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Eurographics Symposium on Rendering 2016
E. Eisemann and E. Fiume
(Guest Editors)
Volume 35 (2016), Number 4
Solid Angle Sampling of Disk and Cylinder Lights
Manuel N. Gamito
Framestore, London, UK
Figure 1: The illumination from a cylinder light, sampled uniformly within its solid angle (above left) and within its area (below left), with 9
samples per pixel. Only the cylinder sectors visible from each illumination point have been sampled by area. The insets show the increase in
sampling quality to be progressively greater, the closer the illuminated surface is to the light.
Abstract
A new unbiased sampling approach is presented, which allows the direct illumination from disk and cylinder light sources to be
sampled with a uniform probability distribution within their solid angles, as seen from each illuminated point. This approach
applies to any form of global illumination path tracing algorithm (forward or bidirectional), where the direct illumination
integral from light sources needs to be estimated. We show that taking samples based on the solid angle of these two light
sources leads to improved estimates and reduced variance of the Monte Carlo integral for direct illumination. This work follows
from previously known unbiased methods for the solid angle sampling of triangular and rectangular light sources and extends
the class of lights that can be rendered with these improved sampling algorithms.
Categories and Subject Descriptors (according to ACM CCS): I.3.7 [Computer Graphics]: Three-Dimensional Graphics and
Realism—Raytracing
1. Introduction
The computation of the direct illumination from a light source on
the reflecting surface of objects is a fundamental part of the solu-
tion of the rendering equation [Kaj86]. All modern image render-
ers, commonly used in the film production industry, use the path
integral formulation of this rendering equation to compute the light
transport in complex scenes [Vea97]. It is therefore important to
solve the direct illumination integral efficiently and with good qual-
ity for any surface point.
For lights that have a Dirac delta distribution (such as point and
spot lights), the illumination integral has a simple analytical so-
lution. For area lights, however, the computation of the direct il-
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2016 The Author(s)
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2016 The Eurographics Association and John
Wiley & Sons Ltd. Published by John Wiley & Sons Ltd.
M.N. Gamito / Solid Angle Sampling of Disk and Cylinder Lights
lumination requires a numerical integration over some appropriate
space. Traditionally, this has been done by distributing point sam-
ples uniformly over the area of the light and using them as part of
a Monte Carlo integration [PH10]. Additional work has shown that
distributing directional samples over the solid angle of an area light,
relative to each illumination point, leads to better Monte Carlo es-
timates, while keeping the algorithm unbiased (see example in Fig-
ure 1). This new solid angle sampling approach has, so far, been
restricted to area lights that are either spherical, triangular or rect-
angular [Wan92,Arv95,UFK13].
In this paper, we extend solid angle sampling to two more
area lights that find use in film production: disk lights and cylin-
der lights. Although rectangular lights (commonly known as quad
lights) are arguably the most ubiquitous area lights in production
scenes, together with HDRI environments, there are still lighting
situations for which disks or cylinders are best suited. It is de-
sirable, therefore, to extend the solid angle formulation, with its
proven convergence qualities, to as many light types as possible.
Central to the solid angle sampling of any finite area light is the
ability to express its solid angle in analytic form. We present the
solid angle formulae for disks and cylinders, as taken from the sci-
entific literature. The common element between these two geome-
tries is that their solid angle requires the evaluation of elliptic inte-
gral functions [AS70]. The next step would be the construction of
a mapping from the space [0,1)2of two-dimensional random num-
bers to the solid angle space of the object, such that equal areas in
[0,1)2map to subsets of the solid angle that have equal measure.
Unfortunately, the integral function formulation for disks and cylin-
ders makes the derivation of such a mapping problematic. We rely
instead on a rejection sampling approach, where the geometries of
interest are bound by simpler objects for which a parameterisation
of their solid angle is available. This rejection sampling technique
remains unbiased and we show that it is efficient, with only a small
number of trial samples being rejected, due to the tightness with
which the intended solid angles can be bound.
2. Background and Previous Work
The direct illumination integral from an area light source can be
expressed in the following way, relative to a point x, with normal
n, on the surface of an object (Figure 2):
L(x,ω
ω
ωo) = L0ZΩ
f(x,ω
ω
ωo,ω
ω
ωi)G(x,ω
ω
ωi)|n·ω
ω
ωi|dω
ω
ωi.(1)
In the above, L0is the constant emission of the light source, Ω
is the solid angle of the light, as seen from x,fis the surface bidi-
rectional reflectance distribution function (BRDF) and G∈ {0,1}
is the binary geometric visibility function on x, along the incoming
direction ω
ω
ωi. The illumination depends on the outgoing direction
ω
ω
ωoonly through the surface’s BRDF.
