What are the development prospects of computer graphics? What are the current research fields?

Computer graphics emerged and developed with computers and their peripheral devices. As an independent branch of computer science and technology, it has experienced nearly 40 years of development. On the one hand, as a subject, computer graphics has made great progress in three aspects: basic graphics algorithms, graphics software and graphics hardware, and has become a technology and tool used to enhance the understanding and transmission of information in almost all contemporary science and engineering technology fields. On the other hand, computer graphics hardware and software have developed into a huge industry in their own right.

1. Active theories and technologies in computer graphics

(1) Fractal theory and applications

Fractal theory is a very active new theory in today's world. As a cutting-edge subject, fractal theory believes that nature is composed of fractals. In the vast world, symmetrical and balanced objects and states are a minority and temporary, while asymmetric and unbalanced objects and states are the majority and long-term. Fractal geometry is the geometry that describes nature. As a new cognitive method for humans to explore complex things, fractals have practical application significance in all fields involving organizational structure and morphogenesis, and have made achievements in oil exploration, earthquake prediction, urban construction, cancer research, economic analysis, etc. Many breakthrough developments. The concept of fractals was first proposed by the American mathematician B.B. Mandelbrot. In 1967, he published a paper in the American "Science" magazine titled "How Long is the British Coastline?" 》 famous paper. As a curve, the coastline is characterized by being extremely irregular and not smooth, showing extremely meandering and complex changes. It cannot be described by conventional, traditional geometric methods. We cannot distinguish any essential difference between this part of the coast and that part of the coast in terms of shape and structure. This almost same degree of irregularity and complexity shows that the coastline is self-similar in morphology, that is, the local morphology and Similarity in overall form. Without buildings or other objects as reference objects, the two photos of a 100-kilometer-long coastline taken from the air and the enlarged photo of a 10-kilometer-long coastline look very similar. Someone once put forward such an obviously absurd proposition: "The length of the British coastline is infinite." The argument goes like this: the coastline is broken and zigzag, and when we measure it, we always use a certain scale to get an approximate value. , for example, set up a benchmark every 100 meters. In this way, what we measure is an approximate value, which is calculated along a polyline. Each segment of this polyline is a straight line segment with a length of 100 meters. If instead a benchmark is set up every 10 meters, what is actually measured is the length of another polyline, with each segment being 10 meters long. Obviously, the length measured later will be greater than the length measured previously. If we continue to shrink the scale, the measured length will become larger and larger. In this way, wouldn't the length of the coastline become infinite? Why is this conclusion reached? Mandelbrot proposed an important concept: fractal dimension, also known as fractal dimension. Generally speaking, the dimensions are integers, straight line segments are one-dimensional graphics, and squares are two-dimensional graphics. In mathematics, the geometric object of Euclidean space can be continuously stretched, compressed, and twisted without changing the dimension. This is the topological dimension. However, this dimensional view cannot solve the problem of coastline length. Mandelbrot described the dimensions of a tetherball like this: when viewed from a long distance, the tetherball can be seen as a point (zero dimension); when viewed from a closer distance, it fills a spherical space (three dimensions). ); get closer, and you can see the rope (one-dimensional); go deeper into the microscopic level, and the rope becomes a three-dimensional column, and the three-dimensional column can be decomposed into one-dimensional fibers. So what about the intermediate states between these observation points? Obviously, there is no exact limit at which a tetherball changes from a three-dimensional object to a one-dimensional object. Why is the UK coastline so inaccurate? Because the Euclidean one-dimensional measure is inconsistent with the dimensions of the coastline. According to Mandelbrot's calculations, the dimension of the British coastline is 1.26. With the concept of fractal dimension, the length of the coastline can be determined. In 1975, Mandelbrot discovered that self-similar forms widely exist in nature, such as rolling mountains and rivers, floating clouds, rock fractures, Brownian particle motion trajectories, tree crowns, cauliflower, cerebral cortex... Mandelbrot called these shapes whose parts are similar to the whole in some way called fractals. The word is derived from the Latin word Frangere, which itself has meanings such as "broken" and "irregular". The highlight of Mandelbrot's research was the discovery in 1980 of the collection that bears his name, in which he discovered that the entire universe was structured in a self-similar manner in an unexpected way. The boundaries of Mandelbrot set graphics have infinitely complex and sophisticated structures. On this basis, the science of studying fractal properties and their applications was formed, called fractal theory or fractal geometry.

