Appearance Sampling for Obtaining A Set of Basis Images
for Variable Illumination
Imari Sato
Takahiro Okabe
Yoichi Sato
Katsushi Ikeuchi
Institute of Industrial Science, The University of Tokyo, Japan
imarik, takahiro, ysato, ki@iis.u-tokyo.ac.jp

Abstracta 3-D linear subspace defined with three input images taken
under linearly independent lighting conditions [9, 15, 18].
However, the assumption of this model would be too restrictive to be used for object recognition in more realistic
settings.

Previous studies have demonstrated that the appearance ofan object under varying illumination conditions can be represented by a low-dimensional linear subspace. A set ofbasis images spanning such a linear subspace can be obtained by applying the principal component analysis (PCA)
for a large number of images taken under different lighting conditions. While the approaches based on PCA havebeen used successfully for object recognition under varying
illumination conditions, little is known about how many images would be required in order to obtain the basis images
correctly. In this study, we present a novel method for analytically obtaining a set of basis images of an object forarbitrary illumination from input images of the object taken
under a point light source. The main contribution of ourwork is that we show that a set of lighting directions canbe determined for sampling images of an object depending on the spectrum of the object’s BRDF in the angularfrequency domain such that a set of harmonic images canbe obtained analytically based on the sampling theorem on
spherical harmonics. In addition, unlike the previously proposed techniques based on spherical harmonics, our method
does not require the 3D shape and reflectance properties ofan object used for rendering harmonics images of the object
synthetically.

Other researchers have reported empirical studies for representing image variation due to varying illumination for human faces and other objects [7, 4, 17]. Interestingly enough,most of the image variation caused by varying illumination
can be explained with a low-dimensional linear subspaceslightly higher than 3D even when images contain a significant amount of shadows. For instance, Hallinan reporteda) that a 5-D subspace would suffice to represent most of
the image variation due to illumination change including extreme cases where a face is lit from its sides, and b) that a
3-D subspace would be sufficient when a face is lit mainly
from its front [7]. Georghiades et al. used a similar observation more specifically for object recognition under varying
lighting conditions [5, 6].
A set of basis images spanning such a linear subspacecan be obtained by applying the principal component analysis to a large number of images of an object taken under
different lighting, e.g., by moving a point light source along
a sphere surrounding the object. One might ask whether acertain set of input images of an object would be sufficient
to fully span the subspace of the object for arbitrary illumination conditions. Previous empirical studies do not necessarily provide enough insight to this important question,and thus it has been a common practice to use as many input images as possible to ensure that the set of input images
span a subspace entirely. Another interesting observation ofHallinan’s early work is that a subspace obtained by PCA
does not vary widely between sparsely and densely sampled
sets of lighting directions in the case of faces [7]. However,it was not known how many images would be sufficient inorder to obtain the basis images correctly.

1. Introduction
It is well known that the appearance of an object changes
significantly under different illumination conditions. For instance, the appearance change of someone’s face often becomes much larger than the difference of two different faces
under the same lighting. Thus, for the task of object recognition, it is very important to be able to predict the variation of
objects’ appearance under varying illumination conditions.
Previous studies have demonstrated that appearancechanges of an object under varying illumination can be represented with a low-dimensional linear subspace. In themost simplistic case of a convex Lambertian object under
distant illumination without attached and cast shadows, theappearance of the object can be completely described with

Recently, it was shown through the frequency-space analysis of reflection that appearance of a convex Lambertian
object can be well represented with a 9-D linear subspacespanned by basis images of the object, called harmonic images, each of which corresponds to an image of the object
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Proceedings of the IEEE International Conference on Computer Vision (ICCV 2003), pp. 800-807, October 2003.

