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Real-time Light Transport in Analytically Integrable
Quasi-heterogeneous Media
Tom´
aˇ
s Iser∗
Supervised by: Oskar Elek†
Faculty of Mathematics and Physics
Charles University
Prague / Czech Republic
Abstract
Our focus is on the real-time rendering of large-scale vol-
umetric participating media, such as fog. Since a physi-
cally correct simulation of light transport in such media is
inherently difficult, the existing real-time approaches are
typically based on low-order scattering approximations or
only consider homogeneous media.
We present an improved image-space method for com-
puting light transport within quasi-heterogeneous, opti-
cally thin media. Our approach is based on a physically
plausible formulation of the image-space scattering kernel
and analytically integrable medium density functions. In
particular, we propose a novel, hierarchical anisotropic fil-
tering technique tailored to the target environments with
inhomogeneous media. Our parallelizable solution en-
ables us to render visually convincing, temporally co-
herent animations with fog-like media in real time, in a
bounded time of only milliseconds per frame.
Keywords: Real-time Rendering, Light Transport, Mul-
tiple Scattering, Participating Media, GPU Shaders
1 Introduction
In computer graphics, one of our major topics is gener-
ating physically plausible images. This process is usu-
ally referred to as photorealistic rendering. It can be used
in architectural or design visualizations, simulators, video
games, or movies. The goal is to create imagery that re-
sembles reality as we perceive it.
To achieve photorealistic results, it is necessary to un-
derstand and simulate how light interacts with the matter
around us. When rendering simple scenes, the physical
processes can be simplified by assuming that light travels
in a vacuum. But once we want to render more advanced
optical phenomena, such as light scattering, one can no
longer rely on simplifications valid only in a vacuum.
Light scattering is a process where the otherwise
straight trajectory of the light can be deviated, which hap-
∗tomasiser@gmail.com
†oskar.elek@gmail.com
Figure 1: Typical outputs rendered with our approach in
different demo scenes. Both scenes contain several point
light sources and emissive textures. They are blurred in an
inhomogeneous fog with an altitude-dependent density.
pens when light travels through air, water, milk, and other
non-metallic participating media [3]. Numerous optical
phenomena, such as the blurring of objects and lights in a
foggy weather, can be explained by the scattering.
Existing Approaches In the case of non-real-time ren-
dering, there exist physically precise, but rather slow
methods. Efficient Monte Carlo approaches are described,
for example, by Jarosz [4]. However, rendering of a single
picture may take several hours or even days. Furthermore,
even a slight adjustment of the medium parameters can
significantly change the rendering time. When we need
our visualizations to run in real time, such as in video
games or simulations, we need very efficient approxima-
Proceedings of CESCG 2018: The 22nd Central European Seminar on Computer Graphics (non-peer-reviewed)
tions to process a single frame in only a few milliseconds.
Current real-time approaches, which we compare later
in Section 6, are mostly based on empirical or single-
scattering approximations [6, 8, 14]. These solutions are
very fast, but ignore multiple scattering of light, which is
reasonable only in optically very thin media. They typi-
cally compute in-scattering from light sources only, disre-
garding emissive materials and reflections. Therefore, they
cannot blur the geometry in the scenes, which is an impor-
tant effect noticeable in real media. Multiple-scattering
real-time solutions exist only for homogeneous media [3]
or we found them too slow for real-time use [12].
Our Approach In this paper, we aim at rendering scenes
that are filled with the medium (Figure 1), i.e., the camera
itself is in the medium. This corresponds to various real-
life situations: foggy weather, sandstorms, when looking
at a mist above a lake, or even when swimming in the lake
itself. Our approach that we propose and implement is
based on novel modifications of a solution originally de-
scribed by Elek et al. [3] for homogeneous media only.
The method we propose is capable of simulating multiple-
scattering effects in inhomogeneous media in real time. In
particular, our key achievements in this paper are:
•We propose a novel multiple-scattering approximation
suitable for quasi-heterogeneous media and introduce
closed-form solutions for the necessary density inte-
grals (Section 3).