Characteristics and theoretical contributions of fractals Mathematically, fractals have the following characteristics: (1) Infinitely fine structure; (2) Proportional self-similarity; (3) Generally, its fractal dimension is larger than The topological dimension of; (4) can be defined by a very simple method and generated by recursion, iteration, etc. The two items (1) and (2) illustrate the inherent regularity of fractal structure. Self-similarity is the soul of fractals, which makes any fragment of the fractal contain the information of the entire fractal. Item (3) illustrates the complexity of fractals, and item (4) illustrates the generation mechanism of fractals. We compare Euclidean geometry, a representative of traditional geometry, with fractal geometry, which takes fractals as the research object, and we can come to the conclusion that Euclidean geometry is a logical system based on axioms, and its research is on rotation, translation, Various invariant quantities under symmetry transformation, such as angles, lengths, areas, and volumes, are mainly applicable to man-made objects; while fractals are generated by recursion and iteration, and are mainly applicable to objects with complex shapes in nature. Fractal geometry is no longer Look at the points, lines, and surfaces in a fractal as separate things, but instead look at them as a whole. We can understand fractal geometry from the characteristics of fractal patterns. Fractal patterns have a series of interesting characteristics, such as self-similarity, invariance to certain transformations, infinite internal structure, etc. In addition, fractal patterns are often associated with certain geometric transformations. Under some changes, the pattern remains unchanged. Starting from an arbitrary initial state, after several geometric transformations, the graphics will be fixed on this specific fractal pattern without any change. Change again. The principle of self-similarity and the principle of iterative generation are important principles of fractal theory. Fractal theory developed the concept of dimensionality. Before the discovery of fractal dimensions, people were accustomed to defining points as zero dimensions, straight lines as one dimension, planes as two dimensions, and space as three dimensions. Einstein introduced the time dimension in the theory of relativity, forming a four-dimensional space-time. By considering many aspects of a certain problem, a high-dimensional space can be established, but all are integer dimensions. Fractal is a new scientific thought and a new perspective on the world that emerged in the 20th century. Theoretically, it is a new development of mathematical thinking and a deepening and promotion of human understanding of concepts such as dimensions and point sets. At the same time, it is closely connected with the real physical world and has become an important tool for studying chaos (Chaos) phenomena. As we all know, the study of chaos phenomena is one of the frontiers and hot spots of modern theoretical physics. Thanks to the study of fractals, people have a better understanding of the dialectical relationship between randomness and determinism. It also has a beneficial impact on the connection between process and state, on the connection between macro and micro, on the transformation between levels, and on the richness and variety of infinity. Fractal theory is also the frontier and important branch of nonlinear science. As a methodology and epistemology, its inspiration is multifaceted: First, the similarity between the fractal whole and the local form inspires people to understand the whole by understanding the part, and to understand the infinite from the finite. Second, fractals reveal new forms and orders between whole and part, order and disorder, complexity and simplicity; third, fractals reveal the universal connection and unity of the world from a specific level.

