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Sampling

Area-Coverage Real-world sensors get a signal based on the area fraction of the sensor “covered” by objects Coverage:Signal:A ray tracer only gets a sample of the geometry (using a ray-geometry intersection point)A scanline renderer projects the entire triangle onto the image plane Testing pixel centers against triangles only uses sample information from the geometry Computing area overlap between triangles and (square) pixels would better mimic real-world sensors

Missing Information Eyes/cameras don't collect all of the information either The staggered spatial layout of real-world sensors means that large regions lack information for certain wavelengths (layered approaches can help to circumvent this) layered approaches can help to circumvent this:

Aliasing Testing only the pixel center (with ray-tracing or scanline rasterization) leads to jagged edges This causes aliasing artifacts (an alias/imposter takes the place of the correct feature)A jagged line appears instead of the correct straight line Anti-aliasing strategies aim to reduce aliasing artifacts (caused by sampling information)

Aliasing: Shaders & Textures Aliased normal vectors can cause erroneous sparkling highlights (top left)Aliasing can occur when texture mapping objects too (top right)

Temporal Aliasing A spinning wheel can appear to spin backwards, when the motion is insufficiently sampled in time (“wagon wheel” effect)

Sampling Rate Artifacts can be reduced by increasing the number of samples (per unit area)This can be accomplished by increasing the number of pixels in the image; but:It takes longer to render the scene (because there are more pixels colors to determine)Displaying higher-resolution images requires additional storage/computation Instead: Optimize the Sample Rate!Use the lowest possible sampling rate that does not result in “noticeable” artifacts What is the optimal sampling rate?

4 samples per period

samples

reconstruction

2 samples per period

samples

reconstruction

1 sample per period

samples

reconstruction Appears to be a different function

2/3 sample per period

samples

reconstruction Appears to be a different function

Aliasing These two cosine waves appear identical to the sample points

Sampling Rate Sampling at too low a rate results in aliasing, where two different signals become indistinguishable (or aliased)Nyquist-Shannon Sampling Theorem If 𝑓(𝑡) contains no frequencies higher than 𝑊 hertz, it can be completely determined by samples spaced 1/(2𝑊) seconds apart That is, a minimum of 2 samples per period are required to prevent aliasing

Anti-Aliasing The Nyquist frequency is defined as half the sampling frequency If the function being sampled has no frequencies above the Nyquist frequency, then no aliasing occurs Real world frequencies above the Nyquist frequency appear as aliases to the sampler Before sampling, remove frequencies higher than the Nyquist frequency

Fourier Transform Transform between the spatial domain 𝑓𝑥 and the frequency domain 𝐹𝑘Frequency Domain: 𝐹𝑘=∫!""𝑓𝑥𝑒!#$%&'𝑑𝑥Spatial Domain: 𝑓𝑥=∫!""𝐹𝑘𝑒#$%&'𝑑𝑘𝑒!"=cos𝜃+𝑖sin𝜃cos𝜃= $!"%$#!"& sin𝜃= $!"'$#!"&!

Constant Function

Low Frequency Cosine

High Frequency Cosine

Narrow Gaussian Narrow Wide

Wider Gaussian Wider Narrower

sum of two different cosine functions

samples

reconstruction Aliasing!

Fourier transform

identify Nyquist frequency bounds

remove the high frequencies

inverse Fourier transform

samples

reconstruction No Aliasing!

Anti-Aliasing Sampling causes higher frequencies to masquerade as lower frequencies After sampling, can no longer untangle the mixed high/low frequencies Remove the high frequencies before sampling (in order to avoid aliasing)Part of the signal is lost But, that part of the signal was not representable by the sampling rate anyways

Blurring vs. Anti-Aliasing blurring jaggies after samplingremoving high frequencies before sampling

Images Images have discrete values (and are not continuous functions)Use a discrete version of the Fourier transform The Fast Fourier Transform (FFT) computes the discrete Fourier transform (and its inverse) in 𝑂𝑛 log 𝑛 complexity (where 𝑛 is the number of samples)Images are 2D (not 1D)A 2D discrete Fourier transform can computed using 1D transforms along each dimension1. Fourier transform (into the frequency domain)Discrete image values are transformed into another array of discrete values2. Remove high frequencies3. Inverse Fourier transform (back out of the frequency domain)

Constant Function

sin2𝜋/32𝑥

sin2𝜋/16𝑥

sin2𝜋/16𝑦

sin2𝜋/32𝑥 ∗sin2𝜋/16𝑦

An obvious star!

lowest frequencies

intermediate frequencies

(larger) intermediate frequencies

highest frequencies (edges)

Convolution Let 𝑓 and 𝑔 be functions in the spatial domain (e. g. images), and 𝐹𝑓 and 𝐹𝑔 be transformations of 𝑓 and 𝑔 into the frequency domain In our prior examples: 𝑓 was the image (to the left), 𝐹𝑓 was the frequency domain version of the image (to the right)Removing higher frequencies of 𝐹𝑓 is equivalent to multiplying by a Heaviside function 𝐹𝑔 (=1 for smaller frequencies, =0 for larger frequencies)Then, the inverse transform 𝐹!((𝐹𝑓𝐹𝑔) gives the final result This entire process is called the convolution of 𝑓 and 𝑔:𝑓∗𝑔=𝐹!((𝐹𝑓𝐹𝑔)

Convolution Integral Convolution can be achieved without the Fourier Transform:𝑓∗𝑔𝑡=3!""𝑓𝜏𝑔𝑡-𝜏𝑑𝜏=3!""𝑓𝑡-𝜏𝑔𝜏𝑑𝜏A narrower 𝑔 makes the integral more efficient to compute A narrower 𝐹𝑔 better removes high frequencies But, they can't both be narrow Recall: the narrower Gaussian had wider frequencies, and the wider Gaussian had narrower frequencies

Box Filter Let 𝑔 have nonzero values in an Nx N block of pixels (surrounding the origin), and be zero elsewhere The discrete convolution (integral) is computed via:overlay the filter 𝑔 on the image, multiply the corresponding entries, and sum the results The final result is (typically) defined at the center of the filter

Filters Most (but not all) High Frequencies 𝐹𝑔𝑔

Wider Box Filter𝐹𝑔𝑔 more expensive convolution integralremoves more of the high frequencies

Super-Sampling Collect extra information/samples (in each pixel), and average the result (e. g. with a box filter)E. g. render a 100 by 100 image with 4 by 4 super-sampling (equivalent to rendering a 400 by 400 image)This properly represents (without aliasing) frequencies up to 4 times higher (than the original image could)Apply a 4 by 4 box filter aiming to remove as much of those extra frequencies as possible Converges to the area coverage integral, as the number samples per pixel increases Efficiency: only super-sample pixels that have high frequencies (e. g. edges)Better to use pseudo-random Monte-Carlo super-sampling strategies (instead of uniform super-sampling)

Super-Sampling Point Sampling4 by 4 Super-Sampling Exact Area Coverage

Super-Sampling Jaggies Anti-Aliased