Since sampling the solid angle Ωis generally considered diffi-
cult, a change of variables leads to an alternative formulation of the
illumination integral, where the integration takes place instead over
the area of the source:
L(x,ω
ω
ωo) = L0ZA
f(x,ω
ω
ωo,ω
ω
ωi)G(x,ω
ω
ωi)|n·ω
ω
ωi||m·ω
ω
ωi|
ky−xk2dy.(2)
Ω
A
x
y
ω
ω
ωi
ω
ω
ωon
m
Figure 2: The direct illumination from an area light.
After this change, Ais the area of the light source that is visible
from xand the integration is performed over points y, with normal
m, on the light. The incoming direction at xis now the direction
towards y:ω
ω
ωi= (y−x)/ky−xk.
The two integrals return the illumination from a single light
source but it is possible to work in a multi-light environment by
randomly selecting a light for integration, out of the full light set
[SWZ96]. Both integrals can be solved with Monte Carlo integra-
tion in the usual way. In the case of (2), area samples yare taken
according to a uniform probability density pA(y) = 1/|A|. In the
case of (1), directional samples ω
ω
ωiare taken according to the prob-
ability density:
pΩ(ω
ω
ωi) = (1/|Ω|if ω
ω
ωi∈Ω,
0 otherwise.(3)
The use of stratified samples is desirable, in order to reduce the
variance of the Monte Carlo integrals. Stratification is achieved if
the mapping from the stratified random numbers in [0,1)2to the
domain being sampled is a bijection. Algorithms for the stratified
sampling of areas are available for many objects such as spheres,
cylinders, disks and triangles [Tur90,SWZ96,SC97]. The sampling
of rectangular polygons is trivial and an algorithm for sampling
general convex quadrilaterals is also known [AN07].
An algorithm for sampling cylinder lights, that is intermediate
between area and solid angle sampling, is possible [Zim95]. The
cylindrical surface is parameterised by (θ,h), where θis the angle
around the cylinder axis and his the axial distance, and each of
these is sampled independently. Contrary to area sampling (where
sampling along θand hwould follow uniform probability distribu-
tions), importance sampling is performed for each of the two pa-
rameters, using probability distributions that depend on the config-
uration of the cylinder, relative to the illumination point.
Stratified sampling in the solid angle domain is easily done
for spheres [Wan92]. Triangles, whose solid angle projection to
the unit sphere leads to spherical triangles, can also be sampled
[Arv95]. This is an important result because it allows the solid an-
gle sampling of any general polygon mesh: the mesh is triangu-
lated, a spherical triangle is then randomly chosen, with a prob-
ability proportional to its individual solid angle, and this is then
sampled. The only drawback is that the initial triangle selection
step breaks the overall sample stratification — stratification is only
possible inside each individual triangle, once it has been chosen.
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M.N. Gamito / Solid Angle Sampling of Disk and Cylinder Lights
A bijection that allows the stratified sampling of spherical rect-
angles has been presented [UFK13]. This contribution successfully
showed the advantage of sampling a rectangular light directly, in-
stead of decomposing it into two spherical triangles, which allows
the stratification to be preserved across the whole light source and
leads to a decrease in the integral estimator variance. In this pa-
per, we use spherical rectangles as the bounding object for the light
sources that we are interested in sampling.
A technique for solid angle sampling of light sources, that
has some similarities to ours, has previously been developed
[TWCC06]. This technique surrounds the light source with a
bounding mesh and samples that mesh, using the aforementioned
algorithms for spherical triangles. The technique, however, is bi-
ased — it is the illumination from the bounding mesh that is actu-
ally computed and not from the original light source. The bias can
be reduced (but not eliminated) by subdividing the mesh into many
finer polygons, making the method more expensive.
Orthogonal to our work, methods that model the illumina-
tion distribution from complex light fixtures have been developed
[KHM09,VADWG15]. These attempt to describe the transport of
light through several layers of reflective and refractive surfaces that
surround the light source. An approximate sampling method for
cylinders, which uses a piecewise linear probability density func-
tion, has been proposed to describe the indirect scattering from
volumes [NNDJ12]. Ray segments inside a volume are spatially
blurred and become cylinders, thus removing a source of singular-
ities from the sampling.
Recently, an analytic solution for the direct illumination from
polygonal mesh lights onto a class of BRDFs, designated as Lin-
early Transformed Cosines, has been presented [HJSD16]. Such
BRDFs are generated from a cosine-weighted hemisphere, which
has been transformed by a matrix that accounts for such linear ef-
fects as rotation, anisotropic scaling and shearing.
