the tasks of visual recognition and reconstruction from perspective views. It is shown that an affine invariant exists between any view and a fixed view chosen as a reference view. This implies that for tasks for which a reference view can be chosen, such as in alignment schemes for visual recognition, projective invariants are not really necessary. We then use the affine invariant to derive new algebraic connections between perspective views. It is shown that three perspective views of an object are connected by certain algebraic functions of image coordinates alone (no structure or camera geometry needs to be involved).

Relative Affine Structure: Canonical Model for 3D from 2D Geometry and Applications

We propose an affine framework for perspective views, captured by a single extremely simple equation based on a viewer-centered invariant we call "relative affine structure". Via a number of corollaries of our main results we show that our framework unifies previous work --- including Euclidean, projective and affine --- in a natural and simple way, and introduces new, extremely simple, algorithms for the tasks of reconstruction from multiple views, recognition by alignment, and certain image coding applications.

Automatically Recovering Geometry and Texture from Large Sets of Calibrated Images

Three-dimensional models which contain both geometry and texture have numerous applications such as urban planning, physical simulation, and virtual environments. A major focus of computer vision (and recently graphics) research is the automatic recovery of three-dimensional models from two-dimensional images. After many years of research this goal is yet to be achieved. Most practical modeling systems require substantial human input and unlike automatic systems are not scalable. This thesis presents a novel method for automatically recovering dense surface patches using large sets (1000's) of calibrated images taken from arbitrary positions within the scene. Physical instruments, such as Global Positioning System (GPS), inertial sensors, and inclinometers, are used to estimate the position and orientation of each image. Essentially, the problem is to find corresponding points in each of the images. Once a correspondence has been established, calculating its three-dimensional position is simply a matter of geometry. Long baseline images improve the accuracy. Short baseline images and the large number of images greatly simplifies the correspondence problem. The initial stage of the algorithm is completely local and scales linearly with the number of images. Subsequent stages are global in nature, exploit geometric constraints, and scale quadratically with the complexity of the underlying scene. We describe techniques for: 1) detecting and localizing surface patches; 2) refining camera calibration estimates and rejecting false positive surfels; and 3) grouping surface patches into surfaces and growing the surface along a two-dimensional manifold. We also discuss a method for producing high quality, textured three-dimensional models from these surfaces. Some of the most important characteristics of this approach are that it: 1) uses and refines noisy calibration estimates; 2) compensates for large variations in illumination; 3) tolerates significant soft occlusion (e.g. tree branches); and 4) associates, at a fundamental level, an estimated normal (i.e. no frontal-planar assumption) and texture with each surface patch.

Geometry and Photometry in 3D Visual Recognition

The report addresses the problem of visual recognition under two sources of variability: geometric and photometric. The geometric deals with the relation between 3D objects and their views under orthographic and perspective projection. The photometric deals with the relation between 3D matte objects and their images under changing illumination conditions. Taken together, an alignment-based method is presented for recognizing objects viewed from arbitrary viewing positions and illuminated by arbitrary settings of light sources.