Symmetric-light Photometric Stereo
Kazuma Minami Hiroaki Santo Fumio Okura Yasuyuki Matsushita
Graduate School of Information Science and Technology, Osaka University{kazuma.minami,santo.hiroaki,okura,yasumat}@ist.osaka-u.ac.jp

Abstract
This paper presents symmetric-light photometric stereo
for surface normal estimation, in which directional lights
are distributed symmetrically with respect to the optic center. Unlike previous studies of ring-light settings that required the information of ring radius, we show that even
without the knowledge of the exact light source locationsor their distances from the optic center, the symmetric configuration provides us sufficient information for recovering
unique surface normals without ambiguity. Specifically, under the symmetric lights, measurements of a pair of scene
points having distinct surface normals but the same albedo
yield a system of constrained quadratic equations about thesurface normal, which has a unique solution. Experiments
demonstrate that the proposed method alleviates the needfor geometric light source calibration while maintaining theaccuracy of calibrated photometric stereo.

1. Introduction
Photometric stereo estimates surface normals from a
set of measurements captured under varying light directions [23, 21]. Because of its capability of estimating thesurface normal at each pixel, photometric stereo is useful for recovering a detailed shape of real-world scenes.
Conventional photometric stereo methods [8, 17], including
recent learning-based methods [14, 12, 7], assume known
light directions obtained by geometric light source calibration [26, 15] for determining surface normals without ambiguity. To eliminate the need for the light source calibration,
uncalibrated photometric stereo [10, 5] has been studied.
However, their surface normal estimates generally sufferfrom ambiguities [2] due to the lack of knowledge about thelight source directions. More recently, a ring-light configuration [27, 20, 3], in which point light sources are located ona circle centered at the camera, has been introduced by exploiting viable constraints in light configurations. While thering-light configuration is useful for better constraining the
problem, their estimates still suffer from ambiguities [27].
In this paper, we show that the subset of ring-light con-Symmetric-light (Ours)

Ring-light⊆
Figure 1. Symmetric- and ring-light configurations. The planes on
which point light sources are placed are parallel to the image plane
and aligned to the optic center.

figuration, namely, a symmetric-light configuration (Fig. 1),
can sufficiently constrain the problem by exploiting thesymmetry of the lights, yielding unique surface normal estimates without ambiguity. In the symmetric-light configuration, two sets of two point light sources are symmetrically placed w.r.t. the x- and y- axes of the camera in thesame interval, respectively. We assume that these four point
light sources are placed infinitely far away from a scene, resulting in symmetric directional lightings. The differencesof the symmetric measurements produce independent constraints over x-, y-, and z- components of surface normals.
By taking two scene points having distinct surface normals
but the same albedo, our method creates a system of constrained quadratic equations about the surface normals thathas a unique solution. The experiments demonstrate thatthe proposed method achieves comparable accuracy to thecalibrated photometric stereo even without the need for directional light source calibration.
The chief contributions of this paper are twofold:
1. We introduce symmetric-light photometric stereo, in
which the light symmetry is exploited for constraining
the problem, for determining surface normals.
2. By being a subset of ring-light configuration, thesymmetric-light configuration is simple and easy to set
up and avoids explicit geometric light source calibration, while it achieves comparable accuracy to fully

Table 1. Summary of (a) calibrated, (b) ring-light, (c) symmetric-light, and (d) uncalibrated photometric stereo.

(a) calibrated [23](b) ring-light [27, 20](c) symmetric-light (ours)

known
?

Light sources

Unique solution
with?

(d) uncalibrated [10, 18]?

?

-

radius of ring /
integrability

calibrated photometric stereo.

