Sampling theory

Sampling and Reconstruction

Digital Image Synthesisg g y Yung-Yu Chuang

10/22/2008 10/22/2008

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

sample values (grid of color pixels) from functions defined over continuous domains (incident radiance defined over the film plane) (incident radiance defined over the film plane) and then using those samples to reconstruct new functions that are similar to the original

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 the original function. Error is called aliasing.

l l

sampling reconstruction

sample value

l i i sample position

Sampling in computer graphics

• Artifacts due to sampling - Aliasing

J i – Jaggies – Moire

Fli k i ll bj t – Flickering small objects – Sparkling highlights

l b h h l ff

– Temporal strobing (such as Wagon-wheel effect)

• Preventing these artifacts - Antialiasing

Jaggies

Retort sequence by Don Mitchell

Staircase pattern or jaggies

Moire pattern

• Sampling the equation equation

) sin(x² + y²

Fourier analysis

• Can be used to evaluate the quality between the reconstruction and the original

the reconstruction and the original.

• The concept was introduced to Graphics by R b t C k i 1986 ( t d d b D Mit h ll) Robert Cook in 1986. (extended by Don Mitchell)

Rob Cook V.P. of Pixar 1981 M.S. Cornell 1981 M.S. Cornell

1987 SIGGRAPH Achievement award

1999 Fellow of ACM

2001 Academic Award with Ed Catmull and Loren Ed Catmull and Loren Carpenter (for Renderman)

Fourier transforms

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

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 Frequency

( ) ( ) ^{i x} Fω = ^∞

∫

Spatial Domain

Frequency Domain

( ) 1 ( )

2

f x F ω ei x^ω dω π

−∞

∞

=

∫

) (x

f F(ω)

2π _−∞

∫

Fourier analysis

spatial domain frequency domain

Fourier analysis

spatial domain frequency domain

Fourier analysis

spatial domain frequency domain

Convolution

• Definition

∫

( ) ( ) ( )

h x = ⊗ = f g ∫ f x g x ′ − x dx ′ ′

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

domain.

f ⊗ ↔ × g F G

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

f g × ↔ ⊗ F G

f g

1D convolution theorem example 2D convolution theorem example

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

∗ ⇒

∗ ⇒

× ⇒ ⇒

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

The delta function

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

and unit area

Sifting and shifting

Shah/impulse train function

frequency domain spatial domain

,

Sampling

band limited

Reconstruction

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

In math forms

) ( ) III( ) )

(

~ ( F

F (F(s) III(s)) Π )(s

F = ∗ ×Π

) ( sinc )

III ) (

~ (

x (x)

x f

f = × ∗

∑

=

i

i f i x x

f ( ) sinc( ) ( )

~

−∞

= i

Reconstruction filters

The sinc filter, while ideal, has two drawbacks:

• It has a large support (slow to compute)

I i d i i i

• It introduces ringing in practice

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

Aliasing

increase sample decrease sample

increase sample spacing in

spatial domain

p spacing in

frequency domain

Aliasing

high-frequency d t il l k i t details leak into lower-frequency regions

regions

Sampling theorem

Sampling theorem Aliasing due to under-sampling

Sampling theorem

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

increase the sampling rate

• However, few of interesting functions in t hi b d li it d i computer graphics are band limited, in particular, functions with discontinuities.

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

Aliasing

• Prealiasing: due to sampling under Nyquist rate

P li i d f i f

• 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 resamplep p g p 3. Nonuniform or stochastic sampling

Antialiasing (Prefiltering)

It is blurred, but better than aliasing

Uniform supersampling

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

of the spectra further apart, potentially reducing the overlap and thus aliasing

R lti l t b l d (filt d)

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

s s

s

Pixel=

∑

Samples Pixel

Point vs. Supersampled

Point 4x4 Supersampled

Checkerboard sequence by Tom Duff

Analytic vs. Supersampled

Exact Area 4x4 Supersampled

Non-uniform sampling

• Uniform sampling

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

– Multiplying the signal by the sampling pattern corresponds to

l i f h h ik (i f )

placing a copy of the spectrum at each spike (in freq. space) – Aliases are coherent (structured), and very noticeable

• Non uniform sampling

• 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 (structurelss) and much less objectionable – Noise is incoherent (structurelss), and much less objectionable

• Aliases can’t be removed, but an be made less noticeable

noticeable.