3. Sampling Disk and Cylinder Lights
The solid angle sampling of disk and cylinders proceeds in an iden-
tical manner and, in its most basic form, can be expressed in Algo-
rithm 1. It takes the geometry Lof the light source and the point
xfor which illumination is required. It returns a direction ω
ω
ωiand a
probability density pi.
Algorithm 1 Sampling of Disks and Cylinders
Require: point x, light L,return direction ω
ω
ωi, PDF pi
Ω←solid angle of L,as seen from x
R ← spherical rectangle bounding L,as seen from x
repeat
(ξ1,ξ2)←random numbers ∈[0,1)2
ω
ω
ωi←sample R(ξ1,ξ2)See [UFK13]
until {x,ω
ω
ωi}intersects L
pi←1/Ω
The algorithm repeatedly tries directions, generated towards the
spherical rectangle, until the semi-infinite ray, with origin at xand
direction ω
ω
ωi, intersects the light source. In what follows, we first
P
O
l
Rd
r1r0
φ
Figure 3: The solid angle of a disk, seen from P.
consider the case of a disk light, due to its simplicity, followed by
the cylinder light. For each light, we begin by presenting the solid
angle formula, relative to x, and proceed with the placement of the
bounding spherical rectangle R.
3.1. Disk Lights
We consider a disk light to have a front and a back face, with only
the front face providing illumination. The formula for the solid an-
gle of a disk of radius Ris derived in [Pax59], for a point Pat a
distance lfrom the plane of the disk and a distance dalong the
plane of the disk (see Figure 3):
Ω=
−2l
r1K(k)−πΛ0(φ,k) + 2πif d<R,
−2l
r1K(k) + πif d=R,
−2l
r1K(k) + πΛo(φ,k)if d>R,
(4)
r2
1=l2+ (d+R)2,
r2
0=l2+ (d−R)2,
k=1−r2
0/r2
1,
φ=arcsin l
r0.
In this equation, K(k)is the complete elliptic integral of the first
kind and Λo(φ,k)is the Heuman lambda function [AS70]. The con-
tinuity between the three cases of d, relative to R, is ensured by the
fact that Λ0(π/2,k) = 1.
3.1.1. Bounding the Disk Light
The spherical rectangle R, in solid angle space, that is used to
bound Ω, corresponds to a square of side 2R, embedded in the plane
of the disk (see Figure 4). The square is oriented along the axis from
Q, the projection of Ponto the plane, to the centre of the disk at O.
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P
d
l
OQ
R
Figure 4: A square bounding the disk.
This orientation is important, in the case d>R, because it ensures
that Qis always outside the square. If this were not so and Pwas
too close to the plane (a small lin Figure 4), there would be an
excessive number of trial directions ω
ω
ωithat would not intersect the
disk. For d<R, the problem does not occur, since Pprojects inside
both the square and the disk, but we maintain the same orientation
for consistency.
3.2. Cylinder Lights
The formula for the solid angle Ωof a cylinder of radius Rand
height Hcan be found in [Try12]. The value of Ωis computed in
a local coordinate system (see Figure 5) where the point Pis at
a distance dfrom the cylinder axis and the base of the cylinder
is offset by a distance h0from the XY plane. We always consider
d>Rbecause cylinder lights do not illuminate their interiors. The
solid angle is the composite of three smaller solid angles Ω1,Ω2
and Ω3, corresponding to the surfaces S1,S2and S3in Figure 5:
Ω=Ω1+Ω2−sgn(h0)Ω3.(5)
The central axial region contributes Ω1to the total. The upper
end cap adds Ω2to this contribution. The lower end cap subtracts
Ω3from the total if the cylinder is above the XY plane (h0>0),
otherwise it adds Ω3to the total when the cylinder extends below
the plane (h0<0). The values for Ω2and Ω3can be computed from
a single formula ΩC(h)for a cylindrical cap, parameterised by the
height halong the Zaxis:
ΩC(h) = 4R
q(d−R)2+h2nΠ(n1,φ|m)−Π(n0,φ|m)o,(6)
n1=h+i(d+R)
h+i(d−R),
n0=(d+R) + ih
(d−R) + ih ,
φ=iarcsinhrd−R
d+R,
m=(d+R)2+h2
(d−R)2+h2.
We then have Ω2=ΩC(h0+H)and Ω3=ΩC(h0). The func-
tion Π(n,φ|m)is the incomplete elliptic integral of the third kind
[AS70]. Details of how equation (6) is evaluated are given in the
Appendix .
h0
c
S2
S3
S1
x
y
z
P
R
H
d
b
Figure 5: The solid angle of a cylinder, expressed in a local coor-
dinate system centered at P.
