Sampling theory

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Sampling and Reconstruction

Digital Image Synthesis Yung-Yu Chuang

10/26/2005

with slides by Pat Hanrahan, Torsten Moller and Brian Curless

Sampling theory

• Sampling theory: the theory of taking discrete sample values (grid of color pixels) from

functions defined over continuous domains (incident radiance defined over the film plane) and then using those samples to reconstruct new functions that are similar to the original

(reconstruction).

Sampler: selects sample points on the image plane Filter: blends multiple samples together

Aliasing

• Reconstruction generates an approximation to the original function. Error is called aliasing.

sample position sample value

sampling reconstruction

Sampling in computer graphics

• Artifacts due to sampling - Aliasing

– Jaggies – Moire

– Flickering small objects – Sparkling highlights

– Temporal strobing (such as Wagon-wheel effect)

• Preventing these artifacts - Antialiasing

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Jaggies

Retort sequence by Don Mitchell

Staircase pattern or jaggies

Moire pattern

• Sampling the equation

) sin(x² + y²

Fourier transforms

• Most functions can be decomposed into a weighted sum of shifted sinusoids.

• Each function has two representations

– Spatial domain - normal representation – Frequency domain - spectral representation

• The Fourier transform converts between the spatial and frequency domain

Spatial Domain

Frequency Domain

( ) ( )

( ) 1 ( )

2

i x

F f x e dx

f x F e d

ω

ω ω

π

∞

−

−∞

∞

−∞

=

∫

) (x

f F(ω)

Fourier analysis

spatial domain frequency domain

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Fourier analysis

Convolution

• Definition

• Convolution Theorem: Multiplication in the frequency domain is equivalent to convolution in the space domain.

• Symmetric Theorem: Multiplication in the space domain is equivalent to convolution in the frequency domain.

f ⊗ ↔ × g F G

f g × ↔ ⊗ F G

( ) ( ) ( )

h x = ⊗ = f g ∫ f x g x ′ − x dx ′ ′

1D convolution theorem example

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2D convolution theorem example

∗

× ⇒

⇒

f(x,y) h(x,y) g(x,y)

F(s_x,s_y) H(s_x,s_y) G(s_x,s_y)

The delta function

• Dirac delta function, zero width, infinite height and unit area

Sifting and shifting Shah/impulse train function

frequency domain spatial domain

,

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Sampling

band limited

Reconstruction

The reconstructed function is obtained by interpolating among the samples in some manner

In math forms

) ( ) III(s) )

(

~ (

s s

F

F = ∗ ×Π

) ( sinc ) III ) (

~ (

x (x)

x f

f = × ∗

∑

^∞

−∞

=

−

=

i

i f i x x

f( ) sinc( ) ( )

~

Reconstruction filters

The sinc filter, while ideal, has two drawbacks:

• It has large support (slow to compute)

• It introduces ringing in practice

The box filter is bad because its Fourier transform is a sinc filter which includes high frequency contribution from the infinite series of other copies.

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Aliasing

increase sample spacing in

spatial domain

decrease sample spacing in

frequency domain

Aliasing

high-frequency details leak into lower-frequency regions

Sampling theorem Sampling theorem

• For band limited function, we can just increase the sampling rate

• However, few of interesting functions in computer graphics are band limited, in particular, functions with discontinuities.

• It is because the discontinuity always falls between two samples and the samples provides no information about this discontinuity.