Antialiasing (nonuniform sampling)

• The impulse train is modified as

∑

⎠

⎜⎜ ⎞

⎝

⎛ ⎟

⎠

⎜ ⎞

⎝

⎛iT + −

x- ξ

δ 2

1

• It turns regular aliasing into noise But random

∑

⎜ ⎠

⎝ ⎝ ⎠

i 2

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

Jittered vs. Uniform Supersampling

4x4 Jittered Sampling 4x4 Uniform

Prefer noise over aliasing

reference aliasing noise

Jittered sampling

Add uniform random jitter to each sample Add uniform random jitter to each sample

Poisson disk noise (Yellott)

• Blue noise

S h ld b i d l k

• Spectrum should be noisy and lack any concentrated spikes of energy (to avoid

h t li i ) coherent aliasing)

• Spectrum should have deficiency of low- frequency energy (to hide aliasing in less noticeable high frequency)

Distribution of extrafoveal cones

Monkey eye cone distribution

Fourier transform cone distribution

Yellott theory

„Aliases replaced by noisep y

„Visual system less sensitive to high freq noise

Example

Aliasing

frequency domain

function (a) function (b) domain

alias=false frequency

Stochastic sampling

Stochastic sampling

function (a) function (b)

Replace structure alias by structureless alias by structureless (high-freq) noise

Antialiasing (adaptive sampling)

• Take more samples only when necessary.

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

be used.

• It only makes a less aliased image, but may not

b ffi i t th i l li

be more efficient than simple supersampling particular for complex scenes.

Application to ray tracing

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

objects, textures and materials

• Good news: we can do sampling easily

• Bad news: we can’t do prefiltering (because we do not have the whole function)

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

pbrt sampling interface

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

B l ti di i tl it t

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

• core/sampling.*, samplers/*

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

lowdiscrepancy.cpp, p y pp

An ineffective sampler A more effective sampler

Main rendering loop

void Scene::Render() {

Sample *sample = new Sample(surfaceIntegrator, volumeIntegrator volumeIntegrator, this);

...

while (sampler >GetNextSample(sample)) {

fill in eye ray info and other samples for integrator

while (sampler->GetNextSample(sample)) { RayDifferential ray;

float rW = camera->GenerateRay(*sample, &ray);

<Generate ray differentials for camera ray>

<Generate ray differentials for camera ray>

float alpha;

Spectrum Ls = 0.f;

if (rW > 0 f) if (rW > 0.f)

Ls = rW * Li(ray, sample, &alpha);

...

camera->film->AddSample(*sample ray Ls alpha);

camera >film >AddSample( sample,ray,Ls,alpha);

...

} ...

...

camera->film->WriteImage();

}

Sample

struct Sample {

Sample(SurfaceIntegrator *surf,

store required information for one eye ray sample

VolumeIntegrator *vol, const Scene *scene);

...

float imageX, imageY;

float lensU, lensV;

float time;^; Note that it stores all samples

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

float **oneD **twoD;

p required for one eye ray. That is, it may depend on depth.

float oneD, twoD;

...

}Sample is allocated once in Render(). Sampler is called to fill in the information for each eye ray The integrator 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. y , g p g g For example, WhittedIntegrator does not need samples.

DirectLighting needs samples proportional to #lights.

Data structure

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

samples, usually 2D samples.

•Sampling BRDF requires 2D samples.

•Selection of BRDF components requires 1D samples.

3 1 2

D t D

n1D n2D 2 2 1 1 2 2 sample

allocate together to avoid cache miss filled in by integrators

oneD twoD allocate together to avoid cache miss

mem

bsdfComponent lightSample bsdfSample

integrator

bsdfComponent lightSample bsdfSample

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

// g p p

int nPtrs = n1D.size() + n2D.size();

if (!nPtrs) {

oneD = twoD = NULL;

oneD = twoD = NULL;

return;

}

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

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 *( ) g ( p sizeof(float));

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

oneD[i] = mem;

mem += n1D[i];

}

for (u int i = 0; i < n2D size(); ++i) { for (u_int i = 0; i < n2D.size(); ++i) {

twoD[i] = mem;

mem += 2 * n2D[i];

} } }

DirectLighting::RequestSamples

void RequestSamples(Sample *sample, Scene *scene) { if (strategy == SAMPLE_ALL_UNIFORM) {

i t Li ht >li ht i () u_int nLights = scene->lights.size();

lightSampleOffset = new int[nLights];

bsdfSampleOffset = new int[nLights];

b dfC tOff t i t[ Li ht ]

bsdfComponentOffset = new int[nLights];

for (u_int i = 0; i < nLights; ++i) { const Light *light = scene->lights[i];

i li h S l int lightSamples =

scene->sampler->RoundSize(light->nSamples);

lightSampleOffset[i] = 2

sample->Add2D(lightSamples);

bsdfSampleOffset[i] =

sample->Add2D(lightSamples);

bsdfComponentOffset[i] = sample->Add1D(lightSamples);

}

lightNumOffset = -1;

