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计算机图形学

Posted on | In Computer Graphics , OpenGL

Course Overview

  • Systems: Write complex 3D graphics programs
    • Real-time scene viewer in OpenGL,GL Shading Language, offline raytracer

  • Theory: Mathematical aspects and algotrithms underlying modern 3D graphics systems

  • Homework
    • HW1: Transformations. Place objects in world, view them simple viewer for a teapot
    • HW2: Scene Viewer. View scene, Lighting and Shading ( with GLSL programming shaders )
    • HW3: Ray Tracer. Realistic images with ray tracing. (two basic approaches: rasterize and raytrace image)
  • Workload
    • 3 programming projects, time consuming
    • Course involve understanding of mathematical, geometrical concepts tought
    • Prerequisites: Solid C/C++/Java programming
    • Linear algebra and basic math skills
  • GPU programming
    • Modern 3D Graphics Programming with GPUs
    • GLSL + Programmable Shaders in HW0,1,2
    • Should be very portable, but need to setup your envrionment, compliation framework

Overview and History of Computer Graphics

  • Computer Graphics
    • Coined by William Fetter of Boeing in 1960
    • First graphic system in mid 1950s USAF SAGE radar data (developed MIT)
  • Text
    • Text itself is major development in computer graphics
    • In Alan Turing’s biography, text is missed out.Manchester Mark I uses LED to display what happened,not text.
  • GUI
    • Xerox Star
      • Invented at Palo Alto Research Center in 1970’s , around 1975
      • Used in Apple Machintosh
    • Windows
  • Drawing
    • Sketchpad (suthrland , MIT 1963)
  • Paint Systems
    • SuperPaint System: Richard Shoup, Alvy Ray Smith (PARC, 1973 - 79)
    • Precursor to Photoshop: general image processing
  • Image PRocessing
    • Filter, Crop, Scale, Composite
    • Add or remove objects
  • Modeling
    • Spline curves, surfaces: 70s - 80s
    • Utah teapot: Famous 3D model
    • More recetly: Triangle meshes often acquired from real objects
  • Rendering
    • 1960s (visibiliy)
      • Hidden Line Algorithms: Roberts(63),Appel(67)
      • Hidden Surface Algorithms: Roberts(63),Appel(67)
      • Visibllity = Sorting: Sutherland(74)
    • 1970s(lighting)
      • Diffuse Lighting: (Gouraud 1971)
      • Specular Lighting: (Phong 1974)
      • Curved Surfaces,Texture: (Blinn 1974)
      • Z-Buffer Hidden Surface (Catmull 1974)
    • 1980s,90s (Global illumination)
      • Ray Tracing - Whitted(1980)
      • Radiosity - Goral, Torrance et al(1984)
      • The Rendering equation - Kajiya(1986)

Basic Math

  • Vectors
  • Matrix

Vectors

点积（Dot Product）

假设二维向量$a=[x_{a},y_{a}]$ 和 $b=[x_{b},y_{b}]$

  • 点积的代数表达为：
\[a·b=x_{a} x_{b}+y_{b} y_{b}\]

由上式可知，点积的结果是标量（Scalar），无方向

  • 假设向量$a$, $b$间的夹角为θ, 点积的几何表达为：
\[a·b=|a||b|cos \theta\]


\[\theta=\text{arccos}(\frac{a·b}{|a||b|})\]
  • 点积的几何意义：
    • 计算向量$a$, $b$间的夹角，判断是否是同一方向以及是否正交
      • $a·b>0$，同向，夹角在0-90之间
      • $a·b=0$，正交，互相垂直
      • $a·b<0$，反向，夹角在90-180之间
    • 向量 $b$在向量$a$上的投影长度 再乘以向量$a$的长度。
      • 向量 $b$在向量$a$上的投影长度表示为:$|b→a |$
      • $|b→a |=|b |cos \theta=\frac{a·b}{|a |}$
      • $b→a=|b→a |\frac{a}{|a |}(\text{unit vector})=\frac{a·b}{|a |^{2}} a$

