Matrices in computer graphics

Solving the initial value problem of discrete geodesics

The Geometry Engine[1] is a special-purpose VLSI processor for computer graphics. It is a four-component vector, floating-point processor for accomplishing three basic operations in computer graphics: matrix transformations, clipping and mapping to output device coordinates. This paper describes the Geometry Engine and the Geometric Graphics System it composes. It presents the instruction set of the system, its design motivations and the Geometry System architecture.

Mesh parameterization is one of the fundamental operations in computer graphics (CG) and computeraided design (CAD). In this paper, we propose a novel local/global parameterization approach, ARAP++, for singleand multi-boundary triangular meshes. It is an extension of the as-rigid-as-possible (ARAP) approach, which stitches together 1-ring patches instead of individual triangles. To optimize the spring energy, we introduce a linear iterative scheme which employs convex combination weights and a fitting Jacobian matrix corresponding to a prescribed family of transformations. Our algorithm is simple, efficient, and robust. The geometric properties (angle and area) of the original model can also be preserved by appropriately prescribing the singular values of the fitting matrix. To reduce the area and stretch distortions for high-curvature models, a stretch operator is introduced. Numerical results demonstrate that ARAP++ outperforms several state-of-the-art methods in terms of controlling the distortions of angle, area, and stretch. Furthermore, it achieves a better visualization performance for several applications, such as texture mapping and surface remeshing.

Efficient piecewise higher-order parametrization of discrete surfaces with local and global injectivity

Given a collection of points in the plane, pick an arbitrary horizontal segment and move it vertically until it hits one of the points (if at all). This form ofsegment-dragging is a common operation in computer graphics and motion-planning, it can also serve as a building block for multidimensional data structures. This note describes a new approach to segment-dragging which yields a simple and efficient solution. The data structure requiresO(n) storage andO(n logn) preprocessing time, and each query can be answered inO(logn) time, wheren is the number of points in the collection. The method is best understood as the end result of a sequence of transformations applied to a simple but inefficient starting solution.

Tracing a ray through a scene and finding the closest intersection with the scene geometry is a fundamental operation in computer graphics. During the last two decades, significant efforts have been made to accelerate this operation, with interactive ray tracing as one of the major driving forces. At the heart of a fast method for intersecting a scene with a ray lies the acceleration structure. Many different acceleration structures exist, but research has focused almost exclusively on a few well-tried and well-established techniques: regular and hierarchical grids, bounding volume hierarchies and kd-trees. Spectacular advances have been made, which have contributed significantly to making interactive ray tracing a possibility. However, despite the success of these acceleration structures, several problems remain open. Handling deforming and dynamic geometry still poses significant challenges, and the local vs. global complexity of acceleration structures is still not entirely understood. One therefore wonders whether other acceleration structures, that leave the beaten path of efficient grids, bounding volume hierarchies and kd-trees, can provide viable alternatives.Next to computer graphics, the ray shooting problem is also studied in computational geometry. Ray shooting queries against a large collection of polyhedra are answered by tracing the ray through a simplicial complex such as a constrained tetrahedralization. This is a well-known technique, see e.g. the chapter Ray shooting and lines in space by Pellegrini in Handbook of Discrete and Computational Geometry. However, relevant work in computational geometry is usually theoretical, and practical implementations and experimental results are typically not available.In this work we explore the idea of accelerating the operation of intersecting a scene with a ray using constrained tetrahedralizations. This is illustrated in figure 1. A constrained tetrahedralization of a scene is a tetrahedralization that respects the faces of the scene geometry. The closest intersection of a ray with a scene is found by traversing this tetrahedralization along the ray, one tetrahedron at a time, until a constrained face is encountered. We show that constrained tetrahedralizations are a viable alternative to state-of-the-art acceleration structures, such as kd-trees, and that constrained tetrahedralizations have a number of interesting and unique properties that set them apart from traditional acceleration structures. Constrained tetrahedralizations are not hierarchical yet adaptive; the complexity of traversing them is a function of local geometric complexity rather than global geometric complexity; constrained tetrahedralizations support deforming geometry without any effort (see figure 2); and they have the potential to unify several data structures currently used in global illumination.Although constrained tetrahedralizations are not a silver bullet, and although they are in general not yet faster than the most optimized kd-trees, constrained tetrahedralizations offer several new perspectives on acceleration structures for ray tracing and deserve attention.