2. Related works
We briefly review previous photometric stereo methods
that do not require explicit geometric light source calibration. This direction has been studied as uncalibrated photometric stereo [10] (Table 1 (d)), which assumes unknown
light directions and aims at simultaneously estimating both
surface normals and light directions. Hayakawa [10] introduced a method based on matrix factorization and showed
there existed a linear ambiguity in the surface normal estimates. Later, it is shown that surface integrability [11]can reduce the linear ambiguity to a three-parameter linear transform, which is known as the generalized basrelief (GBR) ambiguity [2, 25]. To resolve the GBR ambiguity, various external assumptions, such as albedo priors [18], specular spike [6], mutual reflections [4], a certain
type of parametric reflectance models [9], symmetry of reflectance [22, 24], have been exploited.
It has also been studied to constrain the problem of uncalibrated photometric stereo by putting assumptions onlight source distributions. Zhou and Tan [27] have proposeda ring-light configuration (Table 1 (b)), in which the point
light sources are located on a circle centered on the camera. They show that the constraint from the ring lights can
reduce the ambiguity of uncalibrated photometric stereo up
to mirror transformation, rotation, and scaling ambiguities.
To resolve these ambiguities, they use (1) known rotation
and radius of ring lights with equal intensities or intervals,(2) the assumption of surface integrability and scene points
with the same albedo, or (3) multi-view information. Shiradkar et al. [20] also show that ring lights that have an
equal interval and known radius but unknown rotation still
exhibit scaling and rotation ambiguities, which may be resolved by multi-view information. Chandraker et al. [3] use
denser ring lights realized by rotating a point light source
around the camera. From the differences of the measurements when slightly rotating the light source, their method
can estimate the surface normals of a scene with unknown
but isotropic reflectances.
As summarized in Table 1, our method is in between calibrated and uncalibrated photometric stereo. The

constant albedo(two pixels)

constant albedo /integrability

symmetric-light setting (Table 1 (c)) can be regarded as asubset of ring lights; thus the setup is simpler than the ringlight setting. By exploiting the new constraints yielded fromthe light symmetry, we show that the proposed method canuniquely determine the surface normals without ambiguity, even without the knowledge of the radius, assumptionof surface integrability, and multi-view information that isneeded in previous approaches.

3. Proposed method
From four images captured under symmetric lights, ourphotometric stereo uses a pixel pair having the same albedo
to estimate the surface normals for these pixels. In what follows, we describe our formulation and the solution method
starting from the image formation model. We here assume
the appropriate pixel pairs are given, while we later introduce a practical method to select reasonable pixel pairs.
Figure 2 shows the overview of the proposed symmetriclight photometric stereo.

3.1. Image formation model
Suppose that a Lambertian surface is illuminated by directional lights and observed by a fixed camera. The measurement at a pixel m(s) ∈ R+ under s-th light source canbe described by  m^{(s)} = \rho \vn ^\top \vl _s, \label {eq:image_formation} (1)where n ∈ S 2 ⊂ R3 and ρ ∈ R+ are the surface normal
and albedo, and ls ∈ S 2 is the s-th light direction.
The proposed method uses four light sourcesthat are symmetrically placed and have a uniform intensity. The light directions are written as
y

𝐥𝐥2

𝐥𝐥1

  \begin {split} \label {eq:light_distribution} \vl _0 &= [l_x, 0, l_z]^\top , \\ \vl _1 &= [0, l_y, l_z]^\top , \\ \vl _2 &= [-l_x, 0, l_z]^\top , \\ \vl _3 &= [0, -l_y, l_z]^\top , \end {split}

x

𝐥𝐥3

𝐥𝐥0

z

(2)

where lx , ly , lz > 0 are unknown. The light sources canrotate around the z-axis; however, for the simplicity of notation, we explain the method under this setting. The measurements under these four light directions are described by

