2016 23rd International Conference on Pattern Recognition (ICPR)
Cancún Center, Cancún, México, December 4-8, 2016

Separating Reflection Components in Images under
Multispectral and Multidirectional Light Sources
Naoto Kobayashi

Takahiro Okabe

Department of Artificial Intelligence
Kyushu Institute of Technology
680-4 Kawazu, Iizuka, Fukuoka 820-8502, Japan
Email: okabe@ai.kyutech.ac.jp
Abstract—The appearance of an object depends on the color
as well as the direction of a light source illuminating the object.
The progress of LEDs enables us to capture the images of an
object under multispectral and multidirectional light sources.
Separating diffuse and specular reflection components in thoseimages is important for preprocessing of various computer
vision techniques such as photometric stereo, material editing,and relighting. In this paper, we propose a robust method forseparating reflection components in a set of images of an object
taken under multispectral and multidirectional light sources. We
consider the set of images as the 3D data whose axes are thepixel, the light source color, and the light source direction, andthen show the inherent structures of the 3D data: the rank 2
structure derived from the dichromatic reflection model, the rank
3 structure derived from the Lambert model, and the sparsenessof specular reflection components. Based on those structures,
our proposed method separates diffuse and specular reflection
components by combining sparse NMF and SVD with missing
data. We conducted a number of experiments by using bothsynthetic and real images, and show that our method works better
than some of the state-of-the-art techniques.

I.

cameraobject

400

500

600

700

800

900

1000

Fig. 1. Our multispectral light stage termed Kyutech Light Stage I [6]. It
consists of LED clusters, each of which has LEDs with different spectral
intensities.
light source direction
Blight source colorA

I NTRODUCTION

The appearance of an object depends not only on the object
itself but also on the light source illuminating the object.
In particular, it depends on the color, i.e. spectral intensity
as well as the direction of the light source. Conventionally,
the dependencies on lighting color and lighting direction areoften studied in separate fields; color analysis (or multispectral
imaging) studies images under multispectral light sources
whereas shading analysis studies images under multidirectional
light sources.
Recently, the progress of light-emitting diodes (LEDs)
enables us to capture a set of images of an object undermultispectral and multidirectional light sources by using themulti/hyper-spectral light stages [1], [5], [6] which extend the
setup of Debevec et al. [3] as shown in Fig. 1. It is reported
that integrating color analysis and shading analysis by using
images under multispectral and mutidirectional light sources iseffective for raw material classification [5], bidirectional texture function (BTF) classification [8], and surface reflectanceand normal recovery [6].
In general, the reflected light observed on an object surface
consists of a diffuse reflection component and a specularreflection component. Separating those reflection components
is important for preprocessing of various computer vision techniques such as photometric stereo [15], image-based material
editing [9], and relighting [4].
978-1-5090-4846-5/16/$31.00 ©2016 IEEE

3199

C
pixel

Fig. 2. A set of images of an object captured from a fixed viewpoint but
under multispectral and multidirectional light sources are represented as the
3D data whose axes are the pixel p, the light source color c, and the lightsource direction d.

In this paper, we propose a robust method for separatingreflection components in images taken under multispectral andmultidirectional light sources. Such a problem is challenging
as discussed in more detail in Section II. First, the colorbased approach [11] does not work well because the spectral
intensities of LEDs are often narrow-band, and then the colorof a specular reflection component is almost the same as thatof a diffuse reflection component. Second, the approach basedon the low-rankness of diffuse reflection components [12]assumes that the light source color is not variable but fixed.
Accordingly, we consider the images of an object captured
from a fixed viewpoint but under multispectral and multidirectional light sources as the 3D data whose axes are the pixel
p, the light source color c, and the light source direction d as
illustrated in Fig. 2. Then, we reveal the inherent structures ofthe 3D data: the rank 2 structure of pixel values on the slice A
in Fig. 2 derived from the dichromatic reflection model [11],the rank 3 structure of diffuse reflection components on theslice B derived from the Lambert model, and the sparsenessof specular reflection components. Based on those structures,
our proposed method separates diffuse and specular reflection
components by combining sparse NMF [10] and SVD withmissing data [13].
To confirm the effectiveness of our proposed method, we
conducted a number of experiments by using both synthetic
and real images. The experimental results show that taking allof the inherent structures of the 3D data into consideration
is effective for robustly separating reflection components andthat our method works better than some of the state-of-the-art
methods based on color [2] and low-rankness [16].
The main contribution of this paper is threefold. First, weaddress a novel problem of separating reflection components inimages under multispectral and multidirectional light sources.
Second, we consider those images as the 3D data, and reveal
its inherent structure. Third, based on the structure of the 3D
data, we propose a robust method for separating diffuse andspecular reflection components.
II.

