Sampling, Aliasing

Examples of Aliasing

What is a Pixel?• A pixel is not:

– a box– a disk– a tiny little light

• A pixel “looks different” ondifferent display devices

• A pixel is a sample– it has no dimension– it occupies no area– it cannot be seen– it has a coordinate– it has a value

Philosophical perspective

• The physical world is continuous, inside the computer things need to be discrete

• Lots of computer graphics is about translating continuous problems into discrete solutions – e.g. ODEs for physically-based animation, global

illumination, meshes to represent smooth surfaces, antialiasing

• Careful mathematical understanding helps do the right thing

More on Samples• The process of mapping a continuous function to a discrete one

is called sampling

• The process of mapping a continuous variable to a discrete one is called quantization

• To represent or render an image using a computer, we must both sample and quantize – Focus on the effects of sampling and how to overcome them

discrete position

discretevalue

Sampling & reconstructionThe visual array of light is a continuous function1/ we sample it

– with a digital camera, or with a ray tracer– This gives us a finite set of numbers,

not really something we can see– We are now inside the discrete computer world

2/ we need to get this back to the physical world: we reconstruct a continuous function– for example, the point spread of a pixel on a CRT or LCD

• Both steps can create problems– But we’ll focus on the first one (you are not display

manufacturers)

Examples of Aliasing

Examples of Aliasing

Examples of Aliasing

Texture Errors

point sampling

Sampling Density

• If we’re lucky, sampling density is enough

Input Reconstructed

Sampling Density

• If we insufficiently sample the signal, it may be mistaken for something simpler during reconstruction (that's aliasing!)

12

Aliasing insufficient sampling (sampling rate is too small)

• Aliasing: a high-frequency signal masquerading as a low frequency

SamplingInterval

Actual (high-frequency) signal

Sampled (aliased) signal

Discussion

• Two types of aliasing– Edges– Repetitive textures

• More tricky• Harder to solve

Solution?

• How do we avoid the high-frequency patterns distort our image?

• We blur!– For ray tracing: compute at higher resolution, blur,

resample at lower resolution– For textures, we can also blur the texture image

before doing the lookup

15

Aliasing and Line Drawing

• We draw lines by sampling at intervals of one pixel and drawing the closest pixels

SamplingInterval

SamplingInterval

• Results in stair-stepping

16

Anti-aliasing Lines

• Idea:– Make line thicker– Fade line out (removes high frequencies)– Now sample the line

17

Anti-aliasing Lines

Solution – Unweighted Area Sampling:– Treat line as a single-pixel wide rectangle

– Colour pixels according to the percentage of each pixel covered by the rectangle.

18

Solution : Unweighted Area Sampling

• Pixel area is unit square• Constant weighting function • Pixel colour is determined by computing the amount of

the pixel covered by the line, then shading accordingly• Easy to compute, gives reasonable results

Line

One Pixel

19

Solution : Weighted Area Sampling

Treat pixel area as a circle with a radius of one pixel Use a radially symmetric weighting function (e.g., cone):

– Areas closer to the pixel centre are weighted more heavily

Better results than unweighted, slightly higher cost

Line

One Pixel

Re-hashing

• Your intuitive solution is to compute multiple colour values per pixel and average them

• A better interpretation of the same idea is that– You first create a higher resolution image– You blur it (average)– You resample it at a lower resolution

Sampling Theorem

• When sampling a signal at discrete intervals, the sampling frequency must be greater than twice the highest frequency of the input signal in order to be able to reconstruct the original perfectly from the sampled version (Shannon, Nyquist, Whittaker, Kotelnikov)

Filters (convolution kernel)• Weighting function

• Area of influence often bigger than "pixel"

• Sum of weights = 1– Each sample contributes

the same total to image

– Constant brightness as object moves across the screen.

• No negative weights

Filters

• Filters are used to – reconstruct a continuous signal from a sampled signal

(reconstruction filters)– band-limit continuous signals to avoid aliasing during

sampling (low-pass filters)

• Desired frequency domain properties are the same for both types of filters

• Often, the same filters are used as reconstruction and low-pass filters

Pre-Filtering• Filter continuous primitives

• Treat a pixel as an area

• Compute weighted amount of object overlap

• What weighting function should we use?