The value of Ω1is the solid angle of a rectangle, oriented relative
to the coordinate axes as shown in Figure 5:
Ω1=2arctan c(h0+H)
bqb2+c2+ (h0+H)2−
2arctan ch0
bqb2+c2+h2
0
,
(7)
b=d1−(R/d)2,c=Rq1−(R/d)2.
Each of the two terms in (7) is the solid angle of a rectangle that
starts at the height h=0 on the plane. Their difference gives the
desired result for the rectangle from h0to h0+H.
3.2.1. Bounding the Cylinder Light
The cylinder light can be bounded by a rectangle shape Sthat is
very similar to the rectangle S1of Figure 5. To make sure that the
end cap S2is also sampled, the bounding rectangle needs to have its
upper height increased by a sufficient amount. Figure 6shows this
bounding rectangle. The range of heights for Sis [h0,h1], where
h1can be obtained trigonometrically and with a simple algebraic
manipulation:
h1= (1+R/d)(h0+H).(8)
Note that h0+H<h162(h0+H), given that d>R, so the
bounding rectangle will, at most, be twice as high as the cylin-
der. If the cylinder intersects the XY plane (h0<0), we also ex-
tend Sdownwards, along the −Zaxis, by the negative amount
(1+R/d)h0. Similarly to the upper end cap, the downwards ex-
tension is bounded by 2|h0|in absolute value. The rectangle Sbe-
comes, in solid angle space, the spherical rectangle Rthat provides
an optimal bound for the solid angle Ωof the cylinder.
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h0
h1
c
S
x
y
z
P
R
H
d
b
Figure 6: The bounding rectangle S of a cylinder, with a height h1
that extends above h0+H.
3.2.2. Clipping the Cylinder Light
The illumination integral (1) is clamped by the cosine term |n·ω
ω
ωi|.
Any attempt to sample an incoming direction ω
ω
ωifor which (n·
ω
ω
ωi)<0 represents a wasted computational effort and increases the
noise in the integral estimate. For some configurations of a cylinder
light, relative to a surface point P, with normal N, it is possible to
clip the cylinder against the surface tangent plane so as to minimise
the number of light samples taken from below that plane.
Since a general formula for the solid angle of a clipped disk is
not known, the clipping of a cylinder light is only possible when the
surface plane does not intersect the cylinder caps. If the cylinder is
long and thin, with an aspect ratio H/R1, then any intersecting
plane is much more likely to clip the cylinder geometry without
touching any of the end caps, which makes the clipping procedure
very effective. If, on the other hand, the cylinder is short and wide
(H/R1) then clipping is almost certainly not possible and the
sampling of the light proceeds without taking this step.
To clip the cylinder light before sampling, we find the intersec-
tion of the central rectangular surface S1with the tangent plane
(see Figure 7) and work with the smaller surface S0
1⊂S1above the
plane, with a solid angle Ω0
1<Ω1. Let {ha,hb} ∈ [h0,h0+H]be
the two heights where the surface plane clips S1, such that ha6hb.
Also let Nzbe the vertical component of the plane’s normal vec-
tor N. After clipping, we can obtain the total solid angle Ωof the
cylinder and the height range [hA,hB]of the clipped surface S0
1, ac-
cording to Algorithm 2.
For simplicity, we ignore the general trapezoidal shape of S0
1and
treat it instead as a rectangle, by taking either the maximum or the
minimum of {ha,hb}, depending on the sign of Nz(refer to Algo-
rithm 2and Figure 7). It would have been possible to improve the
clipping and further reduce the number of discarded samples by
h0
c
S0
1
x
y
z
N
P
R
H
ha
hb
d
b
Figure 7: The clipping of a cylinder by an upward-facing plane
passing at P, with normal N.
Algorithm 2 Clipping of Cylinders
Require: ha,hb,return Ωand [hA,hB]
[hA,hB]←[h0,h0+H]Initialise clipped range
if Nz>0 & ha>(1+sgn(h0)(R/d))h0then
hA←haClip lower cap
Ω←Ω0
1+Ω2
else if Nz<0 & hb<(1−R/d)(h0+H)then
hB←hbClip upper cap
Ω←Ω0
1−sgn(h0)Ω3
else No clipping was possible
Ω=Ω1+Ω2−sgn(h0)Ω3Equation (5)
end if
evaluating the solid angle of the correct S0
1trapezoid, at the cost
of a more complex clipping algorithm. The clipped solid angle Ω0
1
has an expression similar to equation (7) but using instead the hA
and hBheights. If the clipping is not effected, the total solid angle
remains that of equation (5) and the height range for S0
1is the same
as for S1, effectively making S0
1=S1.
