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A Pixel is not a Little Square!
A Pixel is not a Little Square!
A Pixel is not a Little Square!
(And a Voxel is not a Little Cube!)
Paper by: Alvy Ray Smith
Link to the paper
http://alvyray.com/Memos/CG/Microsoft/6_pixel.pdf
The Little Square Model
- A pixel (picture element) is a geometric square
- Pixel (i, j) = {(x, y) | i-.5 ≤ x ≤i+.5, j-.5 ≤ y ≤j+.5}
- Or "Half Integers": {(x, y) | i≤x≤i+1., j≤y≤j+1.}
What is a pixel?
- A point sample
- Color picture may contain 3 samples per point
- not a square!
What is an image?
- Rectilinear grid of point samples (pixels)
- Viewing such a grid requires us to reconstruct it
- Need an appropriate Reconstruction Filter
- Nyquist-Shannon Sampling theorum tells us how that works
But first...
A (brief) diversion into Digital Image Processing
Nyquist-Shannon Sampling Theorum
- Says that a finite bandwidth continuous function can be perfectly represented by discrete samples
- The sampling rate needs to be twice the maximum frequency
- if all frequencies x(t) are below B, then points only need to be spaced out at intervals of 1/ (2B)
- Most easily seen in CD audio quality
- Humans can only hear up to ~22khz, CD's sample at 44khz
Reconstruction Filter
- Describes how to construct a smooth analog signal from digital input
- A.K.A what a Digital Analog Converter (DAC) uses
- Many different kinds of filters
- Nearest neighbor aka box filter with size of 1
- Bilinear interpolation - basic texture sampling
- Gaussian
Box Filter
- Not to be confused with an aquarium filter
- Weights the neighbors contributions equally
- Usually paired with nearest neighbor interpolation
- Can be used to average out an image
- Example: Mipmaps
Gaussian Filter
- Weight samples based on Gaussian Distribution
- Give most weight to center samples
- Commonly used for blurring
- More 'true to life', eg found in physical processes
Filter comparisons
This is for a single dimension and doesn't define the range of underlying samples
Interpolation comparison
- Colored dots are the digital 'samples'
- Black dot is the output value
Back to the Paper
Figure 1
Figure 1 cont.
Figure 2
Notes
- Footprint is not rectangular
- Filters overlap quite a bit
- Box has 'jaggies' - sharp corners
WhyIs the Little square Model So Persistant?
Two principle reasons
- Geometry based computer graphics
- Render abstract geometry into viewable pixels
- Fastest to assume rectangular viewport with square pixels
- Video magnification of computer displays appear to show it
- Zooming in shows 'bigger' pixels
- It does the cheap box filter to quickly render
- More accurate zooming is expensive
Scanners scan, not pixelate
- The output image may be a rectangular grid of samples
- Scanner head isn't a 'square', its an analog source turned into digital signals
Camera Sensors aren't perfect squares
- Multiple individual color sensors arrayed in a grid
- Means the samples aren't in the same point
CRT's don't display square pixels
- Electron beam's intensity varies over time, is constantly drawing a cross a phosphor layer
- Colors comes from shadow mask
- Several different masks
LCD's kinda do have square pixels
- Pixel colors are addressed as a rectilinear grid
- No electron gun zooming around
- OLED's don't on the other hand
- Use a pentile display
Proposed Discrete to Continuous Mapping
- Put the samples on the integers
- Make the boundaries be (filterwidth/2) away from the outermost integer column boundary
- Assume symmetric filters
- Y increasing downwards is the preferred coordinate frame
Downscaling an image
- Imagine a 640x480 image
- Create the "minimal enclosing rectangle" using the symmetric bicubic filter (ala figure 2)
- This would have boundaries (-2, 641) x (-2, 481)
- Now we can downsample the sample points however we please
- I'd like to imagine how each new sample would absorb the surrounding image
- Contrast this to simply averaging the 2x2 blocks into single pixels, a basic box downscale
Thanks!
Questions?
BONUS SLIDE/LINKS
https://twitter.com/nbonneel/status/1376886994031288323?s=20
Great point about this means