Illumination Distribution from Shadows
Imari Sato
Yoichi Sato
Katsushi Ikeuchi
Institute of Industrial Science, The University of Tokyo
7-22-1 Roppongi, Minato-ku, Tokyo 106-8558, Japan{imarik, ysato, ki}@iis.u-tokyo.ac.jp
Abstract
The image irradiance of a three-dimensional object isknown to be the function of three components: the distribution of light sources, the shape, and reflectance of areal object surface. In the past, recovering the shape andreflectance of an object surface from the recorded image
brightness has been intensively investigated. On the other
hand, there has been little progress in recovering illumination from the knowledge of the shape and reflectance of areal object. In this paper, we propose a new method for estimating the illumination distribution of a real scene fromimage brightness observed on a real object surface in thatscene. More specifically, we recover the illumination distribution of the scene from a radiance distribution insideshadows cast by an object of known shape onto another object surface of known shape and reflectance. By using theocclusion information of the incoming light, we are able
to reliably estimate the illumination distribution of a realscene, even in a complex illumination environment.

1 Introduction
The image irradiance of a three-dimensional object isknown to be the function of the following three components: the distribution of light sources, the shape, and reflectance of a real object surface. From the relationshipamong them, three kinds of analyses on the recorded image
brightness are derived: recovering the surface shape fromthe surface reflectance and illumination of the scene, recovering the surface reflectance from the surface shape and illumination of the scene, and recovering illumination fromthe shape and the reflectance of the surface.
In the past, the first two kinds of analyses, the shape recovery and the reflectance recovery, have been intensively
studied using the shape from shading method [7, 8, 9, 16]
as well as through reflectance analysis research [1, 12, 13,
15, 18] . In contrast, there has been little progress on thesubject of recovering illumination from the knowledge ofthe shape and the reflectance of an object surface. This isbecause real scenes usually include both direct and indirect
illumination distributed in a complex way and it is difficult
to obtain correct illumination models to be used for the estimation. Most of the previously proposed methods relatedwith the first two kinds of analyses aimed to estimate illuminant direction and color in a very specific illumination
condition such as a case where there would be only one direct light source in the scene. Accordingly, those methods
cannot be applied to the images taken under natural illumination environments.
The purpose of this study is to present our progress inrecovering an illumination distribution of a real scene fromthe knowledge of the shape and reflectance of an real object. In the proposed method, we use radiance changes inside shadows rather than appearance changes on the surface
due to the 3D geometry of the surface and the illuminant direction. More specifically, we estimate an illumination distribution of the scene by observing a radiance distribution
inside shadows cast by an object of known shape onto another object surface of known shape and reflectance. Shadows in a real scene are caused by the occlusion of incoming
lights as illustrated in Figure 1, and thus shadows containvarious pieces of information about the illumination of thescene. Nevertheless, in the past, shadows have been used
for determining the 3D shapes and orientations of an object
which cast shadows onto the scene [2, 11, 14, 19], whilevery few studies have focused on the the illuminant information which shadows could provide.
In the proposed method, we are able to reliably estimate
an illumination distribution of a real scene by making use of
the occlusion information of the incoming light. Also, ourmethod is applicable to the images taken under a complex
illumination environment such as images taken in an ordinary room, including reflections from the wall and otherobjects in the scene.

1.1 Overview
Before we describe the proposed method in detail, we
should clarify the basic steps of our method.
First, we take an image of the scene using a color CCD
camera so that shadows of an object appear in the image.
In the rest of the paper, we refer to the image with shadowsas the shadow image and the object of known shape, whichcast shadows onto the scene, as the occluding object. (Atypical example of a shadow image is shown in Figure 3.)
Then, based on the radiance distribution inside shadows,

