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In order to render images with as low noise and as fast convergence as p=
ossible, it is important to use samples that do not clump and do not leave =
large parts of the sampling domain empty. We call such samples <=
em>well stratified.
Left: 256 random 2D samples. Right: 256 well-stratified 2D samples=
.
There are several different ways of generating well-stratified samples; =
RenderMan uses lookups in pre-generated tables of progressive multi-jittere=
d (PMJ) sample sequences which are suitable for progressive rendering and a=
re stratified in 1D, 2D, and 3D. The 1D sample values are between 0 and 1, =
the 2D samples are on the unit square, and the 3D samples are in the unit c=
ube. (Practical tips on how to map to other domains are mentioned bel=
ow.)
In order to look up in one of the pre-generated sample tables one needs =
to specify which table to use and which sample number is requested. R=
enderMan provides a convenient abstraction class and API for this, called <=
code>RixRNG. The API is=
defined in the include/RixRNG.h
header file (and the implemen=
tation of PMJ table lookups are in the include/RixRNGProgressive.h header file). The purpose of the abstraction is that authors of =
Bxdfs and Integrators do not have to worry a=
bout explicit indexing, sample counting, table lookups, etc.
Th=
e SampleCtx struct
A sample context (SampleCtx
) consists of two unsig=
ned integers: patternid
and sampleid
. Patternid
is =
a 32-bit pattern that gets mapped to one of the built-in PMJ sample tables,=
and sampleid
determines which of the samples in the tabl=
e to use. Camera rays are initialized such that samples in a given pi=
xel will have the same patternid
; typically, the first sample =
in each pixel will have sampleid
0, the next sample in th=
e pixel will have sampleid
1, etc.
The RixRNG class
The "RNG" part of RixRNG stands for "Random Number Generator" even though the samples are =
not random at all. The RixRNG
class is basically ju=
st a wrapper around an array of per-shading-point sample contexts (Sa=
mpleCtxArray
) and an integer (numPts
) specifying how ma=
ny sample contexts there are in the array. There is typically one SampleCtx
for each point in a ray shading batch. The <=
code>RixRNG wrapper class makes it convenient to generate sampl=
e points (or new sample domains) for an entire ray shading batch with just =
a single function call.
Gen=
erating samples
There are six different functions to generate samples:
GenerateSample1D()
GenerateSample2D()
GenerateSample3D()
DrawSample1D()
DrawSample2D()
DrawSample3D()
The GenerateSample functions increment the sample context sampleid
; Dra=
wSample
functions do not.
There are also six multipoint functions that generate samples for al=
l points in a RixRNG
:
GenerateSamples1D()
GenerateSamples2D()
GenerateSamples3D()
DrawSamples1D()
DrawSamples2D()
DrawSamples3D()
Example: Bxdf GenerateSample()
functions need 2D=
samples to generate sample directions. The GenerateSample() functions have a RixRNG
input parameter (often =
called "rng") for this purpose. The RixRNG's
<=
code>sampleCtxArray
contains numPts
sample co=
ntexts, where numPts
is the number of shading points to g=
enerate sample directions for. Each sample context keeps track of the=
patternid
and sampleid
for that shadin=
g point. In order to get a 2D sample for each shading point, a single=
call to RixRNG::DrawSamples2D()
is sufficient:
RtFlo=
at2 *xi =3D (RtFloat2 *) RixAlloca(sizeof(RtFloat2) * numPts);
rng->DrawSamples2D(xi); // fill in xi array with numPts 2D samples=
pre>
By convention, Bxdfs call the DrawSamples1D(), DrawSamples2D(), and/or D=
rawSamples3D() functions. These functions do not increment the <=
code>sampleids. This means that an Integrator calling a Bxdf h=
as to do this incrementing after the Bxdf call =E2=80=93 otherwise mul=
tiple bxdf samples will be the same sample value, ie. the same direction. &=
nbsp;Similar for light sampling and indirect illumination sampling: the Int=
egrator has to increment the sampleids
after the samples =
have been used. This is easily done by looping over all the sample co=
ntexts in a RixRNG
like this:
for (in=
t i =3D 0; i < numPts; i++)
sampleCtxArray[i].sampleid++;
Generating new sample domains (new patternids)<=
/h4>
We need different sample sequences for each combination of pixel, ray de=
pth, and domain (bxdf lobes, area lights, indirect illumination, etc.). &nb=
sp;(If the same sample sequence was used for everything, there would be cor=
relation between samples and the image would not converge.) RenderMan=
internally selects sample sequences for pixel position (for anti-aliasing)=
, lens position (for depth-of-field), and time (for motion blur), and sets =
up an initial patternid
(bit pattern) and sampleid<=
/code> for camera ray hit points. All sample sequences used by B=
xdfs and light source sampling need to be set up in the Integrators calling=
them: separate domains for bxdf, light sources, and indirect illumina=
tion derived from the parent domains =E2=80=93 and then new domains again a=
t the next bounce (derived from those at the first bounce), etc. This=
is accomplished by calling one of the NewDomain*()
=
functions. The NewDomain*()
functions create a new =
patternid
based on a hash of the parent RixRNG's pa=
tternid
and a scramble bit-pattern.
