1

Supplementary Material for “Near-light
Photometric Stereo with Symmetric Lights”
Lilika Makabe, Student Member, IEEE, Heng Guo, Hiroaki Santo, Member, IEEE,Fumio Okura, Member, IEEE, and Yasuyuki Matsushita, Senior Member, IEEE✦

To further assess the proposed method, in this supplementarymaterial,
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2)
3)
4)
5)
6)

we add a detailed description on number of linear independencies of constraints,
we provide an additional experimental study on light falloff relaxation,
we evaluate our method for non-Lambertian reflectance,
we discuss the estimation accuracy with varying light-tocamera offsets along z -axis,
we show albedo estimation results on the real-world
dataset, andwe provide further analysis on uniform light radiance inour real-world experiment.

1 L INEAR INDEPENDENCIES OF CONSTRAINTS
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rank (A)rank (A′ 1A )rank (A′ 1B )rank (A′ 2A )rank (A′ 2B )= 2npairs − 3= npairs − 2 (non-ring light)= npairs − 1 (ring light).
= npairs − 2 (xyz -axis offset)= npairs − 1 (z -axis offset)

Elementary operations of adding one row with scaling toanother row do not alter the original matrix rank. First, 2npairs
independent equations are given by Eq. (2). Eq. (4) and Eq. (7)are obtained by taking the difference and summation of these
2npairs equations, respectively, giving npairs independent equations each. Eq. (6) from Eq. (4) uses two basis and Eq. (7) uses
one to eliminate unknowns from equations, leading to npairs − 2and npairs − 1 independent equations, respectively, leading torank (A) to be 2npairs − 3.
Similarly, Eq. (13) and (14) are also obtained by Eq. (11),and use two or one basis, respectively, to eliminate unknowns in
the original equation, leading to rank (A′ 1A ) = npairs − 2 andrank (A′ 1B ) = npairs − 1 in total. Similar analysis can be applied
to rank (A′ 2A ) and rank (A′ 2B ).
In the special case where all lights are on the same line, Eq.
(6) from Eq. (4) needs only one basis to eliminate unknowns fromequations since all the lights span in a line, leading to rank (A)to be 2npairs − 2. With the constraint of A′ 2B , whose rank is•

L. Makabe, H. Guo, H. Santo, F. Okura, and Y. Matsushita arewith Graduate School of Information Science and Technology, Osaka
University, Japan.
E-mail:{lilika.makabe,heng.guo,santo.hiroaki,okura,yasumat}@ist.osaka-u.ac.jp

npairs − 1. In total, 3npairs − 3 constraints can be introduced,and thus light-to-surface distances can be recovered with at least
2npairs , which is the minimum setting for a line light.

2L IGHT FALL - OFF RELAXATION ANALYSISIn Sec. 4.2 in the main paper, we evaluate the effect of the lightfall-off relaxation using npairs = 4 light arrangement from Fig.12
used in the synthetic experiments. While the experiment in themain paper only shows results of normal estimation accuracy byvarying light-to-surface distances, we here analyze the applicability of this relaxation. By rewriting Eq. (8) in the main paper usingd∗ = ∥s∗ − x∥2 , where s∗ and x respectively denote the point
light source and surface positions, our light fall-off relaxation
including the normalization factor is read as:11≈ 2.
d3∗
d∗(1)

Since we solve the homogeneous system to obtain light-to-surface
distances e∗ = ρ−1 d2∗ , the solutions naturally involve ambiguity.
When the nullity of the design matrix is 1, the solutions areobtained up to scale. Namely, the light fall-off approximation
yields a solution equivalent to the one without the approximation
pp T
pif d2 = kd3 holds, where d∗ = [d1 , ..., dN ] , k ∈ R, andN = 2npairs is the number of light sources. Given the two lightindices i, j , the following metric approaches 1 if the relaxation
well approximates the unrelaxed light fall-off term:
d3j
d3i
d2j
d2i

=dj
≈ 1.
di(2)

To assess the error by the relaxation, we evaluate the error r oflight fall-off terms between 1/d2 and 1/d3 case defined as 2di(3)r ≜ meani−1 ,d1where i = 2, ..., 2npairs .
Figure S1 visualizes the relative error r at varying point
locations x with respect to the camera and field of views (FOVs)
associated with their focal lengths. For the visualization purpose,
we only consider a slice of the view volume on the x-z plane.
Other experimental settings are the same as the synthetic experiments in the main paper, e.g., resolution.
If we allow 2% error, i.e., r = 0.02, in the case of focal lengthf = 85 mm, for example, we should keep the depth z ≥ 3.54,

2.