1063-6919/99 $10.00 (c) 1999 IEEEan illumination distribution of a real scene is estimated as acollection of imaginary point light sources distributed overthe entire scene. The key idea of the proposed method is thediscretization of the overall illumination distribution by using the node directions of a geodesic dome. In the proposed
method, we assume that light sources in the scene are sufficiently distant from the objects and thus all light sources
project parallel rays onto the object surface.
By substituting a collection of imaginary point lightsources for the entire illumination, we are able to derive asystem of equations in n unknown radiance values of imaginary point light sources from the image irradiance of the
shadow image. We then solve for a solution set of unknown
radiance values which approximates the illumination distribution of the scene.
The rest of the paper is organized as follows. In Section 2
and Section 3, we explain how to estimate an illumination
distribution of a real scene from the image irradiance of ashadow image. We first obtain a formula which relates an illumination distribution of a real scene with the image irradiance of the shadow image (Section 2). Second, by assigning
the image irradiance of the shadow image to the formula, we
obtain a set of linear equations with unknown illumination
radiance values sampled at an equal solid angle. Finally,
we solve the set of linear equations for an illumination radiance solution set which represents the illumination distribution of the scene (Section 3). Section 4 shows experimental
results of the proposed method applied to real images. To
evaluate the accuracy of the illumination distribution estimated by our method, we superimpose a synthetic object
with the same shape as that of the occluding object onto an
image of the scene. In Section 5, we present concluding
remarks.

2 Formula for Relating Illumination Radiance with Image IrradianceIn this section, we present a formula which relates anillumination distribution of a real scene with the image irradiance of a shadow image. Based on the image formation,
the formula is obtained as follows:
1. Illumination radiance to scene irradiance: find a relationship between the illumination distribution of areal scene and the irradiance at a surface point in thescene.
2. Scene irradiance to scene radiance: compute howmuch of the incoming lights are reflected from the surface toward an image plane.
3. Scene radiance to image irradiance: find a relationship between the reflected light from the surface andthe image irradiance at a corresponding point on theimage plane.

(a)

(b)

Figure 1: Total irradiance: (a) without occluding object (b)with occluding object

2.1 From Illumination Radiance to Scene IrradianceFirst, scene irradiance is computed from the entire illumination of the scene. To take illumination from all directions into account, let us consider an infinitesimal patch of
the extended light source, of size δθi in polar angle and δφiin azimuth as shown in Figure 2.
Seen from the center point A, this patch subtends asolid angle δω = sin θi δθi δφi . Let L0 (θi , φi ) be theillumination radiance per unit solid angle coming fromthe direction (θi , φi ); then the radiance from the patch is
L0 (θi , φi ) sin θi δθi δφi [6], and the total irradiance of thesurface point A isE=

Z π Z π2−π0

L0 (θi , φi ) cos θi sin θi dθi dφi(1)Then occlusion of the incoming light by the occluding object is considered as
E=

Z π Z π2−π0

L0 (θi , φi )S(θi , φi ) cos θi sin θi dθi dφi(2)where S(θi , φi ) are occlusion coefficients; S(θi , φi ) = 0 if
L0 (θi , φi ) is occluded by the occluding object; Otherwise
S(θi , φi ) = 1.

2.2 From Scene Irradiance to Scene RadianceSome of the incoming lights at point A are reflected toward the image plane. As a result, point A becomes a secondary light source with scene radiance, which can be computed from scene irradiance at point A.
The bidirectional reflectance distribution function(BRDF) f (θi , φi ; θe , φe ) is defined as a ratio of the radiance of a surface as viewed from the direction (θe , φe ) to
the irradiance resulting from illumination from the direction (θi , φi ). Thus, by integrating the product of the BRDF
and the illumination radiance over the entire hemisphere,
the scene radiance Rs(θe , φe ) viewed from the direction

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3.1 Approximation of Illuminate Distribution bya Geodesic Dome

n

θ

δθi

A

A

φ
(a)

(b)

δφi

Figure 2: (a)the direction of incident and emitted light rays(b)infinitesimal patch of an extended light source)(θe , φe ) is computed as
Z π Z π2f (θi , φi ; θe , φe )L0 (θi , φi )Rs(θe , φe ) =−π