R ELATED W ORK

Existing methods can be classified into three approaches:(i) the color-based approach, (ii) the approach based on the
low-rankness of diffuse reflection components, and (iii) thepolarization-based approach. In this section, we briefly explain
the point of each approach, and then discuss the limitations of
those approaches when they are used for separating reflection
components in images under multispectral and multidirectional
light sources.
The first approach based on color makes use of the difference between the colors of a specular reflection component anda diffuse reflection component. According to the dichromaticreflection model [11], the former is the same as the light source
color whereas the latter depends on the spectral reflectance ofa surface. The color-based approach has an advantage that it
can be applied even to a single image, i.e. the column C in
Fig. 2.
Unfortunately, however, the color-based approach does notnecessarily work well for images taken under LEDs. This isbecause the spectral intensities of LEDs are often narrowband. It is almost impossible to distinguish the color of a
specular reflection component from that of a diffuse reflection
component in a single image taken under a narrow-band lightsource. In addition, when the color-based approach is used
independently for each image, we cannot capture the inherent
structures of the 3D data described in Sections III.A and III.B.
The second approach is based on the low-rankness ofdiffuse reflection components under varying light source directions. Specifically, the image of a Lambertian object under
an arbitrary directional light source is represented by thelinear combination of three basis images of the object [12]1 .
Therefore, we can extract diffuse reflection components froma set of images of an object taken under multidirectional
1 We mention the effects of attached shadows and/or cast shadows in Section
III.B.

3200

light sources, i.e. the slice B in Fig. 2 via low-rank matrix
completion and recovery [16] for example.
Unfortunately, however, the approach based on the lowrankness assumes that the light source color is not variable
but fixed because the images of an object under multispectral
light sources live in 3D subspaces different from each other.
Therefore, when it is used independently for each slice with afixed light source color, we cannot capture the inherent structure of the 3D data under varying light source colors describedin Sections III.A. In addition, although the approach based on
the low-rankness can extract diffuse reflection components on
the Slice B in Fig. 2, there is no guarantee that the remaining
components are specular reflection components due to noise.
The third polarization-based approach makes use of thedifference of the polarization states of a specular reflection
component and a diffuse reflection component. When we
observe the reflected light from an object surface illuminatedby polarized light, the former is polarized whereas the latter is
unpolarized [14]. The polarization-based approach requires aset of images taken by placing linear polarizing filters in frontof a light source and a camera, and rotating one of them.
The polarization-based approach is not necessarily suitedfor multi/hyper-spectral light stages due to the following two
reasons. First, placing linear polarizing filters in front of lightsources is more burdensome, as the number of light sources
increases. Second, more importantly, capturing a large numberof images by rotating the linear polarizing filter in front ofa camera is time consuming. This is a serious problem for
multi/hyper-spectral light stages using LEDs because LEDs
are usually dim and require long exposure time.
III.

