计算机视觉的书写


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=[xa,ya]b=[xb,yb]

  • 点积的代数表达为:
a·b = xaxb + ybyb

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 θ


θ = arccos ( 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:||ba||
      • ||ba|| = ||b||cosθ=a·b||a||
      • ba=||ba||a||a||(unit vector)=a·b||a||2a

叉积(Cross Product)

假设三维向量a=[xa,ya,za]b=[xb,yb,zb]

  • 叉积的代数表达为:
axb = | i j k xa ya za xb yb zb | = ( yazb - za yb ) i + ( zaxb - xazb ) j + ( xayb - yaxb ) 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*

axb = A*b = [ 0 -za ya za 0 -xa -ya xa 0 ] [ xb yb zb ]

假设向量a, b间的夹角为θ, 叉积的几何表达为:

axb = |a| |b| sin θ
  • 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
      • axb=-bxa

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=uxv
    • 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=a||a||
      • 找到一个和ab构成平面垂直的单位法向量u: u=bxw||bxw||
      • 根据右手规则找到单位向量v:v=wxu

Matrix

  • 矩阵乘法
    • 向量的点积可用矩阵乘法表示:
      • a·b=aTb
    • 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:
      [ -1 0 0 -1 ] [ x y ] = [ -x y ]
    • 转置矩阵
      • (AB)T=BTAT
    • 逆矩阵
      • (AB)-1=B-1A-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>