Calibrating a Non-isotropic Near Point Light Source using a Plane
Jaesik Park1∗ Sudipta N. Sinha2
Yasuyuki Matsushita3
Yu-Wing Tai1
In So Kweon1
1
Korea Advanced Institute of Science and Technology, Republic of Korea
2
Microsoft Research Redmond, USA 3 Microsoft Research Asia, China
Abstract

unmodeled errors that may be present when simpler lightmodels are used.
Near point light source calibration is usually performedin controlled scenes using specialized equipments. In particular, accurate recovery of luminous or radiant intensity
distribution of non-isotropic lights requires special-purpose
equipments such as an Imaging Sphere [17] or a goniophotometer [20]. However, such hardware based approaches
are generally expensive in both cost and time.
In this paper, we explore an image-based approach forcalibrating a near point light source that is rigidly attached
to a conventional camera, that does not require special calibration objects or a special acquisition phase. We address
both geometric and photometric calibration issues. The geometric aspects include estimating the position and orientation of the light whereas photometric calibration involves
estimating the radiant intensity distribution (RID) of thelight, which we assume is radially symmetric with respect
to its dominant direction, i.e., axis of the light source.
We show that under this assumption, the shading produced by a non-isotropic point light on a Lambertian plane
is bilaterally symmetric whereas the shading produced by
an isotropic point light is radially symmetric. Given a calibrated camera and a known plane, detecting the axis of bilateral symmetry or the center of radial symmetry in an image provides linear constraints for estimating the light axis
and position, respectively. We model the RID using a loworder polynomial and show that if the light position and axis
are known, the unknown RID coefﬁcients can be efﬁciently
estimated. Also, non-Lambertian surfaces of a single material can be handled by our method using a more a ﬂexible
reﬂectance model, and we show such an example using abiquadratic BRDF model [19].
Based on these observations, we propose a practicaltechnique to calibrate a light-camera rig using a plane thatrecovers all unknown parameters of the light model by analyzing the shading and specular reﬂections observed in images captured from multiple viewpoints. When combinedwith structure-from-motion [9] for calibrating the cameraand estimating the plane, our method can perform a full selfcalibration of the light-camera rig in an unknown scene.

We show that a non-isotropic near point light source
rigidly attached to a camera can be calibrated using multiple images of a weakly textured planar scene. We prove that
if the radiant intensity distribution (RID) of a light source
is radially symmetric with respect to its dominant direction,then the shading observed on a Lambertian scene plane isbilaterally symmetric with respect to a 2D line on the plane.
The symmetry axis detected in an image provides a linear
constraint for estimating the dominant light axis. The light
position and RID parameters can then be estimated using
a linear method. Specular highlights if available can also
be used for light position estimation. We also extend our
method to handle non-Lambertian reﬂectances which wemodel using a biquadratic BRDF. We have evaluated our
method on synthetic data quantitavely. Our experiments onreal scenes show that our method works well in practice andenables light calibration without the need of a specialized
hardware.

1. Introduction
Calibration of light sources is often an important ﬁrst
step in various photometric methods in computer vision. For example, shape-from-shading [11], photometricstereo [24] and reﬂectometry [16] methods often require
accurate knowledge of light sources for recovering surface
normal maps and reﬂectances. While estimating an incident
light direction from a distant source [27, 23, 1] is well studied and the problem of near light position estimation has
also been analyzed [2, 12, 28], the topic of modeling a near
point light source with a non-isotropic radiance distribution
has received less attention in our ﬁeld. Accurate calibrationof a non-isotropic near light is important for disambiguating
causes behind light attenuation in a scene and can reduce∗ Part of this work was done while the ﬁrst author was visiting Microsoft
Research Redmond as a research intern. This work was also supported
in part by the National Research Foundation grant funded by the Korea
government (MSIP) (No. 2010-0028680). Yu-Wing Tai was supported by
the MSIP, Korea, under the IT/SW Creative research program supervisedby the NIPA (NIPA-2013-H0503-13-1011)