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Aliasing

• Prealiasing: due to sampling under Nyquist rate

• Postaliasing: due to use of imperfect reconstruction filter

Antialiasing

• Antialiasing = Preventing aliasing

1. Analytically prefilter the signal

– Not solvable in general

2. Uniform supersampling and resample 3. Nonuniform or stochastic sampling

Antialiasing (Prefiltering)

It is blurred, but better than aliasing

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Uniform Supersampling

• Increasing the sampling rate moves each copy of the spectra further apart, potentially

reducing the overlap and thus aliasing

• Resulting samples must be resampled (filtered) to image sampling rate

Samples Pixel

s s

s

Pixel =

∑

w Sample⋅

Point vs. Supersampled

Point 4x4 Supersampled

Checkerboard sequence by Tom Duff

Analytic vs. Supersampled

Exact Area 4x4 Supersampled

Distribution of Extrafoveal Cones

Monkey eye cone distribution

Fourier transform

Yellot theory

Aliases replaced by noise

Visual system less sensitive to high freq noise

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Non-uniform Sampling

• Intuition

• Uniform sampling

– The spectrum of uniformly spaced samples is also a set of uniformly spaced spikes

– Multiplying the signal by the sampling pattern corresponds to placing a copy of the spectrum at each spike (in freq. space) – Aliases are coherent, and very noticable

• Non-uniform sampling

– Samples at non-uniform locations have a different spectrum; a single spike plus noise

– Sampling a signal in this way converts aliases into broadband noise

– Noise is incoherent, and much less objectionable

Antialiasing (nonuniform sampling)

• The impulse train is modified as

• It turns regular aliasing into noise. But random noise is less distracting than coherent aliasing.

∑

^∞

−∞

= ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ ⎟

⎠

⎜ ⎞

⎝

⎛ + −

i

iT

x- ξ

δ 2

1

Jittered Sampling

Add uniform random jitter to each sample

Jittered vs. Uniform Supersampling

4x4 Jittered Sampling 4x4 Uniform

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Antialiasing (adaptive sampling)

• Take more samples only when necessary.

However, in practice, it is hard to know where we need supersampling. Some heuristics could be used.

• It makes a less aliased image, but may not be more efficient than simple supersampling particular for complex scenes.

Application to ray tracing

• Sources of aliasing: object boundary, small objects, textures and materials

• Good news: we can do sampling easily

• Bad news: we can’t do prefiltering

• Key insight: we can never remove all aliasing, so we develop techniques to mitigate its impact on the quality of the final image.

Prefer noise over aliasing

reference aliasing noise

pbrt sampling interface

• Creating good sample patterns can substantially improve a ray tracer’s efficiency, allowing it to create a high-quality image with fewer rays.

• Because evaluating radiance is costly, it pays to spend time on generating better sampling.

• core/sampling.*, samplers/*

• random.cpp, stratified.cpp, bestcandidate.cpp,

lowdiscrepancy.cpp,

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An ineffective sampler A more effective sampler

Sampler

Sampler(int xstart, int xend,

int ystart, int yend, int spp);

bool GetNextSample(Sample *sample);

int TotalSamples()

samplesPerPixel *

(xPixelEnd - xPixelStart) * (yPixelEnd - yPixelStart);

Render() in core/scene.cpp,

while (sampler->GetNextSample(sample)) { ...

}

sample per pixel range of pixels

used for generating eye rays

Sample

Struct Sample {

Sample(SurfaceIntegrator *surf, VolumeIntegrator *vol, const Scene *scene);

...

float imageX, imageY;

float lensU, lensV;

float time;

// Integrator Sample Data vector<u_int> n1D, n2D;

float **oneD, **twoD;

...

}

store required information for one eye ray sample

Sample is allocated once in Render(). Sampler is called to fill in the information for each eye ray. The integrator can ask for multiple 1D and/or 2D samples, each with an arbitrary number of entries, e.g. depending on #lights.

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Date structure

3 1 2

mem

oneD twoD

n1D n2D

•Different types of lights require different number of samples, usually 2D samples.

•Sampling BRDF requires 2D samples.

•Selection of BRDF components requires 1D samples.