Here "domain" actually just means a different patternid
&nbs=
p;bit-pattern. The name "domain" was chosen because typically a diffe=
rent patternid
is used for bxdf sampling, light source sa=
mpling, etc., with the bxdf and light sources being different "sample domai=
ns".
First it should be mentioned that there is a special C++ type for scramb=
le patterns. The scramble patterns are (as mentioned above) just 32-b=
it patterns, but these are easy to mistake for other integers. To ena=
ble the compiler to catch incorrect parameter order in various function cal=
ls, the scramble pattern has its own type =E2=80=93 an enum class call=
ed 'Scramble' that is just an alias for unsigned int. Create a Scramb=
le bit-pattern like this example:
RixRNG::Scramble scram =3D static_cast<RixRNG::Scramble>(0xfa1af=
e1)
There are three SampleCtx functions to generate a new SampleCtx based on=
an existing one, but with a different patternid: NewDomain()
,=
NewDomainDistrib()
, and NewDomainSplit()
.
SampleCtx NewDomain(Scramble scramble);
The simplest function is NewDomain()
. Given a (32-bit=
unsigned int) "scramble" bit-pattern and an existing "parent" sample=
Ctx
, it returns a new sampleCtx
with a patte=
rnid
that is different from the existing one (and a sampl=
eid
that is the same as the existing one). Pass a differe=
nt scramble bit-pattern for different sample domains: bxdfs, light source s=
ampling, etc. As mentioned above, the Scramble type is just an unsign=
ed int, but made into a distinct type for type safety.
SampleCtx NewDomainDistrib(Scramble scramble, unsigned newsampleid);=
pre>
The NewDomainDistrib()
function is similar, but <=
span style=3D"letter-spacing: 0.0px;">should be used where the new domain's=
expected number of samp=
les differs from that of the parent and repeated visits may nor may not have the same sample count =
or may consume differing=
numbers of samples: distribution sampling. This function generates a=
n independent sample domain where the samples in that domain are stratified=
with respect to each other, but not with respect to previous or future sam=
ples in the same pixel. The new patternid
depends o=
n the scramble bits, the parent's patternid
, and the current <=
code>sampleid =E2=80=93 including the sampleid
&nbs=
p;ensures that there is a new, independent sample distribution for every it=
eration. The sampleid
of the new SampleCtx is set to the value of the 'newsampleid'
parame=
ter. In normal use, the new domain should be created with newsa=
mpleid =3D 0
, and then the sampleid
should be incr=
emented every time a new sample from that domain has been used.
SampleCtx NewDomainSplit(Scramble scramble, unsigned newnumsamples);=
pre>
The NewDomainSplit()
function is also similar to NewDomain()
, but should be used where every visit will consume=
the same number of samples, and it is expected that all sibling visit=
s will also always result in the drawing of a new domain =E2=80=93&nbs=
p;thus exploring the full space. A fancy term for thi=
s is "trajectory splitting". (If newnumsamples
is 1 this call is the same as NewDomain(scramble).)