P ROPOSED M ETHOD

We assume that the color images of an object of interest
are captured from a fixed viewpoint and under multispectral
and multidirectional light sources by using a multispectral lightstage similar to the existing ones [1], [5], [6] as shown in Fig. 1.
Specifically, the light stage has D LED clusters at different
directions, and each cluster has C LEDs with different spectral
intensities, i.e. C × D LEDs in total.
For the sake of simplicity, we consider the C color images taken under C different light source colors as the 3C
grayscale images taken under 3C different light source colors
without loss of generality. We denote the pixel value at thep-th pixel (p = 1, 2, 3, ..., P ) under the c-th light source
color (c = 1, 2, 3, ..., 3C) and the d-th light source direction(d = 1, 2, 3, ..., D) by ipcd .
A. Varying Light Source Colors and DirectionsFirst, let us consider the slice A in Fig. 2, i.e. the variationof pixel values at a fixed pixel but under varying light source
colors and directions.
According to the dichromatic reflection model [11], theobserved pixel value ipcd is the sum of a specular reflection
component and a diffuse reflection component:
ipcd = s1,c gp1,d + sp2,c gp2,d ,(1)

where the first and second terms are the specular and diffuse
reflection components respectively. The specular reflection

component is described by the product of the spectral term
s1,c and the geometric term gp1,d . The former depends on the
spectral intensity of a light source and the spectral sensitivity ofa camera, and the latter depends on both the viewing and lightsource directions. In a similar manner, the diffuse reflection
component is described by the product of the spectral term
sp2,c and the geometric term gp2,d . The former depends on
the spectral reflectance of a surface in addition to the spectral
intensity and the spectral sensitivity, and the latter depends on
the light source direction.
Therefore, the 3C × D matrix Ip , whose (c, d)-th element
is given by ipcd , is described as⊤⊤.
+ sp2 gp2
Ip = s1 gp1(2)

Here, the c-th element of the 3C-dimensional vector s1 is s1,c ,the d-th element of the D-dimensional vector gp1 is gp1,d ,and so on. The vectors s1 and sp2 represent the colors of the
specular and diffuse reflection components respectively undervarying light source colors. The vectors gp1 and gp2 represent
the intensity profiles of the specular and diffuse reflection
components respectively under varying light source directions.
It is known that the color of the specular reflection component
s1 is the same as the color of the light sources [11], and thatthe intensity profile of the specular reflection component gp1 issparse since specularity is observed only under a small numberof light source directions.
From eq.(2), the matrix Ip is represented as the product of
the 3C × 2 matrix Sp and the 2 × D matrix Gp :
( ⊤ )
gp1
Ip = ( s1 sp2 )
(3)
⊤
gp2
= Sp Gp .
(4)Thus, the matrix consisting of the pixel values at a fixed pixel
but under varying light source colors and directions, i.e. Ip
has rank 2.
Note that the rank of Ip is 2 even when attached shadows
and/or cast shadows are observed at the p-th pixel, because thegeometric terms include them. Therefore, we can make use of
the rank 2 structure of pixel values on the slice A without
paying special attention to those shadows.
B. Varying Pixels and Light Source Directions
Second, let us consider the slice B in Fig. 2, i.e. thevariation of pixel values under a fixed light source color but
at varying pixels and under varying light source directions.
Assuming the Lambert model, the diffuse reflection component, i.e. the second term of the right-hand side of eq.(1) is
described as∫
sp2,c gp2,d = lc (λ)rp (λ)s(λ)dλ n⊤(5)p ld .
Here, lc (λ), rp (λ), s(λ), np , and ld are the spectral intensityof the c-th light source color, the spectral reflectance at the p-th
pixel, the spectral sensitivity of a camera, the surface normal at
the p-th pixel, and the d-th light source direction respectively.
Therefore, the P ×D matrix Dc , whose (p, d)-th element isgiven by sp2,c gp2,d , is represented as the product of the P × 3
3201

matrix and the 3 × D matrix:


..
.
 ∫

⊤ 
Dc = 
 lc (λ)rp (λ)s(λ)dλ np  ( · · · ld..
.