Capturing setup

Symmetric-light

Capture device
LED1
LED2

𝑚𝑚

1

LED0
2
−𝑙𝑙𝑥𝑥

LED3

0

𝑚𝑚

Estimation

Feature extraction

Inputs
𝑚𝑚 2

1

3

𝑙𝑙𝑦𝑦

−𝑙𝑙𝑦𝑦
𝑚𝑚

Constraints :𝐧𝐧 22 = 1 and 𝑙𝑙𝑥𝑥 = 𝑙𝑙𝑦𝑦
0

3

𝑙𝑙𝑥𝑥

𝑎𝑎

×

𝑛𝑛𝑥𝑥

×

×

𝑎𝑎𝑎

pixel 1 : 𝑎𝑎

𝑚𝑚 0 + 𝑚𝑚 1
+𝑚𝑚 2 + 𝑚𝑚 3

𝑚𝑚 1 − 𝑚𝑚 3

𝑚𝑚 0 − 𝑚𝑚 2

𝑛𝑛𝑦𝑦

pixel 2 : 𝑎𝑎𝑎

𝑏𝑏

𝑏𝑏

×

𝑏𝑏𝑏

𝑏𝑏𝑏

×

𝑐𝑐

𝑛𝑛𝑧𝑧 ⨀

𝑐𝑐𝑐

𝑐𝑐

×

𝑐𝑐𝑐

Linear system
𝑨𝑨𝑨𝑨 = 𝒃𝒃
Surface normals

pixel pairFigure 2. Overview of the proposed method. We capture four images under the symmetric-light configuration and extract features using thesymmetry of the measurements. We construct a linear system from the features and constraints and solve it to estimate the surface normals.

Eq. (1) using the unit surface normal n = [nx , ny , nz ]⊤ as  \left \{ \begin {aligned} \begin {split}\label {eq:4eq} &m^{(0)} = \rho \vn ^\top \vl _0 = \rho n_{x}l_x + n_{z}l_z, \\ &m^{(1)} = \rho \vn ^\top \vl _1 = \rho n_{y}l_y+ n_{z}l_z, \\ &m^{(2)} = \rho \vn ^\top \vl _2 = -\rho n_{x}l_x + n_{z}l_z, \\ &m^{(3)} = \rho \vn ^\top \vl _3 = -\rho n_{y}l_y + n_{z}l_z. \end {split} \end {aligned} \right .

on a circle, i.e., lx = ly , we obtain the following constraints:
  \begin {split} \label {eq:xy_ratio} \left ( \frac {b}{a}, \frac {b^{\prime }}{a^{\prime }} \right ) = \left ( \frac {n_{y} l_y}{n_{x} l_x}, \frac {n_{y}^{\prime } l_y}{n_{x}^{\prime } l_x} \right ) = \left ( \frac {n_{y}}{n_{x}}, \frac {n_{y}^{\prime }}{n_{x}^{\prime }} \right ). \end {split} 
(6)

(3)

3.2. Solution method
From the symmetry of the light directions, we obtain thefeatures a, b, and c by linearly combining the measurements
in Eq. (3) as:

By squaring both sides of Eqs. (5) and (6) and plugging
in the unit norm constraints, ||n||22 = 1 and ||n′ ||22 = 1, we
have a system of quadratic equations as  \label {eq:Dx} \underbrace { \begin {bmatrix} a^{\prime 2} & 0 & 0 & -a^2 & 0 & 0\\ 0 & b^{\prime 2} & 0 & 0 & -b^2 & 0\\ 0 & 0 & c^{\prime 2} & 0 & 0 & -c^2\\ b^2 & -a^2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & b^{\prime 2} & -a^{\prime 2} & 0\\ 1 & 1 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 1 & 1 \end {bmatrix} }_\vD \underbrace { \begin {bmatrix} n_{x}^2\\ n_{y}^2\\ n_{z}^2\\ n_{x}^{\prime 2}\\ n_{y}^{\prime 2}\\ n_{z}^{\prime 2} \end {bmatrix} }_\vx = \underbrace { \begin {bmatrix} 0\\ 0\\ 0\\ 0\\ 0\\ 1\\ 1 \end {bmatrix} }_\vb . 

  \begin {split} \label {eq:3eq} \frac {m^{(0)} - m^{(2)}}{2} &= \rho n_{x}l_x \equiv a, \\ \frac {m^{(1)} - m^{(3)}}{2} &= \rho n_{y}l_y \equiv b, \\ \frac {m^{(0)} + m^{(1)} + m^{(2)} + m^{(3)}}{4} &= \rho n_{z}l_z \equiv c. \end {split}(4)