叉积（Cross Product）

假设三维向量$a=[x_{a},y_{a},z_{a}]$ 和 $b=[x_{b},y_{b},z_{b}]$

  • 叉积的代数表达为：
\[a x b=|\begin{bmatrix}i & j & k \\ x_{a} & y_{a} & z_{a} \\ x_{b} & y_{b} & z_{b}\end{bmatrix}|=(y_{a} z_{b}-z_{a} y_{b}) i+(z_{a} x_{b}-x_{a} z_{b}) j+(x_{a} y_{b}-y_{a} x_{b}) k\]

上式可知，叉积的结果是矩阵的行列式的值，是向量，另一种表达方式是使用向量 $a$的对偶矩阵（dual matrix）$A^{\cdot}$

\[a x b=A^{\cdot} b=[\begin{bmatrix}0 & -z_{a} & y_{a} \\ z_{a} & 0 & -x_{a} \\ -y_{a} & x_{a} & 0\end{bmatrix}][\begin{bmatrix}x_{b} \\ y_{b} \\ z_{b}\end{bmatrix}]\]

假设向量$a$, $b$间的夹角为θ, 叉积的几何表达为：

\[a x b=|a||b|sin \theta\]
  • 几何意义
    • 向量 $a$,$b$叉乘的结果向量为为向量 $a$,$b$所构成的平行四边形平面的法向量，法向量方向遵守“右手”定律
    • 向量 $a$,$b$叉乘的模为向量 $a$,$b$所构成的平行四边形面积
      • $a x b=-b x a$

Orthonormal Basic Frames

如何使用向量的点积和叉积创建直角坐标系。

  • 坐标系种类
    • Global， Local， World， Model， Parts of model
  • 关键问题
    • 物体在不同坐标系中的位置和相互关系
  • 坐标系
    • 3D 坐标系
      • 单位向量：$|u |=|v |=|w |=1$
      • 相互正交：$u·v=v·w=u·w=0$
      • 满足叉乘：$w=u x v$
    • p向量在三个方向上的投影
      • $p=(p·u) u+(p·v) v+(p·w) w$
    • 给定向量$a$, $b$（不正交），如何构建一组三维直角坐标系的基底
      • 在a方向上构建单位向量w：$w=\frac{a}{|a |}$
      • 找到一个和ab构成平面垂直的单位法向量u： $u=\frac{b x w}{|b x w |}$
      • 根据右手规则找到单位向量v：$v=w x u$

Matrix

  • 矩阵乘法
    • 向量的点积可用矩阵乘法表示：
      • $a·b=a^{T} b$
    • 向量的叉积可以用向量a的对偶矩阵乘以向量b得到，如上文所示
  • 矩阵变换
    • 矩阵transform时图形学重点,通常是使用一个转换矩阵乘以一条向量（或一个点），得到转换后的结果
      • 例如对某坐标点进行y轴镜像：
\[[\begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix}][\begin{bmatrix}x \\ y\end{bmatrix}]=[\begin{bmatrix}-x \\ y\end{bmatrix}]\] 
- 转置矩阵
	- $(A B)^{T}=B^{T} A^{T}$

- 逆矩阵
	-  $(A B)^{-1}=B^{-1} A^{-1}$
  • < Real-Time Rendering>
  • < Computer Graphics: Principles and Practice>, 2nd Edition (3rd would be released around mid 2013) - < Computer Graphics, C Version>, 2nd Edition (not 3rd or 4th)
  • < Fundamentals of Computer Graphics>, 3rd Edition
  • < Computer Graphics using OpenGL>, 2nd or 3rd Edition*
  • < Interactive Computer Graphics: A Top-Down Approach with Shader-Based OpenGL>, 6th Edition
  • < 3D Computer Graphics: A Mathematical Introduction with OpenGL>