This ensures that you will get consecutive samples from a single sample =
sequence, with 'newnumsamples'
samples for each iteration=
. I.e. you will get the exact same samples values as if you had shot =
'newnumsamples'
as many camera rays and only 1 sample at =
each camera hit. In the NewDomainSplit()
function, =
there is a line with the following: newd.sampleid *=3D newnumsamples<=
/code>. That means skipping ahead in the sequence by the splitting br=
anching factor 'newnumsamples'
, thereby ensuring that the comb=
ined samples are consecutive and non-overlapping.
=
Camera rays through a pixel, hitting a surface and being reflected. &nbs=
p;Left: no trajectory splitting (8 camera rays with 1 reflection ray each).=
Right: trajectory splitting (2 camera rays with 4 reflection rays ea=
ch). Note that the 8 reflection directions are the same in both cases=
.
There are also similar functions in the RixRNG
class t=
hat can fill in new domains (patternids
) for an entire array o=
f sample contexts (RixRNG sampleCtxArray
): NewDomains()<=
/code>, NewDomainsDistrib()
, and NewDomainsSplit()
.
The scramble bit patterns can be any 32-bit pattern, but it is important=
that the scramble bit pattern to generate different new domains are differ=
ent. For example, if an Integrator is generating new domains for bxdf=
and light source sampling, those should use different scramble bit pattern=
s. Otherwise there will be correlation between bxdf and light source =
sampling, leading to visible artifacts and poor convergence =E2=80=93 =
or even no convergence at all! Examples of bit patterns used in the <=
code>PxrPathTracer Integrator are 0x2d96c92b, 0x3917fe2e, and 0=
xdeb189cf; there isn't anything particular about these bit patterns, the ma=
in point is that they are "random" and different. The same scramble b=
it patterns can be used at different ray depths because the parent ray's patternids will differ, so when we generate a new pat=
ternid
based on a different parent patternid
&=
nbsp;and the same scramble, the new patternid
will be dif=
ferent.
Integrators call RixRNG
constructors to set up sample =
domains for bxdfs lobe selection and sampling, light selection and sampling=
, stochastic transmission, volume scattering, and several other things. &nb=
sp;
There are several different RixRNG
constructors.  =
;The simplest ones are simply passed an already allocated sample context ar=
ray and its size, and assigns these to the RixRNG sampleCtxArray and numPts
member variables. The data in th=
e sampleCtxArray
(patternids
and sampl=
eids
) can be filled in before or after the RixRNG
=
constructor call. The more fancy RixRNG
constructor=
s construct a new RixRNG based on an existing one and fills in all the sampleCtxArray
values =E2=80=93 optionally with splittin=
g or distribution. These fancy constructors are convenient when the n=
umber (and order) of shading points in the new RixRNG
&nbs=
p;matches the shading points in the parent RNG. If the number (or ord=
er) is different, then the sample contexts in the new RixRNG
&n=
bsp;has to be initialized individually =E2=80=93 as in the following e=
xample.
Example: Here is an example=
where a sampleCtxArray
is allocated, a RixRNG is constructed (containing that sampleCtxArray
), and t=
he RixRNG's
sampleCtxArray
is initializ=
ed by
by looping over sh=
ading points:
Sampl=
eCtx* samplectxarray =3D new SampleCtx[numPts];
RixRNG rng(parentRng, samplectxarray, numPts);
RixRNG::Scramble scramble =3D static_cast<RixRNG::Scramble>(0x8732f=
9a1)
for (int pt =3D 0; pt < numPts; pt++)
{
int index =3D shadingContext->integratorCtxIndex[pt];
samplectxarray[pt] =3D parentRng=E2=86=92NewDomainSplit(index, scramble=
, 4);
}
Here the scramble bit-pattern is 0x8732f9a1 and the splitting factor is =
4.
Tips for using samples in practice
This section contains a few practical tips for how to use the samples re=
turned by the RixRNG DrawSample*
and GenerateSample=
*
API functions.
