What is a point sample (aka sample)? An evaluation

Aliasing

Jaggies

Demo

What is a point sample (aka sample)?

• An evaluation

• At an infinitesimal point (2-D)
• Or along a ray (3-D)

• What is evaluated

• Inclusion (2-D) or intersection (3-D)
• Attributes such as distance and color

Why point samples?

• Clear and unambiguous semantics

• Matches theory well (as we’ll see)

• Supports image assembly in the framebuffer

• will need to resolve visibility based on distance, and this works well with point samples

• Anything else just puts the problem off

• Exchange one large, complex scene for many small, complex scenes

Fourier theory

Reference sources

• Marc Levoy’s notes

• Ronald N. Bracewell, The Fourier Transform and its Applications, Second Edition, McGraw-Hill, Inc., 1978.

• Private conversations with Pat Hanrahan

• MATLAB

Ground rules

• You don’t have to be an engineer to get this

• We’re looking to develop instinct / understanding
• Not to be able to do the mathematics

• We’ll make minimal use of equations

• No integral equations
• No complex numbers

• Plots will be consistent

• Tick marks at unit distances
• Signal on left, Fourier transform on the right

Dimensions

• 1-D

• Audio signal (time)
• Generic examples (x)

• 2-D

• Image (x and y)

• 3-D

• Animation (x, y, and time)

Fourier series

Fourier series example: sawtooth wave

Sawtooth wave summation

Sawtooth wave summation (continued)

Fourier integral

Basic Fourier transform pairs

Reciprocal property

Scaling theorem

Band-limited transform pairs

Finite / infinite extent

• If one member of the transform pair is finite, the other is infinite

• Band-limited  infinite spatial extent
• Finite spatial extent  infinite spectral extent

Convolution

Convolution example

Convolution theorem

• Let f and g be the transforms of f and g. Then

Sampling theory

Reconstruction theory

Sampling at the Nyquist rate

Reconstruction at the Nyquist rate

Sampling below the Nyquist rate

Reconstruction below the Nyquist rate

Reconstruction error

Reconstruction with a triangle function

Reconstruction error

Reconstruction with a rectangle function

Reconstruction error

Sampling a rectangle

Reconstructing a rectangle (jaggies)

Sampling and reconstruction

• Aliasing is caused by

• Sampling below the Nyquist rate,
• Improper reconstruction, or
• Both

• We can distinguish between

• Aliasing of fundamentals (demo)
• Aliasing of harmonics (jaggies)

Summary

• Jaggies matter

• Create false cues
• Violate rule 1

• Sampling is done at points (2-D) or along rays (3-D)

• Sufficient for depth sorting
• Matches theory

• Fourier theory explains jaggies as aliasing. For correct reconstruction:

• Signal must be band-limited
• Sampling must be at or above Nyquist rate
• Reconstruction must be done with a sinc function

Before Thursday’s class, read

• Before Thursday’s class, read

• FvD 3.17 Antialiasing

• Project 1:

• Breakout: a simple interactive game
• Demos Wednesday 10 October