Stability Issues in Recovering Illumination Distribution from
Brightness in Shadows
Imari Sato
Yoichi Sato
Katsushi Ikeuchi
Institute of Industrial Science, The University of Tokyo{imarik, ysato, ki}@iis.u-tokyo.ac.jp
Abstract
This paper describes a robust method for estimating, in a
reliable manner, the illumination distribution of a real scenefrom shadows in a given image. In general, shadows in ascene are caused by the occlusion of incoming light; image
brightness inside shadows have a great potential for providing distinct clues to the illumination distribution of the scene.
Taking advantage of this fact, we recently proposed to estimate the illumination distribution of a real scene from asingle image of the scene. The proposed method has been
applied successfully to real images with complex illumination distributions. Nevertheless, it was found that, under certain circumstances, the method sometimes failed to provide a
correct estimate of illumination distribution. Those failures
stem from the fact that the method does not take into accountseveral factors regarding the stability of illumination estimation. The purpose of this study is to analyze how muchinformation is obtainable from a given image about the illumination distribution of the scene. In particular, we carefully
examine the source of instability of using shadows obtained
from a single image for the estimation in several aspects:
blocked view of shadows by the object; limited sampling resolution for image brightness inside shadows; and the appropriate light model to approximate the illumination distribution of the scene. Based on this analysis, we propose a new
method that guarantees to estimate the illumination distribution of a scene in a reliable manner regardless of types of
input images such as the shape of an occluding object or acamera position.

1 Introduction
The image brightness of a three-dimensional object is thefunction of the following three components: the distributionof light sources, the shape, and reflectance of a real object
surface [5, 6]. Three major research areas in physics-based
vision are derived from the relationship among them : shapefrom-brightness (with a known reflectance and illumination);reflectance-from-brightness (with a known shape and illumination); illumination from brightness (with a known shapeand reflectance).
The first two kinds of analyses, shape-from-brightness
and reflectance-from-brightness, have been intensively studied using the shape from shading method [4, 7, 8, 18], as wellas through reflectance analysis research [1, 9, 11, 13, 16, 22].

In contrast, relatively limited amounts of research
have been conducted in the third area, illumination-frombrightness [3, 7, 12, 15, 17, 19, 25]. This is because realscenes usually include both direct and indirect illumination
distributed in a complex way and it is difficult to analyze
characteristics of the illumination distribution of the scenefrom image brightness. Most of the previously proposed approaches were conducted under very specific illumination
conditions, e.g. there is only one direct light source in thescene, and difficult to be extended for more natural illumination conditions, or multiple input images taken from different
viewing angles were necessary.
Recently, we turned our attention to the information
which shadows cast by an object in a scene could provide
about the illumination distribution of the scene,1 and introduced a method for estimating the illumination distributionof a real scene from shadows observed in a single image [20].
This method was later extended for more efficient estimationof illumination distribution by incorporating an iterative refinement scheme [21].
Our method has shown promising results for estimating
illumination distribution in a real scene. Nevertheless, thequestion of how to examine whether a given input image provides sufficient information to robustly estimate the illumination distribution of the scene still remains to be addressed.
In fact, it was found that, under certain circumstances, ourmethod sometimes failed to provide a correct estimate of illumination distribution.
The instability of this method stems from the lack of consideration of the fact that shadow pixels observed in a single image provide only a limited amount of the information
about the illumination distribution of a scene. Namely, anobject in a scene occludes part of the view from a camera
taking input images, and only part of the shadows cast by theobject can be observed in the image. Also, the field of viewof a camera taking an input image directly controls what size
portions of shadows cast by an object can be observed in the
input image. Furthermore, since the number of image pixels
is limited, radiance distribution inside shadows can be measured only up to a certain resolution. Without taking those
1 In the past, shadows have been mainly used for determining the 3D
shapes and orientations of an object which cast shadows onto the scene [2,10, 14, 23].

