A Supervised Combination Strategy for Illumination
Chromaticity Estimation
BING LI, Institute of Automation, Chinese Academy of Sciences and Beijing Jiaotong University
WEIHUA XIONG, OmniVision Technologies
DE XU and HONG BAO, Beijing Jiaotong University
Color constancy is an important perceptual ability of humans to recover the color of objects invariant of light information. It is
also necessary for a robust machine vision system. Until now, a number of color constancy algorithms have been proposed in the
literature. In particular, the edge-based color constancy uses the edge of an image to estimate light color. It is shown to be a rich
framework that can represent many existing illumination estimation solutions with various parameter settings. However, color
constancy is an ill-posed problem; every algorithm is always given out under some assumptions and can only produce the best
performance when these assumptions are satisfied. In this article, we have investigated a combination strategy relying on the
Extreme Learning Machine (ELM) technique that integrates the output of edge-based color constancy with multiple parameters.
Experiments on real image data sets show that the proposed method works better than most single-color constancy methods
and even some current state-of-the-art color constancy combination strategies.
Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture/Image Generation—Display algorithms, viewing
algorithms; I.4.8 [Image Processing and Computer Vision]: Scene Analysis—Color
General Terms: Algorithms, Experimentation
Additional Key Words and Phrases: Combination strategy, color constancy, illumination estimation, extreme learning machine
ACM Reference Format:
Li, B., Xiong, W., Xu, D., and Bao, H. 2010. A supervised combination strategy for illumination chromaticity estimation. ACM
Trans. Appl. Percept. 8, 1, Article 5 (October 2010), 17 pages.
DOI = 10.1145/1857893.1857898 http://doi.acm.org/10.1145/1857893.1857898
1.
INTRODUCTION
Color perception is one of the most important approaches for human beings to perceive the world. It is
actually a fascinating series of physical and chemical perception reactions in the eyes and brain. The
perceived color of an object is not just a simple function of the spectral composition of the light reflected
from it, but also depends on the processing done by the brain and the retina. Human visual perception
system has a fundamental ability called color constancy, which can diminish the affect of illumination
This work is partly supported by National Nature Science Foundation of China (No. 61005030, 60825204, 60935002, 60803072,
and 60972145) and China Postdoctoral Science Foundation.
Author’s addresses: Bing Li, National Laboratory of Pattern Recognition (NLPR), Institute of Automation, Chinese Academy
of Sciences, Beijing, 100190, China; email:[email protected]; Weihua Xiong, OmniVision Technologies, Sunnyvale, CA 95014;
De Xu, Hong Bao, Institute of Computer Science & Engineering, Beijing Jiaotong University, Beijing 100044, China.
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DOI 10.1145/1857893.1857898 http://doi.acm.org/10.1145/1857893.1857898
ACM Transactions on Applied Perception, Vol. 8, No. 1, Article 5, Publication date: October 2010.
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and allow people to perceive the physical scene more precisely [Ebner 2007]. With science development
during the recent 100 years, modern imaging technology is going through the conventional chemical
method to digital electronic imaging method. However, the color signal from any imaging device is
affected by three factors: the color of light incident on the scene, the surface reflectance of the object,
and the sensor sensitivity of the camera [Barnard et al. 2002a, 2002b]. Therefore, the same surface
under different light sources will inevitably appear in different colors. Motivated by color perception
principles, a variety of color constancy models and algorithms have been proposed. Many of these models are used for both “human vision” and “machine vision” purposes [Spitzer and Semo 2002]. Models
for “human vision” or human perception color constancy models are mostly designed to mimic human
color constancy performance. These models attempt to describe the physiological visual mechanisms.
The models for machine vision, on the other hand, attempt to extract the spectral properties of objects
or surfaces in an image. They can also reflect the human perceptual color constancy mechanism to a
certain extent.
The color constancy models for machine vision can be roughly classified into color invariant and illuminant estimation procedures [Hordley 2006]. In illuminant estimation procedures, color constancy
is achieved by first obtaining an estimate of the illuminant in a scene from the image data. Once the
scene illuminant is known, the recorded image data can be corrected to discount the color of the scene
light to achieve image color constancy Finlayson et al. [1994]. Color invariant approaches, on the other
hand, represent images by features that are independent of light source to achieve color constancy,
without explicitly estimating the scene illuminant. From machine vision perspective, illumination estimation has many important applications such as object recognition and scene understanding, as well
as image reproduction and digital photography.