· · · ) . (6)Thus, the matrix consisting of the diffuse reflection components under a fixed light source color but at varying pixels
and under varying light source directions, i.e. Dc has rank 32 .
In other words, the image of a Lambertian object under anarbitrary directional light source is represented by the linear
combination of three basis images of the object [12]3 .
Note that the rank of Dc deviates from 3 when attachedshadows and/or cast shadows are observed under varying lightsource directions. As described in Section III.D, we remove
the effects of those shadows so that we can make use of therank 3 structure of diffuse reflection components on the slice
B.
C. Separating Reflection Components
Based on the structures described in Sections III.A and
III.B, we propose a method for separating reflection components in images taken under multispectral and multidirectional
light sources as follows:{ŝp2 , Ĝp } = arg

min1{sp2 ,Gp } 2||Ip − Sp Gp ||22 , ∀p(7)

subject to sp2,c ≥ 0, gp1,d ≥ 0, gp2,d ≥ 0, ∀p, c, d (8)||gp1 ||0 ≤ L, ∀p(9)rank(Dc ) = 3, ∀c.
(10)
Our proposed method assumes that the color of the specularreflection component s1 is known4 , and estimates the color ofthe diffuse reflection component sp2 and the intensity profilesof the specular and diffuse reflection components Gp for all
pixels.
More specifically, based on the rank 2 structure describedin Section III.A, our proposed method decomposes the matrix
Ip into the matrices Sp and Gp for each pixel as eq.(7). Sinceboth the spectral and geometric terms are non-negative, we
impose the non-negativity constraints on the elements of sp2
and Gp as eq.(8). In addition, our method takes the sparsenessof the intensity profile of the specular reflection component
gp1 into consideration as eq.(9). Here, L is the parameterfor controlling the sparseness. Based on the rank 3 structure
described in Section III.B, we impose the rank 3 constraints
on the matrix consisting of diffuse reflection components foreach light source color as eq.(10).
D. Implementation Details
We solve the optimization problem of eq.(7) with theconstraints of eq.(8) to eq.(10) by iteratively using sparse NMF
2 We assume that an object of interest has 3D shape. Degenerate cases
include 1D planar objects and 2D cylindrical objects.
3 It is clear that those three basis images depend on the light source color∫
as lc (λ)rp (λ)s(λ)dλ in eq.(6). Therefore, the images of an object undermultispectral light sources live in 3D subspaces different from each other.
4 We assume that the spectral intensities of light sources and the spectral
sensitivity of a camera are calibrated in advance.

(a)

(b)

(a)

(b)

(c)

(d)

(c)

Fig. 3. The results of our proposed method on synthetic images (sphere): (a)the input images, (b) the diffuse and (c) the specular reflection components
separated by using our method, when σ = 8.

Fig. 4. The comparison with other methods on synthetic images (sphere): thediffuse reflection components separated by using (a) the sparse NMF, (b) the
SVD with missing data, (c) Wu et al. [16], and (d) Akashi and Okatani [2].

(non-negative matrix factorization) on the slice A and SVD(singular value decomposition) with missing data on the slice
B in Fig. 2 as follows.
1)
2)
3)
4)
5)
6)

For each pixel, updating sp2 and Gp by using sparse
NMF with L0 constraints [10].
Converting sp2 and gp2 for all pixels into Dc for all
light source colors.
For each light source color, approximating Dc with arank 3 matrix by using SVD with missing data [13].
⊤Converting Dc for all light source colors into sp2 gp2
for all pixels.
⊤
For each pixel, decomposing sp2 gp2
into sp2 and gp2
by using NMF.
Repeating 1) to 5) until convergence.

Before the step 1), we compute the median of the 3Cdimensional pixel colors observed under D light source directions, and use it for initializing sp2 . In the step 3), we consider
shadowed pixels with smaller pixel values than a thresholdas missing data. We empirically confirmed that the above
optimization is usually converged within a few iterations.
IV.