factors into consideration, estimation of illumination distribution of a real scene from a given image may not be performed reliably.
The purpose of this study is to present a method to analyze how much information is obtainable from a given input
image for a particular direction of the entire illumination distribution. In particular, we carefully examine the source ofinstability of using shadows obtained from a single image for
estimating the illumination distribution of a scene in several
aspects: blocked view of shadows by the object; limited sampling resolution for image brightness inside shadows; and theappropriate light model to approximate the illumination distribution of the scene. Then, based on this analysis, a set oflight sources that approximates the illumination distributionof the scene is defined from the given input image so that radiance values of those light sources can be estimated reliably
regardless of types of input images such as the shape of an
occluding object or a camera position. This new proposed
approach based on the analysis in observed image brightness
enlarges the variety of images to which the method can beapplied.
The rest of the paper is organized as follows. In Section 2, we briefly describe the basic framework of our previously proposed method for estimating illumination distribution of a real scene from a single image. Then, in Section 3, we explain several problems regarding the stability ofthe estimation method. In Section 4, we provide solutions
to overcome those problems, and report experimental results
to demonstrate the effectiveness of the proposed solutions in
Section 5. Finally, in Section 6, we present concluding remarks.

2 Overview of illumination estimation fromshadows
Before we discuss each of the factors causing instabilityof illumination estimation, let us briefly present the overview
of our previously proposed method for estimating illumination distribution of a scene from a given image of thescene [20].
In the following sections, we refer to the image with shadows as the shadow image, to the object which casts shadows
onto the scene as the occluding object, and to the surface
onto which the occluding object casts shadows as the shadow
surface.
The illumination distribution of a scene is first approximated by discrete sampling of an extended light source inour method; whole distribution is represented as a set ofpoint sources equally distributed in the scene as shown in
Figure 1. Then total irradiance E at the shadowsurf ace received from the entire illumination distribution is computed
as
E=

n


Li Si cos θiθφillumination distribution of a scene is approximated by discrete
sampling over the entire surface of the extended light source.

Figure 1: Approximation of the illumination distribution ofa scene(1) Each shadow pixel provides a linear equationn

unknown L i∑ f θ i  ϕ i θ e  ϕ e  Lθ i  ϕ i  S θ i  ϕ i   θ i = Pθ e  ϕ e i =

a  L  + a  L  + a  L  + ... + a n L n = P

coefficients ai

θ e ,ϕe  

image brightness(2) Select m pixels, a set of linear equations is obtained asaL  + a  L  + a  L + ... + a n L n = P aL  + a  L  + a  L + ... + a  n L n = P ...
aL  + a  L  +. a  L + + a n L n = P ..

amL  + a m  L  + a m  L + ... + a mn L n = Pm

Solve for a solution set of
unknown L i

Figure 2: Outline of the previously proposed method

solid angle δω = 2π/n coming from the direction (θi , φi ),and Si are occlusion coefficients. Si = 0 if Li is occludedby the occluding object, and Si = 1 otherwise. Then,
this approximation leads each image pixel inside shadows
to provide a linear equation with unknown radiance of thosesources as
a1 L1 + a2 L2 + a3 L3 + · · · + a1n Ln = P(2)where P is the image brightness of the pixel, and the coefficients ai (i = 1, 2, ...., n) represent how much incoming Li
is reflected on the shadow surface towards the viewing direction. Here, ai includes occlusion coefficients Si , the bidirectional reflectance distribution function (BRDF) and cosine factors as illustrated in Figure 2 (1). These coefficientsare computed from the 3D geometry of a surface point, theoccluding object and the illuminant direction, and the surface reflectance property of the shadow surface that is either
given or estimated.2
Finally, a set of linear equations is derived from thebrightness changes observed in the shadow image and solved
for a solution set of unknown radiance of Li ’s in Equation 3.

(1)

i=1

where Li (i = 1, 2, ...., n) is the illumination radiance per

2 For modeling the shape of an occluding object from a shadow image, amodeling tool called the 3D Builder from 3D Construction Company [26]was used to obtain the experimental results shown in Section 5.