Mappin=
g samples
The samples returned by the DrawSample2D()
and G=
enerateSamples2D()
functions are in the unit square. It i=
s often necessary to map from the unit square to other domains. Example: uniform sampling a disk with a g=
iven radius can be done by picking a radius proportional to the square root=
of a sample between 0 and 1, and an angle between 0 and 2pi, and then comp=
uting the xy position corresponding to that radius and angle:
RtFlo=
at2 sample =3D rng->DrawSample2D(i);
float r =3D radius * sqrt(sample.x);
float angle =3D 2 * M_PI * sample.y;
point.x =3D r * cos(angle);
point.y =3D r * sin(angle);
An arguably better mapping from the unit square to a disk is Shirley and=
Chu's concentric mapping. (See Pete Shirley's blog for the =
most efficient implementation.)
Other common mappings map samples to the surface of a sphere, hemisphere=
, or cylinder, map samples to directions proportional to a glossy bxdf lobe=
, and so on.
Shir=
ley remapping
We often need to combine a selection between sub-domains and generation =
of positions (or directions) on one of those sub-domains. For example=
: select a light source and generate a sample position on that light source=
, or select a bxdf lobe and generate a sample direction proportional to tha=
t lobe.
One way to do that is to get a 3D sample and use the z coordinate to cho=
ose the sub-domain, and the xy coordinates to select the position (or direc=
tion) in that sub-domain. This can be a fine way to do this when the =
samples are well stratified in 3D and their xy coordinates are well stratif=
ied in 2D.
A common =E2=80=93 equally good =E2=80=93 alternative is called "Sh=
irley remapping". It does not rely on 3D stratification, but only on =
2D stratification. Get a 2D sample and first use one of the coordinat=
es of the sample to select the sub-domain, and then map the samples within =
that selected domain (based on the selection probability) back to the unit =
square domain. A figure provides a more intuitive explanation =
=E2=80=93 in this example we choose between two light area sources, with 80=
% chance of selecting the first light and 20% chance of selecting the secon=
d light:
=
p>
To the left are the original samples on the unit square. The middl=
e square contains the samples from the left 80% of the original samples, on=
ly stretched a bit horizontally. Likewise, the right square contains =
the samples from the right 20% of the original samples, but stretched 5x ho=
rizontally. The stretching preserves the stratification properties of=
the original sample points.
This method of using 2D samples to both choose between sample sub-domain=
s and at the same time stretching the samples from the chosen sub-domain to=
provide stratified 2D sample points is implemented in the RixChooseAndRema=
p() function in the RixShadingUtils.h include file.
Path splitting vs distribution ray tracing
Path splitting and distribution ray tracing were mentioned above (in the=
description of the various NewDomain*() functions). But when is it a=
ppropriate to use which? A couple examples should provide some guidan=
ce:
Use path splitting if there is a known, fixed branching factor. Fo=
r example, if all camera ray hits spawn four direct illumination rays then =
NewDomainSplit() can correctly offset into the sample sequence so that all =
samples in the sequence are used exactly once. This way, the direct i=
llumination ray directions from all camera ray hit points for a given pixel=
combine into one well stratified sample pattern.
Resort to distribution ray tracing if there is no fixed branching factor=
. For example, if some specular ray hit points spawn 16 new sample di=
rections but other specular ray hit points (perhaps with a lower throughput=
) only spawn 4 or even 1 new sample directions. In this case there is=
no simple way to compute the offset into a combined sample sequence.  =
;Instead the 16 (or 4) sample directions are stratified with respect to eac=
h other, but not with respect to any other samples for that same pixel. &nb=
sp;(If only 1 new sample direction is spawned then there is no stratificati=
on at all.)
