Study On Line Scan-Conversion, Linear Transformations, And Perspective-Correct In Graph Rendering

With the development of computer and its accessories, the application areas of computer graphics become wider and wider. Meanwhile, the voice of requiring higher rendering speed and realism is becoming louder and louder. There are two methods to match these two requirements—the one is the modeling, the other is the rendering. Graph rendering is a process of producing realistic image efficiently from a given model. How to render fast and get more realistic image are two basic issues of computer graphics rendering. Based on the standard graphics rendering pipeline, studies have been made in the areas of graphics rendering efficiency and realism:1) A fast algorithm of line scan-conversion is presented.In this thesis, by analyzing the Bresenham's integer algorithm, slope 1 is used as a factor to calculate the accumulation errors, whose sign can be used to decide which next pixel can be selected. Compared with the classical method of line scan- conversion—Bresenham's middle algorithm, the proposed algorithm in this thesis also has the advantage of integer computing, at least saves 6 percent of scan-conversion time, and has the same visual effect as that of Bresenham's middle algorithm.2) The problem of unified deriving method of 3D linear transformation is solved by using quaternion algebra. A breakthrough is made on Goldman's statement—linear transformation can not be derived using unified deriving method of quaternion algebra.The matrix form of linear transformation has been popularly used in software and hardware in computer graphics. In 2003, based on vector algebra, Goldman presented a unified driving method. His algorithm can be used to derive linear transformations in 3D computer graphics. However, in the process of solving the unified deriving formula, vector triple multiplications and the perpendicular relation of two given vectors are required.In this thesis, by analyzing the characteristics of linear transformations, a quaternion method of deriving linear transformation is presented. Moreover, examples in 3D computer graphics are also presented. Compared with the vector method, vector triple multiplication and the perpendicular relation of two given vectors are not needed while solving the unified deriving formulas.3) A unified deriving method of linear transformations is presented by usinggeometric algebra.Geometric algebra is more abstract than vector algebra and quaternion algebra, and is not popularly used in computer graphics. By analyzing the deriving methods of matrix, vector algebra and quaternion algebra, a unified deriving formula of linear transformations is presented by using geometrical algebra. The mathematics expressions of using geometric algebra to solve the unified driving of linear transformations are equal to those of quaternion algebra. Similarly, the perpendicular relation of two given vectors is not needed in the process of solving the unified deriving by using geometric algebra. Moreover, the expressions of results in rotation and reflection are simpler than those using vector algebra and quaternion algebra.4) The perspective-correct formula used in Gouraud and Phong shading is proved to be used in a wider graph rendering pipeline, which is a rendering pipeline from the stage of perspective transformation, to the stage of perspective division, till arrive at viewport transformation stage.Low method solved the perspective distortion of Gouraud and Phong shading only in the perspective stage by using similar triangle theory. By analyzing the standard graphics rendering pipeline and existed method of perspective-correct formula, a formula of perspective-correct is obtained from the given pipeline. The proposed formula is the same as that of Low method. Results show that Low method can be used not only in the stage of perspective stage, but also in the whole graphics rendering pipeline from prospective transformation. By applying the proposed formula to Phong shading, perspective-correct curve and experiments are given.5) A perspective-correct method for equal angle interpolation is presented. Line segments length ratio is used in the perspective-correct of Gouraud andPhong shading. Angle ratio is used in the equal angle interpolation shading. A perspective-correct formula for equal angle interpolation shading is presented. The proposed formula also has the advantage of unit normal vector. Finally, a comparison experiment of equal angle interpolation shading is presented by using the proposed formula.