These features represent the relative magnitudes for eachcomponent of the surface normals, nx , ny , and nz , as illustrated in Fig. 2.
Our method uses a pair of scene points (pixels) having
distinct surface normals but the same albedo. Under theassumption of directional lights, all the scene points sharethe same light directions. We compute the ratios of features
in Eq. (4) from a pixel pair with the same albedo ρ = ρ′ :  \begin {split} \label {eq:constat_al} \left ( \frac {a}{a^{\prime }}, \frac {b}{b^{\prime }}, \frac {c}{c^{\prime }} \right ) &= \left ( \frac {\rho n_{x} l_x}{\rho ^{\prime } n_{x}^{\prime } l_x}, \frac {\rho n_{y} l_y}{\rho ^{\prime } n_{y}^{\prime } l_y}, \frac {\rho n_{z} l_z}{\rho ^{\prime } n_{z}^{\prime } l_z} \right ) \\ &= \left ( \frac {n_{x}}{n_{x}^{\prime }}, \frac {n_{y}}{n_{y}^{\prime }}, \frac {n_{z}}{n_{z}^{\prime }} \right ), \end {split} (5)where prime indicates the quantities of the other pixel in thepair. Additionally, by assuming that light sources are placed(7)
Our method treats the squares of unknowns⊤[n2x , n2y , ..., n′2
as a new variable x (≥ 0). Thez ]least squares approximate solution x∗ can be derived byminimizing the objective function:
  \V {x}^* = \argmin _\vx \left \|\vD \vx - \vb \right \|_2^2 \quad \mathrm {s.t.} \quad \vx \geq \V {0}. \label {eq:objective} (8)We use a method of non-negative least squares [13] for obtaining the approximate solution x∗ .
Because lx , ly , lz , ρ, c, c′ > 0 for any pixel in Eq. (4), thefollowing conditions should also be satisfied:  \left \{ \begin {aligned} &n_{x} \geq 0 &(\mathrm {if}\ a \geq 0 )\\ &n_{x} < 0 &(\mathrm {otherwise})\\ \end {aligned} \right . , \left \{ \begin {aligned} &n_{y} \geq 0 &(\mathrm {if}\ b \geq 0 )\\ &n_{y} < 0 &(\mathrm {otherwise})\\ \end {aligned} \right . , \left . \begin {aligned} &n_{z} \geq 0 & \end {aligned} \right . \\ \label {eq:ineq_constraint} (9)

Using the constraints in Eq. (9), we can uniquely determine the surface normals n = [nx , ny , nz ]⊤ andn′ = [n′x , n′y , n′z ]⊤ from the estimated x.

3.3. Solvability condition
For ensuring that the objective Eq. (8) has a unique solution, the coefficient matrix A in Eq. (7) should be full rank.
This implies that the two pixels should have distinct surface
normals in terms of the elevation angles.
Let θ and ϕ as the elevation and azimuth angles
of a surface normal, respectively, and suppose thatthe two pixels have the same elevation angle, i.e.,θ = θ′ . With the angular expression of surface normals n = [nx , ny , nz ]⊤ = [sin θ cos ϕ, sin θ sin ϕ, cos θ]⊤ ,Eqs. (5) and (6) are written as

Surface normal clustering To select a pixel pair having
distinct surface normals in terms of their elevation angles,
we cluster surface normals based on their elevation angles.
While the elevation angles are unknown, let us suppose that
two pixels have elevation angles θ and θ′ ∈ 0, π2 . We
use a metric r for assessing the difference of the elevation
angles defined as  r=|\tan ^2\theta - \tan ^2\theta ^{\prime }|. \label {eq:r_metric} 

(11)

From Eq. (4), we have  \begin {split} \left (n_{x},n_{y}, n_{z} \right ) = \left ( \frac {a}{\rho l_x}, \frac {b}{\rho l_y}, \frac {c}{\rho l_z} \right ). \end {split} \label {eq:n_expression}(12)

Substituting Eq. (12) into the metric r in Eq. (11), we have  r &= \left |\tan ^2\theta - \tan ^2\theta ^{\prime }\right | = \left |\frac {1 - \cos ^2\theta }{\cos ^2\theta } - \frac {1 - \cos ^2\theta ^\prime }{\cos ^2\theta ^\prime }\right | \\ &= \left | \frac {1 - n^2_z}{n^2_z} - \frac {1 -{n^\prime _z}^2}{{n^\prime _z}^2} \right | = \left | \frac {n^2_x + n^2_y}{n^2_z} - \frac {{n^\prime _x}^2 + {n^\prime _y}^2}{{n^\prime _z}^2} \right | \\ &= \frac {l^2_z}{l^2_x} \left |\frac {a^2 + b^2}{c^2} - \frac {a^{\prime 2} + b^{\prime 2}}{c^{\prime 2}} \right |,  \left \{ \begin {aligned} \label {eq:two_pix_const_pol} \frac {a}{a^{\prime }} &= \frac {\cos \phi }{\cos \phi ^{\prime }} \\ \frac {b}{b^{\prime }} &= \frac {\sin \phi }{\sin \phi ^{\prime }} \\ \frac {c}{c^{\prime }} &= 1 \end {aligned} \right . , \quad \left \{ \begin {aligned} \frac {b}{a} &= \frac {\sin \phi }{\cos \phi }\\ \frac {b^{\prime }}{a^{\prime }} &= \frac {\sin \phi ^{\prime }}{\cos \phi ^{\prime }} \end {aligned} \right . .