3



a11
 a21

 a31


am1

a12
a22
a32
···
am2

a13
a23
a33
···
am3

···
···
···
···









P1
L1
a1n
 L2   P2 
a2n     
 L3   P3 
a3n 
= 

.   . 
 ..   .. 
amn
Ln
Pm

A·L =P(3)

3 Stability issues in illumination estimation
Shadows are, in general, caused by occlusion of incoming light by an occluding object; brightness changes insideshadows have a potential for providing important clues tothe illumination distribution of the scene. However, brightness distributions inside shadows vary as a function of thelighting environment of the scene and the shape of an occluding object. Furthermore, the amount of the information
contained in a shadow image about the illumination distribution of a scene changes from image to image, depending on
how much of the shadow surfaces are blocked by the occluding object and how much are covered by the field of view of
the camera taking the shadow image.
Therefore, it is essential to evaluate an shadow image firstin terms of how much information about the illumination distribution of the scene the image can provide. The lack of thisevolution step is the main reason why our previously proposed method for illumination estimation sometimes failed
to recover the illumination distribution of a scene from observed shadows. In this section, we explain two main factorsthat control the stability of the illumination estimation fromshadows: blocked view of shadows; limited sampling resolution for radiance distribution inside shadows. Then, later in
Section 4, we will introduce solutions for overcoming thesetwo problems.

3.1

Distribution solvable from a given shadow image

Let us first consider the ideal case where radiance valuesof light source Li ’s can be estimated without suffering fromany instability problems.
As shown in Figure 2 (2), a set of linear equations is derived from a given shadow image. In Equation 3, the coefficients matrix A of the system A · L = P is consists ofn column vectors ai = (a1i , a2i , . . . , ami ) where n is thenumber of sampling directions, and m is the number of theimage pixels. Here, if the coefficients matrix A consist of nlinearly independent column vectors ai , we numerically obtain enough constraints for solving A for unknown radianceof Li ’s.
As has been noted, the amount of the information thatshadows in a shadow image provide is limited. Under
3 For our current implementation, we solve the problem using the linear least square algorithm with non-negativity constraints (using a standard MATLAB function) to obtain an optimal solution with no negative
components.

some circumstances, we might not be able to derive nlinear independent vectors ai from a given shadow image. Note that the coefficients ai in Equation 2 representf (θi , φi ; θe , φe )Si cos θi , and the most powerful clue to theillumination distribution of a scene among the componentsof ai is Si (either 0 or 1) since the other components do
not change so much from image pixel to pixel. As a consequence, it makes the estimation more stable if we observe thedifference between radiance of two shadow regions for eachlight source: one illuminated and the other not illuminatedby the light source. In other words, the more variations in
Si ’s in shadow regions we have, the more likely it is that wewill obtain n linearly independent column vectors, thereby
making it possible to robustly estimate radiance values ofthe light sources approximating illumination distribution of
the scene.
In the following sections, we describe the main sourcesthat prevent us from observing both shadow and non-shadow
regions for a light source in a particular direction. This tends
to makes the matrix A ill-conditioned and leads to the instability problem of the proposed method.

3.2 Blocked view of shadowsAs has been noted, the visible portion of a shadow surfacechanges from image to image depending on how wide thefield of view of a camera taking the shadow image is, andwhat size portions of the view of the camera are occludedby the occluding object. Only part of the shadows cast by
the occluding object can be usually observed in the shadow
image, with the exception of special cases such as a camera
aimed directly towards a floor onto which shadows are cast.
Consider the case shown in Figure 3 as an example. In
this case, the region behind the occluding object cannot beseen by the camera. More precisely, only the shadow causedby the light source 3 can be observed in the shadow image,and shadows cast by the light sources 1 and 2 are not seen.
Let a1 and a2 as column vectors and L1 and L2 as radiance values for the light source 1 and 2 respectively, thenin the case shown in this example, a1 and a2 become close
to singular. In other words, the occlusion coefficients Si ofa1 and those of a2 are the same for all shadow pixels, andonly differences for distinguishing the light source 1 fromthe light source 2 are in f (θi , φi ; θe , φe )cosθi . As a result,the estimation becomes too sensitive to the image noise, andradiance values of the light source 1 and 2 can not be reliably
estimated.