1

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Various aspects of light calibration have been studied in
the computer vision literature. Debevec [7] used a mirror
sphere to capture an environment map for photo-realistic
rendering and relighting where all incoming distant illumination was modeled. For a single distant light source as inphotometric stereo [24], the problem reduces to that of estimating the light direction. This is possible in a lab setup [5],using reﬂective objects [25, 8] or by self-calibration [4].
Recently, Abrams et al. [1] treat the sun as a distant lightsource and extend photometric stereo to outdoor cameras.
Unlike distant lights, near light sources cause the incident light directions to vary within a scene. Although somemethods model area light sources [27], it is common to assume that the light source is a point thereby reducing the
problem to that of estimating the 3D point position. Objectsof known geometry e.g. spheres are used to further simplify the problem [15, 21, 26]. Spherical probes have also
been used for multiple, mixed (directional and near) lightsources [28]. Ackermann et al. [2] proposes a new technique for accurately modeling reﬂections on a sphere. Other
methods also use either reﬂective spheres [12, 23, 18] or
specially designed planar light probes [3] or analyze shading of known geometric objects [22]. The case of a near
point light rigidly attached to a camera has been analyzed
for near light photometric stereo [10] and the light fall-off
effect due to distance from the light is modeled. Light falloff can provide cues for depth estimation [13]. However,such methods will suffer in accuracy with non-isotropic
lights unless their intensity distribution is also modeled.
Certain aspects of non-isotropic near lights have beenstudied for light source design and optimization [20, 14].
The geometric light source model now requires an additional parameter - the dominant axis of the light; a directionthat provides a reference point for storing the non-uniform
radiant intensity distributon on a sphere centered at the light
position. However, most existing geometric methods thatuse reﬂective calibration objects cannot be used for thismodel. In our method, we propose a general approach forrecovering this light axis from shading cues. Our approach
does not use a speciﬁc calibration object but instead requires
multiple images of a plane. We propose a simple method
based on detecting symmetries in shading as well as specular highlights if available. Once the light source geometry
is known, we show that it is possible to estimate the radiantintensity distribution (RID) from the same input images.
To the best of our knowledge, our work is the ﬁrst to address the estimation of geometric (position and orientationof light source w.r.t. camera) and photometric (RID curve)
properties of a non-isotropic near point light source using
a pure image-based approach. We show that our method
holds promise for calibrating regular cameras that contain a
built-in ﬂash.

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2. Related Work

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Figure 1. Polar graphs of E(φ) radiant intensity distribution
curves. Some common shapes – (a) circular, (b) petal, (c) bell,(d) heart, that can be modeled using cubic polynomials.

3. Shading Model and Properties
Consider the case of a homogeneous textureless plane illuminated by a light source observed by a calibrated camera
as shown in Fig. 2a. Let Π = (n, b) denote the plane, wheren ∈ R3 is its surface normal and b ∈ R is an offset, suchthat any 3D point P ∈ R3 on the plane satisﬁes the equationnT P + b = 0. Let C denote the camera projection centerand I denote the observed image. We denote the light position as L ∈ R3 , its dominant direction as v, and the angle
between the light vector l and v as φ. Vectors n, l and v areunit vectors. This is true for other vectors used elsewhere in
the paper unless speciﬁed otherwise.
Assuming Π to be a Lambertian surface that is illuminated by a nearby, isotropic point light source, the intensity
at the pixel p in the image I is given by
Ip = ρ E

nT l
,
d2

(1)

where ρ is the diffuse albedo, E is the light radiance emitted towards the point P that is observed at pixel p and dis the distance from the light source to P. The image I isrelated to I  , the image projected on the plane Π via thehomography induced by Π [9]. We assume that vignetting,gamma correction and radial distortion correction have been
performed, and image intensities are in a linear space.