During the past decades, many researchers have proposed various illumination estimation algorithms. We divide them into two categories: unsupervised and supervised approaches. Unsupervised
methods use the nature of the image’s color components itself to solve the color constancy problem.
maxRGB [Land and McCann 1971] grey world [Buchsbaum 1980], shades of grey (SOG) [Finlayson
and Trezzi 2004], Grey surface identification [Xiong et al. 2007], color constancy using achromatic surface [Li et al. 2010], and edge-based color constancy [Weijer et al. 2007] belong to this category. Meanwhile, supervised methods build up the relations between image color distribution and illumination
color values through learning methods. It includes Bayesian color constancy [Brainard and Freeman
1997], color by correlation [Finlayson et al. 2001]; [Barnard et al. 2000], neural networks-based algorithm [Cardei et al. 2002], support vector regression (SVR)-based algorithm [Funt and Xiong 2004]
[Xiong and Funt 2006], and others [Xiong et al. 2007; Rosenberg et al. 2001; Sapiro 1999]. A complete
review of color constancy algorithms can be found in Ebner [2007], Barnard et al. [2002a, 2002b], and
Hordley [2006].
However, color constancy is an ill-posed problem; all of the aforementioned algorithms are always
proposed under some assumptions and can only work best when these assumptions are satisfied. To
improve the overall performance, some researchers have developed certain combination strategies to
analyze and integrate the output of multiple illumination estimation algorithms. The first combinational color constancy method was proposed by Cardei and Funt [1999]. In their committee-based color
constancy method, both simple average (SA) and least mean square (LMS) were used to combine the estimation results of maxRGB, grey world, and the neural networks (NN)-based color constancy method.
Gijsenij and Gevers [2007] proposed another color constancy combination method using natural image
statistics, in which different existing color constancy methods were selected or combined according to
the images’ texture characters. More recently, Bianco et al. [2008] proposed a consensus-based framework for color constancy. In this framework, four other color constancy combination schemes were
presented: Nearest2, Nearest-N, No-N-Max, and Median.
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The new method we propose here is similar to previous combinational schemes that aim to recover
the chromaticity of the scene by combing multiple illumination estimations; however, the proposed
method uses a new powerful supervised algorithm, ELM [Huang et al. 2006], which can provide good
generalization performance as the combination strategy. The strategy is based on the edge-based color
constancy solution proposed by Weijer [2007], which assumes that the average of reflectance differences in a scene is achromatic. This color constancy algorithm is also shown to be rich in representing
other algorithms with different parameters. In particular, we apply 150 parameter settings on edgebased color constancy to produce a variety of illumination chromaticity values, and then feed all of the
output into ELM to obtain the final estimation. Experimental results show that our proposed strategy
is better than most single-color constancy solutions and other combinations strategies.
The remainder of this article is organized as follows. In Section 2, we briefly review edge-based
color constancy. In Section 3, the ELM is introduced. Section 4 explains the combination strategy for
illumination estimation using the ELM in details. In Section 5, we present several existing combination
methods compared with our proposed algorithm. Two other alternative strategies, back-propagation
neural networks (BP) and SVR are also discussed for comparison. The experimental results on different
image data sets are presented in Section 6. Section 7 concludes this article.
2.
EDGE-BASED COLOR CONSTANCY
According to the Lambertian reflectance model, the image f = (R, G, B)T can be computed as follows:
f (X) = e(λ)s(X, λ)c(λ)dλ,
(1)
ω
where X is the spatial coordinate, λ is the wavelength, and ω represents the visible spectrum. e(λ)
is the spectral power distribution of the light source, s(X, λ) is the surface reflectance, and c(λ) =
(R(λ), G(λ), B(λ))T is the camera sensitivity function. If we ignore the differences of cameras and consider camera sensitivity as a part of illumination, the goal of color constancy is to estimate illumination
e.