3.3 Limited sampling resolution for radiance distribution inside shadows
Second, the number of image pixels is fixed, and therefore
radiance distribution inside shadow regions can be measured
only up to a certain resolution.
In the basic formulation of the previously proposed
method described in Section 2, it was simply assumed that
we could obtain a large enough number of image pixels
inside shadow regions to provide sufficient constraints for

The

3

by

reg
ion
the blockobj
ect ed1 and 3
1 and 2

1

3

3

2

1 and 2

1 and 3

2

occluding object

2camera1, 2 and 3 are
occluded2

1

3

light sources

Figure 3: Blocked view of shadows

solving a set of linear equation (Equation 3) for unknown radiance values of point light sources. However, this is true
only when we can measure radiance values inside shadowsat any resolution.
To illustrate this point, consider two cases shown in Figure 4. In the first case in (1), illumination distribution ofa scene is represented by coarsely distributed point lightsources 1, 2, and 3. On the other hand, more densely distributed point light sources are used in the other case shownin (2). Shadows are partitioned into smaller regions depending on which light source is occluded by the occluding object. For instance, the shadow in the left case is partitioned
into the following regions in order from left to right: 3, 1&3,
1, 1&2&3, 1&2, and 2.
As we can see in the case in Figure 4 (2), the shadow
does not contain a region where only the point light source
1 is occluded. Also, partitioned regions such as 3 or 2 aresignificantly smaller than those in the case in Figure4 (1).
As a result, we obtain less of a variety of partitioned regions
inside shadows, thus making the estimation less reliable, andthe estimation further becomes more sensitive to image noise
because each partitioned region contains fewer image pixels.
Since the number of image pixels is limited, radiance distribution inside shadows can be measured only up to a certain
resolution. As a result, sampling resolution of illumination
distribution of a real scene is also bound to a certain limit.
For instance, if we use more densely distributed point lightsources for the estimation, we might not even observe any
image pixels of some shadow regions.
It follows from the discussions above that there is achance that the estimation becomes too unstable to provide a
correct estimate of the illumination distribution of the scene
for a given set of point light sources for the estimation. As
a solution to this problem, a new procedure to evaluate an
input shadow image in terms of the information about the illumination distribution of a scene obtainable from the image
is considered in the next section. Using the outputs from thisprocedure, a set of point light sources is defined so that radiance values of those light sources can be estimated reliably

2
1

3

2

1 3
(2)

(1)

Figure 4: Difference in visible combinations of occlusions oflights: (1) coarsely distributed light sources and (2) densely
distributed light sourcesfrom the given shadow images.

4 Techniques for robust illumination estimation
Recall the case shown in Figure 3 where two column vectors a1 and a2 of the light sources 1 and 2 respectively areclose to singular. Here, if the light sources 1 and 2 are combined to form a larger light source for more coarse samplingof illumination distribution, we are able avoid facing the instability problem caused by the light source 1 and the lightsource 2. Similarly, the sampling resolution problem described in Section 3.3 can be avoided if we use the morecoarse sampling instead of dense sampling of illumination
radiance whenever it is decided to be necessary in terms of
the stability of computation for solving the set of linear equations obtained from a given shadow image.
In this section, we will adopt a strategy for changing thesampling density of the illumination distribution dependingon the amount of the information obtainable from a shadow
image for a particular direction of the illumination distribution.

4.1 Selection of illumination distribution samplings
The derived coefficients matrix A from a shadow image
is first examined in terms of the stability of computation forsolving the set of linear equations obtained from a given
shadow image.
Stability of a given system is measured by the condition
number. Some systems are sensitive to errors and others arenot. For a positive matrix A, the condition number c is computed (again using a standard MATLAB function) as
c = σmax /σmin(4)where σmax is the maximum singular value of the matrix A,
and σmin is the minimum singular value.