2 2 1 1 2 2

bsdfComponent lightSample bsdfSample

integrator sample

allocate together to reduce cache miss filled in by integrators

Sample

Sample::Sample(SurfaceIntegrator *surf,

VolumeIntegrator *vol, const Scene *scene) { // calculate required number of samples

// according to integration strategy surf->RequestSamples(this, scene);

vol->RequestSamples(this, scene);

// Allocate storage for sample pointers int nPtrs = n1D.size() + n2D.size();

if (!nPtrs) {

oneD = twoD = NULL;

return;

}

oneD=(float **)AllocAligned(nPtrs*sizeof(float *));

twoD = oneD + n1D.size();

Sample

// Compute total number of sample values needed int totSamples = 0;

for (u_int i = 0; i < n1D.size(); ++i) totSamples += n1D[i];

for (u_int i = 0; i < n2D.size(); ++i) totSamples += 2 * n2D[i];

// Allocate storage for sample values

float *mem = (float *)AllocAligned(totSamples * sizeof(float));

for (u_int i = 0; i < n1D.size(); ++i) { oneD[i] = mem;

mem += n1D[i];

}

for (u_int i = 0; i < n2D.size(); ++i) { twoD[i] = mem;

mem += 2 * n2D[i];

} }

Random sampler

RandomSampler::RandomSampler(…) { ...

// Get storage for a pixel's worth of stratified samples imageSamples = (float *)AllocAligned(5 *

xPixelSamples * yPixelSamples * sizeof(float));

lensSamples = imageSamples +

2 * xPixelSamples * yPixelSamples;

timeSamples = lensSamples +

2 * xPixelSamples * yPixelSamples;

// prepare samples for the first pixel

for (i=0; i<5*xPixelSamples*yPixelSamples; ++i) imageSamples[i] = RandomFloat();

// Shift image samples to pixel coordinates

for (o=0; o<2*xPixelSamples*yPixelSamples; o+=2) { imageSamples[o] += xPos;

imageSamples[o+1] += yPos; } samplePos = 0;

}

Just for illustration; does not work well in practice

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Random sampler

bool RandomSampler::GetNextSample(Sample *sample) { if (samplePos == xPixelSamples * yPixelSamples) {

// Advance to next pixel for sampling if (++xPos == xPixelEnd) {

xPos = xPixelStart;

++yPos; }

if (yPos == yPixelEnd) return false;

for (i=0; i < 5*xPixelSamples*yPixelSamples; ++i) imageSamples[i] = RandomFloat();

// Shift image samples to pixel coordinates

for (o=0; o<2*xPixelSamples*yPixelSamples; o+=2) { imageSamples[o] += xPos;

imageSamples[o+1] += yPos; } samplePos = 0;

}

number of generated samples in this pixel

generate all samples for one pixel at once

Random sampler

// Return next sample point according to samplePos sample->imageX = imageSamples[2*samplePos];

sample->imageY = imageSamples[2*samplePos+1];

sample->lensU = lensSamples[2*samplePos];

sample->lensV = lensSamples[2*samplePos+1];

sample->time = timeSamples[samplePos];

// Generate samples for integrators

for (u_int i = 0; i < sample->n1D.size(); ++i) for (u_int j = 0; j < sample->n1D[i]; ++j) sample->oneD[i][j] = RandomFloat();

for (u_int i = 0; i < sample->n2D.size(); ++i) for (u_int j = 0; j < 2*sample->n2D[i]; ++j)

sample->twoD[i][j] = RandomFloat();

++samplePos;

return true;

}

Random sampling

completely random a pixel

Stratified sampling

• Subdivide the sampling domain into non- overlapping regions (strata) and take a single sample from each one so that it is less likely to miss important features.

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Stratified sampling

completely random

stratified uniform

stratified jittered

turn aliasing into noise

Comparison of sampling methods

256 samples per pixel as reference

1 sample per pixel (no jitter)

Comparison of sampling methods

1 sample per pixel (jittered)

4 samples per pixel (jittered)

Stratified sampling

reference random stratified

jittered

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High dimension

• D dimension means N^D cells.

• Solution: make strata separately and associate them randomly, also ensuring good distributions.

Stratified sampler

if (samplePos == xPixelSamples * yPixelSamples) { // Advance to next pixel for stratified sampling ...