(2)
e = e(λ)c(λ)dλ,
ω
e and s(X, λ) are both unknown, so color constancy is obviously an under-constrained problem, which
cannot be solved without further assumptions.
Recently, Weijer et al. [2007] proposed an edge-based color constancy algorithm by incorporating
higher-order derivatives of an image based on grey-edge hypothesis, which assumes that the average
of reflectance differences in a scene is achromatic. By introducing Minkowski-norm and Gauss filter, it
becomes
1/ p
n σ
∂ f (X) p
dX
= ken, p,σ ,
(3)
∂ Xn where f = (R, G, B)T represents the image color, X is the spatial coordinate again, and f σ = f ⊗ Gσ is
n
the convolution of the image with a Gaussian filter Gσ with a standard deviation σ . ∂∂Xn is an n-order
σ
derivative for f . p is the Minkowski-norm.
Equation (3) is obviously a general framework that can reformulate many existing algorithms by
changing the parameters n, p, σ to reflect the assumptions of the relationship between illumination
and image colors.
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Fig. 1. The architecture of SLFNs.
3.
EXTREME LEARNING MACHINE
A new learning algorithm for Single hidden Layer Feedforward neural Networks (SLFNs) named the
ELM was proposed by Huang et al. [2004, 2006]; Huang et al. [2004]. ELM can give better performance
than traditional tuning-based learning methods for feedforward neural networks such as the BP algorithm in terms of generalization and learning speed Li et al. [2005]. Figure 1 shows the architecture
of SLFNs; the typical function approximation problem of SLFNs and the ELM algorithm is briefed as
follows.
For N arbitrary distinct samples (Aj , D j ) j = 1...N, where Aj = [a j1 , a j2 , . . . , a jn]T ∈ Rn and the
corresponding output D j = [dj1 , dj2 , . . . , djm]T ∈ Rm, a standard SLFNs with L hidden neurons and
activation function g(x) can be mathematically modeled as follows.
L
Ki g(Wi • Aj + bi ) = O j , ( j = 1, 2, . . . , N),
(4)
i=1
where Wi = [wi1 , wi2 , . . . , win]T , i = 1...L is the weight vector connecting the ith hidden node, and the
input nodes, Ki = [ki1 , ki2 , . . . , kim]T , i = 1...L is the weight vector connecting the ith hidden node, and
the output nodes, and bi is the threshold of the ith hidden node. Wi • Aj denotes the inner product of
Wi and Aj .
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N training samples, the standard SLFNs with L hidden nodes with zero error means
For
N
O
j − D j = 0, that is, there exist Ki , Wi , and bi such that
j=1
L
Ki g(Wi • Aj + bi ) = D j , ( j = 1, 2, ..., N).
(5)
i=1
The previous N equations can be written compactly as follows:
HK = D,
where
⎡
⎢
K=⎣
⎡
⎤
g(w1 • A1 + b1 ) · · · g(w L • A1 + bL)
⎢
⎥
..
..
H=⎣
,
⎦
.
···
.
g(w1 • AN + b1 ) · · · g(w L • AN + bL) N×L
⎤
⎡ T ⎤
K1T
D1
⎢ .. ⎥
.. ⎥
,D = ⎣ . ⎦
. ⎦
KLT
L×m
T
DN
(6)
N×m
H is called the hidden layer output matrix of the neural network; the ith column of H is the ith hidden
node output with respect to input A1 , A2 , . . . , AN [Huang et al. 2006]. Therefore, the least-squares
solution K̂ of the general linear system in Equation (6) is as follows.
K̂ = H† D,
(7)
where the H† is the Moore-Penrose generalized inverse of H [Huang et al. 2004, 2006]. A simple learning method for SLFNs called ELM was summarized by Huang et al. [2004, 2006]:
—Step 1. Randomly assign input weight Wi and bias bi , i = 1, 2, ..., L.
—Step 2. Calculate the hidden layer output matrix H.
—Step 3. Calculate the output weight according to Equation (7) K = K̂ = H† D.
The details of ELM can be found in Huang [2003], Huang et al. [2004; 2006], and Li et al. [2005]. The
ELM learning algorithm has received more attention and has correspondingly been applied in many
fields [Zhang et al. 2007; Nizar et al. 2008]. In this article, this simple and extremely fast-learning
algorithm is introduced as a combination strategy for illumination estimation.