If this condition number c is sufficiently small, Equation 3
can be solved without numerical instability. Therefore, thegiven set of light sources representing real illumination distribution is appropriate for the given shadow image. On theother hand, if the condition number c is large, the problem ofillumination estimation for the given set of light sources isclose to ill-conditioned. Therefore, we need to reduce lightsources representing illumination distribution by combining
several light sources into one with a larger solid angle.
Then, our question is how to select those light sources
to be combined for sampling the illumination distribution
more coarsely. This is done by examining the column vector ai = (a1i , a2i , . . . , ami ) corresponding to a light source
Li in Equation 3. If two vectors ai and aj for two lightsources Li and Lj are about the same, Equation 3 becomes
ill-conditioned and brightness distribution inside shadows
does not provide sufficient information for determining radiance of those two light sources. Therefore, those two lightsources need to be combined to form a larger light source formore coarse sampling of illumination distribution. We evaluate the similarity of two vectors ai and aj with their dot
product ai · aj .
Since the entire stability computation depends on the coefficients matrix A and the radiance values Li are estimatedfrom these coefficients as well, it is important to compute aiof the matrix A as accurately as possible. For this purpose,
we further introduce the following two techniques. One is atechnique for computing more accurate shadow coefficients
Si using the more appropriate light model for Li . The otheris a technique for sampling pixels from a shadow image insuch a way as to maximize the information obtainable fromthe image.

4.2

Occlusion test of incoming lights

In the proposed approach, the entire illumination distribution of a scene is represented as a set of point sources
equally distributed in the scene whose solid angle is definedby a sampling density n. Nevertheless, the occlusion tests
are previously performed simply by examining occlusionsof the center points of those light sources.4
The important point to note is that all objects casting
shadows from extended light sources produce three types ofregions: completely illuminated, completely occluded (umbra), and partially occluded (penumbra). As illustrated in
Figure 5, a penumbra surrounds an umbra and there is always
a gradual change in intensity from a penumbra and an umbra. The previous approach can treat only hard edged shadows correctly, and the approximation with point light sources
introduces inaccuracy in illumination estimation in the presence of soft edged shadows, i.e., shadows with umbra andpenumbra. Mixing up the umbra and the penumbra area not
only prevents us from computing accurate coefficients ai in
4 As described in Section 2, occlusion coefficients S in Equation 1 bei
comes 0 if the center point of Li is occluded by the occluding object, and
Si = 1 otherwise.

penumbraumbra

Figure 5: Umbra and penumbra

Figure 6: Hexagonal grids used for supersampling inside one
light source

Equation 2 but also reduces the varieties of occlusions observed in a shadow image.
One approach to reduce the side-effect from this approximation is to increase the number of point light sources used
for approximating illumination distribution and to computeocclusion coefficient Si more accurately between 0 and 1,
cf., 0 or 1.5
In our actual implementation of this super-sampling
scheme, a hexagonally packed grid is defined inside eachlight source and used for computing the occlusion coefficients. The size of a hexagonally packed grid for alight source is determined from the solid angle of the lightsource [24]. Figure 6 shows a hexagonal grid, made up of
19 hexagons, used in our experiments. For each point inside
the hexagonal grid, we determine whether a light ray comingfrom the point is occluded by an occluding object. Then, occlusion coefficient Si for the light source is given as a ratioof unoccluded sampling points inside the hexagonal grid.

4.3 Pixel selection
For evaluating an input shadow image correctly, it is essential to select image pixels from a shadow image to maximize variation of patterns of occlusion of light sources by an
occluding object.
First, the visible part of shadow surface is partitioned into
clusters based on the combinations of occlusion coefficients
5 In theory, any subtle change in illumination distribution can be approximated with sufficient accuracy if an infinite number of point light sources
is used. However, the computational cost would increase prohibitively if
too many light sources were to be used. Moreover, the use of too many
point light sources causes the sampling resolution problem described in Section 3.3.

red

green

blue

Figure 7: Clustering results
cameraSi . In other words, pixels that have the same combinationof coefficients Si are clustered into the same group. Conceptually, those clusters are similar to partitioned regions inshadows in Figure 4. At the same time, the number of pixels
in one group is examined so that we can avoid selecting apixel from a group that contains fewer image pixels.
Figure 7 shows several examples of partitioning based on
the occlusion coefficients Si with pseudo colors. Here eachcolor represents an individual class with a different combination of occlusions of the light sources, and the block region
corresponds to the occluding objects. From these examplesof partitioning, we also see that patterns of partitioning differ
from image to image depending on factors such as the shapeof an occluding object and the camera viewpoint.
After the shadow surface is partitioned into clusters, one
pixel is selected from each of those clusters. By selecting
image pixels in this way, we can maximize variation of patterns of occlusion of light sources by an occluding object,
and therefore, we are able to evaluate the input shadow image in an appropriate manner. In addition, we are able to
avoid selecting redundant pixels, i.e., pixels that provide thesame information about the illumination of the scene as otherpixels.