In addition to the theoretical significance of the application fields of fractals, fractals also show great potential in practical applications. It has been effectively used in many fields, and its application range is wide. The benefits clearly far exceed any predictions made more than a decade ago. At present, a large number of application cases of fractal methods are emerging one after another. These cases cover the fields of life process evolution, ecosystems, digital encoding and decoding, number theory, dynamical systems, theoretical physics (such as fluid mechanics and turbulence), etc. In addition, some people use fractals to make urban rules and earthquake predictions. The application of fractal technology in data compression is a very typical example. The Journal of the American Mathematical Society published Basili's article "Using Fractals for Graphics Compression" in the June 1996 issue. He used fractals for graphics compression in optical disc production. Generally speaking, we always store and process a graphic as a collection of pixels. The most common picture often involves hundreds of thousands or even millions of pixels, thus occupying a large amount of storage space and the transmission speed is also greatly limited. Basili used an important idea in fractals: fractal patterns are associated with certain transformations. We can regard any graphic as the product of repeated iterations of certain transformations. Therefore, to store a graphic, you only need to store information about these transformation processes, without storing all the pixel information of the graphic. As long as this transformation process is found, the graphics can be accurately reproduced without having to store a large amount of pixel information. Using this method, in actual applications, the effect of compressing storage space to 1/8 of the original has been achieved. In recent years, fractal art (FA), developed from fractal theory, has also made breakthrough progress in terms of expression form and understanding of fractal geometry. Fractal art is two-dimensional visual art that is similar to photography in many ways. Fractal image works are generally displayed through computer screens and printers. Another important part of fractal art is fractal music, which is produced by multiple iterations of an algorithm.

Self-similarity is the essence of fractal geometry. Some people use this principle to construct some synthetic music with self-similar segments. The theme is repeated in a repeated cycle of minor keys, and some random changes can be added to the rhythm. Many of our common computer screen savers are also derived from fractal calculations.

Since the 1990s, people have begun to increasingly use this theory to study some issues in the economic field, mainly focusing on the study of financial markets (such as stock markets, foreign exchange markets, etc.). The manipulator can make the stock price change as desired on a micro scale through manipulation at several points in time; from a macro scale of time, making the stock price change as desired requires the manipulator to have considerable economic strength. From a fractal perspective, stock prices have fractal characteristics. On the one hand, the stock price has a complex microstructure; on the other hand, it has scale invariance to time, that is, it has a similar structure under different observation scales. Its structure is the unity of complexity and simplicity, irregularity and order. . It is not difficult for stock price manipulators to influence the stock price at a single point in time. Even if it affects the stock price on a large time scale, it is possible. However, it is necessary to maintain the stock price while affecting the stock price through artificial manipulation. Consistency between micro and macro scales of time would be technically very difficult.

(2) Surface modeling technology. It is an important part of computer graphics and computer-aided geometric design (Computer Aided Geometric Design). It mainly studies the representation, design, display and analysis of curved surfaces in the environment of computer graphics systems. It originated from the appearance lofting technology of aircraft and ships, and the theoretical foundation was laid by masters such as Coons and Bezier in the 1960s. After more than thirty years of development, it has now formed two types of methods, namely parametric feature design and implicit algebraic surface representation, represented by Bezier and B-spline methods, with interpolation and fitting as the main body. , Approximation, these three methods are the geometric theory system of the skeleton. With the increasing requirements for authenticity, real-time and interactivity of computer graphics display, and with the increasingly obvious trend of geometric design objects moving towards diversity, particularity and topological complexity, as the graphics industry and manufacturing industry move forward, With the increasing pace of integration, integration and networking, and with the increasing improvement of three-dimensional data sampling technology and hardware equipment such as laser ranging scanning, surface modeling has made great progress in recent years. This is mainly reflected in the rapid expansion of research fields and the innovation of representation methods.

1. From the perspective of research fields, surface modeling technology has expanded from the traditional study of surface representation, surface intersection and surface splicing to surface deformation, surface reconstruction, surface simplification, surface transformation and surface dislocation. .