The clipped rectangle S0
1is extended into a bounding rectangle
S0by multiplying its height range [hA,hB]with the same factor
(1+R/d)used in Section 3.2.1, based on the following rules: hBis
multiplied only if it is equal to h0+Hand hAis multiplied only if
it is negative and equal to h0. In other words, only an extreme of S0
1
that corresponds to the height of one of the original cylinder caps
is extended to form the bounding rectangle, while an extreme that
results from the clipping procedure stays unchanged.
4. Implementation
As with any rejection sampling method, some care is required to
ensure that it does not become locked in an infinite loop, end-
lessly testing trial samples. In our case, this can happen for very
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small solid angles, as when a disk light is seen almost edgeways
from some surface point. The spherical rectangle sampling algo-
rithm is sensitive to numerical errors for small solid angles and this
can cause the sampled directions never to intersect the light, even
though the spherical rectangle correctly bounds the solid angle of
the latter.
We avoid infinite loops in a two-fold way. Firstly, we skip rejec-
tion sampling entirely if the solid angle Ωof the light is smaller
than some value ΩMIN . Secondly, we break out of the sampling
loop after a maximum number of iterations NMAX . If either of these
two conditions is reached, we fall back to using the conventional
area sampling method for disks or cylinders. This ensures that not
only are lockups avoided but also that the lighting integral esti-
mator remains unbiased, even for those surface points where the
solid angle sampling algorithm fails. In our implementation, we
use ΩMIN =0.001 steradians and NMAX =100 iterations.
A second issue brought about by rejection sampling is that strati-
fied sampling is not possible, because the mapping of random num-
bers from the [0,1)2domain to Ωis no longer a bijection. Not all
values in the unit square map to a valid direction in Ωand this could
lead to a situation where the attempt to take a random number pair
in a stratum of [0,1)2would repeatedly fail, if that stratum maps to
a solid angle that is completely outside of the solid angle Ωof the
light.
Since stratified random numbers cannot be used, we rely instead
on low discrepancy sequences. These are infinite deterministic se-
quences of numbers that are known to be well distributed within the
unit square domain [PH10]. A low discrepancy sequence generates
the quasi-random numbers that drive the rejection sampling algo-
rithm. In this respect, the algorithm can be considered as a form of
deterministic rejection sampling [Wan00]. We use the implementa-
tion of a lattice sampler for our number generator [CKN06].
If desired, the proposed algorithm can be optimised by replac-
ing the numerical evaluation of the elliptic integrals with tabulated
functions, at the cost of introducing bias. More details about the
tabulation of the elliptic integrals, used by disk and cylinder lights,
is given in the Appendix.
The algorithm was developed as a plugin for the Arnold path
tracer [KFC∗10]. This is a fully multi-threaded path tracing so-
lution that allows customisable surface shaders to be written. For
every ray-surface intersection, our surface shaders invoke our own
light sampling functionality, thus bypassing the native light sam-
pling done by Arnold.
5. Results
Figure 8shows the illumination from two back-to-back disk lights.
For clarity, the lights themselves are invisible, to enable a bet-
ter viewing of the surface illumination. The two lights have been
placed vertically and touch the horizontal surface at their lower ex-
tremities. The results are visibly smoother with solid angle sam-
pling. The difference heatmaps, obtained against a ground truth so-
lution, also show that solid angle sampling achieves better conver-
gence for the same number of samples, particularly at short dis-
tances to the lights.
(a) Area sampling. (b) Solid angle sampling.
Figure 8: Disk light illumination on surfaces (top row). The lower
row shows the colour coded difference of the two techniques to a
ground truth image (9 samples per pixel).
The algorithm for cylinder lights has been tested against area
sampling and against an improved version of the cylinder sam-
pling algorithm of [Zim95], which is here called Zimmerman Sam-
pling. The improved algorithm replaces the importance sampling
for the height halong the cylinder axis with equi-angular sam-
pling [KF12]. This is a better technique, which was not available
at the time of the original contribution, that takes samples along a
line segment such that the angle, relative to the illumination point,
has a uniform distribution.
Figure 9shows the results of cylinder light sampling with the
three algorithms, for a cylinder on top of the horizontal plane.