illuminated under harmonic lights whose distributions are
specified in terms of spherical harmonics[10, 11, 1]. Basriand Jacobs successfully used this 9-D subspace defined withharmonic images for face recognition under varying illumination [1]. Other researchers have also used a set of harmonic images for the purpose of efficient rendering of an object under complex illumination [12, 16]. More recently, Ramamoorthi has shown theoretically that, under certain conditions of lighting and object shape and reflectance, a set ofbasis images obtained from PCA on input images taken under varying lighting coincides with a set of harmonic images[13].
While harmonic images are specified analytically, it isdifficult to obtain harmonic images for various kinds of real
objects because harmonic lights consist of both negative andpositive values distributed in a complex way and thus do notexist as real lighting. Therefore, the previously proposed
techniques for object recognition and image synthesis basedon harmonic images require the model of an object about its
3-D shape and reflectance properties such as albedo so that
harmonic images of the object under harmonic light can berendered synthetically. This is not a problem for the caseof synthetic objects because the model of a synthetic object
is given a priori by definition. On the other hand, acquiring
shape and reflectance properties of real objects is not an easy
problem and therefore, most of the previous studies treatedreal objects as Lambertian objects and estimated only albedos by using vision techniques such as photometric stereo.
This motivated us to develop a method for determining a
set of harmonic images of a real object by using only input images of the object under simple lighting such as apoint light source. Lee et al. recently proposed an interesting method to determine a configuration of 9 light source
directions such that input images taken under those lightsource directions approximate a 9-D subspace spanned byharmonic images [8]. They have reported that such a configuration of 9 light source directions does not vary much forsimilar objects, e.g., different faces. However, Lee et al.’s
method still needs a set of harmonic images that are rendered synthetically by using an object’s model in order todetermine a set of 9 lighting directions. Moreover, Lee etal.’s method chooses 9 light directions from a large numberof candidates and therefore, a large number of input images
are required for each new object or new class of objects.
In this study, we present a novel method for analyticallyobtaining a set of basis images of a convex object for arbitrary illumination from input images of the object taken
under a point light source. The main contribution of ourwork is that we show that a set of lighting directions can be
determined for sampling images of an object depending on
the spectrum of the object’s BRDF in the angular frequency
domain such that a set of harmonic images can be obtained
analytically based on the sampling theorem on spherical har-monics [3]. Using those sampling directions determinedfrom the sampling theory, we are able to obtain harmonic
images by using a significantly smaller number of input images than other techniques which do not take into account a
relationship between a spectrum of BRDFs and a sampling
density of illumination directions. In addition, unlike othermethods based on spherical harmonics, our method does not
require the shape and reflectance model of an object used
for rendering harmonics images of the object synthetically.
Thus, our method can be easily applied for determining a setof basis images for representing the appearance change of areal object under varying illumination conditions.
The rest of the paper is organized as follows. We briefly
review the spherical harmonics transformation of a function
over the unit sphere and harmonics images based on spherical harmonics in Section 2. We show details of the sampling
theorem used in our method for obtaining harmonic imagesof real objects in Section 3, and consider issues in obtaining harmonic images of real objects based on this sampling
theorem in Section 4. In particular, we discuss what kind of
artifacts should be expected when input images provide insufficient samplings of an object’s appearance. Finally, weshow experimental results of the proposed method applied toimages of real objects in Section 5, and present concluding
remarks in Section 6.

2 A Set of Basis Images for Variable
Illumination
2.1 Review of Spherical Harmonics
Spherical Harmonics define an orthonormal basis over theunit sphere. Consider the unit sphere in, a unit vector
on the sphere can be described by the polar coordinate sys  in elevation and     
tem  
in azimuth. In this coordinate system, spherical harmonics
         are defined as  (1)   (2)     

where  are the normalized constants, and   are theassociated Legendre functions of degree  and order .
A function    defined over the unit sphere is expanded as a linear combination of spherical harmonics as    

and  denote coefficients in its spherical harmonic expansion computed as 1

          












1 In this study, we consider spherical harmonics in a real form.

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Proceedings of the IEEE International Conference on Computer Vision (ICCV 2003), pp. 800-807, October 2003.