3.1. Isotropic and non-isotropic point lights
An isotropic point light source emits light equally in all
direction. Hence, the term E in Eq. (1) is a constant. The
shading at a scene point depends only on the incident light
direction and the squared distance to the light.
In general, point lights are often non-isotropic, i.e., they
have a dominant light direction or axis and emit different
amounts of light in different directions. Radiant intensity isdeﬁned as the radiant power of the light source emitted in aparticular direction and it is measured in Watts per steradians. Varying radiant intensities of a light source are represented by its radiant intensity distribution (RID) and this isoften visualized as a polar graph to display the angular light

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Figure 2. (a) Geometry of a camera observing a plane illuminated by a near, directional point light source. (b,c) Supporting ﬁgures for
explaining the symmetry properties (in Sec. 3.2).

fall-off from the light’s axis. Figure 1 shows common RID
curves as described in detail in [20].
For point lights, it is common to assume that the RID
curve is radially symmetric (see [14]). As a result the term
E in Eq. (1) becomes a function of the angle φ between thelight vector l and the light’s dominant axis v, and will hence
be denoted as E(φ). The observed image intensities are
now a function of the angular variation of radiance, varying
incident light directions and the distance to the light.

3.2. Symmetry Properties
We now prove a key result concerning symmetry in theshading observed on a plane illuminated by a point light.
Lemma 1 (Bilateral Symmetry) Let X be the point on theplane Π closest to the light L and Y be the point where thelight axis meets Π. If the radiant intensity distribution isradially symmetric about the light axis, then the shading
image generated on the plane will be bilaterally symmetricwith respect to the line through X and Y.
Proof of Lemma 1. Let ΠLXY denote the plane determinedby the three points L, X and Y, and let lXY denote the line
where ΠLXY intersects Π. Since X is the closest point on
Π to L, the line lLX must be perpendicular to Π. Therefore,ΠLXY must be orthogonal to Π (Fig. 2b).
Consider any two points P1 and P2 on Π which areequidistant from lXY and the line segment P1 P2 is perpendicular to lXY . Let, M be the midpoint on P1 P2 (Fig. 2c).
Since ΠLXY and Π are orthogonal to each other, we haveP1 P2 ⊥ LM. Then, triangles LMP1 and LMP2
must be congruent (using the SAS rule for congruent triangles). This gives two equality constraints.
P1 L = P2 L,LP1 M = LP2 M.

(2)
(3)

Similarly, triangles YMP1 and YMP2 are also congruent (result of the SAS rule). Therefore, we haveP1 Y = P2 Y. This leads to the fact that triangles
LYP1 and LYP2 are also congruent (follows from the
SSS rule for congruent triangles). Therefore we have,(4)
P1 LY = P2 LY.
Now, consider the intensities at P1 and P2 according to
the shading model (Eq. (1)). Condition (2) implies that P1
and P2 are equidistant from L, so the denominators d2P1
and d2P2 are equal. Condition (3) implies that the light vectors lP1 and lP2 at P1 and P2 form the same angle withthe plane normal, so the dot products must be equal, i.e.,nT lP1 = nT lP2 . Condition (4) implies that the light vectors at P1 and P2 form the same angle with the light axis,
and as the RID is radially symmetric, we have EP1 = EP2 .
= IP. As this holds
From this, we conclude that IP
1
2
true for any pair of points located equidistant from lXY and
its line segment is perpendicular to lXY , the image I  isbilaterally symmetric with respect to that line. 
Lemma 2 (Radial Symmetry) The shading from the light
L on the plane Π is radially symmetric with respect toX, the closest point on Π from L when either the light isisotropic or the light is non-isotropic but is oriented suchthat its dominant axis is perpendicular to the plane.
Proof of Lemma 2. For an isotropic light, consider pointson Π at a constant distance from L. Any point P in thisset lies on a circle C centered at X. The light vector at P
i.e. the line lLP traces out a right circular cone with axislLX and C as its base. This implies that the terms nT lP
and dP are constant for all points P on C. As E is a global
constant, it implies image intensities at all points on C must
be equal making the image radially symmetric w.r.t. X.
For a non-isotropic light, when the light axis v is perpendicular to the plane, the point Y, where the axis meetsthe plane coincides with X. Points on a circle centered at X
are again equidistant from L and their light vectors form thesame angle with the plane normal. Hence, E(φ) is constantfor all points on the circle which implies that the observed
intensities at these points are equal and the image is radiallysymmetric with respect to the center X. 