(10)Equations in (Eq. (10)) are independent of the elevation angles θ and θ′ , and it indicates that we cannot determine theunique solution under the condition of θ = θ′ . Indeed, thisdegenerate condition introduces rank deficiency in the coefficient matrix A in Eq. (7).
To sum up, for uniquely determining surface normals,
the proposed method requires a pixel pair that has (1) thesame albedo, i.e., ρ = ρ′ and (2) distinct surface normals interms of the elevation angles.

4. Pixel-pair selection
Our method uses a pair of scene points with the same
albedo but distinct elevation angles of their surface normals.
For stably selecting such pixel pairs, we develop a simple
yet effective method based on albedo and surface normal
clustering.
Albedo clustering To select pixels having the same
albedo, we use a chromaticity-based clustering [18]to approximate the albedo clustering. From the four
RGB measurements under our symmetric-light setting foreach pixel, we computethe mean RGB measurementsm̄(R) , m̄(G) , m̄(B) . Using the mean RGB measurements,
we compute the chromaticity as
  \left ( \frac {\bar {m}^{(R)}}{\bar {m}^{(R)} + \bar {m}^{(G)} + \bar {m}^{(B)}}, \frac {\bar {m}^{(G)}}{\bar {m}^{(R)} + \bar {m}^{(G)} + \bar {m}^{(B)}} \\ \right ). 

Once the albedos are clustered, our method selects a pixel
pair that belongs to the same cluster.

with the assumptions that lx = ly and ρ = ρ′ . Since thelight directions lx and lz are constant across a scene, we can
2
2
′2
′2
define a distance r̂ = a c+b− a c+bthat is free from the2′2light directions. Using the metric r̂, we use K-means clustering to obtain clusters of similar surface normals in termsof their elevation angles. Once the clusters are obtained, apixel pair is selected by taking pixels from different clusters.
For robustifying the pixel selection, we avoid selecting
pixels from clusters that have large intra-cluster variances ofthe distance r̂. Namely, we avoid clusters with top 50 percentile intra-cluster variances and randomly sample fromthe remaining clusters. In our implementation, we sample
100 pixel pairs and take the median of those estimates by
vector median filter [1].

5. Experiment
We evaluate the proposed method using synthetic andreal-world data. For comparison, we use a calibrated photometric stereo method by Woodham [23], denoted as “Calibrated.” The estimation accuracy by the calibrated method
can be considered as the upper bound of the accuracy, as
our method only uses less information about the light directions. For the real-world data, we use the DiLiGenT
dataset [19], which is a benchmark dataset for calibrated
photometric stereo, as well as the data captured by our
setup. For the albedo and surface normal clustering, thenumbers of clusters are set to 20 and 50, respectively.
Since the proposed method is designed for grayscale measurements, the color measurements are first converted tograyscale in our implementation.

Albedoclustering

Surface normal
clustering

GT

Normal map
Proposed

Calibrated

Error map
Proposed
Calibrated

7.37 × 10−2

9.64 × 10−3

4.51 × 10−2

2.89 × 10−2

3.67 × 10−1

2.46 × 10−1

7.67 × 10−2

1.09 × 10−2

10.2

10.6

4.84 × 10−1

2.50 × 10−1

1.0 [deg.]

0.0 [deg.]

TEDDY

PLANT

with global illumination

PILLOW

TEDDY

PLANT

without global illumination

PILLOW

Image

Figure 3. Estimation results for our synthetic dataset. For each row, we show one of the input images, results of albedo and surface normal
clustering, the ground-truth and estimated normal maps for each method, and corresponding error maps. For each error map, we show themean angular errors in degrees. For each scene, we have two rendering conditions: with and without global illumination effects.