(3)  of the object from the knowledge of its 3D shapeand reflectance properties. Since     are predefined
functions,  are computed from (3). Most of the previously proposed methods employed this approach to computeharmonic images synthetically.
One might think of observing an object under physically constructed harmonic lights. In this case, each pixel
value directly corresponds to coefficients  , and thus the
3D shapes and reflectance properties of the object need not
be given a priori. However, as pointed out by other researchers [1, 8], harmonic lights are complex diffuse lighting consisting of both negative and positive radiance. Thus,
it is difficult to physically construct such lighting in a realsetting.
In this work, we take an alternative approach of observing a reflection kernel   for each surface point on an
object directly by taking an image of the object under a point
light source located at the direction ( . This approach isbased on the assumption that a point light source used in thismethod is sufficiently distant from the objects and thus, thislight source projects parallel rays onto the object surface.
The reflectance kernel   represents radiance of reflected light due to incoming light with unit radiance fromthe direction  . Thus if we suppose that this point lightsource has unit radiance, the value of  can be obtained simply as an observed brightness of the surface point
when the object is illuminated under a light source from thedirection  .2 Once we determine the values of  
for all directions, we can compute pixel values of harmonic
images as  from (3). In this way, we do not need to synthetically compute the reflection kernel   of the object, nor are the 3D shapes and reflectance properties of theobject required.
Since the function   are continuously distributedon the unit sphere in (3), we first need to approximate its
distribution by a discrete set of the function   so that
we can sample   using a point light source physicallylocated at   direction. Then, the question we have to ask
is how densely   need be sampled in order to correctlycompute coefficients  from them. In other words, wewant to know how densely a point light source needs to bepositioned around an object to obtain harmonics images ofthe object correctly. In the next section, we will consider thisissue in terms of the characteristics of spherical harmonics
  .

2.2 Harmonic Image Representation
The reflectance property of an object is characterized bya bidirectional reflectance distribution function (BRDF)
        , where     and     are incident andreflection directions with respect to the surface normal of theobject surface whose local coordinate is denoted by using  .
Then, the brightness  of the object surface is represented by the global coordinate defined on the unit sphere
as


 

          












      

(4)
where    is light source distribution, and  represents a rotation operator that rotates   into the local
coordinate.
In this paper, we consider the appearance of an object
under variable illumination seen from a fixed viewpoint andtherefore, we represent           by using
the global coordinate as  , and refer to it as the reflection kernel.
Since both the light source distribution and the reflection
kernel are functions defined on the unit sphere, we can represent them as      
   (5)   (6) 

where  and  are coefficients in their spherical harmonic expansion from (3). Assuming spherical light sources
with    radiance in its   direction,  represents the brightness seen under these spherical light sources
called harmonic lights in (3).
From (5), (6), and the orthonormality of spherical harmonics, the surface brightness  in (4) is represented as   

(7)

Here, if we prepare  for all pixels, that is, for all corresponding points on the object surface, images under variable
illumination are represented from (7). Images containing
 are called harmonic images [1].

3.1 Sampling Theorem on Spherical Harmonics Transformation

3 Methods for Obtaining Harmonic
Image

There have been several methods proposed in the field of
applied mathematics to efficiently compute coefficients of a
2 Note that we can determine   only up to some unknown scaling
factor, so it is reasonable to treat a point light source to have unit radiance.