E XPERIMENTS

To confirm the effectiveness of our proposed method, we
conducted a number of experiments by using both synthetic
and real images.
A. Synthetic Images
First, we compared the performance of our proposed
method with those of the following methods by using synthetic
images.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 5. The diffuse reflection components: (a) the ground truth and thoseseparated by using (b) our proposed method, (c) the sparse NMF, (d) the SVD
with missing data, (e) Wu et al. [16], and (f) Akashi and Okatani [2]. They
are closeups from Fig. 3 and Fig. 4.

based only on the rank 3 structure of diffuse reflection
components on the slice B. It has an empirical parameter for removing specular reflection components asoutliers by thresholding.
•

Wu et al. [16]: This method makes use of the lowrankness of diffuse reflection components as wellas the sparseness of specular reflection components
and shadows under varying light source directions5 .
It has an empirical parameter for controlling theirsparseness. As described in Section II, we used it foreach slice with a fixed light source color.

•

Akashi and Okatani [2]: This method makes useof the sparseness of body colors in addition to thedifference between the colors of a specular reflection
component and a diffuse reflection component. It has
an empirical parameter for controlling the sparsenessof body colors. As described in Section II, we used itfor each image.

•

Sparse NMF: This is a building block of our proposed
method, in particular eq.(7) to eq.(9), and is based
only on the rank 2 structure of pixel values and thesparseness of specular reflection components on theslice A. It has an empirical parameter for controllingthe sparseness of specular reflection components.

We synthesized 120 images of a sphere under multispectral
and multidirectional light sources. The sphere has texture
consisting of 4 different spectral reflectances: the red, green,
blue, and yellow from the X-Rite color checker. The numberof light source colors is 6 (C = 6) and the number of lightsource directions is 20 (D = 20). The spectral intensities anddirections of the light sources are the same as those of ourmultispectral light stage. In order to prevent saturated pixels,•

SVD with missing data: This is another building
block of our method, in particular eq.(10), and is

5 We implemented eq.(9) in their paper by using the augmented Lagrangian
multiplier method [7].

3202TABLE I.

T HE RMS ERRORS OF THE SEPARATED REFLECTION COMPONENTS : DIFFUSE ( TOP ) AND SPECULAR ( BOTTOM ) IN EACH FIELD .

Method\Noise level (σ)
Proposed method
based on both the structures on the slices A and B
Sparse NMF
based only on the structure on the slice A
SVD with missing data
based only on the structure on the slice B
Wu et al. [16]based on low-rankness and sparseness
Akashi and Okatani [2]based on color and sparseness

0
0.11
0.11
0.07
0.11
2.74
2.19
2.27
2.27
18.74
18.24

1
0.25
0.24
0.32
0.26
2.84
2.34
2.49
2.37
20.37
19.79

2
0.46
0.46
0.63
0.48
3.04
2.66
2.93
2.52
20.68
20.14

we assume HDR images, i.e. the pixel values are not 8-bit
integers but real numbers. We added Gaussian noise, whosemean and standard deviation are 0 and σ respectively, to eachpixel value, and then used them as input. The parameter ofour method for controlling the sparseness, i.e. L in eq.(9), is
empirically set to 5. The parameters of the other methods, one
for each method, are tuned so that they perform best.

4
0.91
0.93
1.27
0.95
3.63
3.57
4.12
2.87
21.62
21.01

8
1.87
1.80
2.51
1.83
5.65
5.93
6.81
3.62
21.77
20.83

16
4.57
3.50
5.05
3.54
13.19
11.95
12.28
5.11
23.99
21.54

(a)

In Fig. 3, we show some of (a) the input images, (b) the
diffuse and (c) the specular reflection components separated byusing our proposed method, when σ = 8. Here, we converted
HDR images into 8-bit images for display purpose6 . We canqualitatively see that our method works well.
In Fig. 4, we show the diffuse reflection components separated by using (a) the sparse NMF, (b) the SVD with missing
data, (c) Wu et al. [16], and (d) Akashi & Okatani [2] for thecomparison with our proposed method. We can see that (a)the result of the sparse NMF is slightly noisier than that ofour method. In addition, we can see that some artifacts due to
specular reflection components are visible in the results of (b)the SVD with missing data and (c) Wu et al. [16]. It is clearthat (d) Akashi and Okatani [2] does not work well because
the color of a specular reflection component is almost the sameas that of a diffuse reflection component due to narrow-bandLEDs as suggested also in their paper. The effectiveness of
our method is clear from the closeups shown in Fig. 5.
For quantitative comparison, we summarize the RMS (rootmean-square) errors of the separated diffuse and specularreflection components in TABLE I, where σ varies from 0
to 16. We can see that our proposed method works best
except for σ = 0. Specifically, our method based on boththe structures on the slices A and B works better than thesparse NMF based only on the structure on the slice A andthan the SVD with missing data based only on the structureon the slice B. Therefore, we can conclude that taking all of
the inherent structures of the 3D data into consideration iseffective for robustly separating diffuse and specular reflection
components in images under multispectral and multidirectional
light sources. In addition, our method works better than someof the state-of-the-art methods based on low-rankness [16] andcolor [2].
B. Real Images
Second, we compared the performance of our proposed
method with that of the sparse NMF, i.e. the second best
6 The value of σ is defined in 8-bit images and then scaled for HDR images.

