Photometric Stereo with Small Angular Variations
Jian Wang† , Yasuyuki Matsushita‡ , Boxin Shi§ , and Aswin C. Sankaranarayanan††
ECE Department, Carnegie Mellon University, Pittsburgh, PA, USA‡
Osaka University, Osaka, Japan§
Singapore University of Technology and Design, Singapore

Abstract
Most existing successful photometric stereo setups require large angular variations in illumination directions,which results in acquisition rigs that have large spatial extent. For many applications, especially involving mobile devices, it is important that the device be spatially compact.
This naturally implies smaller angular variations in the illumination directions. This paper studies the effect of small
angular variations in illumination directions to photometricstereo. We explore both theoretical justification and practical issues in the design of a compact and portable photometric stereo device on which a camera is surrounded bya ring of point light sources. We first derive the relationship between the estimation error of surface normal and thebaseline of the point light sources. Armed with this theoretical insight, we develop a small baseline photometricstereo prototype to experimentally examine the theory andits practicality.

1. Introduction
Size, weight, and power (SWaP) are key factors in designing a practical 3D acquisition system. The popularity ofcommercial depth cameras such as the Kinect [14], which
is based on time-of-flight technology, and RealSense [10],which is based on stereo and active illumination, can be attributed to their careful SWaP consideration. In contrast tothis, 3D acquisition based on photometric stereo is yet togain widespread commercial adoption. There are many advantages to use photometric stereo — all arising from its
ability to compute surface orientation at the same resolution as the input images [20, 18], which is achieved by determining surface normal per pixel from shading variations
observed under varying lightings. Reliable surface normal
estimates are obtained when the light sources have a large
angular spread [4]; however, lighting with a large angularspread necessarily requires a large space especially when
imaging macroscopic objects, and hence, does not satisfy
small SWaP.

yxz

Figure 1. A compact photometric stereo system with a light ring ofradius r and a camera placed at the center of the ring. We seek toimage a scene whose depth d is significantly larger than the radiusof the light ring, i.e., d  r.

This paper deals with photometric stereo under small angular variations. Our envisioned acquisition setup is as follows. A camera is surrounded by a small light ring. It is
used to capture the surface normal of a scene whose spatial
extent is significantly larger than the system itself. We define the baseline as the greatest distance between every twolights, which is the diameter of the light ring here. To fulfill
the small SWaP requirement, we want the baseline to be as
small as possible.

1.1. Problem setup
In this paper, we analyze the estimation error with regard
to baseline and other system parameters like the number oflights, camera noise level and scene location. Our problem
formulation is as follows.
As illustrated in Fig. 1, our setup consists of n identicalpoint light sources distributed uniformly on a circle of radius r, centered at the camera. We obtain n images, one
for illumination under each light source. The camera is assumed to make intensity measurements corrupted by additive noise with mean 0, variance σ 2 , and i.i.d both acrosspixels and across images. Finally, the baseline of the system is small, which translates to the condition r  d.
We consider a Lambertian scene point P at thelocation p = [0, h, d]> ∈ R3 with a surface normaln ∈ R3 , knk2 = 1 and diffuse albedo ρ. When P is illuminated by a point light source at the location s ∈ R3 , its

intensity is given as>bi = i + ∆ig = l (ρn) + ∆ig ,
klk3

(1)

where l = s − p is the illumination direction, klk3 comesfrom normalization and light fall-off [2], light source intensity at a unit distance is assumed as 1, and ∆ig is themeasurement noise. In the absence of measurement noise,
the baseline of the system can be infinitely small as long as
it is non-zero.1 However, in the presence of measurement
noise, the variations in shading become less obvious withdecreasing baseline, and thus the recovered surface normals
will have larger errors. We note here that the variance of∆ig , σ 2 , accounts for the camera response function. Given
three or more intensity observations obtained under varying lightings l1 , . . . , ln located at s1 , . . . , sn , respectively,b which is the estimate of the albedo-scaled surface normalbρn is given as a least-squares approximate solution asb = (LL> )−1 Lbi,
b

(2)

where
L=

h

l1
kl1 k3l2
kl2 k3ln
kln k3

...

and
bi =



bi1 bi2>. . . bini,

The matrix L is referred to as the light matrix or the lightcalibration matrix. This matrix is often estimated via a calibration procedure, i.e., placing known targets at suitable
depths in the scene.
Traditional photometric stereo systems assume distantlighting — which implies that the light matrix L is identical at all pixels/scene points. In contrast, it is extremely
important in the small angular case to account for variations of the light matrix as the scene point P of interest varies.