4.
COLOR CONSTANCY COMBINATION STRATEGY USING ELM
In this section, we will discuss how to use ELM as a combinational strategy to achieve better illumination estimation results. It focuses on three key issues: the input vectors of ELM, the response vector
construction, and a single parameter, that is, hidden neurons number L.
4.1
Input Vectors of ELM
In this article, contrary to what other researchers have done before, we supply the output of edgebased color constancy using different parameter settings as the input of ELM. To make sure that all
of 0-, 1-, and 2-order derivative image structures are used for illumination estimation, we will select
n ∈ {0, 1, 2}, p ∈ {1, 2,..., 10}, σ ∈ {1, 3, 5, 7, 9}. Therefore, we will obtain 150 illumination estimation
results, denoted as ei = (Ri , Gi , Bi ), i = 1, 2, . . . , 150. ei is then transformed into chromaticity space to
Gi
). Now, the input vector of ELM can
cancel out the effect of shading cei = (ri , gi ) = ( Ri +GRii +Bi , Ri +G
i +Bi
be gained by concatenating the color channels of cei : V = [ce1 , ce2 , . . . , ce150 ]T , so the dimension of the
vector V is 300.
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Fig. 2. Architecture of the proposed algorithm.
4.2
Response Vector Construction
The response vector of ELM is in chromaticity space, so 2-output ELM networks are necessary for illumination chromaticity estimation: One is the illumination r component, and the other is the illumination g component. During the training phase, the response vector is composed by the real illumination
Go
)T of each image, where (Ro , Go , Bo )T is the measured
chromaticity oe = (ro , go )T = ( Ro +GRoo +Bo , Ro +G
o +Bo
illumination color used as ground truth. The architecture of the proposed algorithm for illumination
estimation is shown in Figure 2.
4.3
Cross-Validation for ELM Parameter Selection
There is only one insensitive parameter that is hidden neurons number L in ELM. We allow L to be
chosen from following candidate data: L = {10, 20, 30, . . . , 300}. The sigmoid and sine functions are
used as activation function candidates for ELM [Huang et al. 2006]. Then k-fold cross-validation is
used to choose the best L for illumination estimation. In k-fold cross-validation, the whole training
set is divided evenly into k distinct subsets. For any candidate parameter setting of each activation
function, we conduct the same process k times, during which (k − 1) of the subsets are used to form
a training set, and the remaining subset is taken as the test set. RMS chromaticity distance errors
(defined in Section 6.1) from k trials are computed to represent the error for that candidate parameter setting. The parameter setting leading to the minimum error is then chosen for the final ELM
training.
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Table I. The Best Result of Threefold
Cross-Validation for Each Activation Function
and the Corresponding Parameter Setting
Activation Function
Sigmoid
Sine
L
L = 200
L = 70
RMS Dist(×102 )
2.60
2.81
Cardei’s set of 900 uncalibrated images [Cardei et al. 2002], taken by using a variety of different
digital cameras, is used for parameter tuning and activation function selection with threefold crossvalidation. Table I presents the best L selection result under each activation function. It shows that the
overall performance of the sigmoid activation function with L = 200 performs slightly better than that
of the sine function with L = 70. These two parameter settings are used in the following experiments
for performance evaluation.
5.
OTHER COMBINATION STRATEGIES
In this section, we will introduce other combination strategies to compare with our proposed method.
Besides some existing algorithms [Cardei and Funt 1999; Bianco et al. 2008], we also present two
other potential solutions: BP Neural Networks and SVR Methods. All of these methods can be roughly
divided into two major categories: unsupervised and supervised approaches.
5.1
Unsupervised Combination Approaches
The SA [Cardei and Funt 1999], Nearest2, Nearest-N%, No-N-Max, and Median Method [Bianco et al.
2008] belong to unsupervised category. The details of these methods can be found in Cardei and Funt
[1999] and Bianco et al. [2008]. For the Nearest-N% method, Nearest-10% and Nearest-50% will be
used in the following experiments. For the No-N-Max method, we will use No-10-Max and No-50-Max.