There are several approaches for obtaining harmonic imagesof an object. One approach is to provide a reflection kernel
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Proceedings of the IEEE International Conference on Computer Vision (ICCV 2003), pp. 800-807, October 2003.

function   in its spherical harmonic expansion using
fewer samplings of the function  . We have adapted
one of their theories to enable us to compute harmonic images using fewer input images of objects taken by moving apoint light source to particular locations.
It is common knowledge that the sampling theorem on
the 1D line tells us that a band-limited function can be reconstructed accurately from properly sampled discrete data.
Namely, Fourier coefficients of the function can be determined by weighted sums of the function sampled at evenintervals. In the case of a function defined on the sphere, thesimilar theorem, the sampling theorem on the sphere, hasbeen proved [3]. In this section, we outline the theorem.
Let us assume that the reflection kernel   is bandlimited with bandwidth N, that is,      . Then,consider the “comb” function   with equiangular grid               Æ    Æ    


 

 











   Æ     

(9)

(14)

4.1 Convex Lambertian Surface
Let us start with the simplest case of convex Lambertian objects. It has been shown in the previous studies that the firstnine spherical harmonics with the order    are sufficient
to capture more than  of the reflection energy of a convex Lambertian surface [1, 10, 11].
Accordingly, we can consider that the function  
is band-limited with bandwidth , and this resultsin     samplings of   necessary for computing  correctly. In other words, the coefficients  aregiven as a finite weighted sums of the function   sampled at equiangular grid:            ,          . Namely,  input images of an
object taken by moving a point light source to the directions
specified with ( ,  ) on a sphere around the object are required to compute harmonic images of the object.
Note that what needs to be satisfied is not the number ofsamplings (  ) of the function but rather, the sampling intervals (    ,    ) that this sampling
can provide. For instance, even when a large number of
input images of an object taken under a point light source
are available, there is no guarantee that this set of images(10)             (13)

The sampling theorem described in the previous section tells
us the minimum number of sampling      
to compute spherical harmonics transformation of a bandlimited function with bandwidth . In this section, we consider issues in defining bandwidth of an object and discuss
what kind of artifacts we should expect when the function is
not band-limited within .

Then, from (6) and (10), the product of the reflection kernel and the comb function is written as           



4 Appearance Sampling of Real Objects based on The Sampling Theorem(8)  

(12)



where the Kronecker delta Æ   if    , and Æ  
if    . Here, we can uniquely choose the weight   so that         Æ . Thus, the coefficientsare described simply as   Æ Æ  for degree    .
Equivalently, the comb function is represented by the addition of       and higher-degree terms as                  
where the weight   are analytically given by

 


      

  





    

Hence, the coefficients of the reflection kernel can becomputed accurately by the Fourier transform of     aswhere           are sampling
points on the sphere, and   are weight of the points. We
can compute the coefficients  of the comb function fromdefinition (3). For degree    , we obtain,in this case. Therefore, for degree  is equal to  , that is,(11)

because   is band-limited. The
where degree  second term is known as aliasing introduced by discrete
sampling. However, it is known that the product of thespherical harmonics       is represented as alinear combination of spherical harmonics with a degree
greater than or equal to    . Accordingly, aliasing appears to be a degree greater than or equal to     
4Proceedings of the IEEE International Conference on Computer Vision (ICCV 2003), pp. 800-807, October 2003.

  for  
. In this case, the coefficients  for    degree are correctly computed. Therefore, closer
to  we select , smaller error in  we expect.

can produce correct harmonic images that can span a lowdimensional linear subspace representing the appearance ofthe object under arbitrary illumination unless those intervals
are satisfied.