•We use efficient hierarchical screen-space filtering:
we use the original filtering of Elek et al. [3] and ex-
tend it for inhomogeneous media and by adding new
steps for visual improvements (Section 4).
•We show results indicating that our method is indeed
capable of being executed in real time with a stable
framerate even in Full HD resolutions (Section 5).
2 Fundamentals
In order to describe our approach in detail, we first briefly
explain screen-space approaches in general, and then the
physics of light transport in participating media.
2.1 Screen-space Methods
In this paper, we present a screen-space method [5, 3] in-
tended to be used in a post-processing step, i.e., applied on
an original frame unaffected by any participating medium.
We treat every single pixel of the original frame as an im-
plicit light source, as a source of radiance L incoming to
the camera from a surface on that pixel. Intuitively, the
pixels represent packets of photons traveling from surfaces
towards the camera. By adding a participating medium,
these photons (pixels) are affected, so we need to post pro-
cess the individual pixels of the original frame.
Screen-space methods are popular and beneficial as
their time complexity only depends on the number of pix-
els, i.e., the resolution, and not on the complexity of the
rendered scene. Apart from participating media render-
ing [3, 14, 12], they are used for various other effects such
as depth of field rendering introduced by Lee et al. [5].
2.2 Participating Media
For our purposes, we model a participating medium as
a collection of identical, randomly positioned, and small
particles, such as water droplets or dust. It allows us to
describe media by the following properties, explained in
more details by Elek [2] and Jarosz [4].
Photons travelling through such media can collide with
the particles in the media. When such an interaction oc-
curs, the photon itself can be completely absorbed, or it
can be scattered in a new direction. Furthermore, certain
media emit new photons, which is called emission.
For the position xin a certain medium, let ρ(x)m−3
denote the density of the particles at that location. Each
particle is characterized by three properties: absorption
cross-section Cam2, scattering cross-section Csm2,
and a phase function fp. Both Caand Csare color-channel-
dependent, which will be implicit throughout the rest of
the paper and is taken into account in our implementation.
Absorption Absorption happens when a photon is ab-
sorbed by a particle it hit. We consider the light energy to
be “lost”. Macroscopically, the “lost energy” can be ob-
served as a decrease of the overall intensity of the light.
How much the medium absorbs light is determined by the
absorption coefficient σam−1:σa(x) = Ca·ρ(x).
Scattering If a photon is not absorbed when it hits a par-
ticle, its energy is scattered into a new direction. The new
direction depends on the phase function fpof the medium,
which we can understand as a probability distribution
function. How much the scattering happens is described
by the scattering coefficient σsm−1:σs(x) = Cs·ρ(x).
In a beam of light passing through a medium, the pho-
tons can be scattered out of the path (out-scattering), and
other photons can converge into the beam (in-scattering).
Furthermore, we distinguish between single scattering,
when a single photon can only scatter once, and multiple
scattering. Multiple scattering causes spatial spreading
(Figure 2) and angular spreading, which can be observed
in real media, e.g., as blurring of distant objects in a fog.
Optical Thickness Extinction describes when a photon
is either absorbed or scattered. We define the extinction
coefficient as σtm−1=σa+σs. By integrating the ex-
tinction along a line segment l, we get the optical thickness
(optical depth): τ(l) = Rlσt(x)dx.
Beer–Lambert Law The optical thickness is related to
transmittance T , where T(l) = T(x0,x1)expresses the ra-
diance between two points x0,x1after being reduced by
interacting with the medium: L(x1,ω) = T(l)·L(x0,ω).
According to Beer–Lambert law, the relation between the
transmittance and the optical thickness along lis exponen-
tial: T(l) = exp(−τ(l)).
Proceedings of CESCG 2018: The 22nd Central European Seminar on Computer Graphics (non-peer-reviewed)
3 Our Scattering Approximation
In this section, we propose a novel multiple-scattering ap-
proximation suitable for inhomogeneous media. We first
show an existing approach for homogeneous media (Sec-
tion 3.1). We then propose and derive our novel solution
for quasi-heterogeneous media (Section 3.2).