2.3

P (θe , φe ) =(3)

From Scene Radiance to Image Irradiance

Finally, the illumination radiance of the scene is relatedwith image irradiance on the image plane. Since what we
actually observe is not image irradiance on the image plane,
but rather a recorded pixel value in a shadow image, it isalso necessary to consider the conversion of the image irradiance into a pixel value of a corresponding point in theimage. This conversion includes several factors such as D/A
and A/D conversions in a CCD camera and a frame grabber.
Other studies concluded that image irradiance was proportional to scene radiance [6]. In our method, we calibratea linearity of the CCD camera by using a gray scale chart sothat the recorded pixel values also become proportional to
the scene radiance of the surface. From Equation 3 the pixel
value of the shadow image P (θe , φe ) is thus computed as
Z π Z π2f (θi , φi ; θe , φe )L0 (θi , φi )P (θe , φe ) = k−π0S(θi , φi ) cos θi sin θi dθi dφinXf (θi , φi ; θe , φe )L(θi , φi )S(θi , φi ) cos θi

i=10

S(θi , φi ) cos θi sin θi dθi dφiFirst, the double integral in Equation 4 is approximatedby discrete sampling over the entire surface of the extended
light source.
Node directions of a geodesic dome are used for approximating the illumination distribution of the scene as asummation of illumination radiance sampled at equal solid
angles. Nodes of a geodesic dome are known to be uniformly distributed over the surface of a sphere. Therefore,
by using n nodes of a geodesic dome in a northern hemisphere as a sampling direction, the double integral in Equation 4 is approximated as a sampling at an equal solid angleδω = 2π/n.

(4)where k is a scaling factor between scene radiance and apixel value. Due to the scaling factor k, we are able toestimate unknown L0 (θi , φi )(i = 1, 2, ...., n) up to scale.
To obtain the scale factor k, we need to perform photometric calibration between pixel intensity and physical unit
(watt/m2 ) for the irradiance.

3 Estimation of Illumination Distribution Using Image Irradiance
After obtaining the formula which relates the illumination radiance of the scene with the pixel values of the
shadow image, illumination radiance is estimated based on
the recorded pixel values of the shadow image.

(5)
where L(θi , φi ) is the illumination radiance per solid angleδω = 2π/n coming from the direction (θi , φi ), which also
includes the scaling factor k between scene radiance andpixel values. The number of the nodes n can be adjusted bychanging the sampling frequency of a geodesic dome.
It should be noted that the recorded pixel value P (θe , φe )is computed as a function of the illumination radianceL0 (θi , φi ) and the BRDF f (θi , φi ; θe , φi ) in Equation 5. We
thus take two different approaches depending on whether
BRDF of the surface is given in the following sections. We
explain the case where the BRDF is given in Section 3.2
and Section 3.3, and the other case where the BRDF is not
given in Section 3.4.

3.2 Known Reflectance Properties: LambertianModel
Suppose the surface is a Lambertian surface; BRDF
f (θi , φi ; θe , φe ) for a Lambertian surface is known to bea constant. From Equation 5, an equation for a Lambertiansurface is obtained as
P (θe , φe ) =nX

Kd L(θi , φi )cosθi S(θi , φi )

(6)

i=1

where Kd is a diffuse reflection parameter of the surface.
From Equation 6, a linear equation is obtained for each image pixel of the shadow image asa1 L1 + a2 L2 + a3 L3 + · · · + a1n Ln = P(7)where Li (i = 1, 2, ...., n) are n unknown illumination radiance specified by n node directions of a geodesic dome.
The coefficients ai (i = 1, 2, ...., n) represent Kd cos θi Si in
Equation 6; we can compute these coefficients from the 3D
geometry of a surface point, the occluding object and the1063-6919/99 $10.00 (c) 1999 IEEE