5.2
Supervised Combination Approaches
All approaches belonging to this category always include some parameters to be determined trough
training procedure. Specifically, the LMS-based combinational scheme was proposed in Cardei and
Funt [1999], while the other two are explained in the following.
5.2.1 LMS Method. In the LMS combination strategy [Cardei and Funt 1999], LMS is used to determine the weight matrix M. In this article, M is defined as a 200 × 300 matrix. The final illumination
chromaticity ce LMS can be gained as
ce LMS = M × V = M × [ce1 , ce2 , ..., ce150 ]T .
(8)
5.2.2 BP Method. The BP can also be used as a combination strategy; its architecture is the same
as that of ELM. The big difference between them lies in the learning algorithm during the training
procedure; the former used the BP algorithm, while the latter used the ELM algorithm. As compared
to ELM, learning speed is the major bottleneck of BP. Likewise, ELM has a much better generalization
than the BP algorithm. Different from traditional neural networks algorithms, the ELM algorithm
not only tends to reach the smallest training error but also the smallest norm of weights [Huang
et al. 2006]. Bartlett’s [1998] theory on the generalization performance of feedforward neural networks
states that for those feedforward neural networks that reach a smaller training error, the smaller the
norm of weights is, the better the generalization performance that the networks tend to have. To
facilitate the comparison of BP and ELM, we select the sigmoid activation function and the hidden
neurons number as L = 200.
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Table II. The Best Parameter
Combination of Each Kernel Function for
the SVR-Based Fusing Method
Kernel Function
Linear(r)
Linear(g)
RBF(r)
RBF(g)
γ
—
—
0.05
0.05
C
5
1
20
10
The values of γ and C represent the best parameters
for the SVR models with different kernels.
5.2.3 SVR Method. SVR, a good regression tool, was introduced into illumination estimation by
Xiong and Funt [2006] and Funt and Xiong [2004]. It can be a possible combination strategy alternative. However, SVR has its own disadvantages as compared to ELM. First, SVR is a single-output
regression, which means we have to define two distinct functions for illumination chromaticity estimations, one for r and one for g, respectively. Hence, optimal estimations cannot be reached simultaneously. Second, SVR depends on several parameters that are very difficult to tune. The last one
lies on its computational complexity and long training time. In this article, to compare its performance with that of ELM, we use the 300D chromaticity vector V as the input of SVR. Both linear
kernel function and radial basis function (RBF) are used as kernel functions, respectively. The 3-fold
cross-validation on the 900 uncalibrated image set is also used to determine the parameter selections for SVR. The parameter candidates here are as follows, C ∈ {0.01, 0.1, 1, 5, 10, 15, 20, 25, 30, 50},
γ ∈ {0.0125, 0.025, 0.05, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0, 2.0}. The best parameters for each kernel function
that will be used in the following are listed in Table II.
6.
EXPERIMENTS
We evaluate the proposed algorithm and compare it with some single-color constancy algorithms, including grey world, maxRGB, SoG, grey edge, 2nd-order grey edge and SVR, and other combinational
strategies on three real-image datasets. The first is Barnard’s set of 321 SFU images [Barnard et al.
2002], the second is Cardei’s 900 uncalibrated images from different cameras [Cardei et al. 2002],
and the third is Ciurea’s and Funt’s [2003] real-world images captured from a digital video. The ELM
source code is downloaded from [ELM], and the SVR source code is downloaded from [SVM]. We also
implemented other combinational strategies in MATLAB 7.0.
6.1
Error Measure
Several error measures are used to evaluate performance of the proposed algorithm. For each image,
the distance Ed between the measured actual illumination chromaticity (ra , ga ) and the estimated illumination (re , ge ) is calculated as follows:
Ed = (ra − re )2 + (ga − ge )2 .