3.3. Geometric Constraints
The symmetry properties imply that for a calibrated camera and a known plane illuminated by an unknown light

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Algorithm: Light Calibration

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Figure 3. (a) Half-vector h and angles θh and θd at the point X on
the plane Π. (b) Light camera conﬁguration for the case when C
is not coplanar with L, X and Y. The lines lLC and lXY meet on
Π at an angle ψ (see Sec. 3.4).

source, detecting the axis of bilateral symmetry in the image
provides a plane constraint that is equivalent to one linear
constraint on the coordinates of L. Alternatively, the center of radial symmetry if detected, provides a line constraintthat provides two linear constraints on L. In Sec. 4.2, we describe our algorithm that uses these constraints to estimate
the light position and orientation.

3.4. Non-Lambertian surfaces
We now address the non-Lambertian case and analyze
how the symmetry deviates from the perfect case dependingon the camera-light geometry or the type of BRDF.
We replace ρ(nT l) in Eq. (1) with ρ(θh , θd ), an isotropic
BRDF model which is a function of θd , the angle between
h and lXL and θh , the angle between n and h, where his the half-way vector [16, 19] (Fig. 3a). This biquadratic
BRDF model is known to be appropriate for modeling lowfrequency components of isotropic BRDFs [19].
Consider the plane ΠLC passing through L and C orthogonal to Π. Let lXC denote the line where ΠLC intersects Π (Fig. 3b). For any two points on a line perpendicular to lXC on Π, equidistant from lXC and on opposite
sides of it, their half-vectors form the same angle with theplane normal and their difference angles are also equal. So
the BRDF map will be bilaterally symmetric with respect tolXC . Therefore, if lines lXC and lXY coincide, symmetry is preserved about this line for any isotropic BRDF. Thisoccurs when the four points L, X, Y and C are coplanar.
If there are not coplanar, the lines lXC and lXY do not coincide anymore but intersect at some arbitrary angle ψ at X
(see Fig. 3b). The degree of asymmetry due to the BRDF
with respect to lXY will be maximum when ψ = π2 .
The degree of asymmetry also depends on the ratio ω =LCwhere LC and CC  are distances between L
CC 
and C and between C and its closest point on Π. Consider
the case where the distance between the camera and light
position is similar to the distance of the camera from theplane. In this case, θd will vary more and this may cause alarge variance in the observed BRDF.
To quantify the deviation in symmetry for a reasonable

Input: Images {Ik }, Cameras {Pk }, plane Π
Output: Light position L, direction v in camera
coordinates, parameters of E(φ).
1. Rectify {Ik } using plane-induced homographies H.
2. Detect specular highlights.
3. Recover shading images.
4. Detect axes of symmetry in shading images.
5. Estimate L and v using output of steps 2 and 4.
6. Estimate coefﬁcients of E(φ) using estimated L, v.
range of conﬁgurations values for ψ and ω, we analyzed
pose parameters for the camera-light rig i.e. C and L for1, 1). We considered realisvalues of ψ ∈ (0, π2 ) and ω ∈ ( 50
tic BRDF values from the database used in [19] in this analysis and empirically selected different BRDF functions tomeasure the deviation from symmetry in the average case.
In our analysis, the deviation in symmetry for many BRDFs
is small enough when ω is less than 0.1 i.e. the distance
between L and C is less than one-tenth the approximate
distance to the plane. The details of this analysis can befound in the supplementary material. Thus, we conclude
that except for BRDFs with extreme values, we expect thedeviation in symmetry to be small and that our method will
work accurately in most conﬁgurations.