illuminant direction.1 P is the values of the image pixel
P (θe , φe ).
If we select a number of pixels, say m pixels, a set of
linear equations is obtained as
a11 L1 + a12 L2 + a13 L3 +· · · + a1n Ln= P1

a21 L1 + a22 L2 + a23 L3 +a31 L1 + a32 L2 + a33 L3 +· · · + a2n Ln· · · + a3n Ln

= P2
= P3

···
am1 L1 + am2 L2 + am3 L3 + · · · + amn Ln···= Pm(8)Therefore, by selecting a sufficiently large number of image pixels, we are able to solve for a solution set of unknown Li ’s [17]. Note that, since each pixel consists of 3color bands (R, G, and B), each band of radiance Li is also
estimated from the corresponding color band of the image.2

3.3

Known Reflectance Properties:
Non-Lambertian Model

P (θe , φe ) =i=1

Even in the case where the BRDF is not given, we arestill able to estimate an illumination distribution of a realscene if the surface is a Lambertian surface. The question
we have to consider here is how to cancel the additional
unknown number Kd in Equation 6.
An additional image of the scene taken without the occluding object is used to cancel Kd . We refer to the image as a surface image. The image irradiance of a surface
image represents the surface color of the plane in the casewhere none of the incoming light is occluded. From this,in the case of the surface image, the shadow coefficients
S(θi , φi ) always become S(θi , φi ) = 1. Therefore, using
Equation 6, the image irradiance P 0 (θe , φe ) of the surface
image is computed as
P 0 (θe , φe ) = Kd

Our method is not limited only to the Lambertian reflection model; but it can also be extended to other reflection models. As shown in the previous case, our method
requires a set of linear equations with unknown illumination radiance. Hence, any reflection model is applicableto our method providing such a set of linear equations is
obtained. Take a simplified Torrance-Sparrow reflection
model [15, 20] for example; the pixel value of shadow image P (θe , φe ) is computed asnX

3.4 Unknown Reflectance Properties : Lambertian Model

L(θj , φj )cosθj

(10)

j=1

From Equation 6 and Equation 10, the unknown Kd iscanceled asPn
Kd i=1 L(θi , φi )cosθi S(θi , φi ))P (θe , φe )Pn=
P 0 (θe , φe )Kd j=1 L(θj , φj )cosθjn
X=

L(θi , φi )cosθi S(θi , φi )j=1 L(θj , φj )cosθj

Pn

i=1

2

−γ(θi ,φi )1(Kd cosθi + Ks
e 2σ2
)
cosθe

S(θi , φi )L(θi , φi )

n
X

(11)
(9)

where γ(θi , φi ) is the angle between the surface normal
and the bisector of the light source direction and the viewing direction, Kd and Ks are constants for the diffuse andspecular reflection components, and σ is the standard deviation of a facet slope of the Torrance-Sparrow reflection
model.
From Equation 9, we obtain a linear equation foreach image pixel where L(θi , φi )(i = 1, 2, ...., n)
are unknown illumination radiance, and (Kd cosθi +−γ(θi ,φi )21
e 2σ2 )S(θi , φi ) (i = 1, 2, ...., n) are known coKs cosθeefficients. Again, if we use a sufficiently large number ofpixels for the estimation, we are able to solve for a solution set of unknown illumination radiance L(θi , φi )(i =1, 2, ...., n).
1 We established the correspondence between the 3D world coordinate
system in the scene and the 2D image coordinate system by using the camera calibration algorithm proposed by Tsai [21]. From the calibration process, a plane of z = 0 is also defined on the calibration board, onto which
the occluding object cast shadows.
2 The question of how to select image pixels to obtain a solution set forunknown radiance values seems to be leading to an interesting research
topic. For instance, a similar discussion on this subject can be foundin [10].