(9)
Assuming there are Z test images, we will also report the root mean square (RMS), maximum, and
median distance [Xiong and Funt 2006]. The RMS of the chromaticity distance RMSd is defined as
Z
1 Ed2 (i),
(10)
RMSd = Z
i=1
where Ed(i) is the illumination chromaticity distance error of the ith image. The angular error Ea
between the measured actual illumination color ea = (Ra , Ga , Ba )T and the estimated illumination
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Table III. Comparison of ELM Method to MaxRGB, Grey World, Shade of Grey, Grey Edge,
2nd-order Grey Edge, and SVR, Performance
Method
maxRGB*
Grey World*
SoG*
Grey Edge
2nd-order Grey Edge
SVR(2D)*
SVR(3D)*
ELM(sigmoid)
ELM(sine)
Parameters
−
−
p=6
e1,7,4
e2,7,5
2D
3D
L = 200
L = 70
Median
Angle
6.44
7.04
3.97
3.20
2.74
4.65
2.17
1.12
1.82
Max
Angle
36.24
37.31
28.70
31.57
26.75
22.99
24.66
14.27
15.32
RMS
Angle
12.28
13.58
9.03
8.30
7.75
10.06
8.07
3.04
3.45
Median
Dist (×102 )
4.46
5.68
2.83
2.37
2.03
3.41
3.07
0.91
1.54
Max
Dist (×102 )
25.01
35.38
19.77
20.13
17.74
16.41
16.03
13.12
13.51
RMS
Dist (×102 )
8.25
11.12
6.21
5.81
5.47
7.50
6.30
2.32
2.59
The tests are based on the 321 SFU images. The results of other algorithms marked by “*” are from Xiong [2007].
ee = (Re , Ge , Be )T = (re , ge , 1 − re − ge )T is also used. The angular error function angular(ea , ee ) is
defined as
(Ra , Ga , Ba ) • (Re , Ge , Be )
180◦
.
(11)
)×
Ea = angular(ea , ee ) = cos−1 ( π
Ra2 + Ga2 + Ba2 × Re2 + G2e + B2e
As with the distance measure, we also report the RMS, maximum, and median angular error over the
test set of images.
To evaluate the error distribution difference in the performance of two competing methods, Wilcoxon
signed-rank on the angular errors Ea is applied [Hordley and Filayson 2006]. The threshold for accepting or rejecting the hypothesis is set to 0.01.
6.2
321 SFU Image Dataset
First, the proposed algorithms are tested on an image set of colorful objects under different light
sources [Barnard et al. 2002]. The set consists of 321 images taken under 11 varying light sources of 30
different scenes containing both matte and specular objects. We evaluate the illumination estimation
error using the leave-one-out procedure [Xiong and Funt 2006] for all the supervised combinational
strategies. In the leave-one-out procedure, one image is selected for testing, and the remaining 320
images are used for training. After repeating 321 times, the RMS and median of the 321 resulting
illumination estimation errors are calculated.
6.2.1 Comparison with Single Algorithms. We first compare the ELM combinational method with
some single color constancy algorithms. The experimental results are shown in Tables III and IV. The
parameters of Grey Edge and 2nd-order Grey Edge are determined according to their best performance
as shown in Weijer et al. [2007]. ELM(sigmoid) and ELM(sine) represent the ELM neural networks
with the sigmoid and sine activation functions, respectively. From the results in Tables III and IV,
we can find that the proposed ELM(sigmoid) and ELM(sine) methods outperform all other existing
algorithms. Interestingly, the median angle and chromatic distance of the ELM(sigmoid) method are
only 1.12 and 0.91, respectively, which are reduced by 48.4% and 70.4% compared with the SVR(3D)
algorithm.
6.2.2 Comparison with Unsupervised Combinational Methods. In this part, our proposed method is
compared with SA, Nearest2, Nearest-N%, No-N-Max, and Median unsupervised combinational methods on the 321 images. The experimental results are shown in Tables V and VI. In the tables, SA
represents the SA method; we use #Median to represent the method of the median combinational
strategy to distinguish it from the error measure of median angular or median distance. The results
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Table IV. Comparison of ELM to Other Algorithms via the Wilcoxon Signed-Rank Test
2nd-order
maxRGB Grey-World SoG Grey Edge Grey Edge SVR(2D) SVR(3D) ELM(Sigmoid) ELM(sine)
maxRGB
−
−
−
−
−
−
−
−
Grey World
+
−
−
−
−
−
−
−
SoG
+
+
−
−
+
−
−
−
Grey Edge
+
+
+
−
+
−
−
−
2nd-order Grey Edge
+
+
+
+
+
−
−
−
SVR(2D)
+
+
−
−
−
−
−
−
SVR(3D)
+
+
+
+
+
+
−
−
ELM(sigmoid)
+
+
+
+
+
+
+
+
ELM(sine)
+
+
+
+
+
+
+
−
A “+”’ means the algorithm listed in the row is statistically better than the one in the column. A “−” indicates the opposite, and an “=”’ indicates
that the performance difference of these two algorithms is statistically trivial.