Next we show what happens when we have sufficient
or insufficient samplings of the function   by usingsynthetic data. The function   used here is a reflection kernel for the Torrance-Sparrow reflection model withknown reflection parameters (            ).4 The surface normal and viewing direction
for this reflection kernel are set at the direction     .
We computed the coefficients  up to the degree     for this reflection kernel from discrete samplingsof   at different sampling intervals:   ( 
sampling),( samplings),  (100 samplings),  (36 samplings). The computed coefficientsare shown in Figure 1a) d). Spherical harmonic coefficientfor given degree  and order  are represented using a single index     . The horizontal axis represents theindex , and the vertical axis represents the computed coefficients  . The number of samplings is shown for eachgraph, and a red arrow represents the upper bound determined from the number of samplings.
Figure 1e) h) are visualization of the reflection kernel  reconstructed from the computed coefficients in (6) up to the degree    for the cases of ,,  , and  , respectively. Figure 1i) showsa reflection kernel reconstructed from the coefficients up to . In these figures,the degree    for the cases of
the upper half of   is visualized in a polar coordinate
system with radius indicating    , and angle
indicating     as illustrated in Figure 1j).
From Figure 1a) and b), we can see that the coefficients
 obtained from     samplings are almost thesame as those obtained from   samplings, and  is reconstructed correctly in both cases. Thistheshows that a set of samplings forare sufficient tocapture the appearance of a surface with this reflection kernel under varying illumination. This follows what the sampling theorem tells us.
  and 
On the other hand, in the cases ofwhere is smaller than   , the computed coefficients
suffer from aliasing. It is important to note that the coeffi(the left side of the red arrow in Figcients  for  
ure 1c) and d)) closely match the coefficients obtained from
sufficient samplings (Figure 1a)) while the coefficients for  (the right side of the red arrow) differ. As
a result, distinct ringing effects are evident in the reflection
kernel reconstructed from the coefficients up to the degree   (Figure 1g) and h)). On the other hand, the reflec-

4.2 Complex Appearance beyond LambertianRamamoorthi and Hanrahan analytically derived the bandwidth of reflection kernels of objects that have more complex appearance than convex Lambertian surfaces such as
the Phong reflection model and the Microfacet BRDF [10]. 3
For instance, this study shows that the bandwidth of the Mi   . Thus, if thecrofacet BRDF is approximately
surface roughness of an object is predicted even roughly, it
should help us to find the bandwidth of the reflection kernelof the object.
There are a large number of previous studies on BRDF
measurements and BRDF databases which show reflectanceparameters of various kinds of surface materials. Thisknowledge should be useful for estimating bandwidths ofreflection kernels of various objects based on the analysis
presented in [10].

4.3 Aliasing Caused by Insufficient Sampling
There certainly is a situation where bandwidth of reflection
kernels of an object is difficult to predict, or where only alimited number of samplings of the function are obtainabledepending on a particular hardware set-up used for acquiring
input images. Hence it is important to consider what kindsof artifacts we should expect when the function   hasa lower or a higher bandwidth than determined from thenumber of samplings   based on the sampling theorem.
Let  denote actual bandwidth of the function. Driscolland Healy proved that the error in spherical harmonics transformation, generally known as aliasing, is confined to coefficients  of degree greater than or equal to     in[3]. This can be confirmed by replacing the bandwidth of
the reflection function with  in (11).
From this, in the case where the function has bandwidth lower than (  ), there is no aliasing for degree lessthan
 for    . Since we compute coefficientsfor, those coefficients are correctly computed. Inthis case, all of the coefficients  for    become.
In the case where the function   has a bandwidth higher than (  ), error due to insufficient samplingof the function is confined to the coefficients of degree  
3 [14] is also a good reference to get insights into the bandwidth of areflection kernel of various types of objects in CURET database. This work
discusses appropriate sampling resolutions to model appearance of thoseobjects based on their experiments as well.

4  and  are constants for the diffuse and specular reflection components, and  is the standard deviation of a facet slope of a simplified
Torrance-Sparrow reflection model.

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Proceedings of the IEEE International Conference on Computer Vision (ICCV 2003), pp. 800-807, October 2003.

m
l

R

m
l

R

2

N = 200 (4N=16000)

02

N = 11 (4N=484)

0
0

20

40

60

80

100 120

r

0

20

40

(a)
m
l

R

60

80

100 120

r

(b)
m
l

R

2

N = 5 (4N=100)

2

N = 3(4N=36)

0

0
0

20

40

60

(c)

80

100 120

r

0

20

40

60

80

100 120r(d)Figure 2: Image acquisition set-up

(e)

(f)

(g)

