Introduction The idea for this Appendix arose from our perception of a frustrating situation faced by vision researchers. For example, one is interested in some aspect of the theory of perspective image formation such as the epipolar line. The interested party goes to the library to check out a book on projective geometry filled with hope that the necessary mathematical machinery will be directly at hand. These expectations are quickly dashed. Upon opening the book, the expectant reader finds the presentation dominated by endless observations about harmonic relations and a few chapters which explore the minutiae of Pappus' theorem. Finally, as a last cruel twist of irony, the book ends in triumph with a rather exhilarating discourse on the conic pencil. All of the material is presented in the form of theorems defined on points, lines and conics without the use of coordinates, except perhaps for a quick pause to define barycentric coordinates just to taunt the reader. Dejected, the vis

The pattern of retinal binocular disparities acquired by a fixating visual system depends on both the depth structure of the scene and the viewing geometry. This paper treats the problem of interpreting the disparity pattern in terms of scene structure without relying on estimates of fixation position from eye movement control and proprioception mechanisms. We propose a sequential decomposition of this interpretation process into disparity correction, which is used to compute three-dimensional structure up to a relief transformation, and disparity normalization, which is used to resolve the relief ambiguity to obtain metric structure. We point out that the disparity normalization stage can often be omitted, since relief transformations preserve important properties such as depth ordering and coplanarity. Based on this framework we analyze three previously proposed computational models of disparity processing; the MLH model (Mayhew and Longuet-Higgins 1982), the deformation model (Koend...

The geometry of binocular projection is analyzed in relation to the primate visual system. An oculomotor parameterization that includes the classical vergence and version angles is defined. It is shown that the epipolar geometry of the system is constrained by binocular coordination of the eyes. A local model of the scene is adopted in which depth is measured relative to a plane containing the fixation point. These constructions lead to an explicit parameterization of the binocular disparity field involving the gaze angles as well as the scene structure. The representation of visual direction and depth is discussed with reference to the relevant psychophysical and neurophysiological literature. © 2008 Optical Society of America OCIS codes: 330.1400, 330.2210. 1.

Abstract. Since uncalibrated images permit only projective reconstruction, metric information requires either camera or scene calibration. We propose a stratified approach to projective reconstruction, in which gradual increase in domain information for scene calibration leads to gradual increase in 3D information. Our scheme includes the following steps: (1) Register the images with respect to a reference plane; this can be done using limited scene information, e.g., the knowledge that two pairs of lines on the plane are parallel. We show that this calibration is sufficient for ordinal reconstruction- sorting the points by their height over the reference plane. (2) If available, use the relative height of two additional out-of-plane points to compute the height of the remaining points up to constant scaling. Our scheme is based on the dual epipolar geometry in the reference frame, which we develop below. We show good results with five sequences of real images, using mostly scene calibration that can be inferred directly from the images themselves. 1

Abstract. Binocular information about the structure of a scene is contained in the relative positions of corresponding points in the two views. If the eyes rotate, in order to fixate a different target, then the disparity at a given image location is likely to change. Quite different disparities can be produced at the same location, as the eyes move from one fixation-point to the next. The pointwise variability of the disparity map is problematic for biological visual systems, in which stereopsis is based on simple, short-range mechanisms. It is argued here that the problem can be addressed in two ways; firstly by an appropriate representation of disparity, and secondly by learning the typical pattern of image correspondences. It is shown that the average spatial structure of the disparity field can be estimated, by integrating over a series of binocular fixations. An algorithm based on this idea is tested on natural images. Finally, it is shown how the average pattern of disparities could help to put the images into binocular correspondence. 1

The visual system must generate a reference frame to relate retinal images in spite of head and eye movements. We show how a reference frame for storing the visual direction and depth of points can be composed from the angles and changes in angles between pairs and triples of points. The representation has no unique origin in 3-D space nor a unique set of cardinal directions (basis vectors). We show how this relative representation could be built up over a series of fixations and for different directions of translation of the observer. Maintaining gaze on a point as the observer translates helps in building up this representation. In our model, retinal flow is divided into changes in eccentricity and changes in meridional angle. The latter, called `polar angle disparities' for binocular viewing (Weinshall, 1990. Computer Vision Graphics and Image Processing, 49 222 -- 241), can be used to recover the relief structure of the scene in a series of stages up to full Euclidean structure. We show how the direction of heading can be recovered by a similar series of stages. 2001 Elsevier Science Ltd. All rights reserved.

sequence of images. However, a (relative) depth map of a surface may not be a good representation for storage and recognition; a more concise representation seems necessary.

The discontinuities and the large image displacements pose some of the hardest problems in flow estimation. This paper uses a set of filters that change shape to avoid blending of the constraints across discontinuity boundaries. This is done by using an incompatibility measure of the constraints of neighbouring pixels. The algorithm is embedded in a coarse to fine multigrid scheme to address the problem of large displacements. We report results on real images which show that the algorithm works very well. The support of the NSERC (App. No. OGP0046645) is gratefully acknowledged. 1.

Nonmetric multidimensional scaling (MDS) allows one to derive aquantitative representation from a set of qualitative data which satisfy certain simple constraints. As a tool for vision, MDS combines the advantages of both qualitative and classical approaches, by relying, on the one hand, on an ordinal-scale input representation, and by supporting, on the other hand, the extraction of metric information. The proposed approach is illustrated on the example of stereopsis, although it is applicable also to the processing of other visual cues, as well as to the integration of several cues within a common computational framework.

is provided in screen-viewable form for personal use only by members