5.1. Evaluation with synthetic data
We generate our synthetic dataset by Blender1 using thescenes PILLOW, TEDDY, and PLANT2 rendered with theLambertian reflectance. The scenes are rendered with andwithout the global illumination effect, i.e., cast shadow andinterreflections, to assess the robustness of the methods. As
for the light directions, we set the elevation angles to 80
degrees, i.e., lx = ly = 0.1736, lz = 0.9848.
Figure 3 shows the visual results for our synthetic dataset
along with the mean angular errors of the estimated normal
maps in degrees. The proposed method shows almost com1 Blender 2.83 LTS, https://www.blender.org, last accessedon Aug. 17, 2021
2 CGtrader, https://www.cgtrader.com, last accessed on Aug.
17, 2021

parable accuracy with the calibrated photometric stereo,
while for some objects with slightly larger errors. In the
PILLOW scene, the proposed method has a relatively large
error at the boundary of the surface texture due to the errorof the albedo clustering. On the other hand, the PLANT
scene shows the applicability of the proposed method forscenes that do not satisfy the surface integrability.

5.2. Evaluation with real-world data
Figure 4 shows the results for the DiLiGenT dataset.
Since the DiLiGenT dataset uses 96 light directions roughly
arranged in a grid manner, we choose four light directions3that approximately satisfy the symmetric-light assumption,
3 The results with different choices of the light directions are shown inthe supplementary material.

Albedoclustering

Surface normal
clustering

GT

Normal map
Proposed

Calibrated

Error map
Proposed
Calibrated

30 [deg.]

BALL

Image

3.63

10.8

8.63

19.9

19.5

22.3

8.70

0.0 [deg.]

CAT

POT 2

BEAR

4.83

Figure 4. Estimation results for the DiLiGenT dataset. For each row, we show one of the input images, results of albedo and surface normal
clustering, the ground-truth and estimated normal maps for each method, and corresponding error maps. For each error map, we show themean angular errors in degrees.

camerasymmetric lights

LED arrays

objectFigure 5. Our capturing setup. We use four symmetric LEDs forour method, and 100 calibrated LEDs for obtaining the groundtruth of the surface normals.

all lights
symmetric lights

as shown in the left figure. Since
we focus on the Lambertian-like
scenes, we use BALL, BEAR,
POT2, and CAT scenes out of ten
scenes in the dataset.
Although the light configuration does not strictly satisfy oursymmetric-light assumption (discussed in Sec. 6 later), for
the BALL, BEAR, and POT2 scenes, the proposed method
still shows almost comparable accuracy to the calibrated
method even though the proposed method does not use theinformation of the light source calibration. For the CAT
scene, the proposed method has a larger error compared tothe calibrated method, due to the failure of the albedo clustering. The detail of the failure mode will be discussed inSec. 6.
Next, we show the results of scenes captured by our setup
with the symmetric-light configuration in Fig. 5. We use a
CCD camera (FLIR BFS-U3-28S5-C) with the image resolution of 1464 × 1936 pixels with 12 bit color depth and alens with the focal length of 25 mm. To obtain the groundtruth, we use 100 LEDs on the LED arrays, which are calibrated [16]. We use objects with simple (SPHERE andKITTEN) and complicated (LIZARD) shapes for scenes
with uniform albedo. SHEEP and CHALK scenes contain spatially-varying albedos. Figure 6 shows the estimated results for our dataset. Overall, the proposed method
achieves comparable results to the calibrated method. For
the CHALK scene, the proposed method exhibits a larger
error compared to the calibrated method in the concave regions since these regions are clustered as different albedo
due to cast shadows. In contrast, the other regions withspatially-varying albedos are correctly clustered and surface

Albedoclustering

Surface normal
clustering

GT

Normal map
Proposed

Calibrated

Error map
Proposed
Calibrated

SPHERE

Image

3.23

4.05

3.38

5.72

3.51

6.38

3.81

5.44

3.19

0.0 [deg.]

CHALK

SHEEP

LIZARDKITTEN

3.60

20 [deg.]

