How do we represent pictures?
As a set of measurements in the plane
Signal processing: 1D version (easier)
Sampling
Discrete set of samples – continuous underlying phenomena
Try to understand what the problems are
Sample at discrete points (point sample)
Note that we can miss a small object
Note that we can’t localize an edge
Is the little square better?
Don’t know where small object is (or how small)
If it’s on an edge, gets way to blown up
Average over region? – where this is going
Basic idea:
If something is too small – or changes too fast, can’t capture it
How small an object? Depends on how well we sample
If we allow objects that are too small – many inputs to same representation
Aliasing – many inputs map to same output
What to do about it?
Limit small objects
Blur (Filter) – to get rid of sharp edges (quick changes, small things)
Why Care?
Miss small details
Downsample Checkerboard
All black/all white
Get weird patterns
Jaggies
Where line crosses through box
Where line is close enough to sample point
One pixel per column (or row – depends on quadrant)
Darkness issues
Crawlies
Make most effective use of what samples you have
Light Path Sampling
Sampling Basics
Three basic problems:

Sampling

Reconstruction

ReSampling (reconstruct + sampling)
Sampling: how many samples do you need?
Depends on how small an object (fast a change to capture)
Need to formalize this notion of small / fast
Reconstruction
What signal could have created these points
Many possibilities
Need to agree – need to know what to reconstruct
Smoothest signal: no extra wiggles
Can’t have a wiggle between samples
Limits the smallest wiggle allowed
Reconstruct: what signal interpolates, but has no wiggles smaller
Sample: if no wiggles, then samples are enough
What if signal has too many wiggles?
Need to get rid of extra wiggles BEFORE sampling
Example 1: Shader
http://research.cs.wisc.edu/graphics/Courses/559f2008/Main/ShaderAntiAliasing

Hard cutoff = jaggies

Smooth step (get rid of wiggles) – too smooth = blurry

Blur the right amount (about 1 pixel) = crisp, not jaggy
Example 2: PreFiltering
Checkerboard – get white/black
if you sample before filtering, you’re hosed
Once you’ve aliased – you’re stuck
Need to do averaging first
Could “blur” (average) beforehand
Could put averaging as part of the probe
Example 3: Reconstruction Filter
Start with a set of sample values (at discrete locations)
Want to know what signal was at other points
Various forms of interpolation:
Nearest neighbor (get piecewise constant)
Linear interpolation
Smoother
Kernel – multiply and sum
Box filter (too big, too small, just right) – averaging
Tent filter (too big, too small, just right) – lerp via weighted average
0 0 1 0 0 – could be “sharper” (not ideal)
Example 4: Downsample / Upsample
Take image
Divide in half (be sure to prefilter!)
Double in size – will be blurrier
Some Theory
How do we talk about “wiggles”, “too fast”, “remove wiggles”
Fourier Transform:
Any signal can be made up of a series of sin waves of various frequencies
Modulo a few catches (might need phase / …)
Even aperiodic signals (but those get messy)
Idea:
Faster change, needs a faster sin wave (higher frequency)
Signals with fast changes: have high frequencies
BandLimited Signal
Maximum frequency
Has a “fastest change” or smallest feature
Must sample fast enough to capture smallest feature
Nyquist frequency
Need > 2x samples
Filter –
Remove high frequencies
Multiply in frequency domain = convolution in time domain
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