4. Proposed method
In this section, we describe our method for calibrating apoint light rigidly attached to a camera using multiple images of a plane. See Algorithm 1 for an overview. We referto the input images as {Ik }, the camera projection matrices
as {Pk } and camera centers as {Ck } respectively. For eachcamera position Ck , the corresponding light position in theworld coordinate system is denoted Lk .
Camera Calibration. We use structure from motion [9]to calibrate the cameras and then estimate the plane by robustly ﬁtting a plane to the reconstructed 3D points. We
assume that the scene is planar, has a single material andhas enough texture for automatic feature matching to work.
Specular Highlight Detection. We detect highlights in images by analyzing pixel intensity distributions. Images containing specular highlights tend to have long-tail distributions as pixels at and around the highlight are much morebrighter than pixels in the rest of the image. On the other
hand, image without highlights tend to have short-tail distributions. Hence, we construct a histogram of image intensities using all image pixels and ﬁt a 1D Gaussian N (μ, σ)
to the histogram values. Bright pixels, whose intensities exceed the value 6σ are classiﬁed as specular highlights andwe ﬁt a 2D ellipse to those pixels and use the ellipse center
for light position estimation.

p∈P

(5)
where f (·) is gradient clipping function deﬁned as
x if x2 < τ
f (x) =
0 otherwise.

(6)

and O is the input image. This problem can be formulated
as a discrete Poisson problem which can be solved efﬁciently using a sparse linear solver. In Eq. (5), the ﬁrst term
encourages the gradients in I to match the clipped gradientof O whereas the second term encourages the intensities tobe similar. The second term is also modulated by a per-pixel
weight wp ∈ [0, 1] that accounts for outliers arising fromstrong gradients in the albedo map or specular highlights.
We deﬁne wp as follows.
wp = sp (1 − |Op − G ∗ Op |),(7)

where G is a ρ×ρ gaussian kernel (ρ = 100), ’*’ is the convolution operator and sp is a binary variable (zero when the
pixel is classiﬁed as a specular highlight). wp is inversely
proportional to per-pixel values in |Op − G ∗ Op |, the highpass ﬁlter response on O. Parameters λ and τ are set to
0.1 and 0.5 respectively for I ∈ [0, 255]. An example of arecovered shading image is shown in Fig. 4.
Ortho-Rectiﬁcation: Using the known plane and camera
parameters, we compute the plane-induced homographies
for each image and rectify the shading image Iˆk computedin the previous step to remove all perspective effects andcreate rectiﬁed images required by our method.

4.1. Bilateral Symmetry Detection
We now describe our method for detecting the symmetry
axis in the rectiﬁed shading image using RANSAC in a typical hypothesize-and-verify framework. We use the fact, thatfor a least squares quadratic ﬁt g(z) = az 2 + bz + c to samples of a 1D function f (z) symmetric about z ∗ , evaluated atpoints sampled at uniform steps in z ∗ ±D, the function g(z)will attain an extremum at z ∗ . However, this is not always
true for an asymmetric 1D function.
To evaluate any line l for symmetry, we select a point
xj on l and sample points along lj⊥ , the line perpendicular
to l through xj at uniform steps along lj⊥ . We consider theintensity proﬁle Ij⊥ at these sampled points between points
zj+ and zj− that are equidistant from xj but in opposite directions, such that either zj+ or zj− is on the image boundary.

(a)
300G
pixel intensityG

Gradient Filtering. Since our method requires a shading
image, we ﬁrst need to remove high frequency gradients in
the input image that correspond to albedo edges. We recover the shading image Iˆ by minimizing the following objective function2  ∂Ip
∂Op
2
ˆ
− f(
) +λwp (Ip −Op ) ,I = argmin
∂x
∂x
I

(b)
OriginalG
FilteredG

G

200G
100G

0G G
0G 100G 200G 300G 400G 500G 600G
x axis of imageG(c)(d)
Figure 4. (a) Input image. (b) Corresponding shading image computed by gradient ﬁltering (Image contrast stretched for visualization). (c) Intensity plots for the red and blue scanlines. (d) Perpixel weights wp used in Eq. (7).

