Last Lectures

• Viewing
  • View + Projection + Viewport
• Rasterizing triangles
  • Point-in-triangle test
  • Aliasing

Today

• Antialiasing
  • Sampling theory
  • Antialiasing in practice
• Visibility / occlusion
  • Z-buffering

Sampling is Ubiquitous in Computer Graphics

• Rasterization = Sample 2D Positions
• Photograph = Sample Image Sensor Plane
• Video = Sample Time

Sampling Artifacts in Computer Graphics

• Artifacts due to sampling - “Aliasing”
  • Jaggies (Staircase Pattern) - sampling in space
  • Moire 摩尔纹 - under-sampling images → Skip odd rows and columns
  • Wagon wheel effect - sampling in time
  • [Many more] ...
• Behind the Aliasing Artifacts
  • Signals are changing too fast (high frequency), but sampled too slowly

Frequency Domain

Frequencies $cos2\pi fx$

• $f = \frac{1}{T}$

Fourier Transform

• Represent a function as a weighted sum of sines and cosines → 任何一个周期函数，都能写成y一系列正弦和余弦函数的线性组合以及一个常数项
• 方波:
  • $f(x) = \frac{A}{2}+\frac{2Acos(t\omega)}{\pi}$
  • $f(x) = \frac{A}{2}+\frac{2Acos(t\omega)}{\pi}-\frac{2Acos(3t\omega)}{3\pi}$
  • $f(x) = \frac{A}{2}+\frac{2Acos(t\omega)}{\pi}-\frac{2Acos(3t\omega)}{3\pi}+\frac{2Acos(5t\omega)}{5\pi}$
  • $f(x) = \frac{A}{2}+\frac{2Acos(t\omega)}{\pi}-\frac{2Acos(3t\omega)}{3\pi}+\frac{2Acos(5t\omega)}{5\pi}-\frac{2Acos(7t\omega)}{7\pi}+...$
• Higher frequencies need faster sampling