Surface deformation (Deformation or Shape Blending): The traditional non-uniform rational B-spline (NURBS) surface model only allows adjusting control vertices or weight factors to locally change the surface shape, and at most uses hierarchical refinement of the model. Direct operations at specific points on the surface; some simple surface design methods based on parametric curves, such as sweeping, skinning, rotation and stretching, only allow the generated curves to be adjusted to change the surface shape. The computer animation industry and the solid modeling industry urgently need to develop deformation methods or shape allocation methods that are independent of surface representation, so the free deformation (FFD) method was born, which is a deformation method based on physical models (principles) such as elastic deformation or thermoelastic mechanics. Surface deformation technologies such as deformation methods based on solving constraints, deformation methods based on geometric constraints, and surface shape allocation technologies based on polyhedral correspondence or based on Minkowski sum operations in image morphology. Recently, the author and his student Liu Ligang pioneered the new idea of ??movable local spherical coordinate interpolation, which gave a complete mathematical description of the intrinsic variables of the space point set. From the perspective of geometric intrinsic solutions, they designed a set of fast methods for shape allocation of three-dimensional polyhedrons and free-form surfaces. Effective algorithms, smooth graphics, and real-time interaction have achieved breakthroughs in the technical problems of three-dimensional surface deformation.

Surface reconstruction (Reconstruction): In the animation production of exquisite car body design or face-like sculpture surface, clay modeling is often used, and then three-dimensional value point sampling is performed. In medical image visualization, CT slices are also commonly used to obtain three-dimensional data points on the surface of human organs. Recovering the geometric model of the original surface from partial sampling information on the surface is called surface reconstruction. Sampling tools are: laser ranging scanners, medical imagers, contact detection digitizers, radar or seismic exploration instruments, etc. According to the form of the reconstructed surface, it can be divided into two categories: functional surface reconstruction and discrete surface reconstruction.

Surface simplification: Like surface reconstruction, this research field is currently one of the international hot spots. The basic idea is to remove redundant information from the 3D reconstructed discrete surfaces or the output results of the modeling software (mainly triangular meshes) while ensuring the accuracy of the model, so as to facilitate the real-time performance of graphics display and the economy of data storage. and the speed of data transfer.

For multi-resolution surface models, this technology is also conducive to establishing a hierarchical approximation model of the surface, performing hierarchical display, hierarchical transmission and hierarchical editing of the surface. Specific surface simplification methods include: mesh vertex elimination method, mesh boundary deletion method, mesh optimization method, maximum plane approximation polygon method and parametric resampling method.

Surface conversion (Conversion): The same surface can be expressed in different mathematical forms. This idea not only has theoretical significance, but also has practical significance for industrial applications. For example, although parametric rational polynomial surfaces such as NURBS include all the advantages of parametric polynomial surfaces, they also have the limitations of cumbersome and time-consuming differential operations and the inability to control errors in integral operations. In surface splicing and physical property calculations, these two operations are inevitable. This raises the problem of converting a NURBS surface into an approximate polynomial surface. The same requirements are reflected in the data transfer between the NURBS surface design system and the polynomial surface design system and the paperless production process. For another example, in the intersection operation of two parametric surfaces, if the NURBS form of one of the surfaces is converted into implicit, the numerical solution of the equation can be easily obtained. In recent years, the research on surface conversion in the international graphics community has mainly focused on the following aspects: the algorithm and convergence of approximating NURBS surfaces with polynomial surfaces; the implicitization of Bezier curves and surfaces and their inverse problems; the Ball of CONSURF aircraft design system Comparison and mutual conversion of various generalization forms of curves to high dimensions; reduced-order approximation algorithm and error estimation of rational Bezier curves and surfaces; rapid transformation of NURBS surfaces in the triangular domain and the rectangular domain, etc.

Surface offset (Offset): also known as surface isometric property, it is widely used in computer graphics and processing, and has become one of the hot topics in recent years. For example, the tool path design of CNC machine tools requires studying the isometric properties of curves. However, it is easy to see from the mathematical expression that, generally speaking, the isometric curve of a plane parametric curve is no longer a rational curve, which exceeds the scope of use of the general NURBS system, causing complexity in software design and numerical calculations. of instability.