Table V. Comparison of ELM Method to SA, Nearest2, Nearest-N%, No-N-Max and Median Unsupervised
Fusing Methods
Method
SA
Nearest2
Nearest-10%
Nearest-50%
No-10-Max
No-50-Max
#Median
ELM(sigmoid)
ELM(sine)
Median Angle
4.28
3.45
3.45
3.29
4.10
4.04
4.28
1.12
1.82
Max Angle
28.49
32.76
32.77
32.77
29.43
29.59
29.02
14.27
15.32
RMS Angle
8.77
9.72
9.72
9.54
8.75
8.80
8.79
3.04
3.45
Median Dist (×102 )
3.17
2.36
2.38
2.49
3.07
2.95
3.15
0.91
1.54
Max Dist (×102 )
18.75
27.93
27.93
25.61
19.18
19.16
18.98
13.12
13.51
RMS Dist (×102 )
6.3
7.15
7.15
6.95
6.24
6.24
6.3
2.32
2.59
The tests are based on the 321 SFU images.
Table VI. Comparison of ELM to Other Algorithms via the Wilcoxon Signed-Rank Test
SA
Nearest2
Nearest-10%
Nearest-50%
No-10-Max
No-50-Max
#Median
ELM(sigmoid)
ELM(sine)
SA Nearest2 Nearest-10% Nearest-50% No-10-Max No-50-Max #Median ELM(sigmoid) ELM(sine)
−
−
−
=
=
=
−
−
+
=
=
+
+
+
−
−
+
=
=
+
+
+
−
−
+
=
=
+
+
+
−
−
=
−
−
−
=
=
−
−
=
−
−
−
=
=
−
−
=
−
−
−
=
=
−
−
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
−
The labels “+” “−” and “=” have same meanings as those shown in Table 4.
tell us that ELM(sigmoid) and ELM(sine) outperform all other unsupervised combinational methods.
Nearest-N% methods perform better than No-N-Max methods on this image set, and the SA method is
comparable to the No-N-Max methods.
6.2.3 Comparison with Supervised Combinational Methods. The performance comparison among
supervised combinational methods including LMS, BP, and SVR is listed in Tables VII and VIII. We use
#SVR to represent the SVR-based strategy method in order to avoid confusing it with the SVR singlecolor constancy algorithm. The tables show that the performances of SVR-based combinational methods with different kernel functions are comparable with one another and are even slightly better than
ELM(sigmoid). However, the learning speed of ELM is much faster than that of SVR. ELM(sigmoid)
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A Supervised Combination Strategy for Illumination Chromaticity Estimation
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Table VII. Comparison of ELM Method to LMS, BP Neural Networks, SVR (linear) and SVR (RBF)
Method
LMS
BP
#SVR (Linear)
#SVR (RBF)
ELM (Sigmoid)
ELM (Sine)
Median Angle
1.80
2.90
1.25
1.07
1.12
1.82
Max Angle
44.49
60.96
8.46
12.27
14.27
15.32
RMS Angle
12.67
10.59
2.38
2.11
3.04
3.45
Median Dist (×102 )
1.32
2.22
0.91
0.81
0.91
1.54
Max Dist (×102 )
55.8
46.8
5.94
7. 18
13.12
13.51
RMS Dist (×102 )
5.19
9.76
1.82
1.75
2.32
2.59
The tests are based on the 321 SFU images.