Using least squares, we then ﬁt a quadratic function Qj (z)to Ij⊥ and compute the distance
dj = zj∗ − xj ,

where z ∗ = argmax Qj (z).

(8)

z

To measure the overall degree of asymmetry, we perform
quadratic ﬁts at n positions
along l and calculate the averdj
age distance d(l) = nj for all the positions. After generating random line hypotheses 1 , we select the hypothesis
ˆl with the smallest value of d(l). We further reﬁne the estimate ˆl using local optimization. In near degenerate cases,
bilateral symmetry degrades to radial symmetry and then,multiple line candidates may have small values of d(l). The
center of symmetry can then be found by robustly intersecting multiple such symmetric lines found using RANSAC.

4.2. Light Position and Axis Estimation
Each detected symmetry axis provides a plane constraint
on the light position in world coordinates (CW ). We transform these planes to camera coordinates CC and derive a
linear constraint on the coordinates of L in CC . These linear constraints πk L = 0 can be stacked into a matrix of theform AL = b. It may appear that a solution of this linear
system will provide an estimate of L. Unfortunately, theselinear equations are not independent even when the cameras
are in general positions. This is because the planes ΠLXY
for different light positions always form a bundle or pencilof planes and pass through a 3D line. In fact, that 3D line isthe axis of the light, denote by R in CC . This implies that
we cannot directly compute L but it is possible to estimate
R by ﬁnding the null space of the linear system described
1 In practice, a 1-point RANSAC is used when a prior guess for the light
position L is available, since the projection of L on Π lies on line l andonly one other point is needed.

Alternate method. When specular highlights are absent(as in Lambertian scenes), it is possible to estimate L indirectly. If the 3D line R is known, we can parameterize theunknown position L = L0 + λu, where L0 is a point on R
closest to the camera center C, u is a unit vector parallel to
R and λ is an unknown scalar. For each k-th plane, we canﬁnd the point Yk where R intersects it. Then, the distancefrom L to Yk denoted by dk , can be written in terms of λ.
To compute λ, we can substitute various terms for points
Yk into Eq. (1), namely the intensities IYk , known valuesof nT l and expressions for dk in terms of λ. As these points
lie on the light axis, i.e., φ = 0, the term E(φ = 0) is
unknown but constant. Taking ratios between pairs of equations eliminates ρ and E(φ = 0). Multiple equations provide a set of sample estimates of λ after which the medianvalue λ∗ is selected and used to compute L = L0 + λ∗ u.

4.3. Radiant Intensity Distribution Estimation
Once the light position and axis is known, for each image
pixel, we can substitute I and d2 as well as values for φ, θh ,
θd and d into Eq. (1) where the product of the polynomials
E(φ) and ρ(θh , θd ) is expanded in terms of the unknown
coefﬁcients q1 , . . . , q6 and p1 , . . . , p4 .
ρ(θh , θd )E(φ) = (q1 c2h + q2 ch + q3 )(q4 c2d + q5 cd + q6 )(p1 φ3 + p2 φ2 + p3 φ + p4 ).

(9)Here, ch = cos(θh ) and cd = cos(θd ) as proposed in [19].
On expanding ρ(θh , θd )E(φ), we get a higher-order polynomial f (φ, θh , θd ) with 36 unknown coefﬁcients. These
coefﬁcients of f , denoted by vector f can be estimated bysolving a least squares problem of the following form.
Af = Id2 ,(10)

where A has 36 columns and a row corresponding to a pixel.
To recover p1 , . . . , p4 from f , we create a 9 × 4 matrix F ,F = [f1 p1 f2 p2 f3 p3 f4 p4 ],

(11)