// Generate stratified samples for (xPos, yPos) StratifiedSample2D(imageSamples,

xPixelSamples, yPixelSamples, jitterSamples);

StratifiedSample2D(lensSamples,

xPixelSamples, yPixelSamples, jitterSamples);

StratifiedSample1D(timeSamples,

xPixelSamples*yPixelSamples, jitterSamples);

// Shift stratified samples to pixel coordinates ...

// Decorrelate sample dimensions

Shuffle(lensSamples,xPixelSamples*yPixelSamples,2);

Shuffle(timeSamples,xPixelSamples*yPixelSamples,1);

samplePos = 0;

}

Stratified sampling

void StratifiedSample1D(float *samp, int nSamples, bool jitter) {

float invTot = 1.f / nSamples;

for (int i = 0; i < nSamples; ++i) {

float delta = jitter ? RandomFloat() : 0.5f;

*samp++ = (i + delta) * invTot;

} }

void StratifiedSample2D(float *samp, int nx, int ny, bool jitter) {

float dx = 1.f / nx, dy = 1.f / ny;

for (int y = 0; y < ny; ++y)

for (int x = 0; x < nx; ++x) {

float jx = jitter ? RandomFloat() : 0.5f;

float jy = jitter ? RandomFloat() : 0.5f;

*samp++ = (x + jx) * dx;

*samp++ = (y + jy) * dy;

} }

n stratified samples within [0..1]

nx*ny stratified samples within [0..1]X[0..1]

Shuffle

void Shuffle(float *samp, int count, int dims) { for (int i = 0; i < count; ++i) {

u_int other = RandomUInt() % count;

for (int j = 0; j < dims; ++j)

swap(samp[dims*i + j], samp[dims*other + j]);

} }

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Stratified sampler

// Return next _StratifiedSampler_ sample point sample->imageX = imageSamples[2*samplePos];

sample->imageY = imageSamples[2*samplePos+1];

sample->lensU = lensSamples[2*samplePos];

sample->lensV = lensSamples[2*samplePos+1];

sample->time = timeSamples[samplePos];

// what if integrator asks for 7 stratified 2D samples

// Generate stratified samples for integrators for (u_int i = 0; i < sample->n1D.size(); ++i)

LatinHypercube(sample->oneD[i], sample->n1D[i], 1);

for (u_int i = 0; i < sample->n2D.size(); ++i)

LatinHypercube(sample->twoD[i], sample->n2D[i], 2);

++samplePos;

return true;

Latin hypercube sampling

• Integrators could request an arbitrary n samples. nx1 or 1xn doesn’t give a good sampling pattern.

A worst case for stratified sampling LHS can prevent this to happen

Latin Hypercube

void LatinHypercube(float *samples,

int nSamples, int nDim) {

// Generate LHS samples along diagonal float delta = 1.f / nSamples;

for (int i = 0; i < nSamples; ++i) for (int j = 0; j < nDim; ++j)

samples[nDim*i+j] = (i+RandomFloat())*delta;

// Permute LHS samples in each dimension for (int i = 0; i < nDim; ++i) {

for (int j = 0; j < nSamples; ++j) { u_int other = RandomUInt() % nSamples;

swap(samples[nDim * j + i], samples[nDim * other + i]);

} } }

note the difference with shuffle

Stratified sampling

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Stratified sampling

1 camera sample and 16 shadow samples per pixel

16 camera samples and each with 1 shadow sample per pixel

This is better because StratifiedSampler could generate a good LHS pattern for this case

Low discrepancy sampling

When B is the set of AABBs with a corner at the origin, this is called star discrepancy

set of N sample points a family of shapes

volume estimated by sample number

real volume

1D discrepancy

Uniform is optimal! However, we have learnt that Irregular patterns are perceptually superior to uniform samples. Fortunately, for higher dimension, the low- discrepancy patterns are less uniform and works reasonably well as sample patterns in practice.