�FOVLightCamera

�

�
��

5%

which supports the result in Sec. 4.2 in the main paper. For ashorter focal length, we need to put the target object further away
to ensure that the relative error r ≤ 0.025 at all the points within
the FOV, e.g., z ≥ 2.73 for f = 50 and z ≥ 2.53 for f = 25.

R ESULTS ON NON -L AMBERTIAN SURFACE

Calibrated [1]Ours

Calibrated [1]Normal map

Ours

5.0
4.5

Calibrated
Ours

4.0
3.5
0.000

0.025

0.050

0.075

0.100

0.125

0.150

10◦

Error map

Calibrated

Ours°°

Fig. S3: Estimation results on our synthetic dataset with adding
an error β to the light-to-camera positions. The plot shows mean
angular errors by the proposed method (Ours) and the comparison
method (Calibrated) with varying error β . Below the plot, we show
the corresponding estimation of the surface normal, along withthe angular error map by both methods. The calibrated method
is significantly affected by the inaccurate calibration, while theproposed method is not affected by the light calibration results.

roughness 0.5

Input image

roughness 1.0

5.5

Noise level on light calibration �Fig. S1: The left-hand side illustrates the lights and 3D point xin the camera coordinate system looked from the bird-view. Theblue area shows the field of view (FOV) of the camera, and θ is anangle between the optical axis and a camera ray going through the3D point x. The right-hand side shows the heatmap of the lightfall-off approximation error r. The dotted lines indicate FOV for
focal lengths of f = 25, 50, 85 mm.

36.0

Error map Normal map

�� = 85 5Error map Normal map��Mean angular error [deg]6.5

Calibrated. Although in the roughness = 0.5 case, our method
fails to estimate the correct normal map due to the deviation fromthe Lambertian assumption, we still obtain reasonable estimationin the roughness= 1.0 case, even though the modeling of thediffuse reflectance is different from that of Lambertian.

4

R ESULTS ON VARYING LIGHT- TO - CAMERA OFF -SET0
9.759

5.171

13.398

5.057

Fig. S2: Estimation results with varying roughness values. We
show the estimated surface normal together with angular errormaps, under varying roughness values in Disney Principled BSDFmodel. [2]To examine the effect of a different reflectance model, we evaluate our method with a scene rendered using Disney’s Principled
BSDF model [2]. We ignore the clearcoat term and set the specularvalue to 0.5 and roughness values to 0.5 and 1.0. Figure S2
shows the results of the estimated normal maps from ours andIn a geometric light source calibration, the light position along
the z -axis in the camera coordinates, i.e., light-to-camera offset
sz , is known to be sensitive to noise due to the narrow-baseline
triangulation [3], [4].
In this section, we evaluate how the light’s geometric calibration error impacts the surface normal estimation by the calibrated
photometric stereo method [1], while it does not affect our method
at all.
We render synthetic images, where the light-to-surface andlight-to-camera distances are set to 6 and 0, respectively, anduse npairs = 4 lights as shown in Fig. 5. Figure S3 shows theestimated results of the calibrated method with Gaussian noise
added with varying standard deviations β to the light source

3S HEEP

SV-C ATS AILOR S ANTAB UNNYB UDDHAD ONUT

BALL

Calibrated [1]Ours

Input image

C AT

Relative intensity

Fig. S4: Estimated albedo maps for our real-world dataset. For each scene, we show one of the input images and estimate albedo mapsby the proposed method (Ours) and Calibrated.
1.2

61.0

Light radiance is assumed to be uniform for all light sources as weemploy a constant current circuit in our device. To further verify
that, we use a CCD camera and Lambertian calibration board,and measure the light intensities of two LED devices fixed in thesame position, by swapping the device. Figure S5 presents themeasured intensities of LED1, LED2, and ambient illumination,providing a visual representation of their stability over time. The
mean intensities over 10 seconds are 1.00 and 1.01, respectively,
where the intensities are scaled so that the mean intensity ofLED1 should be 1. Also, the corresponding standard deviationsare 8.33×10−4 and 1.56×10−3 . These statistics indicate that theoscillation in intensity over time and between devices is negligible.

0.80.6

Light 1
Light 2Black frame

0.4
0.2
0.0
0.0

1.0

2.0

3.0

4.0 5.0 6.0
Time [sec.]

7.0

8.0

9.0 10.0

Fig. S5: The intensities of two different LED devices, LED1 andLED2, and ambient illumination. Each data point represents themean intensity of a 100 × 100 pixel patch taken from the centerof the captured image.

positions. While the calibrated method estimates accurate surface
normals with accurate light calibration, if calibration becomesinaccurate, the shape errors increase rapidly. The varying light-tocamera offset does not affect our estimation results, highlighting
one of the advantages of the proposed method that does not relyon light source calibration.