1.2. Main results
b in (2), the contriGiven b = ρn and the estimated b
butions of this paper are in analyzing the expected errorb − bk2 ]. We summarize these results below.
E[kbTheorem 1 In the small-baseline scenario, i.e., d  r,b − ρnk2 ] = σ 2 (d2 + h2 )3E∆ig [kb22(2d + h ),nr2 d2(3)

or equivalently,b − ρnk2 ] = σ 2 (d2 + h2 )2E∆ig [kbTheorem 1 predicts the accuracy of the estimate in (2)as a function of measurement noise and the location of thescene point. Intuitively, we expect that the estimation error
increases as we move away from the optical axis of the camera since the solid angle subtended by the light ring reduces
to a 1D line with increasing height h. This is reflected implicitly in (3) as well as in (4) where the dependence on thesubtended solid angle is explicitly characterized. The proofof Theorem 1 is presented in Section 3.
b is dependent on the light
Also note that the estimate b
matrix L which relies on knowledge of the scene point P
and hence, depth d. In practice, this matrix is estimated via
a calibration procedure. So, this reduces to approximate theb Our second contribution
unknown depth d with a value d.
is bounding the error due to use of incorrect depth in thebestimation of b.
Theorem 2 (Sensitivity to calibration error) Consider ascene point that is actually at depth d, but we assume that it
is at depth db (d, db  r). Further, if we assume the surface
normal at this point, n, to be uniformly distributed, then.

2

where Ω is the solid angle subtended at scene point by thelight ring and θ = atan(h/d) is the angle between surface
normal of the light ring and the line from center of light ringto the scene point.

4π
2 cos θnΩ 1+cos
2θ

,

(4)

1 When the baseline is zero, the problem reduces to shape-from-shading
which is known to be severely ill-posed [9].


ρ2
2
b
En [kb−ρnk] = (λ−1)2 2(λ2 + λ + 1)2 + (λ + 1)2 ,3(5)bwhere λ = d/d.
The proof of Theorem 2 is presented in Section 4.
From Theorem 1, we observe that a practical photometric stereo system faces a trade-off between small SWaP and
the accuracy. From Theorem 2, we observe that the decrease in estimation accuracy caused by depth mismatch is
gradual on each side of the calibrated depth, and there is a
depth range where error is tolerable, which relaxes the exact
placement of the target scene. Based on these observations,
we develop a small baseline photometric stereo prototype toexperimentally examine the theory and its practicality.

2. Related work
Many practical systems based on photometric stereo
have been proposed since the early work of Woodham [20]and Silver [18]. For example, Hernández et al. [7] usedcolored lights to capture images under multiple illumination directions simultaneously; this enabled estimation of
surface normals of moving objects under a snapshot acquisition system. Vlasic et al. [19] built a large light stage tocapture surface normal of actors from nearly 360◦ . However, both systems are too large to be portable. Compact
acquisition devices were proposed by Higo et al. [8], Zhou

and Tan [22], Zhou et al. [23], Shiradkar et al. [17], andJohnson et al. [11]; however, all of these devices were used
to image scenes with spatial extent that was comparable tothe size of the light path [8] or light ring [22, 23, 17, 11].
In contrast, the primary focus of this paper is to understanda more extreme setup where we seek to use a portable device to capture scenes that are much larger than the device(see Fig. 1). Jones et al. [12] used a device similar to ours
to capture expressions of a human face, but did not provide
a detailed error analysis including a characterization of thedependence of the normal estimation error on the baseline.
Closely related to this paper is the work of Drbohlav andChantler [4] where an error analysis of Lambertian photometric stereo is presented. However, a key difference is theassumption of distant lighting which is only applicable tolarge baseline systems. The analysis in [4] can be seen as a
special case when h = 0 in our setting.
Our small baseline photometric stereo setup is similar tomulti-flash cameras [16, 5]. However, the goal of such cameras is to recover depth discontinuities. Further, our focus islargely a theoretical analysis of the estimation error which
was not considered in these papers.