3.1 Spreading in Homogeneous Media
We can think of light scattering as a stochastic process
with many realizations. The central limit theorem allows
us to assume the spatial distribution of radiance to be
Gaussian (Figure 2). This approach was used by Premoˇ
ze
et al. [9] in their multiple-scattering approximations.
For a collimated pencil of light between two points with
the distance sin a homogeneous medium, Premoˇ
ze et al.
derived a formula for the standard deviation W of the
Gaussian spatial-spreading distribution:
W(s) = s1
22σa
3s+4
s3σs(1−g)−1
,(1)
where g∈[−1,1]is the scattering asymmetry factor de-
scribing the directions in which the photons scatter (e.g.,
for g>0, scattering in forward directions is favored).
sensor
Gaussian
x
s
Figure 2: Spatial spreading of radiance caused by multiple
scattering. We approximate the spreading width detected
on the sensor (camera) to have a Gaussian distribution.
3.2 Proposed Quasi-heterogeneous Media
Unfortunately, Eq. 1 assumes the medium parameters to
be spatially invariant along a light path. We propose a con-
cept of quasi-heterogeneous media with a smooth density
function ρ(x). The density is therefore similar in a local
neighborhood of a light ray, so we do not need a com-
plicated anisotropic kernel. It remains to explain how we
propose to take the spatially-varying density into account.
Idea: Density-dependent Distance Scaling When light
travels along a line segment with the length D, the spread-
ing width W(D)depends on the scattering interactions oc-
curring along the segment. By looking at the light scat-
tering as a stochastic process, we propose to assume that
W(D)does not depend on where exactly the particles are
accumulated along the line segment. We are only inter-
ested in how many particles along the path in total could
have interacted with the light.
When a pencil of light collides with a higher number
of particles, it spreads more than when it collides with less
particles, even if the distance Dis the same. Hence we pro-
pose to scale the distances w.r.t. the density accumulated
along the line segments, which corresponds to integrating
the density along the segment.
Mathematical Derivation We want to find a new
length D0of a line segment lin a homogeneous
medium that would “match” the behavior in a quasi-
heterogeneous medium. Assume a homogeneous medium
with a constant density ρ0and an optical thickness
τ(l) = Rlσt(x)dx=σt·D. The density of the quasi-
heterogeneous medium is ρ(x) = ρ0ρ00(x), where ρ00 (x)
is an arbitrary spatially-varying ratio between the homo-
geneous and heterogeneous density. The optical thick-
ness in the quasi-heterogeneous medium is by definition
τ(l) = Rlσt(x)dx=Ct·Rlρ(x)dx.
We want the optical thickness along the quasi-
heterogeneous medium (right side of Eq. 2) to match the
thickness along a scaled homogeneous medium (left side),
hence we get:
σt·D0=Ct·Zl
ρ(x)dx
σt·D0=Ct·ρ0·Zl
ρ00(x)dx
D0=Zl
ρ00(x)dx.(2)
Eq. 2 enables us to approximate a quasi-heterogeneous
medium by a homogeneous one. The equation holds true
in case the quasi-heterogeneous medium is in fact homo-
geneous. In that case, the density ratio would be ρ00(x) = 1
for all x∈R3(we compare the exactly same media). Solv-
ing Eq. 2 would give us: D0=Rlρ00(x)dx=Rl1 dx=D,
which obviously holds true as the length remains the same.
Spatial Spreading We found out that simply assuming
ρ00 ≈ρ, i.e., ρ0=1, leads to very good visual results. It
essentially corresponds to fixing a reference homogeneous
medium and only setting the ratio, which is what we want.