}

DirectLighting::RequestSamples

else {

// Allocate and request samples for sampling one light

light

lightNumOffset = sample->Add1D(1);

lightSampleOffset = new int[1];

lightSampleOffset[0] = sample->Add2D(1);

bsdfComponentOffset = new int[1];

bsdfComponentOffset[0] = sample->Add1D(1);

bsdfSampleOffset = new int[1];

bsdfSampleOffset[0] = sample->Add2D(1);p [ ] p ( );

} }

PathIntegrator::RequestSamples

void PathIntegrator::RequestSamples(Sample *sample, const Scene *scene)

{

for (int i = 0; i < SAMPLE_DEPTH; ++i) { lightNumOffset[i] = sample->Add1D(1);

lightNumOffset[i] = sample->Add1D(1);

lightPositionOffset[i] = sample->Add2D(1);

bsdfComponentOffset[i] = sample->Add1D(1);

bsdfDirectionOffset[i] = sample->Add2D(1);

outgoingComponentOffset[i] = sample->Add1D(1);

outgoingDirectionOffset[i] = sample->Add2D(1);

} }

Sampler

Sampler(int xstart, int xend,

int ystart int yend int spp);

range of pixels int ystart, int yend, int spp);

bool GetNextSample(Sample *sample);

int TotalSamples() l i l

int TotalSamples()

samplesPerPixel *

(xPixelEnd xPixelStart) *

sample per pixel

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

Random sampler

RandomSampler::RandomSampler(…) { ...

//

Just for illustration; does not work well in practice // 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

// p p p p

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

// Shift image samples to pixel coordinates // Shift image samples to pixel coordinates

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

imageSamples[o+1] += yPos; }

private copy of the current pixel position imageSamples[o+1] += yPos; }

samplePos = 0;

}

current pixel position

#samples for current pixel

Random sampler

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

//

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

xPos = xPixelStart; number of generated samples in this pixel ++yPos; }

if (yPos == yPixelEnd) return false;

samples in this pixel

;

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

generate all samples for one pixel at once imageSamples[i] = RandomFloat();

// Shift image samples to pixel coordinates for (o=0; o<2*xPixelSamples*yPixelSamples; o+=2) for (o=0; o<2*xPixelSamples*yPixelSamples; o+=2) { imageSamples[o] += xPos;

imageSamples[o+1] += yPos; }

l 0

samplePos = 0;

}

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

// p g

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();

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();

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

++samplePos;

return true;

}

Random sampling

a pixel

completely random random

Stratified sampling

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

miss important features.

Stratified sampling

completely random

stratified uniform

stratified jittered

random uniform jittered

turns aliasing into noise into noise

Comparison of sampling methods

256 l i l f

256 samples per pixel as reference

1 sample per pixel (no jitter)

Comparison of sampling methods

1 l i l (ji d) 1 sample per pixel (jittered)

4 samples per pixel (jittered)

Stratified sampling

reference random stratified

ji d

jittered

High dimension

• D dimension means N^D cells.

S l i k l d i

• 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);p , y p , j p );

StratifiedSample1D(timeSamples,

xPixelSamples*yPixelSamples, jitterSamples);

// Shift stratified samples to pixel coordinates ...

// Decorrelate sample dimensions // Decorrelate sample dimensions

Shuffle(lensSamples,xPixelSamples*yPixelSamples,2);

Shuffle(timeSamples,xPixelSamples*yPixelSamples,1);

l 0

samplePos = 0;

}

Stratified sampling

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

/ n stratified samples within [0..1]

float invTot = 1.f / nSamples;

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

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

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

}

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

}

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

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

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

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 jx = jitter ? RandomFloat() : 0.5f;

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

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

* ( j ) * d

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

} }

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]);

} }

d-dimensional vector swap

}

Stratified sampler

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

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

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

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

sample->time = timeSamples[samplePos];

// h t if i t t k f 7 t tifi d 2D l // what if integrator asks for 7 stratified 2D samples

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

l P ++samplePos;

return true;

Latin hypercube sampling

• Integrators could request an arbitrary n samples.

nx1 or 1xn doesn’t give a good sampling pattern nx1 or 1xn doesn t give a good sampling pattern.

A worst case for stratified sampling 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;p [ j] ( ()) ; // Permute LHS samples in each dimension

for (int i = 0; i < nDim; ++i) { note the difference with shuffle

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

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

swap(samples[nDim * j + i]

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

} } } }

Stratified sampling

Stratified sampling

1 l d 16 h d l i l

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

1 camera sample and 16 shadow samples per pixel

16 camera samples and each with 1 shadow sample per pixel

Low discrepancy sampling

• A possible problem with stratified sampling

• Discrepancy can be used to evaluate the quality of patterns

of patterns

Low discrepancy sampling

set of N sample points a family of shapes

p p maximal difference

volume estimated by sample number

real volume

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

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 discrepancy patterns are less uniform and works reasonably well as sample patterns in practice.

