Position Estimation of Near Point Light Sources
using a Clear Hollow Sphere
Takahito Aoto† , Takafumi Taketomi† , Tomokazu Sato† ,Yasuhiro Mukaigawa‡ , Naokazu Yokoya††Nara Institute of Science and Technology, Japan, ‡ISIR, Osaka University, Japan†{takahito-a, takafumi-t, tomoka-s, yokoya}@is.naist.jp, ‡mukaigaw@am.sanken.osaka-u.ac.jp
Abstract
Light source

We present a novel method for estimating 3-D positions of near light sources by using highlights on theoutside and inside of a single clear hollow sphere. Conventionally, the positions of near light sources havebeen estimated by using observed highlights on multiple reference objects, e.g. mirror balls.

Clear hollow sphere

Camera1. Introduction
The estimation of 3-D positions of near light sources
is one of the fundamental problems in photometric computer vision research, e.g. BRDF estimation, virtual object re-lighting and shape-from-shading. Conventional
studies on the estimation of the positions of near lightsources can be grouped into several categories. Unlike
these approaches, geometric calibration of multiple reference objects is not required for our method, and thisresults in easy setup for measuring the 3-D positions oflight sources. In the experiments, the accuracy of estimated light positions by the proposed method is evaluated using both simulation and real data.
The most straight-forward approach is to use a calibrated stereo camera system. Sato et al.[4] used stereo
cameras with fisheye lenses to estimate the positions ofnear light sources that are widely distributed in a target
scene. To reduce the complexity of the stereo camera
system, reference objects are often used. Some methods use multiple reference objects to estimate 3-D light
positions by triangulation of multiple rays corresponding to highlights from the reference objects[2, 3, 5]. To
avoid a process of geometric calibration between multiple reference objects, some methods use a single reference object. Takai et al.[6] used a single diffuse sphereto estimate the 3-D position of a light source from thedistribution of an isophote on the sphere. They also proposed a skeleton cube to use the self-shadow as an indicator of 3-D information [7]. These methods[6, 7],

Figure 1. Clear hollow sphere. Left: setup
for estimating the 3-D positions of thelight sources. Right: an example of thecaptured image.
which use a single reference object, estimate 3-D positions based on the distribution of intensities over an
input image. This sometimes gives unstable results owing to difficulty on estimating an accurate isophoto andself-shadow.
To achieve easy setup and accurate estimation ofmeasuring positions for near light sources, we propose amethod that uses highlight positions on the outside andinside of a single clear hollow sphere. Figure 1 shows
an example of highlight positions on the clear hollowsphere. The primary contributions of our work are asfollows; (1) Geometric calibration for multiple reference objects is not required., (2) Positions of near lightsources can be accurately estimated by minimizing reprojection errors., (3) Corresponding pairs of highlight
positions under multiple light sources can be easily determined.

2. Position estimation of near light sources
2.1. Problem setting
We estimate near light source positions under the following conditions. The intrinsic camera parameters andsize of a clear hollow sphere are known. The relative
3-D positions of the sphere and a camera have alreadyFigure 2. Rays reflected on a clear hollowsphere.
been estimated, e.g. by the method[8]. We ignore interreflection and refraction. Multiple light sources can betreated simultaneously except for the critical conditions
detailed in Section 2.4.

2.2. Position estimation of near light sources
The proposed method estimates the 3-D positions ofnear light sources in a camera coordinate system from asingle image using a single clear hollow sphere. Figure
2 illustrates the geometric relationship of rays emitted
from a light source L and rays reflected at the positions(P1 , P2 ) on the sphere. These reflected rays are observed as two highlights of the positions (v1 , v2 ) on theimage plane. As shown in this figure, the 3-D position
L of the light source can be determined by the triangulation of two different paths of rays reflected from theinside and outside of the sphere.
Initially, four intersection points Pn (n = 1, · · · , 4)of two rays (v1 , v2 ) from the camera center and thesphere are computed. The directions of the light ln(n = 1, · · · , 4) from Pn are then computed by:
vmvmn− 2(Nn ·) · Nn , m =  , (1)ln = vm  vm 2
where Nn is a unit normal vector of the sphere at Pn .
Using the four rays from Pn in the direction ln , candidates for the light position L can be computed as thefour intersection points of these rays. True 3-D positionof L can be selected from these candidates using following constraints: (i) the light source does not exist in
the opposite direction of ln from Pn ; (ii)the light source
does not exist in the clear hollow sphere.

2.3. Minimization of re-projection errors
For computing light source position L using triangulation of rays, there is the problem that typically
two corresponding rays do not actually intersect in 3-D
space owing to errors of quantization, calibration, anddetection. Although minimization of re-projection errors is regarded as a reasonable way of computing a 3-D
position in computer vision field, this approach has notbeen employed in conventional methods that use mirror

Figure 3. Re-projection errors.
balls. This is due to the difficulty of direct computation of a 3-D reflected position Pn on the sphere fromthe fixed 3-D light position. In this study, we exploited
the ability to directly minimize re-projection errors using the interesting characteristic of epipolar geometryon this system. More specifically, as shown in Fig. 1,
all the highlights corresponding to a single light source
are observed on the epipolar line that is the intersection
line between the image plane and the epipolar plane thatpasses through the light source, camera center, and center of the clear hollow sphere in 3-D space. By using
the constraint of the epipolar line as show in Fig. 3, thesum of re-projection errors E is defined as follows
E = d21 + d22 ,di = Ui − Ûi ,

i = 1, 2,(2)where Ui is the observed reflected light position on theimage plane and Ûi is the nearest positions from Ui on
the epipolar line. E can be deformed as a function of θ,which is the angle of the epipolar line from a scan line,
as follows:( Un − C  sin(θ − θn ))2 ,(3)E(θ) =n=1,2

where C is the center position of the sphere in an input
image and θi is the rotation angle of the vector Ui − Cfrom a scan line. In our system, the re-projection error
can be minimized using a one dimensional search for
θ. After determining optimal θ, the 3-D position of thelight source can be computed as a unique intersection
point of the two rays Ûi .

