We consider the problem of real-time GPU rendering of algebraic surfaces defined by Bézier tetrahedra. These surfaces are rendered directly in terms of their polynomial representations, as opposed to a collection of approximating triangles, thereby eliminating tessellation artifacts and reducing memory usage. A key step in such algorithms is the computation of univariate polynomial coefficients at each pixel; real roots of this polynomial correspond to possibly visible points on the surface. Our approach leverages the strengths of GPU computation and is highly efficient. Furthermore, we compute these coefficients in Bernstein form to maximize the stability of root finding, and to provide shader instances with an early exit test based on the sign of these coefficients. Solving for roots is done using analytic techniques that map well to a SIMD architecture, but limits us to fourth order algebraic surfaces. The general framework could be extended to higher order with numerical root finding.

We present a method for resolution independent rendering of paths and bounded regions, defined by quadratic and cubic spline curves, that leverages the parallelism of programmable graphics hardware to achieve high performance. A simple implicit equation for a parametric curve is found in a space that can be thought of as an analog to texture space. The image of a curve's Bézier control points are found in this space and assigned to the control points as texture coordinates. When the triangle(s) corresponding to the Bézier curve control hull are rendered, a pixel shader program evaluates the implicit equation for a pixel's interpolated texture coordinates to determine an inside/outside test for the curve. We extend our technique to handle anti-aliasing of boundaries. We also construct a vector image from mosaics of triangulated Bézier control points and show how to deform such images to create resolution independent texture on three dimensional objects.

Article Blue screen matting Share on Authors: Alvy Ray Smith Microsoft Corporation, One Microsoft Way, Redmond, WA Microsoft Corporation, One Microsoft Way, Redmond, WAView Profile , James F. Blinn Microsoft Corporation, One Microsoft Way, Redmond, WA Microsoft Corporation, One Microsoft Way, Redmond, WAView Profile Authors Info & Claims SIGGRAPH '96: Proceedings of the 23rd annual conference on Computer graphics and interactive techniquesAugust 1996 Pages 259–268https://doi.org/10.1145/237170.237263Published:01 August 1996 249citation2,665DownloadsMetricsTotal Citations249Total Downloads2,665Last 12 Months77Last 6 weeks2 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

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This panel will discuss the issues and the problems associated with the simulation of natural phenomena. This is a difficult area of research since it generally involves complex data bases and in many instances time variant phenomena. The computational loads can become enormous as one considers the physics or the mathematical modeling of structures.

The study of the physical process of light interacting with matter is an important part of computer image synthesis since it forms the basis for calculations of intensities in the picture. The simpler models used in the past are being augmented by more complex models gleaned from the physics literature. This paper is another step in the direction of assimilating such knowledge. It concerns the statistical simulation of light passing through and being reflected by clouds of similar small particles. (It does not, however, address the cloud structure modeling problem). By extension it can be applied to surfaces completely covered by dust and is therefore a physical basis for various theories of diffuse reflection.

The mathematical description of three dimensional surfaces usually falls in one of two classifications: parametric and algebraic. The form is defined as all points which satisfy some equation: F(x,y,z)=0. This form is ideally suited for image space shaded picture drawing, the pixel coordinates are substituted for x and y and the equation is solved for z. Algorithms for drawing such objects have been developed primarily for first and second order polynomial functions. This paper presents a new algorithm applicable to other functional forms, in particular to the summation of several gaussian density distributions. The algorithm was created to model electron density maps of molecular structures but can be used for other artistically interesting shapes.

The mathematical description of three-dimensional surfaces usually falls into one of two classifications: \nparametric and implicit. An implicit surface is defined to be all points which satisfy some \nequation F (x, y, z) = 0. This form is ideally suited for image space shaded picture drawing; the pixel \ncoordinates are substituted for x and y, and the equation is solved for z. Algorithms for drawing such \nobjects have been developed primarily for fLrst- and second-order polynomial functions, a subcategory \nknown as algebraic surfaces. This paper presents a new algorithm applicable to other functional \nforms, in particular to the summation of several Gaussian density distributions. The algorithm was \ncreated to model electron density maps of molecular structures, but it can be used for other artistically \ninteresting shapes.