The Gaussian spatial-spreading deviation Wcan now be
found using Eq. 1 by using the scaled distances:
W(D0) = WZl
ρ00(x)dx≈WZl
ρ(x)dx.(3)
3.3 Solving Density Integrals Analytically
General Approach The integral in Eq. 3 corresponds to
integrating a density along a path from a camera to a pixel,
i.e., along the camera ray. Solving such integrals analyt-
ically is shown by Qu´
ılez [10, 11], who derived closed-
form solutions for simple exponential and spherical media.
Consider Figure 3: we express the ray rbetween the
camera origin c∈R3and the pixel position x∈R3para-
metrically. For a parameter tand direction ω, we get:
r(t) = c+t·ω. In our case, t∈[0,kx−ck]=[0,D],
Proceedings of CESCG 2018: The 22nd Central European Seminar on Computer Graphics (non-peer-reviewed)
where Dis the true distance between the camera and the
pixel. This approach enables us to integrate the density for
the parameter t, as shown in the example below (Eq. 4).
Example: Parametrized Exponential Medium Unlike
Qu´
ılez, in this paper, we derive a closed-form solution for
a more general arbitrarily rotated and translated exponen-
tial medium. By the exponential medium we understand a
medium where the particle density can be modeled by an
exponential function, which is closely related to real me-
dia according to the barometric formula [1].
As in Figure 3, we suppose that a fixed density ρ0ex-
ponentially decreases with a parameter b>0 in a normal-
ized direction n∈R3and is offset by o∈R3. Hence the
spatially-varying ρ(x):
ρ(x) = ρ0·exp(−b· hx−o,ni),
where h·,·i denotes an inner product. We integrate the den-
sity ρ(x) = ρ(r(t)) along the camera ray r(t)as explained:
P=ZD
0
ρ(r(t))dt=ZD
0
ρ(c+t·ω)dt
=ρ0ZD
0
exp(−b· hc+t·ω−o,ni)dt
=ρ0ZD
0
exp(−b·(hc,ni+thω,ni−ho,ni))dt,(4)
where the last equality uses the bilinearity of the inner
product. The terms hc,ni,hω,ni, and ho,niare constant
for a fixed ray, so the definite integral can be expressed as:
P=ρ0·−exp(−b·(hc,ni+thω,ni−ho,ni))
b· hω,niD
0
.
In case hω,ni=0, when we cannot divide, we simply
solve Eq. 4 directly with hω,ni=0. Otherwise, we get:
P=ρ0·exp(−b· hc−o,ni)·(1−exp(−b·D· hω,ni))
b· hω,ni.
%
c
x2D
x
origin o
!
n
Rx
c%(x0) dx0
r
Figure 3: A general exponential medium. The density de-
creases exponentially along the direction n. A projected
pixel x2D corresponds to the 3D position x. We integrate
the density along the camera ray r.
4 Our Filtering Technique
Our screen-space algorithm is based on the technique pre-
sented by Elek et al. [3], but we generalize their approach
for inhomogeneous media and propose new steps for elim-
inating artifacts. On the input, we expect an HDR RGB
color buffer L:N2→R3and its corresponding distance
buffer D:N2→R, i.e., the world-space distances from
the camera. The images are then preprocessed, hierarchi-
cally filtered, and finally we compose a single attenuated
and scattered image L0:N2→R3. For an overview of our
whole approach, see the full diagram in Figure 4.
4.1 Preprocessing
Integrated Density and Spread Space From Section 3,
we suppose that the densities can be integrated analyti-
cally. A density buffer P:N2→Ris constructed, rep-
resenting the integrated densities from the camera towards
the pixel 3D position. We also build an additional spread-
space buffer W:N2→Raccording to Eq. 3, which corre-
sponds to the spreading widths of the pixels.
P(x) = Zpixel x
camera
ρ(x0)dx0,W(x) = W(P(x)).
Attenuated and Scattered Radiance Beer-Lambert law
(Section 2) defines how the input radiance Lis attenuated
w.r.t. the optical thickness τ(l), which we get as follows:
τ(l) = Zl
σt(x)dx=Zl
(Ca+Cs)ρ(x)dx=CtZl
ρ(x)dx.