3. Error due to measurement noise
We analyze the error in estimation of albedo-scaled surface normals with regard to the baseline and the solid angle
subtended at scene point by the light ring, and in the process, provide the proof for Theorem 1.
There are many sources of noise in the process of imaging including photon, thermal, and dark-current noise. For
simplicity, we model the measurement noise as zero mean
and bounded variance. Specifically, we assume an intensity
measurement made at any pixel can be written asî = i + ∆ig ,Here, we further express trace (LL> )−1 as a function of
the radius of light ring r, the number of lights n, and scene
point’s location by using the following four steps.

Step 1 — Location of light sources. Because lights are
uniformly distributed on a ring, their locations can be written as
S = [s1

=

s2

...

sn ]

r cos( 2πn 1)r sin( 2π
n 1)

...
...
...

0


r cos( 2πn n),r sin( 2π
n n)
0

(8)

where each column si is 3D location of each light.

Step 2 — Expression for L. Given the definition of S, we
can derive the light matrix L as
L = [l1 l2 . . . ln ]s2 − psn − ps1 − p...
=ks1 − pk3 ks2 − pk3
ksn − pk3(a)1s2 − p . . . sn − p] (9)≈3 [s1 − p2(d + h2 ) 2
 


0
1
S −  h  1>  ,=3(d2 + h2 ) 2d(a)where ≈ uses the small baseline criteria, r  d, so that1ksi − pk ≈ (d2 + h2 ) 2 .

(6)where i is the noise-free measurement and ∆ig is the additive measurement noise with mean 0 and variance σ 2 . We
assume the measurement noise ∆ig is i.i.d. across different
pixels in an image and across images.
Given the albedo-scaled surface normal b = ρn and its
2b in (2), we define the error metric el = kb−bkbestimate b.
2
The expected value of el2 can be written as
E [el2 ] = σ 2 trace (LL> )−1 ,(7)
following the derivation of (33) in [4].

Step 3 — Expression for LL> and its inverse.


 

 >
0
0
1
S −  h  1>  S −  h  1> 
LL> = 2(d + h2 )3
d
d
 n 2

0
0
2r
1
n 2
2
 0
ndh 
= 2
2 r + nh(d + h2 )3
0
ndh
nd2
(10)

3.1. Dependence of error on baseline
We derive the error expectation with regard to baselineand other system parameters like measurement noise level,
number of lights and the 3D location of the target.
From (7), we observe that E [el2 ] is the productof thevariance of measurement noise, σ 2 , and trace (LL> )−1 .


(LL> )−1 = (d2 + h2 )3 

2
nr 2

0
0

0
2
nr 2
2h
− ndr
2

0



2h

− ndr
2
r 2 +2h2
nr 2 d2(11)

Step 4 — Expression for trace(LL> )−1 .

trace (LL> )−12
r2 + 2h2
2
+
+
)
nr2
nr2
nr2 d2
r2 + 4d2 + 2h2= (d2 + h2 )3
nr2 d22(a)2(2d + h2 )≈ (d2 + h2 )3,
nr2 d2= (d2 + h2 )3 ((12)

Figure 2. Solid angle subtended at point P by the light ring.

(a)where ≈ follows that r  d.
Replacing (12) in (7), we obtain
E [el2 ] = σ 2 (d2 + h2 )32(2d2 + h2 ).
nr2 d2(13)This proves the first part of Theorem 1.