Figure 6. Estimation results for our dataset. For each row, we show one of the input images, results of albedo and surface normal clustering,
the ground-truth and estimated surface normal maps for each method, and corresponding error maps. For each error map, we show themean angular errors in degrees.

normals are accurately estimated.

discuss the source of errors in albedo clustering, surface
normal clustering, and the symmetric-light configuration.

6. Discussions
This paper studies the conditions derived from thesymmetric-light configuration for surface normal recovery.
As a result, we present the symmetric-light photometricstereo, in which directional lights are symmetrically distributed w.r.t. the camera. Using a pixel pair that has thesame albedo but different elevation angles of the surface
normals, the proposed method can uniquely determine thesurface normal without ambiguities. Unlike previous methods that use a ring-light, the symmetric-light alleviates theneed for assuming a known radius of lights and surface integrability. Our experiments demonstrate that the proposed
method works as good as calibrated photometric stereo even
without the need for the geometric light source calibration.
For improving the surface normal estimation accuracy,
there are a few venues that can be investigated. Here we

Albedo clustering For albedo clustering, our method
uses chromaticities as an approximation of albedos, because the true albedo is inaccessible. To observe the error caused by this approximation, we compute true albedos from the ground truth surface normal, apply the sameclustering method, and assess the surface normal estimation accuracy. Figure 7 shows the estimated results with andwithout true albedos. This result indicates that the proposed
method can be improved by a better albedo approximation
and albedo clustering.
Surface normal clustering We assess the effect of surface normal clustering to the surface normal estimation accuracy. Figure 8 shows the results of the synthetic scenes,
PLANT and CUBE, with and without surface normal clus-

GT normal map

Albedo clusteringwith GTwithout GTGTnormal map

Image

Surface normal
clustering

CUBE

PLANT

Image

Proposedwith GT

Calibrated
PLANT
CUBE
w/o clustering
w/ clustering w/o clustering
w/ clustering

22.3

10.9

8.70

0.0 [deg. ]Figure 7. Comparison of surface normal estimation accuracies
with and without true albedos (denoted as GT) for albedo clustering. The numbers below the error maps indicate the mean angularerrors in degrees.

tering. Since PLANT scene has a complex shape and various surface normals, even without surface normal clustering, we could obtain a reasonable result by random sampling, while surface normal clustering still improves the accuracy. On the other hand, the CUBE scene has only three
distinct surface normals, resulting in a larger error without
surface normal clustering. It indicates that accurate surface
normal clustering contributes to a better surface normal estimation accuracy.
Deviation of symmetric-light configuration We evaluate the effect of deviations of light directions from thesymmetric-light assumption in the surface normal estimation. Figure 9 shows the results of the synthetic experimentswith adding noise to the light directions. We use the samesettings in Sec. 5.1 and render the scenes without the global
illumination effects. The Gaussian noise with varying thenoise level is added to the elevation and azimuth angles ofthe light directions, respectively. The results indicate thatthe accuracy remains comparable to the calibrated method
even when the light directions are off from the symmetriclight assumption.
Limitations One of the limitations of the proposed
method is a distant light assumption. To make photomet-

1.38 × 10−1

4.51 × 10−2

8.02

1.34 × 10−2

0.0 [deg.]

0.1 [deg.]

Figure 8. Estimation results for the PLANT and CUBE scenes by
the proposed method with and without surface normal clustering.
The numbers below the error maps indicate the mean angular errors in degrees. “w/o clustering” randomly selects the pixel pairsthat have the same albedo.

Mean angular error [deg.]

Error map

30 [deg. ]Error map

Normal map

Normal map

without GT

17.5

PILLOW
PLANT
TEDDY

15.0
12.5
10.0

Proposed
Calibrated

7.5
5.0
2.5
0.0

0

1

2

3

4

5

6

7

Standard deviation of noise [deg.]89Figure 9. Estimation results for our synthetic datasets with adding
noise to the light directions. The solid line is the proposed method,
and the dotted line is the calibrated method. Each data point is anaverage of 30 trials.

ric stereo more practical, we typically want to put the lightsources close to a target scene, resulting in a near light setting. One of our future directions is to extend the proposed
method for the near light setting.
Acknowledgments This work was supported by JSPS
KAKENHI Grant Number JP19H01123.

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