By combining this equation with transmittance, we
compute the following two buffers. The attenuated image
Lat :N2→R3corresponds to the radiance that was not ab-
sorbed. The scattered image Lsc :N2→R3corresponds to
the radiance that was scattered but not absorbed.
Lat =exp(−Ct·P)·L,
Lsc =exp(−Ca·P)·(1−exp (−Cs·P)) ·L.
4.2 Hierarchical Filtering
We want to scatter (spatially spread) the Lsc pixels. A sin-
gle pixel x∈N2corresponds to radiance Lsc(x). Assuming
the radiance spreads spatially according to Section 3, the
pixel’s value is distributed onto its neighbors by a Gaus-
sian distribution centered in xwith a standard deviation
W(x). As different pixels correspond to different devia-
tions W, the Gaussian kernels are spatially varying. Un-
fortunately, filtering with spatially varying kernels is not
possible with a fast two-dimensional convolution [3].
Na¨ıve Non-Hierarchical Approaches Two similar ap-
proaches can be used for spatially-varying filtering: splat-
ting and gathering [3, 12]. The splatting approach iterates
over all pixels and adds their distribution to their neigh-
bors, requiring multiple writes into a single pixel, which is
Proceedings of CESCG 2018: The 22nd Central European Seminar on Computer Graphics (non-peer-reviewed)
camera matrix
medium
inputs
color buffer
distance buffer
integrated densities
spread space
densities
scattering
densities (separated)
scattering (separated)
MIP chains
1×1
1×1
1×1
1×1
1×1
interpolations
emissions +
not interacted
output
Figure 4: Our filtering technique expects an HDR input buffer and a distance buffer. We then integrate the densities
according to the distances. The filtering is hierarchical and several MIP chains are built throughout the process. Finally,
we interpolate the data from the MIP chains and get a final output.
not optimal for GPU shaders. The gathering is less effec-
tive, but can be implemented in GPU shaders, as for every
pixel, it gathers distributions from their neighbors. We im-
plemented a gathering filter for our method as a filtering
reference. Processing of a single 1920 ×1080 frame took
more than a minute, unsuitable for real-time rendering.
Hierarchical Gaussian Filtering Fortunately, spatially-
varying Gaussian filtering can be approximated using a
fast hierarchical approach with MIP maps [5, 3]: the origi-
nal image Lsc =L[0]
sc corresponds to level 0, the subsequent
levels are a power of two smaller than the previous, the last
level being Krepresenting a 1×1 image L[K]
sc .
The key idea in the hierarchical filtering is to convolve
the subsequent levels using the previous and the same
Gaussian kernel. The levels k∈ {1,...,K}are:
L[k]
sc =L[k−1]
sc ∗G,
where Gis a discrete normalized Gaussian kernel, e.g., us-
ing weights {0.13,0.37,0.37,0.13}for a 4 ×4 filter [3].
The width of the filter doubles with every level in the hi-
erarchy [5, 3]. The relation between the deviation Wand
the level kcan be approximated as W≈c·2k−1, where cis
ascaling constant (set to 0.86 in our implementation). As
the standard deviation increases exponentially, the level it-
self has to increase logarithmically. Elek et al. [3] propose
to approximate the MIP level kas:
k(P)≈clamplog2
W(P)
c,0,K∈[0,K].
The final filtered scattered image is L0
sc =L[k(P)]
sc (com-
posed pixel-by-pixel). The level can be decimal: one-
dimensional interpolation is used for a level and two-
dimensional interpolation for a pixel. Both benefit from
a fast interpolated texture access on a GPU hardware.
4.3 Eliminating Filtering Artifacts
Luminance Weighting Imagine a very bright object
with a low standard deviationW, e.g., the yellow lantern in
Figure 5b. When there is a dark object in the background
and its standard deviation Wis much higher, the luminance
of the bright pixels leaks into their neighborhood.
When building the MIP chain, the luminance of the
bright object is propagated into high MIP levels, even
though the standard deviation of the object is small. When
fetching the colors for the dark background, the bright lu-
minance incorrectly leaks into the dark area. This problem
is called illumination leaking [3] and can be seen in Fig-
ure 5bc as an incorrectly huge yellow blur of the lantern.
