Estimation of Illumination Distribution
by Using Soft Shadows

Imari Sato

Yoichi Sato

Katsushi Ikeuchi

November 1998
No. 3

Center for Spatial Information Science
The University of Tokyo
4-6-1 Komaba, Meguro-ku, Tokyo 153-8904

Copyrightc1998 by Imari SatoAbstract

The image irradiance of a three-dimensional object is known to be thefunction of three components: the distribution of light sources, the shape,and reflectance of a real object surface. In the past, recovering the shapeand reflectance 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 andreflectance of a real object. In this paper, we propose a new method for
estimating the illumination distribution of a real scene from image brightness observed on a real object surface in that scene. More specifically, werecover the illumination distribution of the scene from a radiance distribution inside shadows cast by an object of known shape onto another object
surface of known shape and reflectance. By using the occlusion information of the incoming light, we are able to reliably estimate the illumination
distribution of a real scene, even in a complex illumination environment.

CSIS Discussion Paper No. 3

1

1 Introduction
The image irradiance of a three-dimensional object is known to be the function of thefollowing three components: the distribution of light sources, the shape, and reflectanceof a real object surface. From the relationship among them, three kinds of analyses
on the recorded image brightness are derived: recovering the surface shape from thesurface reflectance and illumination of the scene, recovering the surface reflectancefrom 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, 17] . In contrast, there
has been little progress on the subject of recovering illumination from the knowledgeof the shape and the reflectance of an object surface. This is because 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 related with the first two kinds of analyses aimed toestimate illuminant direction and color in a very specific illumination condition suchas 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 in recovering an illumination
distribution of a real scene from the knowledge of the shape and reflectance of an real
object. In the proposed method, we use radiance changes inside shadows rather thanappearance 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 arecaused by the occlusion of incoming lights as illustrated in Figure 1, and thus shadows
contain various pieces of information about the illumination of the scene. Nevertheless,in the past, shadows have been used for determining the 3D shapes and orientations ofan object which cast shadows onto the scene [2, 11, 14, 18], while very 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 distributionof a real scene by making use of the occlusion information of the incoming light. Also,
our method 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.

2

1.1

Overview

Before we describe the proposed method in detail, we should clarify the basic steps ofour method.
First, we take an image of the scene using a color CCD camera so that shadowsof an object appear in the image. In the rest of the paper, we refer to the image withshadows as the shadow image and the object of known shape, which cast shadows onto
the scene, as the occluding object. (A typical example of a shadow image is shown in
Figure 3.)
Then, based on the radiance distribution inside shadows, an illumination distribution of a real scene is estimated as a collection of imaginary point light sources distributed over the entire scene. The key idea of the proposed method is the discretizationof 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 light sources for the entire illumination, we are able to derive a system of equations in n unknown radiance values ofimaginary point light sources from the image irradiance of the shadow image. We thensolve 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 irradianceof a shadow image. We first obtain a formula which relates an illumination distributionof 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 a unique illumination radiancesolution 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 an illumination distribution of a realscene with the image irradiance of a shadow image. Based on the image formation, theformula is obtained as follows:

CSIS Discussion Paper No. 3

3

1. Illumination radiance to scene irradiance: find a relationship between the illumination distribution of a real scene and the irradiance at a surface point in thescene.
2. Scene irradiance to scene radiance: compute how much 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 and the 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 takeillumination from all directions into account, let us consider an infinitesimal patch of
the extended light source, of size δθi in polar angle and δφi in azimuth as shown in
Figure 2.
Seen from the center point A, this patch subtends a solid angle δω = sin θi δθi δφi .
Let L0 (θi , φi) be the illumination radiance per unit solid angle coming from the direction (θi , φi); then the radiance from the patch is L0 (θi , φi ) sin θi δθi δφi [6], and the total
irradiance of the surface point A isE=

Z π Z π2

−π0L0 (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

−π0L0 (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 theoccluding object; Otherwise S(θi , φi) = 1.

4

n

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)

2.2

From Scene Irradiance to Scene RadianceSome of the incoming lights at point A are reflected toward the image plane. As aresult, point A becomes a secondary light source with scene radiance, which can becomputed 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 theirradiance resulting from illumination from the direction (θi , φi ). Thus, by integrating
the product of the BRDF and the illumination radiance over the entire hemisphere, thescene radiance Rs(θe , φe ) viewed from the direction (θe , φe ) is computed as
Rs(θe , φe ) =2.3

Z π Z π2

−π0f (θi , φi ; θe , φe )L0 (θi , φi)S(θi , φi) cos θi sin θi dθi dφi(3)From Scene Radiance to Image Irradiance

Finally, the illumination radiance of the scene is related with image irradiance on theimage 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 is also necessary to consider the conversion of the image irradiance into a pixel value of a corresponding pointin the image. 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 calibrate a 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 computedas
P (θe , φe ) = k

Z π Z π2

−π0f (θi , φi ; θe , φe )L0 (θi , φi)CSIS Discussion Paper No. 3
S(θi , φi ) cos θi sin θi dθi dφi5(4)where k is a scaling factor between scene radiance and a pixel value. Due to the scaling
factor k, we are able to estimate unknown L0 (θi , φi )(i = 1, 2, .., n) up to scale. To obtainthe scale factor k, we need to perform photometric calibration between pixel intensityand 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 withthe pixel values of the shadow image, illumination radiance is estimated based on therecorded pixel values of the shadow image.