(h)
ǰ

(i)

for obtaining the input images of the objects; an array oflight sources is mounted on a turntable. These light sources
are equally spaced in elevation, and the set of light sources
is rotated around the objects in azimuthal.
In the case of the sheep and the Venus examples, sincethose objects have an appearance similar to that of a Lambertian surface,  input images of them are taken undera point light source positioned at equiangular grid points:    ,            . For the fish
example, since it has a more complex appearance,  input
images are taken under a point light source at equiangular         .
,   grid points :    Based on the sampling theorem, coefficients  are computed up to the degree    from the  images, and up to
the degree    from the  images.
The first nine harmonic images obtained from those input
images are shown in Figure 3. In spite of given discretesets of the appearance of the objects, the obtained harmonic
images have complex and smooth shading that reflects the
distribution of harmonic lights in these examples.
To evaluate the accuracy of the recovered harmonic images, we took images of those objects under normal lighting
conditions in our laboratory, including direct light sources
such as fluorescent lamps. Coefficients   in (5) representing the illumination condition are computed from an omnidirectional image of the scene taken by a camera with a fisheye lens in the same manner as described in [12]. Then appearance of the objects under this illumination condition is
synthesized from (7).
In Figure 4, the left column shows the real images of theobjects and the right column shows the synthesized appearance. The synthesized appearance of the objects resembles
that of the objects in real images, and this shows that the recovered harmonic images provide a good representation of
the appearance of the objects. The shoulder of the plaster
figure in the middle right image appears darker than that in
the real image. This is due to severe cast shadows observedin the input images that do not follow our method’s assump-Ǿ(j)Figure 1: Experimental results using synthetic data: coefficients are computed from a discrete set of    sampled
at different sampling intervals.

tion kernel reconstructed from the coefficients up to the degree    for  shown in Figure 1i) does not showsuch ringing effects while high frequency components suchas specularity are missing at the center.
It follows from these examples that it is appropriate tocompute  for  only when   samplings of thefunction are available. In this way, we can avoid annoying
 in the reconstructedringing effects caused by  forfunction  .

5 Experimental Results
We have tested the proposed method using real images ofseveral objects taken by moving a point light source toequiangular grid points defined by the sampling theorem.
Figure 2 shows an overview of our hardware set-up 5 used
5 Surface Reflectance Sampler, TechnoDream21 corporation

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Proceedings of the IEEE International Conference on Computer Vision (ICCV 2003), pp. 800-807, October 2003.

Figure 3: Obtained harmonic images: positive values are shown in green, and negative values are shown in red.
tion for convex objects.
To demonstrate how well the recovered harmonic images
represent the appearance of those objects under varying illumination, we synthesize their appearance from the recovered
harmonic images under several natural illumination conditions provided by high-dynamic range light probe measurements by [2]. Figure 5 shows the results. In this figure,synthesized appearance changes dynamically depending on
characteristics of the illumination, and one can say from thisthat the proposed method succeeded in providing a set ofbasis images representing appearance of those objects undervarying illumination.

6 Summary and Conclusions
In this study, we have presented a novel method for analytically obtaining a set of basis images of an object for arbitrary illumination from input images of the object taken
under a point light source. The main contribution of ourwork is that we have shown that a set of lighting directions
can be determined for sampling images of an object depending on the spectrum of the object’s BRDF in the angularfrequency domain such that a set of harmonic images canbe obtained analytically based on the sampling theorem on
spherical harmonics. The future directions of this study include extending our method for: (1) modeling appearanceof object seen from arbitrary viewing directions, and (2)extending our method further for reducing aliasing effects
caused by insufficient sampling of appearance of objects.

Figure 4: Comparison between real images and synthesizedimages under complex illumination

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This research was supported in part by Grands-in-Aid for Scientific
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Proceedings of the IEEE International Conference on Computer Vision (ICCV 2003), pp. 800-807, October 2003.

Figure 5: Synthesized images of objects under natural illumination. The first row shows illumination maps. The second,
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Proceedings of the IEEE International Conference on Computer Vision (ICCV 2003), pp. 800-807, October 2003.