16

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-1000

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5

-2000

12

14
3
611

2

-1000
13

20

0
18

15

-2000
9

1000

Radiant intensity

1
10

y

above. Let L1 and L2 denote 3D points in CC whose linear
combination deﬁnes points on the line R.
In practice, we need at least 2 images, each of which
provides one plane constraint. As long as these planes do
not all coincide, there is no degeneracy. Multiple input images with random 3d camera orientations makes the degenerate case very unlikely. The presence of camera translation
avoids the degeneracy in the initial SfM stage and the 3D
plane ﬁtting step.
If specular highlights are present, we can use them toestimate L. Using the ellipse center as the image of thereﬂection of L, we compute the corresponding incident light
ray in CW using the law of reﬂection. We transform this ray
into CC and obtain a line constraint on L, which gives twolinear constraints on its coordinates. We estimate L using
least squares assuming cameras in general position.

-3000

1

0.5

0
2000

8

-4000
-1000

0

1000

2000

x

0

20

40
60
angle (deg)

80

(a)
(b)
Figure 5. (a) Synthetic dataset example. (b) RID curves with different angular fall-offs used for evaluation (best seen in color).

where fi ∈ R9 are coefﬁcients for pi . On noise-free data, F
would be a rank-1 matrix. Hence, we factorize F = U DV T
using singular value decomposition (SVD) and the values of
p1 , . . . , p4 are obtained from the ﬁrst column of V .

5. Experiments
We have tested our method with both synthetic and realdata. These results are now described.

5.1. Synthetic evaluation
We generated 50 synthetic datasets with 20 images each.
Each set has randomly generated cameras approximately
on a hemisphere oriented roughly towards the origin. The
scene plane is ﬁxed and passes through the origin. Suitable intrinsic parameters are chosen randomly as well. The
light position and orientation was generated randomly in thecamera coordinates. Each dataset was assigned one of the
RID curves shown in Fig. 5b. Images were rendered at 320
×240 resolution and a Blinn-Phong model [6] was used togenerated specular highlights2 . Image noise was added tothe synthetic images. The level of noise added was varied
from 1 to 8 on a scale of [0,255] on all 50 datasets to generate results for 400 different runs; these are now discussed.
We ﬁrst present the symmetry axis detection resultsin Fig. 6a where the detected line is compared with theground truth line using the Hausdorff distance. In general
the estimation is stable but the median errors across different runs increases slightly with noise. As expected the detection is consistently more accurate for lights that have astronger angular fall-off in their RID (blue curves).
The light axis estimation which depends on the accuracy of the previous step also appears to be quite stable ingeneral (Fig. 6b). The estimated direction is within 1◦ ofground truth for the blue curves and is at most 2.3 degrees
for the light which has the most uniform distribution. The
position estimation using specular highlights clearly outperforms the alternative method that seems to be affected by
2 We use a very high value of the shininess parameter (= 50) and when-ever the specular highlight is observed within the image, the view vectors
are not at grazing angles to the plane. We conﬁrmed that the error introduced by the Blinn-Phong model which is approximate, is quite small.

Line to Line distanceLight direction error

Light position error

Light position error

5
0

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E(I)

E(I)

(a)
(b)
(c)
(d)
Figure 6. Median errors across multiple runs for (a) symmetry axis detection and for estimating (b) light direction, (c) light position using
specular highlights and (d) light position using the alternative method. Results are color-coded by the RID curves shown in Fig. 5b.

0.9
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Figure 7. Ground truth and estimated RID for each of the ﬁve light models (Fig. 5b). The medians of the estimated coefﬁcients from 80
runs are used to plot the curve. The error bars show the deviation in the curve due to variance of each estimated coefﬁcient.
DatasetI0 I1 Est.
X
Y
Z MAE
C AMERA -L ED 39 3 Pos. 5.57 9.02 0.94 8.97
Dir. 0.09 -0.05 0.99
SLR-F LASH
17 3 Pos. 0.29 26.35 17.38 15.35
Dir. 0.02 -0.25 0.97
Table 1. Statistics for real datasets: #images used for calibration(I0 ) and held-out for evaluation (I1 ). Estimated light position andorientations are listed (position reported in cm. from camera centerand direction reported as a unit vector).

