Acquiring Non-parametric Scattering Phase Function from aSingle Image
Yuki Minetomo

Nara Institute of Science andTechnologyHiroyuki Kubo

Nara Institute of Science andTechnologyMikio Shinya

Takuya Funatomi

Nara Institute of Science andTechnologyYasuhiro Mukaigawa

Toho University / UEI Research

Nara Institute of Science andTechnology

Figure 1: Workflow for measuring the non-parametric phase function. (a) The scattering distribution is obtained from a single
camera image. (b) The integral Φ1 is calculated from the total flux of the scattered light Ψ1 over a circle of radius r on theacquired image. (c) According to our theory, partial derivative of Φ1 with respect to r is proportional to the phase function. (d)Rendering image of the media using the measured phase function.

ABSTRACTKEYWORDS

Acquiring accurate scattering properties is critical for renderingtranslucent materials such as participating media. In particular, thephase function, which determines the distribution of scattering
directions, plays a significant role in the appearance of the material.
While there are many techniques to acquire BRDF, there are only afew methods for the non-parametric phase function. We propose a
distinctive scattering theory that approximates the effect of single
scattering to acquire the non-parametric phase function from asingle image. Furthermore, in various experiments, we measured
the phase functions from several real diluted media and rendered
images of these materials to evaluate the effectiveness of our theory.

phase function, scattering, measurement

CCS CONCEPTS• Computing methodologies → Appearance and texture representations;
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SA ’17 Technical Briefs, November 27–30, 2017, Bangkok, Thailand© 2017 Copyright held by the owner/author(s).
ACM ISBN 978-1-4503-5406-6/17/11.
https://doi.org/10.1145/3145749.3149424

ACM Reference format:
Yuki Minetomo, Hiroyuki Kubo, Takuya Funatomi, Mikio Shinya, and Yasuhiro Mukaigawa. 2017. Acquiring Non-parametric Scattering Phase Function from a Single Image. In Proceedings of SIGGRAPH Asia 2017 TechnicalBriefs, Bangkok, Thailand, November 27–30, 2017 (SA ’17 Technical Briefs),
4 pages.
https://doi.org/10.1145/3145749.3149424

1

INTRODUCTION

Participating media exists everywhere in our surroundings. To
reproduce a realistic computer graphics of a medium, the scattering
phenomena occurring inside the material needs to be describedaccurately. The rendering of the scattering material depends on
several of its physical characteristics. The phase function is one
such characteristic and determines the distribution of scattered
light direction. Therefore, measuring the phase function of a real
object is important in portraying its appearance realistically.
Similar to the phase function, the bidirectional reflectance distribution function (BRDF) also describes the distribution of reflected
light. Although, recent studies on measuring BRDF adopt a nonparametric representation, phase-function measurements usuallyadopt parametric models such as the Henyey-Greenstein (HG)SA ’17 Technical Briefs, November 27–30, 2017, Bangkok, Thailand

Y. Minetomo et al.

Table 1: Symbols used in equations.
r̂, r, zŝ, sσtσsp(s)position (r̂ = r + zẑ)
direction (ŝ ≃ s + ẑ)
extinction coefficientscattering coefficientphase function

phase function[Henyey and Greenstein 1941], in many cases. The
parametric model, in an approximate manner, can easily reproduce phenomena from a small number of parameters. However,it sacrifices some physics plausibility and there are limitations in
its capacity to generate correctly a realistic appearance of actual
media. Nevertheless, Hawkins et al.[2005] proposed a method tomeasure the phase function directly. However, it requires extensive
measurements using special equipment to acquire the variety ofangular data.
We hereby propose a method for acquiring the non-parametricphase function of a real media from a single image. Our method
is based on a distinctive formulation of the relationship between
the single scattered field and the phase function. To validate ourmethod, we prepare a projector-camera optical system that enables
the phase function to be measured from a single image captured
with simple equipment.
In summary, the contributions of this paper are as follows:• A formulation of single scattering fields is proposed.
• A simple method is proposed to measure the non-parametricphase function from just a single image.
• Experiments using simple equipment have been performed
demonstrating the technique.