Discretization and reconstruction are fundamental operations in computer graphics, enabling the conversion between sampled and continuous representations. Major advances in signal-processing research have shown that these operations can often be performed more efficiently by decomposing a filter into two parts: a compactly supported continuous-domain function and a digital filter. This strategy of “generalized sampling” has appeared in a few graphics papers, but is largely unexplored in our community. This survey broadly summarizes the key aspects of the framework, and delves into specific applications in graphics. Using new notation, we concisely present and extend several key techniques. In addition, we demonstrate benefits for prefiltering in image downscaling and supersample-based rendering, and analyze the effect that generalized sampling has on the noise due to Monte Carlo estimation. We conclude with a qualitative and quantitative comparison of traditional and generalized filters.

Computer graphics is receiving much attention in the development of interactive educational software, multimedia systems, and many other applications. It not only adds a new dimension to such applications but also makes them more exciting and dynamic. Furthermore, the use of computer graphics is already well accepted in computer science education. If skillfully and relevantly used, it can be an important component of computer-assisted instruction, which is an educational application area with tremendous potential.

Logarithmic arithmetic (LA) is a very fast computational method for real numbers. And its computation precision is much better than a floating point arithmetic of equivalent word length and range. This paper shows a method of fixed point number computations by LA—just to do numeric conversion before and after LA computations. It is used to handle discrete coordinate addresses and pixel intensity data of digital images. The geometrical transformation is a typical application of the method. Linear (affine) and non-linear transformations with three interpolations of "nearest neighbor," "bi-linear" and "cubic convolution" in LA are demonstrated. It is the processing of coordinate addresses and pixel intensities of fixed point numbers. Experiments by 16-bit personal computer program showed that quality and speed are surprisingly high. The latter is comparable to that by using the floating point hardware chip. Some other examples of the applications are shown—curve drawing, three dimensional computer graphics and fractal image generation—all excellent.

Unlike textbook images that are static and television images that permit only passive viewing, computer graphics can be drawn dynamically and interactively on the display screen. For years, most computer games have featured animated, interactive computer graphics. Now computer-based tutorials on a wide range of topics are available that present new information to learners via dynamic, interactive graphics. One of the most recent developments in the area of computer graphics is the educational use of graphics applications software—the visual equivalent of word processing software. With it, learners can create their own computer graphics on the display screen and then use the computer to review the images in a variety of ways. Images can be enlarged, flipped, rotated, animated, duplicated, colored, sized up or down, moved around the screen, deleted, and more—all without programming. It is this dynamic and interactive aspect of computer graphics that is so appealing to learners and holds great promise for improving instruction.

Image downscaling via co-occurrence learning

Line segment clipping is a basic operation of the visualization process in computer graphics. So far there exist four computational models for clipping a line segment against a window, (1) the encoding, (2) the parametric, (3) the geometric transformation, and (4) the parallel cutting. This paper presents an algorithm that is based on the third method. By making use of symmetric properties of a window and transformation operations, both endpoints of a line segment are transformed, so that the basic cases are reduced into two that can be easily handled, thus the problems in NLN and AS where there are too many sub-procedure calls and basic cases that are difficult to deal with are tackled. Both analytical and experimental results from random input data show that the algorithm is better than other developed ones, in view of the speed and the number of operations.