noise levels (Fig. 6c and d). The results of the ﬁrst method
also does not depend on the RID curve, which is anotheradvantage.
Figure 7 shows the ground truth RID curves and the estimated curves. In each case, the coefﬁcients of the RID
curve were estimated from multiple runs and the curve corresponding to the median of these estimates is shown. The
deviation due to the 25-th and 75-th percentiles is indicatedusing error bars. In general, the synthetic evaluation showsthat our method successfully estimates the RID curves withhigh accuracy. Difﬁcult random camera conﬁgurations wasa source of error in less than 5% of the cases – either the
optical axis was at a grazing angle to the plane or the angle between the light vector and the optical axis was quite
large. In the second case, our method suffers in accuracy
due to the limited ﬁeld of view (FoV).

5.2. Evaluation on Real datasets
To test on real data, we captured two datasets with 42
and 20 images of whiteboards (some of which contained
specular highlights) in two different dark rooms. The ﬁrst
dataset C AMERA -L ED was captured using a bright Cree
XM-L LED. rigidly attached to the top of a point-and-shootcamera (Panasonic DMC-LX5). The second dataset SLRF LASH was captured with a Canon Mark III-1D camera

with an auxiliary ﬂash. Even though the ﬂash violates theassumption of a point light source, we evaluate the performance of our method on this dataset to assess to what extent
it can handle traditional camera ﬂashes. After obtaining the
SfM reconstruction, we resolved the scale ambiguity using
prior knowledge of the scale. For these datasets, we randomly held out three images for evaluating estimated parameters. We generated images using estimated parametersand compared them to the held-out images using the mean
absolute error (MAE) 3 as the accuracy metric.
Figure 8 shows results on the C AMERA -L ED and SLRF LASH datasets. For the Cree XM-L LED, our RID estimate appears to be in reasonable agreement with the manufacturer’s proﬁle within a cone of 25◦ from the light axis.
For angles greater than 25◦ , the insufﬁcient horizontal ﬁeldof-view (70◦ approx.) of the camera causes fewer pixels
with larger angular values to be observed. Having the dominant axis of the light parallel to the camera’s optical axis
also adds to the error for the same reason. Therefore, the fullrange of light emission may not be calibrated. However, calibrating the partial range is useful for post-processing images captured with the same camera and ﬂash and that task
does not require calibration of the full range of light emission. The MAE for the C AMERA -L ED dataset was 8.97.
The estimated geometric parameters of the light source isreported in Table 1. For SLR-F LASH, our method recovers
a more ﬂat RID curve as expected, since its built-in diffusercauses it to behave like an isotropic light source. The MAE
in this case is 15.35. A likely reason for this is that the ﬂash
in this case is a near area-light source. Nevertheless, the geometric parameters are accurately estimated (Table 1) – theestimated light axis is almost parallel to the optical axis andthe light position appears accurate to within 1-2 cms basedon our best guess for the camera center.
3 for image intensities that lie in the range [0,255]

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(a)
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(c)
(d)
(e)
Figure 8. Results on C AMERA -L ED and SLR-F LASH (top and bottom row resp.): (a) Camera-light rig. (b) Input image. (c) Shading image
generated using model. (d) Residuals and MAE between rendered and gradient-ﬁltered image (not shown). (e) The recovered RID curve.

6. Conclusions
We have presented a practical method for calibrating a
non-isotropic near point light rigidly attached to a camera.
Our method uses images of a weakly textured planar scenefrom multiple viewpoints and estimates light position andorientation using shading cues and specular highlights. The
results on real data are encouraging and it seems that themethod can be feasible in situations where special hardware/objects cannot be used. We expect that higher accuracy can be obtained by jointly optimizing all model parameters in a ﬁnal stage. The investigation of more accurate
models for non-point near light source is another interesting
direction of future work.

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