Finally, we obtain a linear equation for each image pixel
i ,φi )(i = 1, 2, ...., n) are unknowns,
where Pn L(θL(θ ,φ )cosθ
j=1

j

j

j

cosθi S(θi , φi ) (i = 1, 2, ...., n) are computable coefficients,
e ,φe )and PP0(θ(θe ,φe ) is a right-hand side quantity. Again, if we use
a sufficiently large number of pixels for the estimation, weare able to solve the set of linear equations for a solution set
i ,φi )(i = 1, 2, ...., n).
of unknown Pn L(θL(θ ,φ )cosθ
j=1

j

j

j

We should point out that the estimated radiance fromthese equations is a ratio of the illumination radiance in
one direction
L(θi , φi ) to scene irradiance at the surface
Pnpoint j=1 L(θj , φj )cosθj . Hence, without knowing theratio of the scene irradiance among color bands, there is no
way to relate the estimated radiance over the color bands.
Our method avoids this problem because of the initial camera calibration. Since we use a white board with regularly
spaced dots as a calibration board, the recorded color of theboard directly shows the ratio of the scene irradiance amongcolor bands.

4 Experimental Results
We have tested the proposed method by using real images of indoor environments. To evaluate the accuracy of
the illumination distribution estimated by our method, we

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superimpose a synthetic object with the same shape as thatof the occluding object onto an image of the scene taken
without the occluding object, and compare the shadows of
the synthetic object with those of the occluding object in the
shadow image. Section 4.1 explains how to superimpose asynthetic object onto the real scene by using the estimatedillumination distribution. In Section 4.2, we describe experimental results in the case where reflectance propertiesof a reflected surface are known. Then, in Section 4.3, we
describe experimental results in the case where reflectanceproperties of the surface are unknown.

4.1

Superimposing a Synthetic Occluding Object
onto the Scene

The ray casting algorithm is used to superimpose a synthetic object. If the ray generated from camera projection
center through the image pixel intersects a synthetic object,
we compute a color to be observed at the surface point using
a simplified Torrance-Sparrow reflection model from Section 3.3. From the model, a color to be observed at thesurface point Rs(θe , φe ) is computed using the estimatedillumination distribution of the real scene as
Rsc (θe , φe ) = Kd,cnX

Lc (θi , φi )cosθi

i=1

+Ks,c

n
X
i=1

Lc (θi , φi )−γ(θi ,φi )2
1
e 2σ2
cosθe

(12)
c = R, G, B

where Lc (θi , φi ) (i = 1, 2, ...., n)are the estimated illumination radiance values.
Otherwise, the influence of the synthetic object onto thereal object surface is considered. In other words, we create
shadows cast by the synthetic object onto the surface.
First, we compute total irradiance E1 at the surface point
using the estimated illumination distribution in the casewhere a synthetic object does not occlude any incoming
light (Figure 1.a).
E1,c =nX

Lc (θi , φi )cosθic = R, G, B

(13)

i=1

where L(θi , φi ) (i = 1, 2, ...., n) are the estimated illumination radiance values.
Second, we compute total irradiance E2 at the surface
point in the case where the synthetic object occludes someof the incoming light (Figure 1.b).
E2,c =nX
i=1

Lc (θi , φi )cosθi S(θi , φi )c = R, G, B

(14)
where S(θi , φi ) = 0 if the synthetic object occludes illumination radiance L(θi , φi ); otherwise, S(θi , φi ) = 1.

Then, we compute the ratio of E2 to E1 , which represents how much of the irradiance at the intersection would
still be preserved if the synthetic object were placed in thescene. Finally, by multiplying the ratio E2 /E1 to the observed color of the image pixel I, we obtain the color I 0
that would be the color of the image pixel if there were asynthetic object in the scene.
Ic0 = Ic

E2,c
E1,c

c = R, G, B

(15)