Non-stratified samples similar to e.g. drand48()
can b=
e obtained by calling the HashToRandom()
function. =
It computes a repeatable random float between 0 and 1 given two unsigned in=
t inputs, for example patternid
and sampleid
=
. Using the same patternid
and continuously increme=
nting sampleid
gives a sequence of samples with good stat=
istical variation =E2=80=93 similar to drand48()
. T=
he HashToRandom()
function is repeatable (i.e. the same t=
wo inputs always give the same output), and has no multi-threaded contentio=
n (whereas drand48()
has notoriously bad cache line conte=
ntion, hampering multi-threaded performance). HashToRandom() is located in the RixRNGInline.h
include file.<=
/p>
Advanced topic: Details of PMJ table lookup =
implementation
In the introduction above we wrote that patternid
is u=
sed to choose the PMJ table, and sampleid
is used to choo=
se the sample in that table. If we had 2^32 pre-generated PMJ tables =
it really would be as simple as that. However, in practice we have on=
ly 384 different PMJ tables, so we have to be a bit inventive to make it lo=
ok like we have many, many different tables even though we don't. The=
trick =E2=80=93 implemented in include/RixRNGProgressive.h =E2=80=93 is to further "randomize" the samples in the tables. &nbs=
p;This randomization is done with scrambling of the sample va=
lues and shuffling of the sample order. First the patt=
ernid
is mapped to one of the 384 PMJ tables with a repeatable =
hash function. Then the sampleid
is shuffled a bit =
by carefully swapping neighbor samples and groups of four samples. (S=
uch local shuffling does not affect the convergence properties of each samp=
le sequence, but decorrelates sample sequences such that they can be safely=
combined with each other.) Finally the mantissa bits of the floating=
-point sample values are scrambled using the patternid
=
=E2=80=93 this scrambling changes the sample values while preserving their =
stratification.
Another practical detail is that even though PMJ sequences theoretically=
have infinitely many samples, in practice we limit each PMJ table to 4096 =
samples to keep memory use reasonable. When more than 4096 samples pe=
r pixel are used =E2=80=93 i.e. sampleid
values high=
er than 4096 =E2=80=93 we look up from the beginning of another PMJ ta=
ble (also with 4096 samples). So the samples beyond 4096 are still st=
ratified, just not quite as well stratified with respect to the first 4096 =
samples as if we had had larger tables. For more than 8188 samples pe=
r pixel we move to yet another table, and so on. In the unlikely case=
that more than 196608 samples per pixel are used, we run out of tables&nbs=
p;=E2=80=93 in this case the samples revert to unstratified, uniform (=
but deterministic) pseudorandom samples generated with the HashToRandom() f=
unction described above.
Advanced topic: Overriding the RixRNG implementa=
tion
It is possible to override the default PMJ samples if other sample seque=
nces are desired. This is scary stuff, but can be useful for experime=
nting with e.g. primary-space Metropolis rendering algorithms or adaptive p=
rogressive photon mapping. The Generator class provides a way to inte=
rcept sample generation: when a custom Generator has been specified, it aut=
omatically gets called when e.g. a Bxdf or light sampling calls one of the =
GenerateSample*()
or DrawSample*()
func=
tions. Some of the RixRNG
constructors can be passe=
d an explicit pointer to a Generator.
For example, for debugging purposes it can be useful to have all samples=
generated by drand48()
. (Note that drand48() is very bad for regular use: convergence is slow since its values a=
re not stratified, and it has cache line contention for access to its inter=
nal state, which slows down multi-threaded executions.) Here is a sni=
ppet of code from a Generator that calls drand48()
:
class R=
andomSampler : public RixRNG::Generator
{
...
// Generate a uniformly distributed pseudorandom sample in [0,1)
virtual float Sample1D(const RixRNG::SampleCtx &sc, unsigned i) con=
st // both params are ignored
{
float x =3D drand48();
if (x >=3D 1.0f) x -=3D 1.0f; // this can happen due to roundi=
ng double to float
return x;
}
... similar Sample2D() and Sample3D() functions ...
=20
// Fill in float array 'xis' with pseudorandom samples in [0,1)
virtual void MultiSample1D(unsigned n, const RixRNG::SampleCtx &sc,=
float *xis) const
{
for (int i =3D 0; i < n; i++)
{
float x =3D drand48();
if (x >=3D 1.0f) x -=3D 1.0f; // this can happen due to ro=
unding double to float
xis[i] =3D x;
}
}
... similar MultiSample2D() and MultiSample3D() functions ...
=20
}
For more (very nerdy!) details about the properties of PMJ samples pleas=
e refer to the following paper:
"Progressive multi-jittered sample sequences",
Per Christensen, Andrew Kensler, and Charlie Kilpatrick,
Computer Graphics Forum (Proceedings of the Eurographics Symposiu=
m on Rendering), 37(4):21-33, 2018.
Paper and presentation slides are available here: graphics.pixar.com/library/Progress=
iveMultijitteredSampling
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