2incident ray

RELATED WORKIn the field of graphics, acquiring scattering properties from real
media has been studied. Jensen et al.[2001] obtained scattering properties using the diffusion approximation. The method assumes thatmultiple scattering is occurring with the optically dense medium,so that certain characteristics of the medium from scattering properties can be estimated but not the phase function.
Narasimhan et al.[2006] measured scattering properties using thesingle scattering approximation. Their method estimates scattering
properties using the minimizing error function about image formulation model of single scattering and observation. This method
represents the phase function as a HG phase function. We believe
the HG function is insufficient in reproducing the properties ofactual media[Gkioulekas et al. 2013a].
Gkioulekas et al.[2013b] measured scattering properties by solving the problem of appearance matching. They proposed to represent the phase function as a convex combination of a tent function
and a weight factor. This function is more flexible than the HG
phase function. To ensure the estimation is accurate, it requires
many observations using a special optical system that requires acomplex calibration process.
Whereas the above methods adopted the parametric model for
representing the phase function, other methods adopt a non-parametricapproach. Hawkins et al.[2005] measured scattering properties byz’(t)zviewing rayr’(t)rtr

^θss^

z
scattering medium^

Figure 2: Coordinate system and experimental arrangement.

developing a laser-scanning system. Because there is no model forthe phase function, their system directly measures the phase function from a wide range of directions. Although they built special
equipment with a conical mirror, scattering from some directions
were unable to be measured, specifically scattering angles ≈ 0 and πof the phase function. In these instances, an optical element used in
the measurement blocks the path of the ray. Moreover, it is not easy
to split the measurement into scattering and direct components.

3

FORMULATION OF SCATTERED FIELDSIn this section, we derive a direct and simple relationship between
single scattering fields and the phase function, which then permits
a one-shot measurement of phase functions. The main symbols
used in the equations are listed in Table 1.
Consider the situation shown in Fig. 2; a perpendicular incident
ray scatters in an uniform participating medium. For a simpler
formulation, let us assume:• the medium is optically thin, and single scattering is dominant,• the medium produces forward scattering, and• the phase function of the medium is axial-symmetrical depending only on the angle between the directions of incident
and scattered light.
The first assumption allows the single-scattering approximation
to be imposed. This can be satisfied by controlling the density ofthe medium during measurements. From the second, the angle θ
between the scattered light and the ray axis ẑ can be regarded tobe small, and the cosine can be approximated asŝ · ẑ = cos θ ≈ 1.
As reported in previous works [Gkioulekas et al. 2013b; Narasimhan
et al. 2006], most participating media were found to be dominantly
forward-scattering, and thus comply with this assumption. The
third assumption relates to the symmetry of the scattering andis commonly made in both measurements and image generation.
There is though no significant lost in generality.

Acquiring Non-parametric Scattering Phase Function

SA ’17 Technical Briefs, November 27–30, 2017, Bangkok, Thailand

The direct light I 0 measured at position r , depth z, in the directions is given by
I 0 (z, r , s) = δ (r )δ (s)e −σt z ,(1)where δ () denotes the Dirac delta function. Note that I 0 (z, r, s) , 0
where r = 0 and s = 0, which is along the incident light. The single
scattering field I 1 can then be calculated by a line integral along aviewing ray denoted byz ′ (t)=z − t(ẑ · ŝ) ≃ z − t(2)′=r − ts.

(3)

r (t)
Therefore,I 1 (z, r , s)

==

∫ z
∫
σse −σt t
I 0 (z ′ (t), r ′ (t), s ′ )p(|s − s ′ |)ds ′dt
4π 0Ω∫
σs −σt z zeδ (r − ts)p(|s |)dt,(4)4π0where Ω represents a unit sphere, and the phase function p(|s − s ′ |)is normalized to 4π .
∫p(|s − s ′ |)ds = 4π
(5)Figure 3: Experimental setup.

4 EXPERIMENT
4.1 Experimental setup
In this section, we describe the experimental setup that we erectedfollowing the requirements of the formulation proposed for measuring the non-parametric phase function of real media.