Geometric transformations are most commonly represented as square matrices in computer graphics. Following simple geometric arguments we derive a natural and geometrically meaningful definition of scalar multiples and a commutative addition of transformations based on the matrix representation, given that the matrices have no negative real eigenvalues. Together, these operations allow the linear combination of transformations. This provides the ability to create weighted combination of transformations, interpolate between transformations, and to construct or use arbitrary transformations in a structure similar to a basis of a vector space. These basic techniques are useful for synthesis and analysis of motions or animations. Animations through a set of key transformations are generated using standard techniques such as subdivision curves. For analysis and progressive compression a PCA can be applied to sequences of transformations. We describe an implementation of the techniques that enables an easy-to-use and transparent way of dealing with geometric transformations in graphics software. We compare and relate our approach to other techniques such as matrix decomposition and quaternion interpolation.

After the long period of mandatory distance learning, either synchronous or asynchronous, caused by the COVID-19 pandemic, the existence of learning tools to support students in similar scenarios is welcomed in many curriculum subjects. Geometric transformations are relevant in mathematics and an omnipresent topic in the computer graphics curriculum that needs plenty of visual tools and learning-by-doing environments. In this work, we propose a framework based on an extended Translation-Rotation-Scale (TRS) pattern to generate composite transformations to support a specific style of learning exercises that we call “before and after”. The pattern, together with specific constraints to reduce the domain into a finite set of cases, allows the automatic exercises generation, evaluation, and real-time feedback. Two pieces of software, called GTVisualizer and GTCards, were developed to empirically test the proposed framework embedding it in a gaming environment. The first tool supports the initial instruction and introduces the formal matrix-based methodology to specify composite transformations; the second one provides a ludic environment where students can drill and reinforce the subject by playing. Our initial results suggest that these tools are suitable for learning geometric transformations; they helped students in simulated self-study mode achieve comparable results to those students receiving the regular lecture materials and explanations in the classroom. Moreover, the usage of the tools appears to favor a similar increasing effect on students’ visual-spatial abilities, similar to the obtained by taking an in-person course on computer graphics, as measured with the Purdue Spatial Visualization Test.

Few fields remained untouched by the advances in computer graphics since the 1970s, but hardly any were affected so profoundly as that of typography. As typography dissolves into the field of graphic design, many of the subtleties of letters are forgotten and ignored, especially when designers have myriad fonts to choose from, many of them illegible and over-stylized. I endeavored to make a teapot glyph that accentuates the spirit each in font I used, but also functions legibly and seamlessly within the text. I chose the fonts after much deliberation; they reflect the statements that the text makes and represent commonly used fonts that are archetypes of their class. Beaming with personality, Bodoni MT continues to be an extremely popular and elegant font. Adobe Jenson Pro based on completely pen-formed letters; its imperfections and inconsistencies recall the humanist calligraphy of the Renaissance. ITC New Baskerville is elegant in its modeled balance and uprightness, showing a true break in type and handwriting and Futura is the quintessential geometric sans serif. Lucida Console has its roots in early digital typography and is used in various other software applications. Adobe Garamond Pro and other interpretations of Garamond's type are the quintessential font for the setting of books and titles alike. The text itself tells a history of teapots in broad strokes, leaving it up to the viewer to make the connection to fonts. That said, the text is not intended to be the focus of the piece by any means. With the x-height measuring about a quarter-inch, the scale of the letters calls for the shapes of the letters to be seen as abstract shapes and for the rhythms of whitespace to be much more tangible. When seen at a distance, each typeface clearly creates a different color, as it's commonly referred to, on the page. When seen close up, the citation of each font calls attention to the juxtaposition of teapots and letters and symbols. Not only do the citations give credit to the designers whose creations account for more than 99% of my piece, but they are insights into the historical context of the letterforms. Because of their prominent position in culture and history, not to mention their anthropomorphic nature and symbolic status, teapots have unique personalities that operate within their given form. This piece asserts that, for the same reasons, typefaces have the same.