2. From the perspective of representation methods, the discrete modeling characterized by grid subdivision (Subdivision) has a great potential for innovation compared with the traditional continuous modeling. Moreover, this surface modeling method is very suitable for the design and processing of vivid and lifelike feature animations and sculptural surfaces, and has been highly used.

Among the films that won the Oscar in 1998, there was a short film on the list, which was "Geri's Game" selected by the famous American animation film studio Pixar. The cartoon describes a humorous story about an old man named Geri who plays chess with himself in the park and tries every possible means to win. The characters and scenery in the picture are detailed and vivid, integrating seamlessly with the storyline, giving the audience real aesthetic enjoyment. The designer in the production of this cartoon is the author of the above paper, the famous computer graphics scientist T. DeRose. The paper DeRose reported at the SIGGRAPH'98 conference talked about the use of C-C subdivision surfaces as the background of the Geri old man's characteristic modeling model. He pointed out that although NURBS has long been used by the international standards organization ISO as a STEP standard to define industrial product data exchange and has been widely used in industrial modeling and animation production, it still has limitations. A single NURBS surface, like other parametric surfaces, is limited to representing surfaces that are topologically equivalent to a piece of paper, a cylinder, or a torus, and cannot represent surfaces with any topological structure. In order to express more complex shapes in feature animation, such as a human head, human hands or human clothing, we faced a technical challenge. Of course, we can use the most common modeling methods of complex smooth surfaces, such as NURBS trimming (Trimming) to deal with it. Indeed, there are already some commercial systems, such as Alias-Wavefront and SoftImage, that can do this, but they will at least encounter the following difficulties: first, pruning is expensive and has numerical errors; second, it requires The seams of the surfaces remain smooth, and even approximate smoothness is difficult because the model is active. Subdivision surfaces have the potential to overcome the above two difficulties. They do not need to be trimmed, there are no seams, and the smoothness of the active model is automatically guaranteed. DeRose successfully applied C-C's subdivision surface modeling method, and at the same time invented the practical technology for constructing smooth variable-radius contours and composites, proposed an effective new algorithm for collision detection in clothing models, and constructed a method for subdivision surfaces. Smooth factor field method. With these mathematical and software foundations, he realistically represented Old Man Geri's skull, fingers and clothes, including jacket, trousers, tie and shoes. These are difficult to achieve with traditional NURBS continuous surface modeling. So, how is the C-C subdivision surface constructed? It is similar to the traditional Doo-Sabin subdivision surface. It starts from a polyhedron called a control mesh (the mesh can mostly be input from a manual model using a laser), and recursively calculates each vertex on the new mesh. These vertices are the weighted average of certain vertices on the original mesh.

If a face of a polyhedron has n sides, after subdividing it once, the face will become n quadrilaterals. As the subdivision continues, the control mesh is gradually polished, and its ultimate state is a free-form surface. It is seamless and therefore smooth, even when the model is active. This approach significantly reduces the time required to design and build an original model. More importantly, it allows the original model to be locally refined. This is where it is superior to the continuous surface modeling method. C-C subdivision is based on quadrilaterals, while Loop surfaces (1987) and butterfly surfaces (1990) are based on triangles. They are all equally reused by today's graphics workers.

(3) Computer-aided design and manufacturing (CAD/CAM). This is one of the most extensive and active application areas. Computer Aided Design (CAD) is a method that uses the powerful computing functions and efficient graphics processing capabilities of computers to assist knowledge workers in the design and analysis of projects and products to achieve ideal goals or achieve innovative results. kind of technology. It is an emerging discipline formed by integrating the latest developments in computer science and engineering design methods. The development of computer-aided design technology is closely related to the development and improvement of computer software and hardware technology, and the innovation of engineering design methods. The use of computer-aided design is an urgent need for modern engineering design. CAD technology has been widely used in all aspects of the national economy, and its main application areas are as follows.