3.1

Approximation of Illuminate Distribution by a Geodesic DomeFirst, the double integral in Equation 4 is approximated by discrete sampling over theentire surface of the extended light source (Section 3.1).
Node directions of a geodesic dome are used for approximating the illumination
distribution of the scene as a summation of illumination radiance sampled at equal
solid angles. Nodes of a geodesic dome are known to be uniformly distributed overthe surface of a sphere. Therefore, by using n nodes of a geodesic dome in a northernhemisphere as a sampling direction, the double integral in Equation 4 is approximatedas a sampling at an equal solid angle δω = 2π/n.
P (θe , φe ) =nXf (θi , φi ; θe , φe )L(θi , φi )S(θi , φi ) cos θi

(5)

i=0

where L(θi , φi ) is the illumination radiance per solid angle δω = 2π/n coming fromthe direction (θi , φi), which also includes the scaling factor k between scene radianceand pixel values. The number of the nodes n can be adjusted by changing the sampling
frequency of a geodesic dome.
It should be noted that the recorded pixel value P (θe , φe ) is computed as a functionof the illumination radiance L0 (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 givenin 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: Lambertian Model

Suppose the surface is a Lambertian surface; BRDF f (θi , φi ; θe , φe ) for a Lambertiansurface is known to be a constant. From Equation 5, an equation for a Lambertian6
surface is obtained as
P (θe , φe ) =nX

Kd L(θi , φi)cosθi S(θi , φi )

(6)

i=0

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 the 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 obtainedas
a11 L1 + a12 L2 + a13 L3 + · · · + a1n Ln
a21 L1 + a22 L2 + a23 L3 + · · · + a2n Ln
a31 L1 + a32 L2 + a33 L3 + · · · + a3n Ln···
am1 L1 + am2 L2 + am3 L3 + · · · + amn Ln

= P1
= P2
= P3
···
= Pm

(8)

Therefore, by selecting a sufficiently large number of image pixels, we are able tosolve for a unique solution set of unknown Li ’s. 2 Note that, since each pixel consistsof 3 color bands (R, G, and B), each band of radiance Li is also estimated from thecorresponding color band of the image.

3.3

Known Reflectance Properties: Non-Lambertian Model

Our method is limited not 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 applicable to our method providing such a set of linear equations is obtained.
1
We established the correspondence between the 3D world coordinate system in the scene and the2D image coordinate system by using the camera calibration algorithm proposed by Tsai [20]. From thecalibration 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 unique solution set for unknown radiancevalues seems to be leading to an interesting research topic. For instance, a similar discussion on thissubject can be found in [10].

CSIS Discussion Paper No. 3

7

Take a simplified Torrance-Sparrow reflection model [15, 19] for example; the pixel
value of shadow image P (θe , φe ) is computed as
n
X

n
X

−γ(θi ,φi )2
1
P (θe , φe ) = Kd
L(θi , φi )S(θi , φi )cosθi + Ks
L(θi , φi )S(θi , φi )
e 2σ2
cosθ
r
i=0
i=0

=

n
X

(Kd cosθi + Ks

i=0−γ(θi ,φi )21
e 2σ2 )S(θi , φi )L(θi , φi )cosθrwhere θr is the angle between the surface normal and the viewing direction, 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 and specular reflection components, and σ is the standard deviation of a facet slope of the TorranceSparrow reflection model. From Equation 9, we obtain a linear equation for each
image pixel where L(θi , φi )(i = 1, 2, .., n) are unknown illumination radiance, and−γ(θi ,φi )2

1(Kd cosθi + Ks cosθe 2σ2 )S(θi , φi) (i = 1, 2, .., n) are known coefficients. Again,r
if we use a sufficiently large number of pixels for the estimation, we are able to solve
for a unique solution set of unknown illumination radiance L(θi , φi )(i = 1, 2, .., n).