There is an improvement from area sampling to Zimmerman sam-
pling and then to solid angle sampling. The different heatmaps also
make this improvement clear. Zimmerman sampling takes sam-
ples independently on the (θ,h)parametric domain of the cylin-
der and this leads to a solid angle probability density of pΩ(ω
ω
ω) =
pΘ(θ)pH(h)/|J(θ,h)|, where pΘand pHare the independent distri-
butions for each parameter and J(θ,h)is the Jacobian of the trans-
formation from (θ,h) space to solid angle space. Although pΘand
pHare optimal, in their own sub-domains, their product with the
inverse Jacobian isn’t and leads to a solid angle probability density
that is not uniform. Solid angle sampling overcomes this problem
by taking samples directly in the solid angle space Ωof the light
with the uniform probability density function pΩ(ω
ω
ω).
The implementation of Multiple Importance Sampling (MIS) is
straightforward for the proposed algorithms [VG95]. When sam-
pling the incoming direction from the light, the corresponding
BRDF sampling probability is used for MIS weighting. When
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(a) Area sampling. (b) Zimmerman sampling. (c) Solid angle sampling.
Figure 9: Cylinder light illumination on surfaces (top row). The lower row shows the colour coded difference of the three techniques to a
ground truth image (9 samples per pixel).
sampling the incoming direction from the surface BRDF, the cor-
responding light sampling probability, for the purpose of MIS
weighting, is the inverse of the solid angle (3). Figure 10 shows
the results for cylinder sampling with MIS. Thanks to the BRDF
sampling component of MIS, the sampling noise is reduced along
the ground plane, where it touches the cylinder light, relative to the
results of Figure 9. The improvement is more significant for area
sampling and Zimmerman sampling and less noticeable for solid
angle sampling, which indicates that the latter is already providing
good quality sampling results at close distances to the light.
Figure 11 tests the clipping of a cylinder light that takes place
before sampling. The cylinder is vertical and intersects the ground
plane in such a way that one half of the cylinder is above the plane
and the other half is below. By clipping the light, no light sam-
ples are taken from below the plane, thereby improving the render
quality. For this scene, the rendered result is the same as that for a
cylinder light that has only half the height of the original and rests
above the plane.
The convergence curves, including the implementation of MIS,
are shown in Figure 12, in terms of root mean square error against
a ground truth image. Solid angle sampling converges faster than
area sampling. Zimmerman sampling, in the case of cylinder lights,
shows intermediate convergence. The ground truth was rendered
with 1024 samples per pixel and 10 samples per surface point.
The proposed algorithms can also be applied to light sampling
in volumes. The lighting integral for a point xin a volume is very
similar to that for surfaces, with the surface BRDF being replaced
with the volume phase function p(x,ω
ω
ωo,ω
ω
ωi):
L(x,ω
ω
ωo) = σsL0e−RTσtdt ZΩ
p(x,ω
ω
ωo,ω
ω
ωi)V(x,ω
ω
ωi)dω
ω
ωi
=σsL0e−RTσtdt ZA
p(x,ω
ω
ωo,ω
ω
ωi)V(x,ω
ω
ωi)|m·ω
ω
ωi|
ky−xk2dy.(9)
The constant σsis the scattering coefficient of the volume.
The visibility function V(x,ω
ω
ωi)∈[0,1]now includes attenuation
through the volume. The illumination L(x,ω
ω
ωo)is attenuated by the
factor e−RTσtdt , where Tis the distance travelled along ω
ω
ωoand σt
is the, potentially varying, extinction coefficient of the volume. Fig-
ure 13 shows volume rendering results with both disk and cylinder
lights. Similar to the surface rendering results, render quality is im-
proved with solid angle sampling and particularly when the volume
is close to the light source.
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M.N. Gamito / Solid Angle Sampling of Disk and Cylinder Lights
(a) Area sampling. (b) Zimmerman sampling. (c) Solid angle sampling.
Figure 10: Cylinder light illumination on surfaces with Multiple Importance Sampling (top row). The lower row shows the colour coded
difference of the three techniques to a ground truth image (9 samples per pixel).
Figure 11: A vertical cylinder light, intersecting the ground plane,
without clipping (left) and with clipping (right).
The acceptance rate of trial samples is shown in Table 1for both
types of light sources and for surface and volume rendering. These
statistics are based on the previously shown images that were ren-
dered at 512 ×512 resolution. The acceptance rate for a disk light
matches well with the π/4 area ratio of a disk inscribed in a square,
as should be expected.
The acceptance rate for the cylinder lights is nearly optimal, with
every light sample almost always being accepted. The tested cylin-
ders have an aspect ratio H/Rof 40 for surface rendering and 15
for volume rendering. Such long cylinders have a solid angle that is
mostly dominated by the central spherical rectangle, with the end
caps having only a small contribution. In such cases, a trial sample
is most likely to fall on the rectangle and is immediately accepted.