Post-Filtering

• Filter samples• Compute the weighted average of many samples• Regular or jittered sampling (better)

The Ideal Filter• Unfortunately it has

infinite spatial extent– Every sample contributes

to every interpolated point

• Expensive/impossible to compute spatial

frequency

Problems with Practical Filters

• Many visible artifacts in re-sampled images are caused by poor reconstruction filters

• Excessive pass-band attenuation results in blurry images

• Excessive high-frequency leakage causes "ringing" and can accentuate the sampling grid (anisotropy)

frequency

Gaussian Filter

• This is what a CRTdoes for free!

spatial

frequency

Box Filter / Nearest Neighbour

• Pretending pixelsare little squares.

spatial

frequency

Tent Filter / Bi-Linear Interpolation

• Simple to implement• Reasonably smooth

spatial

frequency

Bi-Cubic Interpolation• Begins to approximate

the ideal spatial filter, the sinc function

spatial

frequency

Ideal sampling/reconstruction

• Pre-filter with a perfect low-pass filter– Box in frequency– Sinc in time

• Sample at Nyquist limit– Twice the frequency cutoff

• Reconstruct with perfect filter– Box in frequency, sinc in time

• And everything is great!

Difficulties with perfect sampling

• Hard to prefilter• Perfect filter has infinite support

– Fourier analysis assumes infinite signal and complete knowledge

– Not enough focus on local effects

• And negative lobes– Emphasizes the two problems above– Negative light is bad– Ringing artifacts if prefiltering or supports are not

perfect

Supersampling in graphics

• Pre-filtering is hard– Requires analytical visibility– Then difficult to integrate

analytically with filter

• Possible for lines, or if visibility is ignored

• usually, fall back to supersampling

35

Solution: Super-sampling

• Divide pixel up into “sub-pixels”: 22, 33, 44, etc.

• Sub-pixel is coloured if inside line• Pixel colour is the average of its sub-pixel colours• Easy to implement (in software and hardware)

No anti-aliasing Anti-aliasing (22 super-sampling)

36

Many Types of Supersampling

GridRandom

Poisson Disc Jittered

37

Polygon Anti-aliasing• To anti-alias a line, we treat it as a rectangle• Anti-aliasing a polygon is similar.• Some concerns:

– Micro-polygons: smaller than a

pixel

– Super-sampling: There may still be polygons that “slip between the cracks”

Uniform supersampling

• Compute image at resolution k*width, k*height• Downsample using low-pass filter

(e.g. Gaussian, sinc, bicubic)

Uniform supersampling• Advantage:

– The first (super)sampling captures more high frequencies that are not aliased

• Issues– Frequencies above the (super)sampling limit are still

aliased

• Works well for edges• Not as well for repetitive textures

Jittering

• Uniform sample + random perturbation• Sampling is now non-uniform• Signal processing gets more complex• In practice, adds noise to image• But noise is better than aliasing Moiré patterns

Jittered supersampling

• Regular, Jittered Supersampling

Jittering

• Displaced by a vector a fraction of the size of the subpixel distance

• Low-frequency Moire (aliasing) pattern replaced by noise

• Extremely effective• Patented by Pixar!

Reconsider Uniform supersampling

• Problem: supersampling only pushes the problem further: The signal is still not bandlimited

• Aliasing still happens

• We need a better solution for high frequency sections of the image

Adaptive sampling• Adjust your sampling rate and/or bucket size on-the-fly,

depending on results of previous samples

– attempt to concentrate more samples in high frequency areas (e.g., edges of objects) and fewer samples in low frequency areas (constant regions)

– get an approximate view of the function by first sampling with a low sampling rate and/or large bucket size over the entire domain

– then use that approximation to estimate where more samples should be taken by focusing on regions of high variance

– generally a recursive process

• Much better than naïve supersampling

Adaptive supersampling

• Use more sub-pixel samples around edges

Adaptive supersampling

• Use more sub-pixel samples around edgesCompute colour at small number of sample

If variance with neighbour pixels is high

Compute larger number of samples

Adaptive supersampling

• Use more sub-pixel samples around edgesCompute colour at small number of sample

If variance with neighbour pixels is high

Compute larger number of samples