3.2. Dependence of error on solid angle
We now investigate the dependence of the estimate on
the solid angle subtended by the light ring at a scene point.
Intuitively, the solid angle is more meaningful from a geometric perspective since it better captures the light configuration as seen at a scene point (see Fig. 2).
The solid angle Ω subtended by the light ring at a scene
point p = (0, h, d)> can be written as
Z 2π Z rldΩ=(14)
3 dldφ,22
2
00 (l + h + d + 2hl sin φ) 2
but the closed form of the double integration is analyticallyintractable [6]. We instead approximate it asΩ≈

πr2 cos θ,
h2 + d2(15)

where we employ two approximations: first, the numerator, which is the area of an ellipse with semi-major axis r
and semi-minor axis r cos θ, is approximated as πr2 cos θ,and second, the denominator, which is the distance from thepoint to the ellipse, as the distance of the point p to the center of the light ring.
Substituting (15) in (13), we obtain:
E [el2 ] = σ 2 (d2 + h2 )24π.
2 cos θnΩ 1+cos2θ

(16)This provides the proof for the second part of Theorem 1.
In the expression, factor (d2 + h2 )2 is caused by light fall2 cos θ
off; 1+cos
2 θ decreases from 1 to 0 when θ increases from0 to π/2, thus it can be viewed as discount rate of the solid angle Ω; in other words, error is inversely proportional
2 cos θto “discounted” solid angle (Ω 1+cos
2 θ ). In particular, it isnoteworthy that the error is not just inversely proportionalto Ω, but also dependent on the angle θ.

Figure 3. Illustration of error due to mis-calibration. Surface normal of point P is computed using light matrix of point Pb.

4. Error due to incorrect calibration
In this section, we derive the error due to our lack ofknowledge of the exact depth at which the target is placed.
Recall that, our results hold for scenarios where the distant lighting assumption is violated and hence, we need toaccount for spatial variations in the direction of the incident illumination. However, this requires knowledge of thescene depth which is often the goal of photometric stereo. While we can often simply use an approximate value
for depth and hope to get reasonable results, we provide atheoretical characterization of the error incurred due to this“mis-calibration”.
Suppose that the true scene point is at a location P and
we assume it to be, incorrectly, at Pb (see Fig. 3). Whenscene point is at P , intensities i are L> b where L is the lightmatrix. When using light matrix associated with the point
Pb to compute the albedo-scaled surface normal, according
to (2), we obtain the estimateb = (LbLb > )−1 LL
b > b,
b(17)

b is light matrix of point Pb.
where L4.1. The hypothesized light ring
We first introduce the so-called “hypothesized lightring” and use it to obtain an approximation for the termbLb > )−1 LL
b > . Subsequently, we derive expected value of(L
el2 both in the absence and presence of measurement noise.
Specifically, we assume that the light ring as seen at apoint P can be replaced by a “hypothesized light ring”,

Combining (20) and (21) together, we havebLb > )−1 LL
b > equal to
(L


1
0
0
db3 
2 d
d 
2
0
cos θ + sin θ b− sin θ cos θ(1 − b) .
ddd3d0 − sin θ cos θ(1 − b)sin2 θ + cos2 θ db
dFigure 4. Illustration of the hypothesized light ring, which is orthogonal to the line from the camera to the scene point.

with radius r0 and distance d0 , that is orthogonal to theline from camera to the scene point (see Fig. 4). Clearly,
d0 = d/ cos θ. An expression for r0 is obtained by ensuringthat the original light ring and the hypothesized light ringshould generate equal error. Specifically, the expression in(13) for a scene point (0, h, d)> with a light ring of radius r
should be identical to that of a scene point at (0, 0, d0 )> anda light ring with radius r0 . This gives us
r0 =

2 cos2 θ
1 + cos2 θ

 21
r.