3.4

Unknown Reflectance Properties : Lambertian Model

Even in the case where the BRDF is not given, we are still able to estimate an illumination distribution of a real scene 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 case where 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 ) = Kdn
X

L(θj , φj )cosθj

(9)

j=0

From Equation 6 and Equation 9, the unknown Kd is canceled asP

P (θe , φe )Kd ni=0 L(θi , φi )cosθi S(θi , φi))=P
P 0(θe , φe )Kd nj=0 L(θj , φj )cosθjn
X
L(θi , φi )=cosθi S(θi , φi )Pn
j=0 L(θj , φj )cosθj
i=0
(10)

8
i ,φi )
Finally, we obtain a linear equation for each image pixel where Pn L(θL(θj ,φj )cosθjj=0(i = 1, 2, .., n) are unknowns, cosθi S(θi , φi ) (i = 1, 2, .., n) are computable coefficients,e)
is a right-hand side quantity. Again, if we use a sufficiently large number
and PP0(θ(θee,φ,φe )of pixels for the estimation, we are able to solve the set of linear equations for a unique
i ,φi )
solution set of unknown Pn L(θ(i = 1, 2, .., n).
L(θj ,φj )cosθjj=0

We should point out that the estimated radiance from these equations is a ratio ofof the illumination radiance in one direction L(θi , φi ) to scene irradiance at the surfacePpoint nj=0 L(θj , φj )cosθj . Hence, without knowing the ratio of the scene irradianceamong 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 the board directly shows the ratio of the scene irradiance among color 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 superimpose a synthetic object with the same shape as that of 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 a synthetic object onto the real scene by using the estimated illumination distribution. In Section 4.2, we describe experimental results in thecase where reflectance properties of a reflected surface are known. Then, in Section 4.3,
we describe experimental results in the case where reflectance properties 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-Sparrowreflection model from Section 3.3. From the model, a color to be observed at the surface
point Rs(θe , φe ) is computed using the estimated illumination distribution of the realscene asRsc (θe , φe ) = Kd,c

n
X
i=0

Lc (θi , φi )cosθi + Ks,c

n
X
i=0

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

c = R, G, B
where Lc (θi , φi ) (i = 1, 2, .., n)are the estimated illumination radiance values.

CSIS Discussion Paper No. 3

9Otherwise, the influence of the synthetic object onto the real 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 case where a synthetic object does not occlude any incoming
light (Figure 1.a).
E1,c =nX

Lc (θi , φi )cosθic = R, G, B

(11)

i=0

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 thesynthetic object occludes some of the incoming light (Figure 1.b).
E2,c =nX

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

(12)

i=0

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
the scene. 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 werea synthetic object in the scene.
Ic0 = Ic

4.2

E2,c
E1,c

c = R, G, B

(13)

Experimental Results for Known Reflectance Property

An image of a surface with an occluding object called a shadow image was taken under
the usual illumination environment in our office, including direct light sources such as
fluorescent lamps and windows to the outside, as well as indirect illumination such asreflections 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, we superimposed anothersynthetic object of a different shape onto the scene in Figure 4(d). The number ofnodes of a geodesic dome used for the estimation is shown under the resulting image.
We found through our experiments that, the larger number of nodes we used, themore the shadows of the synthetic object resembled those of the occluding object inthe shadow image. Especially in the case of 521 nodes, the shadows of the synthetic

10
object are indistinguishable from those of the occluding object in the shadow image:this shows that the estimated illumination distribution gives a good presentation of thatof the real scene.
Figure 5 numerically shows the improvement of the accuracy by increasing thenumber of samplings. The vertical axis represents average error in pixel values inside
the shadow regions in the synthesized images compared with those in the shadow image. Here, the initial difference in pixel values of shadow regions between the surface
image and original image is set to 100 %. The horizontal axis represents the numberof nodes of a geodesic dome used for the estimation. From the plot in the figure, we
can clearly see that the accuracy improves rapidly as we use more imaginary point lightsources.
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
in the shadow regions compared with the real shadows of the occluding object in the
shadow image.
Also, the resulting images indicate that it is required to adjust the number of nodesof a geodesic dome depending on 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 reflectance properties 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 boththe shadow image and the surface image were used for estimating the illumination
distribution of the scene 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 the surface 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 estimatedillumination distribution gives a good representation of the characteristics of the realscene.
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.

CSIS Discussion Paper No. 3

115 Conclusions
In this paper, we have proposed a new method for estimating an illumination distribution of a real scene from a radiance distribution inside shadows cast by a real object ofknown shape onto other object surface of known shape and known 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 areal scene in the field of augmented reality research [3, 4, 5]. However, those methods
tended to measure 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 thescene, 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.

12

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CSIS Discussion Paper No. 3

13[18] S. A. Shafer and T. Kanade, “Using Shadows in Finding Surface Orientations,” Computer Vision,
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14

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

number of nodes

Figure 5: Error Analysis: known reflectance property

CSIS Discussion Paper No. 3

15

Figure 6: Input images : (a) surface image (b) shadow image (c) calibration image(a) number of nodes : 89(b) number of nodes : 193(c) number of nodes : 521(d) number of nodes : 521

Figure 7: Synthesized images: unknown reflectance property
average error

number of nodes

Figure 8: Error Analysis: unknown reflectance property