Masking idea Suppressing the illumination leaking is
explained by Elek et al. [3]. When building the MIP chain,
we introduce a new auxiliary mask Mand build the MIP
chain in a modified way:
M[k]=smoothstepT,(1+ε)·T,W[k−1],
L[k]
sc =M[k]·L[k−1]
sc ∗G.
The masking threshold distance Tand width εcontrol
the masking, where T=c·2k−1and ε≥0 should be set
according to the scene (ε=2 in our implementation).
Spread-space MIP chain The spread-space W[0...K]
should correspond to the standard deviation Waveraged
for neighboring pixels. Elek et al. [3] weight the pixels
with their luminance y, where Uis a uniform distribution:
Y[k]=y(L[k−1]
sc ),W[k]=Y[k]·W[k−1]∗U
Y[k]∗U.
Blurring The filtered L[k(W)]
sc may suffer from noticeable
discontinuities. They can be suppressed by applying the
Gaussian MIP map blurring to our densities P. We build
Proceedings of CESCG 2018: The 22nd Central European Seminar on Computer Graphics (non-peer-reviewed)
another MIP chain P[0...K]and then fetch as proposed by
Elek et al. [3]:
P[k]=P[k−1]∗G,k0=kP[k(P)],L0
sc =L[k0]
sc .
Optional Proposal: Radiance Separation In HDR im-
ages, certain pixels may be much brighter than the rest.
When the density p1∈Pof a bright foreground is much
lower than the density p2∈Pof a neighboring dark back-
ground, the blurring step smoothes this large discontinuity.
This can completely eliminate the foreground from high
levels of P[0...K]. It essentially means that the dark area of
our image remains dark and the blur of a bright foreground
is “cut” at the edge (Figure 5d).
To heuristically separate bright pixels from the original
input L, we split the input Linto two different scattering
images: Lsc denotes the scattering image without the sep-
arated pixels and Lssc with the separated pixels:
Lssc =e−Ca·P·1−e−Cs·P·M·L,
Lsc =e−Ca·P·1−e−Cs·P
| {z }
original weight of Lsc
·(1−M)·L,
M=smoothstepTy,Ty+εy,Y[0]
| {z }
high luminance
·1−smoothstepTd0−εd0,Td0,P[0]
| {z }
low integrated density
,
where Mis based on smoothly selecting the pixels with
high absolute RGB luminance y∈Y[0]and low integrated
density p∈P[0]and the parameters are thresholds.
The image Lsc is filtered using the same luminance
weighting technique described earlier. However, we build
the MIP chain L[0...K]
ssc differently as the image is mostly
black and only contains a few separated pixels. To prior-
itize the densities of the bright separated pixels, we con-
struct P[0...K]
ssc based on luminance weighted averaging:
L[k]
ssc =L[k−1]
ssc ∗G,Y[0...K]
ssc =y(L[0...K]
ssc ),
P[k]
ssc =Y[k]
ssc ·P[k−1]
ssc ∗U
Y[k]
ssc ∗U
.
Result The process to building the final L0
ssc is similar to
L0
sc, except we need to take special care of choosing the
correct level. This heavily depends on how many pixels
were actually separated in our scene. We introduce an ar-
bitrary parameter kssc (0 ≤kssc ≤K) set to kssc =0.7·Kin
our implementation. The fetching equation is:
k0=`P[kssc]
ssc ,L0
ssc =L[k0]
ssc .
(f)
Gathering algorithm
(reference)
(b)
Naive MIP filtering
(c)
Improved filtering
with density blurring
(d)
Improved filtering
with density bluring,
luminance weighting
(e)
Result (density
blurring, luminance
weighting, separation)
(a)
Input image L
Figure 5: Comparison of filtering the input image (a) with
different approaches (b)-(e), where (e) is the final result.