Next, we introduce methods specifically designed for , p y g generating low-discrepancy sampling patterns.

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)g g g p [ , )

inline double RadicalInverse(int n int base) { inline double RadicalInverse(int n, int base) {

double val = 0;

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

while (n > 0) { while (n > 0) {

int d_i = (n % base);

val += d_i * invBi;

/

n /= base;

invBi *= invBase;

}

return val;

}

van der Corput sequence

• The simplest sequence

R i l li 1D li i h lf l

• Recursively split 1D line in half, sample centers

• Achieve minimal possible discrepancy

High-dimensional sequence

• Two well-known low-discrepancy sequences

H lt – Halton – Hammersley

Halton sequence

• Use relatively prime numbers as bases for each dimension recursively split the dimension

dimension recursively split the dimension into p_d parts, sample centers

• Achieve best possible discrepancy for N-Dp p y

• 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

Hammersley sequence

• Similar to Halton sequence.

Sli h l b di h H l

• Slightly better discrepancy than Halton.

• Needs to know N in advance.

Folded radical inverse

• Add the offset i to the ith digit d_i and take the modulus b

modulus b.

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

Radial inverse

Halton Hammersley

Better for that there Better for that there are fewer clumps.

Folded radial inverse

Halton Hammersley

The improvement is The improvement is more obvious

Low discrepancy sampling

t tifi d jitt d 1 l / i l stratified jittered, 1 sample/pixel

Hammersley sequence, 1 sample/pixel

Best candidate sampling

• Stratified sampling doesn’t guarantee good sampling across pixels

sampling across pixels.

• Poisson disk pattern addresses this issue. The

P i di k tt i f i t ith

Poisson disk pattern is a group of points with no two of them closer to each other than some

ifi d di t

specified distance.

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

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

Best candidate sampling

stratified jittered best candidate It avoids holes and clusters It avoids holes and clusters.

Best candidate sampling

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

rolled torus).

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

Best candidate sampling

t tifi d jitt d 1 l / i l stratified jittered, 1 sample/pixel

best candidate, 1 sample/pixel

Best candidate sampling

t tifi d jitt d 4 l / i l stratified jittered, 4 sample/pixel

best candidate, 4 sample/pixel

Comparisons

reference low-discrepancy best candidate

Reconstruction filters

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

following to compute pixel values.

1. reconstruct a continuous function L’ from samples

2 filt L’ t f hi h th

2. prefilter L’ to remove frequency higher than Nyquist limit

3 sample L’ at pixel locations 3. sample L’ at pixel locations

• Because we will only sample L’ at pixel l ti d t d t li itl locations, we do not need to explicitly

reconstruct L’s. Instead, we combine the first

t t

two steps.

Reconstruction filters

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

of discontinuity in rendering. We choose nonuniform sampling, trading off noise for aliasing There is no theory about ideal aliasing. There is no theory about ideal reconstruction for nonuniform sampling yet.

I t d id i t l ti bl

• Instead, we consider an interpolation problem

∑ ∑

= ⁱf x xⁱ y yⁱ L xⁱ yⁱ y

x

I ( , ) ( , )

) ,

( ( )

filter sampled radiance

∑

y x

I( , ) ( , ) (x,y)

) , (x_i y_i final value

Filter

• provides an interface to f(x,y)

il t i t t filt d it t

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

Filt Filt (fl t fl t )

width, half of support Filter::Filter(float xw, float yw) Float Evaluate(float x, float y);

x y is guaranteed to be within the range;

) (x y f

• filters/* (box, gaussian, mitchell, sinc, x, y is guaranteed to be within the range;

range checking is not necessary

) , (x y f

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

Box filter

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

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

no need to normalize since the weighted

{

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));( , y (y));

}

Gaussian filter

• Gives reasonably good results in practice

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

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

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.

this, the value at the end is calculated and subtracted.

Mitchell filter

• parametric filters, tradeoff between ringing and blurring

and blurring

• Negative lobes improve sharpness; ringing starts t t th i if th b l

to enter the image if they become large.

Mitchell filter

• Separable filter

• Two parameters, p , B and C, B+2C=1 suggestedgg

FFT of a cubic filter.

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

Windowed sinc filter

Lanczos

τ π

τ π

/ / ) sin

( x

x x

w =

sinc

Comparisons

box

Mitchell

Comparisons

windowed sinc

Mitchell

Comparisons

box Gaussian Mitchell