5 Experimental results
We have tested our proposed method described in Section 4 by using real images taken in both a laboratory and an
office. First, we have tested our proposed method by using
images taken under a relatively simple illumination environment so that we can examine the performance of our proposed method carefully. Figure 8 shows our experimental
setup with three light sources with different colors. The input images taken by using this setup with different occluding
objects are shown in Figure 9 (1).
The results of illumination estimation with coarse samplings are shown in Figure 9 (2). Here, the estimated illumination radiance is visualized by mapping their values onto
a dome shown in Figure 1. As we can see in these results,
illumination distribution was correctly estimated for this setof light source samplings. However, after we increased thenumber of light sources for sampling illumination distribution, the estimation became unstable. Figure 9 (3) showsdensely distributed light sources and in fact, we can see thatthe results of estimation shown in this column are erroneous

Figure 8: Experimental setup with three area light sources
with different colors

especially around the blue and green light sources.
On the other hand, Figure 9 (4) shows the result of illumination estimation with our proposed method. Their stability measure is represented in the left of Figure 9 (4). Here,brighter light sources represent more reliable sampling regions which are not required to be merged to form a larger
light source for more coarse sampling. Unlike the estimation
results in Figure 9 (3), illumination distribution due to threelight sources with different colors was correctly estimated in
the right of Figure 9 (4).
Based on the estimated illumination distribution from the
input shadow image shown Figure 10 (1), several synthetic
objects were superimposed onto a synthetic surface in thebottom row of Figure 10 (1). Here, real objects with thesame shape as that of the synthetic objects and shadows castby those objects are shown in the top row for the comparison
purpose.
In the proposed approach, since the entire illumination
distribution of a scene is represented as a set of area lightsources whose solid angle is adjusted depending on its stability, the distribution of shadows is a little different fromthose of real shadows cast by the real objects. However, it isfound through our experiments that if we instead used a setof point light sources and examined only occlusions of thecenter points of those light sources, the entire estimation became unstable unless the method happened to find the correct
locations for those three light sources with coarse sampling.
We have also tested our proposed method by using real
images taken in an ordinary office environment. In the bottom row of Figure 10 (2), several synthetic objects were alsosuperimposed onto the surface using the illumination distribution estimated from the input shadow image shown in thisfigure. It is worth noting that in this example, a relatively
large area of the shadow surface is occluded by the occluding object, and it is often difficult to provide a correct estimate of the illumination distribution in such case. Evenin this challenging case, our proposed approach could reliably estimate the illumination distribution of the scene byinput shadow image(1)coarsesamplings

(2)

dense
samplings(3)

considering stability
high stability
region(4)

Figure 9: Illumination estimation without and with considering stability
taking stability issues into considerations. Shadows cast by
those synthetic objects resemble well those cast by the real
objects, and this shows that the estimated illumination distribution gives a good presentation of that of the real scene.

6 Conclusions
In this paper, we have proposed a method for estimating
illumination distribution of a real scene from shadows in agiven image in a reliable manner. First, we examined thesource of instability of our previously proposed method [20]carefully in several aspects: invisible regions of shadows;
limited sampling resolution for radiance distribution insideshadows; and approximation of illumination distribution asa collection of point light sources. Then, based on that analysis, we proposed methods to overcome those problems regarding the instability of the illumination estimation fromshadows. For estimating the illumination distribution of ascene reliably, we adopted a strategy for changing the sampling density of the illumination distribution depending on
the amount of the information obtainable from a shadow image for a particular direction of the illumination distribution.
For using radiance distribution inside penumbra of shadows
correctly, we introduced a super-sampling scheme for examining occlusion of incoming light from each light source. We
also explained the optimal sampling of image pixels and theselection of illumination distribution samplings for more stable computation. All of these extensions contribute to improve stability and accuracy of illumination estimation fromshadows. Unlike the previously proposed method for illumination estimation, illumination distribution can be estimatedin a reliable manner with these proposed improvements regardless of types of input images such as the shape of an
occluding object or a camera position.

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