The gathering algorithm (f) can be understood as a filtering
reference but is too slow for real-time use.
4.4 Final Compositing
The final step is to compose the output L0. It is based
on three images that we have computed during the pre-
vious steps: Lat corresponds to the radiance that reaches
the camera without any interactions, L0
sc and L0
ssc together
represent the radiance scattered on the way to the camera.
The final output corresponds to adding the results together:
L0=Lat +L0
sc +L0
ssc.
5 Implementation
We implemented the filtering in GPU shaders in our cus-
tom engine based on bgfx1, where we also measured the
performance. To prove that the method itself can be added
to existing frameworks and engines, we have also success-
fully implemented our post-processing filtering in Unity2
and used it in existing scenes without any difficulties.
In our implementation, floating-point GPU buffers are
used as we need an HDR input and floating-point precision
intermediate results. It is beneficial to use RGBA buffers,
where the alpha channel can store the integrated densities.
1https://github.com/bkaradzic/bgfx
2https://unity3d.com/
Proceedings of CESCG 2018: The 22nd Central European Seminar on Computer Graphics (non-peer-reviewed)
Resolution Filter size
Interpolation technique Separation (optional)
Linear-bilinear Linear-bicubic Cubic-bicubic Linear-bicubic
1280 ×720
2×2 0.9 ms 1.1 ms 1.6 ms +0.6 ms
4×4 1.9 ms ♥2.1 ms 2.6 ms +1.0 ms
6×6 3.7 ms 3.9 ms 4.3 ms +2.3 ms
1920 ×1080
2×2 2.0 ms 2.5 ms 3.4 ms +1.2 ms
4×4 4.3 ms ♥4.8 ms 5.6 ms +2.1 ms
6×6 8.3 ms 8.7 ms 9.7 ms +3.6 ms
Table 1: Total times for post-processing a single frame using our method on NVIDIA GeForce GTX 660. In practice,
the 4 ×4 linear-bicubic filtering (♥) offers the best compromise between price and quality. The times of the optional
separation step (last column, see Section 4.3) have to be added to the total if the step is enabled.
In animated scenes, it is critical to use a smooth interpo-
lation method for fetching from the MIP maps, i.e., at least
a linear-bicubic interpolation. In theory, we need 16 tex-
ture reads for a bicubic interpolation, but we accelerated
the interpolation to 4 reads benefiting from the hardware
bilinear filtering as shown by Sigg and Hadwiger [13].
Furthermore, not computing all MIP levels down to 1×1,
but stopping sooner, is acceptable in a lot of smaller scenes
and can improve the performance.
Performance Let us note that our demo application is a
prototype and the fragment shaders are not primarily op-
timized for performance. In production, lot of branching
and parameters could be removed and/or optimized.
The total times to process a single input image and ren-
der the participating effects using our method are detailed
in Table 1. The times were measured using scenes with a
lot of point lights and emissive textures as can be seen on
the screenshots throughout the paper (Figures 1, 6).
Rendering the participating effects takes only a few mil-
liseconds even for the Full HD resolution using a rather old
GPU. The processing times of 2.1 ms and 4.8 ms (♥) for
HD and Full HD resolutions respectively correspond to the
performance measured by Elek et al. [3] for homogeneous
media. The framerate remained stable and the application
was fully responsive and usable in real time.
Figure 6: The same scene rendered using our method. The
ambient color in the 4 pictures was influenced by changing
the sky color and mixing the absorption with emissions.
6 Discussion
6.1 Comparison to Existing Solutions
There exist various empirical approaches to rendering
participating media. Commonly, simple alpha blending
with a constant fog color is used [10, 14]. It is not physi-
cally based, cannot blur the light sources and the geometry
and does not conserve energy. More advanced approaches
use billboards and particle effects [8], but they are only
useful for local effects such as smoke and have the same
disadvantages as noted above.
Our solution is built on a physically-based post-
processing method of Elek et al. [3] capable of rendering
multiple-scattering effects in real time, but only in homo-
geneous media. In this paper, we extended their method
for inhomogeneous media and by adding additional steps
for eliminating the filtering artifacts with HDR inputs.