1． Application in manufacturing industry

CAD technology has been widely used in manufacturing industry, among which the most extensive and profound applications are in manufacturing industries such as machine tools, automobiles, aircraft, ships, and spacecraft. As we all know, the design process of a product goes through several main stages such as conceptual design, detailed design, structural analysis and optimization, and simulation.

At the same time, modern design technology introduces the concept of concurrent engineering into the entire design process, and comprehensively considers the entire life cycle of the product during the design stage. Current advanced CAD application systems have integrated a series of functions such as design, drawing, analysis, simulation, and processing into one system. The more commonly used software now include CAD application systems such as UG II, I-DEAS, CATIA, PRO/E, and Euclid. These systems mainly run on graphics workstation platforms. The main CAD application software running on the PC platform include Cimatron, Solidwork, MDT, SolidEdge, etc. Due to various factors, AutoCAD of Autodesk currently occupies a considerable market share among the two-dimensional CAD systems.

2. Application in engineering design

The application of CAD technology in the engineering field has the following aspects:

(1) Architectural design, including scheme design, three-dimensional modeling, and architectural rendering design , graphic design, architectural structure design, community planning, sunlight analysis, interior decoration and other various CAD application software.

(2) Structural design, including finite element analysis, structural plane design, frame/frame structure calculation and analysis, high-rise structure analysis, foundation and foundation design, steel structure design and processing, etc.

(3) Equipment design, including water, electricity, heating equipment and pipeline design.

(4) Urban planning, urban transportation design, such as urban roads, elevated roads, light rails, subways and other municipal engineering designs.

(5) Municipal pipeline design, such as tap water, sewage discharge, gas, electricity, heating, communications (including telephone, cable TV, data communications, etc.) various types of municipal pipeline line design.

(6) Traffic engineering design, such as roads, bridges, railways, aviation, airports, ports, docks, etc.

(7) Water conservancy project design, such as dams, canals, river and sea projects, etc.

(8) Other project design and management, such as real estate development and property management, project budget estimates, construction process control and management, tourist attraction design and layout, intelligent building design, etc.

3． Applications in electrical and electronic circuits

CAD technology was first used in the design of circuit schematics and wiring diagrams. At present, CAD technology has been extended to the design of printed circuit boards (wiring and component layout), and has shown its talents in the design and manufacturing of integrated circuits, large-scale integrated circuits and very large-scale integrated circuits, and has thus greatly promoted microelectronics technology and Developments in computing and technology.

4. Simulation and animation production

The application of CAD technology can truly simulate the processing of mechanical parts, aircraft takeoff and landing, ships entering and exiting ports, object force damage analysis, flight training environment, combat policy system, and accident scenes. Reappearance etc. In the cultural and entertainment industry, computer modeling has been widely used to simulate realistic primitive animals, aliens, and various scenes that are not found in the real world, and the animation, actual backgrounds, and actors' performances are seamlessly combined. In film production technology It shines brightly on the Internet and produces exciting blockbusters one after another.

5. Other applications

In addition to the applications in the above fields, CAD technology is also used in light industry, textiles, home appliances, clothing, shoemaking, medical treatment and medicine, and even sports

The CAD standardization system has been further improved; system intelligence has become another technical hotspot; integration has become a major trend in the development of CAD technology; scientific computing visualization, virtual design, and virtual manufacturing technology are new trends in the development of CAD technology in the 1990s.

After a stage of computer graphics study, I have a certain understanding of the basic graphics generation algorithm in graphics. In-depth study of graphics requires advanced mathematical knowledge, and each direction of refinement requires different knowledge. Graphics is an active frontier subject in computer science and technology, and is widely used in biology, physics, chemistry, astronomy, geophysics, materials science and other fields. I deeply feel that the breadth of the fields involved in this subject is astonishing, and it can be said to be profound and profound.