Radical inverse

• A positive number n can be expressed in a base b as

• A radical inverse function in base b converts a

nonnegative integer n to a floating-point number in [0,1)

inline double RadicalInverse(int n, int base) { double val = 0;

double invBase = 1. / base, invBi = invBase;

while (n > 0) {

int d_i = (n % base);

val += d_i * invBi;

n /= base;

invBi *= invBase;

}

return val;

}

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van der Corput sequence

• The simplest sequence

• Recursively split 1D line in half, sample centers

• Achieve minimal possible discrepancy

• Use relatively prime numbers as bases for each dimension

• Achieve best possible discrepancy for N-D

• Can be used if N is not known in advance

• All prefixes of a sequence are well distributed so as additional samples are added to the sequence, low discrepancy will be maintained

Halton sequence

recursively split the dimension into p_d parts, sample centers

Hammersley sequence

• Similar to Halton sequence.

• Slightly better discrepancy than Halton.

• Needs to know N in advance.

Folded radical inverse

• It can be used to improve Hammersley and Halton, called Hammersley-Zaremba and Halton-Zaremba.

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Radial inverse

Halton Hammersley

Better for that there are fewer clumps.

Folded radial inverse

Halton Hammersley

The improvement is more obvious

Low discrepancy sampling

stratified jittered, 1 sample/pixel

Hammersley sequence, 1 sample/pixel

(0,2)-sequences

• A useful low-discrepancy sequence in 2D is to use the van der Corput sequence in one

dimension and a Sobol sequence in the other.

• It is stratified in a very general way.

• To generate different sequences for different pixels, pbrt scrambles the (0,2)-sequence by permuting the original sequence.

• Divide the square into half, swap two halves with 50% probability. Repeat until below numerical precision.

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(0,2)-sequences Implementation of (0,2)-sequences

• We use binary base; scramble equals XOR

• Assume the same scramble decision for the same level

(0,2)-sequences

void Sample02(u_int n, u_int scramble[2], float sample[2]) { sample[0] = VanDerCorput(n, scramble[0]);

sample[1] = Sobol2(n, scramble[1]);

}

float VanDerCorput(u_int n, u_int scramble) { n = (n << 16) | (n >> 16);

n = ((n&0x00ff00ff) << 8) | ((n&0xff00ff00) >> 8);

n = ((n&0x0f0f0f0f) << 4) | ((n&0xf0f0f0f0) >> 4);

n = ((n&0x33333333) << 2) | ((n&0xcccccccc) >> 2);

n = ((n&0x55555555) << 1) | ((n&0xaaaaaaaa) >> 1);

n ^= scramble;

return (float)n / (float)0x100000000LL;

}

float Sobol2(u_int n, u_int scramble) {

for (u_int v = 1<<31; n != 0; n >>= 1, v ^= v >> 1) if (n & 0x1) scramble ^= v;

return (float)scramble / (float)0x100000000LL;

}

LDSampler

• pbrt uses (0,2)-sequence instead of

Hammersley because it is prone to aliasing.

• LDSampler uses (0,2)-sequences for position and lens, van der Corput with scramble for time.

// Generate low-discrepancy samples for pixel

LDShuffleScrambled2D(1, pixelSamples, imageSamples);

LDShuffleScrambled2D(1, pixelSamples, lensSamples);

LDShuffleScrambled1D(1, pixelSamples, timeSamples);

LDShuffleScrambled1D(sample->n1D[i], pixelSamples, oneDSamples[i]);

LDShuffleScrambled2D(sample->n2D[i], pixelSamples, twoDSamples[i]);copy to oneD and

twoD of Sample

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LDSampler

void LDShuffleScrambled1D(int nSamples, int nPixel, float *samples) {

u_int scramble = RandomUInt();

for (int i = 0; i < nSamples * nPixel; ++i) samples[i] = VanDerCorput(i, scramble);

for (int i = 0; i < nPixel; ++i)

Shuffle(samples + i * nSamples, nSamples, 1);

Shuffle(samples, nPixel, nSamples);

}

void LDShuffleScrambled2D(int nSamples, int nPixel, float *samples) {

u_int scramble[2] = { RandomUInt(), RandomUInt() };

for (int i = 0; i < nSamples * nPixel; ++i) Sample02(i, scramble, &samples[2*i]);

for (int i = 0; i < nPixel; ++i)

Shuffle(samples + 2 * i * nSamples, nSamples, 2);

Shuffle(samples, nPixel, 2 * nSamples);

}

Best candidate sampling

• Stratified sampling doesn’t guarantee good sampling across pixels.