The acceptance rate degrades for short cylinders, where the con-
tribution of the end caps to the solid angle is more important. We
have tested a cylinder with an aspect ratio of 0.1 and found the
acceptance rate to be 17.68%, or roughly 1 in every 5 trial sam-
ples. Fortunately, short cylinders are seldom needed in production
scenes, in contrast to long cylinders that are frequently used to rep-
resent fluorescent lighting. In the limit, the acceptance rate would
drop to 0% as H/R→0. This is avoided, in practice, by the mech-
anism discussed in Section 4, whereby the algorithm switches to
uniform area sampling when the solid angle falls below a specified
threshold.
Table 1also shows the render times, obtained on a 8-core ma-
chine. The use of tabulated values for the elliptic integral functions
has a clear impact in lowering render times, especially for volume
rendering. Tabulation introduces bias, however, and this can be con-
trolled by the size of the function tables and by the table interpola-
tion scheme used. We have used sizes of 256 for K(k), 90 ×90 for
Λ0(k,φ)and 64 ×64 ×64 for Π(n,φ|m).
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M.N. Gamito / Solid Angle Sampling of Disk and Cylinder Lights
1 10 100 1,000
0.001
0.01
0.1
1
Samples per Pixel
RMSE
Disk Light
Area Sampling
Solid Angle Sampling
1 10 100 1,000
0.001
0.01
0.1
1
Samples per Pixel
RMSE
Cylinder Light
Area Sampling
Zimmerman Sampling
Solid Angle Sampling
Figure 12: Convergence for disk sampling (top) and cylinder sampling (bottom) with MIS, for increasing number of samples.
Render Time
Sample
Acceptance
Function
Evaluation
Function
Tabulation
Disk Surface (Fig. 8) 78.57% 3.55s 2.61s
Volume (Fig. 13) 78.81% 18.96s 9.16s
Cylinder Surface (Fig. 9) 97.24% 3.86s 2.97s
Volume (Fig. 13) 92.24% 19.40s 9.89s
Table 1: Rendering statistics.
Figure 14 shows a frame from a film production where multiple
cylinder lights have been used to illuminate the interior of a space-
ship. This illustrates the use of the proposed algorithms in produc-
tion environments. Each light is assigned a sampling probability,
relative to each illumination point, with a simple heuristic that es-
timates the influence of the light to that point. A specific light is
then randomly selected and the light’s probability is used to scale
the solid angle probability density (3).
6. Future Work
The proposed sampling algorithm is quite general and can be used,
beyond disk and cylinder lights, to perform solid angle sampling of
any light geometry. The only two requirements are as follows:
•An analytical formula for the solid angle of the light must be
known.
•The light must be bounded by simpler geometries whose solid
angle can be sampled.
Bounding geometries can either be spheres, spherical rectangles
or preferably small sets of large spherical triangles, depending on
which can bound the geometry of the light more efficiently.
An example in mind is the sampling of a mesh light. Although
triangle meshes can already have their solid angle sampled, the
sampling is of poor quality if the mesh is made of many small
triangles, due to the loss of sample stratification (refer to the dis-
cussion in Section 2). Bounding the mesh with a sphere or a few
large spherical triangles will improve the sampling quality. Solid
angle sampling, with this addition of mesh lights, could then be
performed on all the lights most commonly used for film produc-
tion, leading to an overall improvement of rendering quality.
It would also be interesting to look at the solid angle sam-
pling of lights that have an emission texture. Such lights will no
longer be characterised by a single emission constant L0, as has
been done in this work. It is conjectured that importance sam-
pling of both the light texture and the light solid angle will require
combined sampling techniques such as MIS or importance resam-
pling [VG95,TCE05].
7. Conclusions
Sampling lights within their solid angle space, relative to an illumi-
nation point, is known to produce better results for global illumina-
tion rendering algorithms, with less noise and faster convergence,
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M.N. Gamito / Solid Angle Sampling of Disk and Cylinder Lights
(a) Area sampling. (b) Solid angle sampling.
Figure 13: Volume rendering results with a disk light (top) and a
cylinder light (bottom). Rendered with 9 samples per pixel and 8
samples per ray.
when compared against simpler area sampling algorithms. Such
solid angle sampling algorithms could only previously be applied to
a small number of production light geometries, namely spheres and
rectangles. We have extended this desirable class of algorithms to
include disks and cylinders. We have also suggested how the same
technique can be extended to triangular meshes, without the need
for an intermediate random triangle selection step that reduces the
quality of the sample distribution.