Shinya et al. [12] recently (2016) presented a novel
physically-based post-processing solution built on a differ-
ent background. They also approximate multiple scatter-
ing in inhomogeneous media. They even claim to achieve
more precise results than Elek et al. when compared to a
path-tracing reference. Unfortunately, according to their
article, rendering a single 640 ×480 frame on a compa-
rable hardware takes 100-110 ms, which is approximately
two orders of magnitude slower than our approach.
Wronski [14] presented a solution successfully used in
AAA video games, e.g., in Assasin’s Creed IV. Their post-
processing volumetric fog runs smoothly on game con-
soles such as Xbox One. They represent the camera frus-
tum in a low-resolution 3D texture. Unlike our method,
their solution accumulates single-in-scattering from light
sources directly, so it does not blur the geometry and does
not handle emissive materials. However, it supports volu-
metric shadows and arbitrary heterogeneous media as they
use ray marching to handle the densities.
Finally, we mention a completely different approach of
Mitchell [6] used for fast screen-space rendering of vol-
umetric shadows in homogeneous media. Unfortunately,
it only supports a single light source (e.g., sun) and is not
intended for multiple scattering in the whole scene.
Proceedings of CESCG 2018: The 22nd Central European Seminar on Computer Graphics (non-peer-reviewed)
6.2 Limitations and Future Work
Temporal Coherence Our method provides good tem-
poral coherence in animated scenes, which can be veri-
fied in a supplemental video. However, it still suffers from
a common limitation of screen-space approaches: it can
only work with information present in the inputs.
When certain objects get occluded or move out of the
image, they may pop and disappear in subsequent frames
causing noticeable flickering. This occurs especially with
distant light sources in sub-pixel regions. Approaches
such as deep screen space by Nalbach et al. [7] could be
implemented and evaluated to overcome these problems.
Participating Effects Before Reflections We only pro-
cess the radiance that has already been reflected from sur-
faces, i.e., on the path between surfaces and the camera.
However, in real media, the attenuation and scattering also
happen on the paths between light sources and surfaces.
For a future work, one could try implementing the partici-
pating effects in the light shaders as well.
Ambient Term The overall atmosphere of the scenes
may be influenced by the total environmental illumination
caused by all light sources together. In Figure 6, we sim-
ulate the ambient color by adding an emissive term to the
medium. For a future work, the ambient term could be
calculated dynamically from the scattering buffers.
Volumetric Occlusions Similarly to the methods of
Elek et al. [3] and Shinya et al. [12], we ignore volumetric
occlusions, hence cannot render phenomena such as cre-
puscular rays. For a future work, additional steps could be
introduced to detect volumetric collisions in screen-space,
e.g., similarly to Mitchell [6].
Media Simplifications We only consider media with an-
alytically integrable and smooth density functions. The
filtering, however, accepts any integrated densities. For a
future work, one could try animating the media using ran-
dom noise. Furthermore, a different approximation based
on angular spreading could provide better results.
6.3 Conclusion
We have presented a novel physically-based approach for
real-time rendering of participating media. Unlike the ex-
isting solutions, ours offers fast multiple scattering in in-
homogeneous media. We have verified that our approach
can be implemented in existing engines and enables us to
render visually convincing animations in real time.
Acknowledgment
I would like to thank Oskar Elek for all the valuable advice
he offered during the creation of this work. I also thank the
Kenney Group, Jim van Hazendonk, and Christoph Peters
for releasing their 3D assets to the Public Domain, which
allowed us to use them in our application and screenshots.
This project has received funding from the Euro-
pean Union’s Horizon 2020 research and innovation pro-
gramme, under the Marie Skłodowska-Curie grant agree-
ment No 642841 (DISTRO), and the Czech Science Foun-
dation grant 16-18964S.
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Proceedings of CESCG 2018: The 22nd Central European Seminar on Computer Graphics (non-peer-reviewed)