 
0
1 
L = 3 R S(r0 ) −  0  1> 
q
q

(18)



(19)

1

where q is (d2 + h2 ) 2 , S(r0 ) is the hypothesized lights’
locations which is obtained by changing r of (8) to r0 , andR is the coordinate rotation matrix given as


1
 0
0


0
sin θ  .
cos θ

0
cos θ
− sin θbL
b > as
We can now express L
 n 022r
1bLb> = R  0
L
qb6
0

0
n 02
2r
0


0
0  R>nbq2bLb > )−1 LL
b > , (17) can be written as
Denoting D = (Lb = Db, and el = kbb − bk2 = (bb − b)> (bb − b) =b2>>>(Db − b) (Db − b) = b (D − I) (D − I)b.
To obtain an expression for el2 that is independent of thesurface normal, we assume a uniform distribution on thesurface normals, described in its Euler angles. Specifically,letting>n = [ρ sin Θ cos Φ, ρ sin Θ sin Φ, ρ cos Θ] ,1
4π sin(Θ), Θbfrom 0 to π and Φ from 0 to 2π, and λ = dd , we can computewith probability density function fΘ,Φ =
1 2
ρ (λ − 1)2 2(λ2 + λ + 1)2 + (λ + 1)2 .
3(22)This completes the proof for Theorem 2. The result is quite
surprising because it only relates to λ and does not depend
on r, n and h.
Finally, in the presence of measurement noise, where theimage intensities bi is given as (L> b + ∆ig ), we can writeEΦ,Θ [el2 ] =In contrast to derivation by foreshortening (r cos θ), the derived error estimation by r0 matches much better with simulations.
The light matrix of hypothesized lights L can be expressed asb = (LbLb > )−1 L(Lb > b + ∆ig ) = Db + (LbLb > )−1 L∆i
b g.
b
(23)
Now, the expression for E[el2 ] can be derived ashib − b)> (bb − b)
E [el2 ] = EΘ,Φ,∆ig (b>bLb > )−1 L∆ib g=EΘ,Φ,∆ig (D − I)b + (LibLb > )−1 L∆ib g(D − I)b + (L=EΘ,Φ b> (D − I)> (D − I)b +hibLb > )−1 L∆ib g +2EΘ,Φ,∆ig b> (D − I)> (LhibLb > )−1 L∆ib g )> (LbLb > )−1 L∆ib g
E∆ig ((L
h
i


(a)
bL
b > )−1=EΘ,Φ b> (D − I)> (D − I)b + σ 2 trace (L

(20)

(24)
(a)

1

b > aswhere qb is (db2 + bh2 ) 2 , and LL
 n 022r
b > = 1 R 0
LL
qb3 q 3
0

0
n 02
2r
0


0
0  R> .
nb
qq

d

(21)

where = follows from the second term being 0 and (7). The
final error is a linear combination of error due to incorrect
placement (the first term in the sum), denoted as E1 below,
and error due to measurement noise (the second term). E1
is only related to the albedo ρ and the ratio λ, and not related
to n, r and h. When db = d, i.e., λ = 1, E1 is zero. When

0.01

5

h

-3

3

1

(a) Relative error of Theorem 1 when n = 8, d =2000, and r = [200, 100, 40, 10]

0.1

0.2

/

0.3

0.03
0.02
0.01
0
1500

0.5

0.4

r = 200
r = 100
r = 40r = 10

0.04

2

0
0

10 x 104

0.05

r/d = 0.1
r/d = 0.05
r/d = 0.02
r/d = 0.005

4

0.005

0
0

x 10

Relative error5

r = 200
r = 100
r = 40r = 10Relative error

Relative error

0.015(b) Relative error of approximation of solid angle
in Theorem 1 when r/d = [0.1, 0.05, 0.02, 0.005]

2000

2500

3000

(c) Relative error of Theorem 2 when n = 8,
2000, and r = [200, 100, 40, 10]=

Figure 5. Relative error of the approximations in theorems.

