Michael J Shantz

Adaptive forward differencing for rendering curves and surfaces

An adaptive forward differencing algorithm is presented for rapid rendering of cubic curves and bicubic surfaces. This method adjusts the forward difference step size so that approximately one pixel is generated along an ordinary or rational cubic curve for each forward difference step. The adjustment involves a simple linear transformation on the coefficients of the curve which can be accomplished with shifts and adds. This technique combines the advantages of traditional forward differencing and adaptive subdivision. A hardware implementation approach is described including the adaptive control of a forward difference engine. Surfaces are rendered by drawing many curves spaced closely enough together so that no pixels are left unpainted. A simple curve anti-aliasing algorithm is also presented in this paper. Anti-aliasing cubic curves is supported via tangent vector output at each forward difference step. The adaptive forward differencing algorithm is also suitable for software implementation.

Surface definition for branching, contour-defined objects

Recent work on mapping polygonal surface mosaics onto contour defined objects has resulted in minimum flow algorithms which do a good job of mapping a surface between two single contours. This report extends these algorithms to handle contour defined objects which are highly branched and have holes.For branching contours where n contours in section I are connected to m contours in section i+1, the surfaces are mapped by first concatenating the section I contours into a single large contour using a minimum number of minimum distance links, similarly concatenating the section I+1 contours, then performing the one to one mapping between the resulting composite contours.Capping off a single arbitrarily shaped contour may be done by computing the medial axis transform of the contour, constructing the medial axis contour and mapping a surface from the contour to the medial axis contour.Medial axis transforms of the section I+1 contours may also be projected onto section I and concatenated with the section I contours to handle the mapping of convoluted, highly branched objects. These algorithms have applications to three-dimensional CAT scanner data, topographical map data and object representation for knowledge guided vision systems.

Shading bicubic patches

We present several techniques for implementing Phong shading in hardware for bicubic patches. Patches are shaded, not by subdividing into polygons, but by drawing many curves close together leaving no pixel gaps. Each curve is drawn using an adaptive forward difference algorithm which generates the coordinates as well as the shading parameters as cubic functions incrementally evaluated along the curve. The forward difference step size is adaptively adjusted so that it generates approximately one pixel along the curve per forward difference step. The hardware implements Phong shading directly with a surprisingly simple configuration built from general purpose compute units and look-up tables. Two new methods are presented for deriving bicubic approximations to the shading parameters over a bicubic patch. One method uses two Coons patches to approximate the unnormalized N·L, and N·H, and a third Coons patch for N·N, where N is the surface normal, L is the light direction, and H is the direction of maximum highlight. In this case the hardware performs the normalization per pixel. The second method uses two Coons patches to approximate the normalized dot products N·L, and N·H. The method is suitable for both hardware and software implementations.

Two-Pass Warp Algorithm For Hardware Implementation

A common requirement in remote sensing and cartography applications is that of warping an image to register with a reference image. A second order polynomial coordinate transformation on two-dimensional images embodies translation, rotation, scaling, and several types of warping. To perform this operation efficiently and with minimal hardware, the transformation is separated into a line operation followed by a column operation. These two transformations are computed using a least square fit algorithm. Resampling with bilinear or cubic convolution interpolation is also separable into the two-pass technique. This algorithm is suitable for hardware implementation of current image processing systems.

Global Optimization In Binocular Vision

Correlation techniques for obtaining depth profiles from stereo pairs have been studied extensively yet continue to have problems with generality and computational efficiency. A new method is presented which develops the camera model constraints required to ensure that binocular disparity shifts only shift in the x direction. An optimal mapping algorithm is then applied which gives a globally optimal depth profile for a given line of the stereo pair. The algorithm performs well in cases which are difficult for correlation based algorithms namely, repetitive patterns, depth discontinuties and large, relatively featureless regions. With currently available IC chips this algorithm could be implemented in hardware to achieve video rate performance.