3203

(b)

(c)

(d)

(e)

Fig. 6. The results on real images (wooden bread): (a) the input images,(b) the diffuse and (c) the specular reflection components separated by using
our method, and (d) the diffuse and (e) the specular reflection components
separated by using the sparse NMF.

(a)

(b)

Fig. 7. The diffuse reflection components separated by using (a) our proposed
method and (b) the sparse NMF. They are closeups from Fig. 6(b) and (d).

method in TABLE I by using real images. Since it is not easy
to acquire the ground truth by using the polarization-based
approach as described in Section II, we qualitatively discuss
the experimental results.
We used 120 images of target objects captured by using ourmultispectral light stage termed Kyutech Multispectral Light
Stage I [6] in Fig. 1. The number of light source colors is
6 (C = 6) and the number of light source directions is 20

structure of diffuse reflection components at varying pixels
and under varying light source directions, and the sparsenessof specular reflection components. We conducted a numberof experiments by using both synthetic and real images, andshow that taking all of the inherent structures of the 3D data
into consideration is effective for robustly separating reflection
components, and that our method works better than some ofthe state-of-the-art techniques.

(a)

The future work of this study includes the extension toother components such as interreflection and fluorescence. The
application to computer vision techniques such as photometricstereo, material editing, and relighting is another direction of
the future work.
(b)

(c)

ACKNOWLEDGMENTS
This work was supported by JSPS KAKENHI Grant Numbers JP25280057 and JP16H01676.
R EFERENCES
[1]

(d)

(e)

Fig. 8. The results on real images (ceramic hippopotamus): (a) the input
images, (b) the diffuse and (c) the specular reflection components separated
by using our method, and (d) the diffuse and (e) the specular reflection
components separated by using the sparse NMF.

[2]

[3]

(D = 20). The target objects are a wooden bread and a ceramic
hippopotamus. We empirically set the parameter L in eq.(9)to 3.
In Fig. 6, we show some of (a) the input images, (b) the
diffuse and (c) the specular reflection components separated
by using our proposed method, and (d) the diffuse and (e)the specular reflection components separated by using thesparse NMF. We can find that some spot-like artifacts due to
specular reflection components are visible in (d) the diffuse
reflection components separated by using the sparse NMF.
This is because the sparse NMF is applied for each pixel,
and does not take the rank 3 structure of diffuse reflection
components at varying pixels and under varying light source
directions. On the other hand, we can see that (b) the diffuse reflection components separated by using our method issmooth. The effectiveness of our method is clear from thecloseups shown in Fig. 7. This result also shows that taking allof the inherent structures of the 3D data into consideration iseffective for robustly separating diffuse and specular reflection
components in images under multispectral and multidirectional
light sources. Fig. 8 shows that our method works well for adifferent material.

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V.

C ONCLUSION AND F UTURE W ORK

In this paper, we proposed a robust method for separating
diffuse and specular reflection components in a set of images ofan object taken under multispectral and multidirectional lightsources. Specifically, our proposed method considers the setof images as the 3D data, and then makes use of the inherent
structures of the 3D data: the rank 2 structure of pixel values
under varying light source colors and directions, the rank 3
3204

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