This paper presents a scan line algorithm for drawing picture of parametrically defined surfaces. A scan line algorithm is characterized by the order in which it generates the picture elements of the image. These are generated left to right, top to bottom in much the same was as a picture is scanned out on a TV screen. Parametrically defined surfaces are those generated by a set of bivariate functions defining the X, Y and Z position of points on the surface. The primary driving mechanism behind such an algorithm is the inversion of the functions used to define the surface. To keep the algorithm general enough to apply to a wide variety of functional forms, this inversion is done numerically. It is only required to provide mechanism for evaluating the function and its derivatives at any parametric location.

Computer generated shaded images have reached an impressive degree of realism with the current state of the art. They are not so realistic, however, that they would fool many people into believing they are real. One problem is that the surfaces tend to look artificial due to their extreme smoothness. What is needed is a means of simulating the surface irregularities that are on real surfaces. In 1973 Ed Catmull introduced the idea of using the parameter values of parametrically defined surfaces to index into a texture definition function which scales the intensity of the reflected light. By tying the texture pattern to the parameter values, the texture is guaranteed to rotate and move with the object. This is good for showing patterns painted on the surface, but attempts to simulate rough surfaces in this way are unconvincing. This paper presents a method of using a texturing function to perform a small perturbation on the direction of the surface normal before using it in the intensity calculations. This process yields images with realistic looking surface wrinkles without the need to model each wrinkle as a separate surface element. Several samples of images made with this technique are included.

Clipping is the process of determining how much of a given line segment lies within the boundaries of the display screen. Homogeneous coordinates are a convenient mathematical device for representing and transforming objects. The space represented by homogeneous coordinates is not, however, a simple Euclidean 3-space. It is, in fact, analagous to a topological shape called a “projective plane”. The clipping problem is usually solved without consideration for the differences between Euclidean space and the space represented by homogeneous coordinates. For some constructions, this leads to errors in picture generation which show up as lines marked invisible when they should be visible. This paper will examine these cases and present techniques for correctly clipping the line segments.

In the production of computer generated pictures of three dimensional objects, one stage of the calculation is the determination of the intensity of a given object once its visibility has been established. This is typically done by modelling the surface as a perfect diffuser, sometimes with a specular component added for the simulation of hilights. This paper presents a more accurate function for the generation of hilights which is based on some experimental measurements of how light reflects from real surfaces. It differs from previous models in that the intensity of the hilight changes with the direction of the light source. Also the position and shape of the hilights is somewhat different from that generated by simpler models. Finally, the hilight function generates different results when simulating metallic vs. nonmetallic surfaces. Many of the effects so generated are somewhat subtle and are apparent only during movie sequences. Some representative still frames from such movies are included.

Homogeneous coordinates have long been a standard tool of computer graphics. They afford a convenient representation (for) various geometric quantities in two and three dimensions. The representation of lines in three dimensions has, however, never been fully described. This paper presents a homogeneous formulation for lines in 3 dimensions as an anti-symmetric 4x4 matrix which transforms as a tensor. This tensor actually exists in both covariant and contravariant forms, both of which are useful in different situations. The derivation of these forms and their use in solving various geometrical problems is described.

In 1974 Ed Catmull developed a new algorithm for rendering images of bivariate surface patches. This paper describes extensions of this algorithm in the areas of texture simulation and lighting models.The parametrization of a patch defines a coordinate system which is used as a key for mapping patterns onto the surface. The parametric values within each picture element are input to a pattern definition function. A weighted average of the values of this function over the picture element scales the intensity of that picture element. By suitably defining the pattern function, various surfaces textures can be simulated. The shape and size of this weighting function is chosen using digital signal processing theory.The other problem addressed here concerns lighting models. The patch rendering algorithm allows accurate computation of the surface.