• Poisson disk pattern addresses this issue. The Poisson disk pattern is a group of points with no two of them closer to each other than some specified distance.

• It can be generated by dart throwing. It is time-consuming.

• Best-candidate algorithm by Dan Mitchell. It randomly generates many candidates but only inserts the one farthest to all previous samples.

Best candidate sampling

stratified jittered best candidate It avoids holes and clusters.

Best candidate sampling

• Because of it is costly to generate best candidate pattern, pbrt computes a “tilable pattern” offline (by treating the square as a rolled torus).

• tools/samplepat.cpp→sampler/sampledata.cpp

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Best candidate sampling

best candidate, 1 sample/pixel

Best candidate sampling

best candidate, 4 sample/pixel

Comparisons

reference low-discrepancy best candidate

Some recent progresses

• Fast Poisson Disk Sampling

• Recursive Wang Tiles for Real-Time Blue Noise

• Good topic for your final project

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Fast Poisson-Disk Sampling Fast Poisson-Disk Sampling

Recursive Wang Tiles for Blue Noise Reconstruction filters

• Given image samples, we can do the following to compute pixel values.

1. reconstruct a continuous function L’ from samples 2. prefilter L’ to remove frequency higher than

Nyquist limit

3. sample L’ at pixel locations

• Because we will only sample L’ at pixel locations, we do not need to explicitly

reconstruct L’s. Instead, we combine the first two steps.

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Reconstruction filters

• Ideal reconstruction filters do not exist because of discontinuity in rendering. We choose

nonuniform sampling, trading off noise for aliasing. There is no theory about ideal reconstruction for nonuniform sampling yet.

• Instead, we consider an interpolation problem

∑ ∑

−

= −

i i i

i i i i i

y y x x f

y x L y y x x y f

x

I ( , )

) , ( ) , ) (

,

( (x,y)

) , (x_i y_i final value

filter sampled radiance

Filter

• provides an interface to f(x,y)

• Film stores a pointer to a filter and use it to filter the output before writing it to disk.

Filter::Filter(float xw, float yw) Float Evaluate(float x, float y);

• filters/* (box, gaussian, mitchell, sinc, triangle)

width, half of support

x, y is guaranteed to be within the range;

range checking is not necessary

Box filter

• Most commonly used in graphics. It’s just about the worst filter possible, incurring postaliasing by high-frequency leakage.

Float BoxFilter::Evaluate(float x, float y) {

return 1.;

}

no need to normalize since the weighted sum is divided by the total weight later.

Triangle filter

Float TriangleFilter::Evaluate(float x, float y) {

return max(0.f, xWidth-fabsf(x)) * max(0.f, yWidth-fabsf(y));

}

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Gaussian filter

• Gives reasonably good results in practice

Float GaussianFilter::Evaluate(float x, float y) {

return Gaussian(x, expX)*Gaussian(y, expY);

} Gaussian essentially has a infinite support; to compensate this, the value at the end is calculated and subtracted.

Mitchell filter

• parametric filters, tradeoff between ringing and blurring

• Negative lobes improve sharpness; ringing starts to enter the image if they become large.

Mitchell filter

• Separable filter

• Two parameters, B and C, B+2C=1 suggested

FFT of a cubic filter.

Mitchell filter is a combination of cubic filters with C⁰and C¹ Continuity.

Windowed sinc filter

sinc Lanczos

τ π

/ / ) sin

( x

x x

w =

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Comparisons

box

Mitchell

Comparisons

windowed sinc

Mitchell

Comparisons

box Gaussian Mitchell