ΩThe total flux Ψ1 is then obtained by integrating Eq. 4 over s:∫Ψ1 (z, r ) =I 1 (z, r, s) cos θds,Ω∫ ∫σs −σt z z=eδ (r − ts)p(s)dsdt,(6)4π0∫ Ω∞
σs 1 −σt ze=p(u ′ )du ′,(7)4π |r |
|r |/zwhere u = 1/t and u ′ = |r |u. Note that the flux Ψ1 depends onthe distance from the axis, |r |, and is constant on a circle |r | =constant. To remove the apparent singularity from the factor 1/|r |,we integrate Ψ1 over the circle, which yields∮Φ1 (z, |r |) =|r |Ψ1 (z, r )dr|r |=∫σs −σt z ∞ep(u ′ )du.
2|r |/z(8)This states an important fact that Φ1 is proportional to an integralof the phase function. By taking the derivative with respect to |r |on both sides, we obtain ∂Φ1σs|r |= − e −σt z p.
(9)
∂r2zz
Note that |r |/z represents the scattering angle.
The flux integral expressed by Eq. 7 can be optically realizedusing a diffusing screen. Therefore, Eq. 9 suggests the following
simple one-shot measurement of the phase function,(1) set a diffusing screen on the boundary of the medium,(2) set a camera on the incident ray axis and image the scattering
field,(3) calculate the circular integral (Eq. 8) from the image, and(4) calculate its derivatives.

4.2

Setup

A schematic illustration and a photograph of the experimental
setup is presented in Fig. 3. In the formulation, the incident ray isconsidered to be a narrow beam. To ensure the beam is narrow, weused an LED projector (Optoma ML750) and two pinholes of thesame size (0.4 mm). We setup the projector to direct a beam of whitelight, through both pinholes and mount them over a water tankensuring that the beam is incident perpendicularly to the surfaceof the participating media. A diffuse screen is set under the tank tointegrate the flux of scattered light. A DSLR camera (NIKON D5300)
is placed under the screen to image the scattered light. In total, ourmethod requires only a projector, a camera, a diffuse screen, a tank,
two pinholes, and holders for each of them. Preparing the setup isvery easy compared with setups of previous studies measuring thenon-parametric scattering phase function.
We confirmed that the setup functioned correctly by testing
several media (milk, apple juice, and chardonnay wine) representing
different types of characteristics (optically thick and optically thin).
To comply with the assumption that single scattering is dominant,the media was diluted in water. The diluted media was poured into
the tank to a depth of 50 mm.

4.3

Experimental Results

The measured phase function is presented in Fig. 4. The RGB color
channels of the captured image are independently used for calculating the phase function. The measured phase functions are different
for each medium; differences arising from color for one medium
are relatively small but still can not be ignored.
There are a few limitations in our method. The major limitationis that, as found with other methods, it is difficult to measure thephase function of the scattering angle ≈ 0. In real settings, direct
light is much stronger than scattered light and causes bleeding oflight on the screen. Thus, the scattered light near the center part isbiased and affects the phase function as well.

SA ’17 Technical Briefs, November 27–30, 2017, Bangkok, Thailand

MilkY. Minetomo et al.

Apple juice

Chardonnay wine

Figure 4: Measured phase functions.

Milk

Apple juice

Chardonnay wine

Figure 5: Rendered images.

4.4

Rendering image

Figure 5 presents the rendered image of measured medium in theglass. We use the path tracing algorithm for rendering. Image sizeis 1024 × 1024 and using 20,000 photon samples. These images arerendered by using measured non-parametric phase function as tabulated data. The scattering and extinction coefficients are manually
determined. Furthermore, since we can not measure whole direction, we complemented the missing value of the phase function
from the measured data for the rendering purpose.

5CONCLUSION

We have proposed a method that acquires the non-parametric phasefunction from a single image. Our setup for the measurement requires neither special equipment nor complicated calibrations. Also,
we have presented rendered images of several different media using
the measured phase functions.
Our proposed model still has a few limitations. Because we ignored the effect of higher-order scattering, we diluted the media todecrease the scattering effect. We also assumed that the scattering
distribution is concentrated in the forward direction so that thephase function of the media corresponding to back scattering neednot be measured.
While we can now measure the phase function of a wide rangeof media, in rendering the image, it is necessary nevertheless tomeasure other scattering characteristics such as the coefficientsof extinction and scattering. Our scattering model presumes thesecoefficients are known. The objective, which remains as an open
problem, is to measure the scattering properties of a wide range ofmedia and to summarize them all in a single data set.

ACKNOWLEDGMENTS
This work is partly supported by JSPS KAKENHI JP15K16027,
JP26700013 and JP15H05918.

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