10

10

10

10

10

10

10

Approximation

10(a) Compare expression in Theorem 1 with theground truth when n ∈ [3, 24], d = 2000, r ∈ (0,1200], ∈ 0,10 , andGround truth

Ground truth

Ground truth

10
10

10
10

10

10

Approximation

10(b) Compare approximation of the solid angle
with the ground truth when r/d ∈ (0, 0.1], and
∈ 0,

10

3

10

10

Approximation

1010(c) Compare expression in Theorem 2 with the groundtruth when n ∈ [3, 24], = 2000, ∈ [1500, 3000], r∈ (0, 200], ∈ 0, 2000), and1

Figure 6. Comparison of the approximations in theorems with the ground truth. Solid line is “y = x.”

db deviates from d, E1 increases gradually. Such analysis isparticularly useful to compute a range of depth values where
error is tolerable.
Limitations of Theorems 1 and 2. It is often more meaningful to analyze the accuracy of a photometric stereo sysb>
tem in terms of angular error e∠ = arccos bb b for surkbkkbk
face normal estimates. However, while e∠ is physicallymeaningful, analyzing it analytically is significantly harderthan el2 which enjoys closed-form expressions. A second
limitation is that we employ approximations in the derivation of both theorems. Next, we show using a wide range ofsimulations that in spite of these approximations, the error
expressions are extremely precise.
Verification of the approximations. We compare thetheoretical predictions of expected error in Theorems 1 and
2 to simulation results. Recall that our system is completely
defined using five parameters: the radius of the light ring, r;
the number of point light sources, n; the depth of the scene
point, d; the height of the scene point h; and the variance ofmeasurement noise, σ 2 . In addition to these, db is the depth

at which we calibrate the light sources. We note that unlessotherwise stated it is to be taken that db = d.
Given a system configuration, denoted by the values of{r, n, d, h, σ 2 }, we compute the expected error from the expressions in Theorem 1 as well as from simulations. Using
the simulations as ground truth, we compute the relative error defined as|ground truth − approximation|.
ground truth
Figure 5(a) verifies Theorem 1 by comparing relative error as a function of h and r for fixed values of d = 2000,
and n = 8 (σ 2 does not matter here). Figure 5(b) verifies the approximation of solid angle by (15) as a functionof r/d and θ. Figure 5(c) verifies Theorem 2, assuming
no measurement noise, by presenting accuracy in estimating calibration error as a function of the true depth d from1500 to 3000 when we have incorrectly calibrated lights at
db = 2000. As expected, when d = db = 2000, the error due
to mis-calibration is zero.
Next, in Fig. 6, we show scatter plots that compare theground truth to the theoretical expressions in the theorems.
The proximity of the scatter plot to the “y = x” line in-LED
CameraFigure 7. Mean angular error maps when σ 2 = 2, n = 8, r ={20mm, 40mm, 60mm}, and the true depth and calibrated depth
both equal to 2000mm. Notice how the angular error is radiallysymmetric — a consequence of the circular light ring. Angularerror of the region enclosed by white line is smaller than 10◦ .
1.5

Baseline = 80 mm

Figure 9. Our portable surface normal sensor with a light ring ofradius r = 40mm (diameter of 80mm).

10 mm

0.75
0

‫ܖ‬ଷ
‫ܔ‬

0.75
1.5 1

2

3

‫ܖ‬ଶ

10 mm

‫ܖ‬ସ

‫ܖ‬ଵ

60

50

40

30

20

Figure 10. Light calibration method.

10

0Figure 8. Mean angular error maps when σ 2 = 2, n = 8,
r = {20mm, 40mm, 60mm}. The system is calibrated fora depth db = 2000mm while we vary the true depth d ∈[1000mm, 3000mm]. The plots shown are a function of d (horizontal axis) and h (vertical axis). Angular error of the region
enclosed by white line is smaller than 10◦ .

dicates that our theoretical predictions are very precise andbroadly independent of system parameters.

5. Experiments
We showcase the predictions of the Theorems 1 and 2
by building an experimental prototype and comparing its
performance to our theoretical predictions.
Choice of baseline. Before we build our prototype with afixed baseline, it is first instructive to look at the theoretical predictions. Let us consider a system with eight lights,
n = 8, and measurement noise with variance σ 2 = 2. We
assume calibration for a depth db = 2000mm. We vary theradius of the light ring r ∈ {20mm, 40mm, 60mm} andconsider two scenarios. First, in Fig. 7, we assume that thescene is at the calibrated depth d = db = 2000mm and lookat mean angular error as a function of h. Second, in Fig. 8,
we vary the true depth d ∈ [1000mm, 3000mm] when thesystem is calibrated for db = 2000mm and plot the error