4.2 Experimental Results for Known Reflectance
Property
An image of a surface with an occluding object called ashadow image was taken under the usual illumination environment in our office, including direct light sources suchas fluorescent lamps and windows to the outside, as well as
indirect illumination such as reflections from a wall (Figure 3).
First, an illumination distribution of the scene was estimated using the image irradiance inside shadows in the
shadow image as explained in Section 3.2. Then a synthetic
object with the same shape as that of the occluding object
was superimposed onto an image of the scene taken without
the occluding object, called the surface image. Synthesized
results are shown in Figure 4 (a), (b), and (c). Also, wesuperimposed another synthetic object of a different shape
onto the scene in Figure 4(d). The number of nodes of ageodesic dome used for the estimation is shown under the
resulting image.
We found through our experiments that, the larger number of nodes we used, the more the shadows of the synthetic object resembled those of the occluding object in the
shadow image. Especially in the case of 521 nodes, theshadows of the synthetic object are indistinguishable fromthose of the occluding object in the shadow image: thisshows that the estimated illumination distribution gives agood presentation of that of the real scene.
Figure 5 numerically shows the improvement of the accuracy by increasing the number of samplings. The vertical axis represents average error in pixel values inside theshadow regions in the synthesized images compared withthose in the shadow image. Here, the initial difference inpixel values of shadow regions between the surface image
and the shadow image is set to 100 %. The horizontal axis
represents the number of nodes of a geodesic dome used forthe estimation. From the plot in the figure, we can clearly
see that the accuracy improves rapidly as we use more imaginary point light sources.
Also the small pictures right next to the plot shows error
distributions inside shadow regions in the synthesized images. Darker color represents larger error in a pixel value inthe shadow regions compared with the real shadows of theoccluding object in the shadow image.

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average error
100
90
80

(a)

(b)

(c)

70
60

Figure 3: Input images : (a) surface image (b) shadow image (c) calibration image

50
40
30
20
10
0
0

(a) number of nodes : 89(b) number of nodes : 193(c) number of nodes : 521 (d) number of nodes : 521

Figure 4: Synthesized images: known reflectance property

Also, the resulting images indicate that it is required toadjust the number of nodes of a geodesic dome dependingon the complexity of a scene to obtain a reasonably good
estimation for less computational cost. We are currently
extending our work so that an appropriate number of nodes
is automatically selected for the estimation, depending on
the scene complexity.

4.3

Experimental Results for Unknown Reflectance Property

We also applied our method to the case where reflectanceproperties of a surface were unknown. The input images
used in this experiment are shown in Figure 6. Since the reflectance properties of the surface were unknown, the image
irradiance of both the shadow image and the surface image
were used for estimating the illumination distribution of thescene as explained in Section 3.4.
In the same way as in the previous case, a synthetic occluding object was superimposed onto the surface of thesurface image. Synthesized results are shown in Figure 7.
Again, in the case of 521 nodes, the shadows in the resulting
image strongly resemble those of the occluding object in the
shadow image. This shows that the estimated illumination
distribution gives a good representation of the characteristics of the real scene.

50

100

150

200

250

300

350

400

450

number of nodes

Figure 5: Error Analysis: known reflectance property

We concluded from our experiments that the proposed
method is effective for providing an illumination distribution which can be used as a substitution for a real illumination distribution.

5 Conclusions
In this paper, we have proposed a new method for estimating an illumination distribution of a real scene from aradiance distribution inside shadows cast by a real object ofknown shape onto other object surface of known shape andknown reflectance. By using the occlusion information of
the incoming light, we could estimate an illumination distribution of a real scene reliably even for the images taken
in a complex illumination environment.
There have also been several methods proposed for measuring illumination of a real scene in the field of augmented
reality research [3, 4, 5]. However, those methods tended tomeasure the illumination directly from images of the scene
and therefore, they suffered from two technical problems:
how to capture a wide field of view of the scene, and how
to record high dynamic range of the scene. In the proposed
method, since we observe shadows and not the illumination
itself, no effort to overcome these problems is required.
To demonstrate the effectiveness of the proposed
method, we have successfully tested our method by using
sets of real images taken in our office with different surface
materials of shadow regions.

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(a)

(b)

(c)

60

Figure 6: Input images : (a) surface image (b) shadow image (c) calibration image

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