2.4. Position estimation of multiple light sources
As mentioned in the previous section, reflected lightsfrom a single light source are observed on a uniqueepipolar line in the proposed system. In this case,searching for corresponding pairs of highlights is quite
easy even under multiple light sources, i.e. we canfind pairs of highlights on an epipolar line that passes
through these highlights and center of the sphere C. It
should be noted that if there exist multiple light sources
on a single epipolar plane, four or more highlights will
be observed on a single epipolar line. Although it may

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(d) ratio of errors ( (b) / (c) )Figure 4. A map of median errors for variable 3-D positions of a light source. The triangle anddouble circle indicate the position of the camera and the sphere.
possible to find pairs even in this case with some constraints, we do not discuss this problem in this paper.

3. Experiments
To evaluate the performance of the proposed method,
we conducted three experiments. First, we quantitatively evaluated the accuracy of the estimated 3-D positions using the proposed method in both simulation
and real scenes. The feasibility of our method was thendemonstrated in real scene using a desk lamp. For all
the experiments, we used a calibrated camera with a resolution of 1024 × 768 pixels, and a clear hollow sphereof radius 100 mm. The highlight positions on an input
image were determined automatically by detecting thecentroids of the highlighted regions.

3.1. Quantitative evaluation in a simulation
We evaluated the accuracy of the estimated resultsfor variable light positions. In this experiment, thesphere was placed 500 mm from the camera, and thecenter of the sphere on the optical axes of the camera. The 3-D position of the single light source waschanged on the Y-Z plane of the camera coordinatesthat passed through the camera position and the centerof the sphere. Input images were generated using ray
tracing software POV-Ray[1] for each position of thelight source at variable distances from the center of thesphere (100 mm to 800 mm, at 10 mm intervals) andvariable angles from the optical axis (0◦ to 180◦ , at 1◦intervals).
Figure 4 shows the median of the estimation errors,
which is computed as the distance between the estimated position and ground truth, over 10,000 trials foreach light position. (a) and (b) indicate the results fromminimizing the re-projection errors with different levelsof detection errors that are the Gaussian errors of thestandard deviation (σ[pixels]). From (a) and (b), we seethat the proposed method can accurately estimate the 3D light positions for large parts of the positions close
to the sphere except for the regions around the opticalaxis (A) and the lines that connects the camera positionand surface of the sphere (B). We confirmed that thecorresponding highlight positions were too close in theimage for region A, and the estimation of the normal
vector N of the highlight position was unstable aroundthe boundary of the sphere for region B. (c) shows theresult from computing the 3-D position as the middle
point of the rays. (d) shows the ratio of the estimation
errors for (b) and (c). As we can see in (d), minimizing
the re-projection errors has an advantage over conventional methods that use the middle position of the rays
as the 3-D position.

3.2. Quantitative evaluation in a real situation
The accuracy of the estimated light source positions
was from the optical axis evaluated in a real situation.
In this experiment, a halogen light source was placed ateight different distances from the center of the sphere.
All the positions were placed roughly on the line thatpassed through the center of the sphere about 70◦ fromthe optical axis. Figure 5 shows the relationship between the estimation errors and the distance from thecenter of the sphere. In this experiment, ground truths
for the light positions were measured using Total Station. The relationship between these errors and distances was similar to that evaluated in the simulation.
We confirmed that we can estimate accurate light positions if they are not too far from the sphere.

Distance error [mm]The distance between the center ofthe sphere and the light source position[mm]Figure 5. Position errors in real scene

3.3. Light position estimation for desk lamp
Our method has the significant advantage that wecan find corresponding pairs of highlights using epipolar lines. To validate the effectiveness of this feature, weconducted a 3-D reconstruction of the linear light sourceon the desk lamp. Figure 6 shows the input image andappearance of the target scene. In this experiment, byfinding pairs of highlights on epipolar lines, we have estimated 3-D position of the light for each epipolar plane.
Figure 6 (c) shows the estimated 3-D light source of thedesk lamp that is reconstructed as clouds of point lightsources.

(a) Input image(b) Appearance of the target situation

4. Conclusions
In this study, we have proposed a method for estimating positions of near light sources using a clear hollow
sphere. In the experiments, the performance of the proposed method was quantitatively evaluated using both
simulation and real data. We demonstrated the advantage of our method in a real situation by reconstructing the 3-D shape of a straight desk lamp. In future
work, we will consider inter-reflection and refraction toachieve a more stable and accurate estimation.

Acknowledgment
This research was supported by MEXT (Japan)
KAKENHI 23240024 and JSPS through the ”NEXTProgram,” initiated by CSTP.

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(c) Estimated light positions

Figure 6. Light position estimation for thedesk lamp.
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