as a function of both d and h. In both cases, we mark the10◦ equi-error contour in white. An important observationis that around r = 40mm we get a sufficiently large region
where the error is smaller than 10◦ . This suggests that we
can have a system that can be fit onto a smart-phone.
Prototype. We used a Point Grey camera FL3-U313E4C-C and Cree XLamp XM-L LEDs to build a small
baseline photometric stereo device (see Fig. 9) with the goalof recovering surface normals and depth maps of scenes
placed nearly 2000mm away. We fixed the number of lightsources on the ring as 8 for all experiments. Note that the
number of lights in the rings controls the trade-off between
accuracy (1/n as detailed in Theorem 1) and acquisition
time (proportional to n). While more lights lead to less error, they also result in longer acquisition time.
Light calibration. Due to the small baseline, precise calibration of the light matrix is very important. We observedthat the traditional method of using specular and diffuse
spheres provided calibration results were not sufficiently accurate. To resolve this, we propose a novel light calibration
technique using a diffuse checkerboard pattern. Figure 10
illustrates our light calibration method. We use the Camera Calibration Toolbox [1] for estimating the location andsurface normal of the checkerboard pattern. By observing
the checkerboard at a certain 3D location under a certain
lighting condition, we can recover light matrix by varying
the orientation of the checkerboard. Once calibration is performed at certain locations, we interpolate to obtain the light150

ight ring (mm)200

Mean angular error (degree)

0

22
20
16
12
8
4
0

50

100

150

200

Radius of the light ring (mm)Figure 11. Mean angular error as a function of the baseline.

r = 20mm
r = 40mm
r = 100mm
r = 200mm
Figure 13. Surface normals of a real scene for varying baseline.
Note that there is only marginal improvement in the quality of thesurface normals beyond r = 40mm.

6. Conclusions(a) One of the input(b) Surface normal map

Left/right views
Top/bottom views(c) Reconstructed surface

Figure 12. Surface normal map and reconstructed surface of a sculpture scene of 1m × 0.6m.

matrix at the remaining locations.

Results. We used the same diffuse checkerboard pattern
for light calibration to quantitatively test our device. We rotated the checkerboard at 20 poses and computed the difference between the recovered surface normal and the groundtruth (obtained by the use of the Camera Calibration Toolbox for Matlab [1]). The change of mean angular error withradius of the light ring is shown in Fig. 11. The error whenbaseline is 40mm is smaller than 10◦ .
We show the surface normals and reconstructed surfacesof real scenes captured by our device in Fig. 12. The radiusof the light ring was set to r = 40mm. In spite of this small
baseline, it is encouraging to see reliable normal estimates.
Finally, in Fig. 13, we obtain surface normal estimates ofa scene while varying the radius of light ring. We observethat for a radius of 20mm, the surface normal estimates arenoisy. Beyond a radius of 40mm, the gains due to increasing baseline seem to be minimal. These results confirm thetheoretical predictions outlined in Figs. 7 and 8.

Photometric stereo with a small SWaP can be immensely
useful. In this paper, we provide the theoretical scaffolding
for understanding the dependence of estimation error as afunction of various system parameters including the radiusof the light ring, the number of light sources, measurement
noise level as well as the dependence of the error on the location of the scene point. The systems we consider are thephotometric duals of micro baseline stereo [13, 21, 15, 3].
However, to the best of our knowledge, we are the first toaddress the small baseline problem in the context of photometric stereo. We believe that our analysis will be useful inthe design of compact and mobile photometric stereo.
Acknowledgments. J. W. and A. C. S. were supported, inpart, by the NSF grant CCF-1117939. Y. M. was partially
supported by JSPS KAKENHI Grant Numbers 26540085a
and 15H06345. B. S. was partially supported by the Singapore MOE Academic Research Fund MOE2013-T2-1-159
and the SUTD Digital Manufacturing and Design (DManD)
Centre which is supported by the Singapore National Research Foundation.

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