5A LBEDO ESTIMATION RESULTS ON THE REAL -WORLD DATASET

Figure S4 shows the estimated albedo maps corresponding to thesurface normal estimations shown in Sec. 5 in the main paper. For
visualization, we scale the albedo values using the mean image of
all eight input images, so that the median of the estimated albedo
map should be equal to that of the mean image. Although boththe calibrated and proposed methods suffer from some specularreflection, overall, our albedo estimations are comparable to thecalibrated method.

L IGHT RADIANCE ANALYSISA PPENDIXProposition 1. the point x ∈ Rn which has the known ratio
d1 , d2 ∈ R (d1 > d2 ) of distances to the fixed points p1 , p2 ∈d2 p −d2 p
Rn lies in the sphere whose center and radius are 1 d22 −d22 1 and11
d1 d2∥p1 − p2 ∥2 , respectively.
d21 −d222

Proof of Proposition 1. We can rewrite the condition on the distances between xand fixed points as:
11∥x − p1 ∥2 : ∥x − p2 ∥2 = d1 : d2 .

(4)

From the above equation, we transform into one sphere equationas:TT

d21 (x − p2 ) (x − p2 ) = d22 (x − p1 ) (x − p1 )i.f.f. x −d21 p2 − d22 p1
d21 − d222=2

d21 d2222(d21 − d22 )d2 p −d2 pwhose center and radius are 1 d22 −d22 112∥p1 − p2 ∥2 ,1d2
and dd21−d2 ∥p1 − p2 ∥2 .
12Proposition 2. When the radius of the symmetric lights r∗ ofevery pair are the same, all the possible Apollonius spheres are
centered on a line, whose direction is [−y ′ , x′ , 0].

4Apollonius sphere from , light pairs

�

�

�

�

�

Intersection solvability
Arrangement
Apollonius spheres

�The centers span in 1D line
→ × Intersection has an ambiguity.

� lies on the Apollonius sphereby , light.

Arrangement

Apollonius spheres

The centers span in 2D space→ ✓ Unique intersection

Fig. S6: Given the position and estimated light-to-surface distances of arbitrary two lights, the corresponding sphere can be drawn
where the surface point x lies. x can be recovered as the intersection of at least 3 spheres. When every light shares the same radii, thecenters of corresponding Apollonius spheres span in a 1D line, leading to an ambiguity in the intersection.

Proof of Proposition 2. Let us define a light position and scaled
distance using a light index e.g., si = sθrii , ei = eθrii , and thecenter of the Apollonius sphere by light indices i, j as cij for asimplicity. We show that the directional vector between two of theApollonius spheres is always the same, regardless of the choice ofthe light indices.
Given ith and j th light, center of the Apollonius sphere cij
can be represented as:ei (sj − so ) − ej (si − so )ei − ejT′T ′x x + 2x′ so (sj − si ) + ri2 sj − rj2 si,=ri2 − rj2 − 2x′T (sj − si )cij =by substituting the definition of the scaled distances in Eq (10) in
the main paper. Especially when ri = rj = r, the center cij canbe further simplified as :Tcij =Tx′ x′ + 2x′ so + r2
(sj − si ) .
−2x′T (sj − si )When all the lights’ radii ri , rj , rk , rl are the same as each other,
the directional vector between two of the Apollonius spheres
center cij , ckl can be obtained as follows:!
(sj − si )(sl − sk )cij − ckl ∝(5)−x′T (sj − si ) x′T (sl − sk )T

∝ [−y ′ , x′ , 0] ,(6)which is independent of the selection of light. Figure S6 illustrates the valid and invalid light arrangements with corresponding
Apollonius spheres.

R EFERENCES[1] Y. Quéau, B. Durix, T. Wu, D. Cremers, F. Lauze, and J.-D. Durou, “Ledbased photometric stereo: Modeling, calibration and numerical solution,”
Journal of Mathematical Imaging and Vision, vol. 60, 2018.
[2] B. Burley and W. D. A. Studios, “Physically Based Shading at Disney,” inProceedings of SIGGRAPH, 2012.
[3] J. Ackermann, S. Fuhrmann, and M. Goesele, “Geometric point lightsource calibration,” in Proceedings of Vision, Modeling, and Visualization,
2013.
[4] H. Santo, M. Waechter, W. Lin, Y. Sugano, and Y. Matsushita, “Lightstructure from pin motion: Geometric point light source calibration,”International Journal of Computer Vision (IJCV), vol. 128, no. 7, pp.
1889–1912, 2020.