The proposed algorithms are unbiased and fit easily into the di-
rect illumination calculation of any path tracer. The cost of evalu-
ating the elliptic integrals, present in the solid angle of disks and
cylinders, may justify implementing a biased version of the algo-
rithm that uses tabulated functions. We have also shown that the
rejection sampling nature of the algorithms is efficient, with trial
samples that are very quickly accepted, due to a tight bounding of
the light’s solid angle with a spherical rectangle, with the only ex-
ception of pathological cases of cylinders that are short and broad.
An improvement to the sampling of lights would include the sur-
face cosine term in the probability distribution: a light would no
longer be sampled according to its solid angle but according to its
projected solid angle. Given that the solid angle formulae for disks
and cylinders are already complex and that equivalent projected
solid angle formulae are not known, it seems that solid angle sam-
pling is the best algorithm that can be implemented for these two
lights. This is in contrast to rectangular lights, where the projected
solid angle has the same expression of the point-to-polygon form
factor of the radiosity algorithm and for which a projected solid
angle sampling algorithm may be feasible in the future [CW93].
8. Acknowledgements
The author would like to thank Martin Preston and Nathan Walster
for providing the opportunities of performing research work in the
middle of an always busy production schedule. The author would
also like to thank Josh Bainbridge for implementing the low dis-
crepancy sampler that was used in this paper.
Appendix: Evaluating Elliptic Integral Functions
We evaluate the elliptic integrals K(k)and Λ0(k,φ), required by disk lights,
with numerical functions provided by the GNU Scientific Library (GSL)
[GSL16]. The Heuman Lambda function Λ0(k,φ)is evaluated in terms of
other elliptic integrals, although we have since found that Boost provides
a direct implementation [Boo16]. The two functions K(k)and Λ0(k,φ)can
be easily tabulated, given that 0 6k<1 and 0 6φ<π/2, and, in fact, such
tables are usually found in the literature [AS70].
For ease of reference, we repeat here the equations from Section 3.2 for
the solid angle of a cylinder cap of radius R, at a height hand along distance
dfrom the illumination point towards the axis of the cylinder:
ΩC(h) = 4R
q(d−R)2+h2nΠ(n1,φ|m)−Π(n0,φ|m)o,(A.1)
n1=h+i(d+R)
h+i(d−R),
n0=(d+R) + ih
(d−R) + ih ,
φ=iarcsinhsd−R
d+R,
m=(d+R)2+h2
(d−R)2+h2.
The difficulty with this integral function is that it has complex argu-
ments and numerical solutions for Π(n,φ|m)in the complex domain are not
easily available, unless commercial mathematics packages are used. The
value of ΩC(h)is real, nevertheless, because φis a pure imaginary number,
while n1and n0form a complex conjugate pair. This causes Π(n1,φ|m)and
Π(n0,φ|m)to have the same imaginary value and so their difference in (A.1)
cancels out. At the same time, the real parts of Π(n1,φ|m)and Π(n0,φ|m)
have the same absolute value and opposite signs. These symmetries indi-
cate that there is scope for simplification in (A.1) and that a brute force
evaluation of two elliptic integrals with complex arguments is not required.
The best approach is to employ the definition of Π(n,φ|m)in (A.1), re-
place the expressions for the function parameters, change the integration
variable from the imaginary to the real line and perform some algebraic
simplification, to arrive at the integral:
ΩC(h) = 16R2h
(d−R)2+h23/2
Z|φ|
0
sinh2ϕdϕ
1+2nsinh2ϕ+msinh4ϕp1+msinh2ϕ,
(A.2)
n=d2−R2+h2
(d−R)2+h2,
where the parameters φand mare as previously in (A.1). The value of ΩC(h)
can be evaluated, using a numerical integrator from the GSL. To tabulate
this function, we use a change of variables to write ΩC(u,v,w).
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M.N. Gamito / Solid Angle Sampling of Disk and Cylinder Lights
Figure 14: A spaceship interior from the film The Martian, illustrating the use of several cylinder lights, among other types of light source,
to illuminate interior scenes (Copyright Twentieth Century Fox).
The new variables 0 6u,v,w<1 are such that:
u=sd−R
d+R,
v=s(d−R)2+h2
(d2−R2) + h2,
w=s(d−R)2+h2
(d+R)2+h2,
(A.3)
and consequently:
|φ|=arcsinhu,
n=1/v2,
m=1/w2.
(A.4)
A table for ΩC(u,v,w)can be built by evaluating (A.2) for triples in the
[0,1)3domain. The choice of square roots for vand wwas found to be
desirable because it removes the quadratic distortion in those parameters
and spreads the values of Ωc(u,v,w)more uniformly within its domain.
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