Automatic on-line signature verification is an intriguing intellectual challenge with many practical applications. I review the context of this problem and then describe my own approach to it, which breaks with tradition by relying primarily on the detailed shape of a signature for its automatic verification, rather than relying primarily on the pen dynamics during the production of the signature. I propose a robust, reliable, and elastic local-shape-based model for handwritten on-line curves; this model is generated by first parameterizing each on-line curve over its normalized arc-length and then representing along the length of the curve, in a moving coordinate frame, measures of the curve within a sliding window that are analogous to the position of the center of mass, the torque exerted by a force, and the moments of inertia of a mass distribution about its center of mass. Further I suggest the weighted and biased harmonic mean as a graceful mechanism of combining errors from multiple models of which at least one model is applicable but not necessarily more than one model is applicable, recommending that each signature be represented by multiple models, these models, perhaps, local and global, shape based and dynamics based. Finally, I outline a signature-verification algorithm that I have implemented and tested successfully both on databases and in live experiments.

Reports on some simple experiments with a pointwise spatiotemporal correlator. The correlator is demonstrated to be potentially useful for edge and corner detection in single image frames, and for generating image motion fields from temporal image sequences. In contrast to previous proposals, the correlations described are computed for single pixels rather than for image windows. This avoids unnecessary smoothing, which for motion field estimation is particularly undesirable across occluding edges. Another novel feature of this work is that the pointwise correlations between a pixel in one image frame and those in an adjoining frame are not computed over a uniform image grid: the grid is finer for smaller image displacements, and coarser for larger ones.>

Results previously derived by the author are used to investigate the implication of bilateral symmetry in line drawings. It is shown that the line drawing of an orthographically protected surface of revolution exhibits bilateral symmetry about the projection of its axis of revolution irrespective of the viewing direction. Barring one exception, a bilaterally symmetric line drawing is necessarily the orthographic projection of a local surface of revolution whenever its symmetry axis continues to be the projection of a fixed line in space under perturbation of viewpoint; the axis of revolution is the invariant preimage of the symmetry axis. Various line-drawing causes are detailed which facilitate the deduction of invariant preimages of symmetry axes, and consequently of local surfaces of revolution.>

The author reports on the progress toward a mathematical theory of line-drawing interpretation. A working framework is developed, and a variety of tools and techniques are demonstrated within it. He reviews related work and discusses his assumptions. He presents a projective mapping which he expects will find use in the extension of results derived for orthographic projection to perspective projection. H. Whitney's definition (1955) of an excellent mapping is reviewed and it is shown that both orthographic and perspective projections of piecewise C/sup 3/ surfaces are excellent under general viewpoint. It is thus demonstrated that there are only two types of local singularities, folds and cusps, associated with the projection of a C/sup 3/ surface. The results render unnecessary the assumption of stability under perturbation of the viewed surface, which was required in previous work based on Whitney's results. The modes of interaction between viewpoint-independent and viewpoint-dependent edges are dealt with and the continuity constraints imposed on the scene by its line drawing investigated.>

Line drawings of man-made scenes often exhibit instances of straight lines and conic sections, i.e. ellipses, parabolas, and hyperbolas. Constraints imposed on the scene by such instances are investigated, under the assumption of general viewpoint, i.e. the mapping of the viewed surface onto the line drawing is stable under perturbation of the viewpoint within some open set. Both orthographic and perspective projection are considered. The viewed surfaces are assumed to be piecewise C/sup 3/. It is shown that straight lines and conic sections in line drawings are projections of scene edges which are also straight lines and conic sections, respectively. It is also shown that scene events which project onto straight lines or conic sections cannot be combinations of view-point-independent and viewpoint-dependent edges. Further, continuous-surface-normal depth discontinuities which project onto straight lines can be locally described by developable surfaces, and those which project onto conic sections can be locally described by nondevelopable quadric surfaces. Each of these quadric surfaces is determined up to four degrees-of-freedom by its projection.>

Most step-edge detectors are designed to detect locally straight edge-segments which can be isolated within the operator kernel. While it can easily be demonstrated that a cross-sectional support of at least 4 pixels is required for the unambiguous detection of a stepedge, edges which cannot be isolated within windows having this width can nevertheless be resolved. This is achieved by preceding the stepedge detection process by image-intensity interpolation. Although resolution can be improved in this fashion, the step-edge position and intensity estimates thus determined may be subject to systematic biases. Also, the higher resolution performance is accompanied by lower robustness to noise.

An edge in an image corresponds to a discontinuity in the intensity surface of the underlying scene. It can be approximated by a piecewise straight curve composed of edgels, i.e., short, linear edgeelements, each characterized by a direction and a position. The approach to edgel-detection here, is to fit a series of one-dimensional surfaces to each window (kernel of the operator) and accept the surfacedescription which is adequate in the least squares sense and has the fewest parameters. (A one-dimensional surface is one which is constant along some direction.) The tanh is an adequate basis for the step-edge and its combinations are adequate for the roof-edge and the line-edge. The proposed method of step-edgel detection is robust with respect to noise; for (step-size/noise) > 2.5, it has subpixel position localization (position < 3) and an angular localization better than 100; further, it is designed to be insensitive to smooth shading. These results are demonstrated by some simple analysis, statistical data, and edgel-images. Also included is a comparison of performance on a real image, with a typical operator (Difference-of-Gaussians). The results indicate that the proposed operator is superior with respect to detection, localization, and resolution.