Table VIII. Comparison of ELM to Other Algorithms via the Wilcoxon Signed-Rank Test
LMS
LMS
BP
#SVR(linear)
#SVR(RBF)
ELM(sigmoid)
ELM(sine)
BP
+
−
+
+
+
=
+
+
+
+
#SVR(Linear)
−
−
=
−
−
#SVR(RBF)
−
−
=
−
−
ELM(Sigmoid)
−
−
+
+
ELM(sine)
=
−
+
+
+
−
The labels “+” “−” and “=” have same meanings as those shown in Table IV.
Table IX. Comparison of ELM to maxRGB, Grey-World, Shade of Grey, Grey Edge, 2nd-Order Grey Edge and
SVR Performance
Method
maxRGB*
Grey World*
SoG*
Grey Edge
2nd-order Grey Edge
SVR(2D)*
SVR(3D)*
ELM(Sigmoid)
ELM(Sine)
Median
Angle
2.96
4.34
3.02
3.27
3.34
2.40
2.02
2.24
2.30
Parameters
−
−
p=6
e1,7,4
e2,7,5
2D
3D
L = 200
L = 70
Max
Angle
27.16
31.44
19.71
31.76
34.98
20.43
17.46
23.46
19.03
RMS
Angle
6.39
6.65
4.99
5.79
5.85
4.47
3.94
3.81
3.86
Median
Dist (×102 )
2.17
3.17
2.19
2.44
2.53
1.74
1.40
1.48
1.69
Max
Dist (×102 )
22.79
29.99
15.96
28.15
31.08
18.40
15.42
20.14
21.54
RMS
Dist (×102 )
4.75
5.26
3.80
4.40
4.48
3.27
2.94
2.87
2.88
The tests are based on the 900 uncalibrated images. The results of other algorithms marked by “*” are from Xiong [2007].
outperforms LMS and BP-based combinational algorithms. Tables III through VIII show that combinational algorithms generally perform better than single color constancy solutions, the supervised
combination method is always the best.
6.3
900 Uncalibrated Image Dataset
We next consider Cardei’s set of 900 uncalibrated images captured by various digital cameras. In this
section, the leave-one-out procedure is also used in the dataset to evaluate the proposed algorithm.
6.3.1 Comparison with Single Algorithms. The experimental results of the comparison with single
algorithms are tabulated in Table IX. Table X summarizes the Wilcoxon test among several algorithms.
From Tables IX and X, we can see that the proposed ELM algorithm still outperforms the maxRGB,
Grey World, SoG, Grey edge, 2nd-order Grey Edge, and SVR(2D) methods and is comparable to
SVR(3D). The ELM with sigmoid activation function is also comparable to the one with the sine activation function. Since the images in this data set are taken from different cameras, the comparison
also shows that the proposed ELM methods have good generalizations.
6.3.2 Comparison with Unsupervised Combinational Methods. The experimental results of the
comparison with unsupervised combinational methods on the 900 uncalibrated image set are
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B. Li et al.
Table X. Comparison of ELM to Other Algorithms via the Wilcoxon Signed-Rank Test
maxRGB
maxRGB
Grey World
SoG
Grey Edge
2nd-order Grey Edge
SVR(2D)
SVR(3D)
ELM (sigmoid)
ELM (sine)
−
=
−
−
+
+
+
+
Grey-World SoG
+
=
−
+
+
−
+
−
+
+
+
+
+
+
+
+
Grey 2nd-order
Edge Grey Edge SVR(2D)
+
+
−
−
−
−
+
+
−
=
−
=
−
+
+
+
+
+
+
+
+
+
+
+
SVR(3D)
−
−
−
−
−
−
−
−
ELM(Sigmoid) ELM(sine)
−
−
−
−
−
−
−
−
−
−
−
−
+
+
=
=
The labels “+” “−” and “=” have same meanings as those shown in Table IV.
Table XI. Comparison of ELM Method to SA, Nearest2, Nearest-N%, No-N-Max and Median Unsupervised
Fusing Methods
Method
SA
Nearest2
Nearest-10%
Nearest-50%
No-10-Max
No-50-Max
#Median
ELM (sigmoid)
ELM (Sine)
Median Angle
2.88
3.12
3.12
3.10
2.83
2.86
2.86
2.24
2.30
Max Angle
20.7
24.63
24.64
24.64
22.23
22.07
19.83
23.46
19.03
RMS Angle
4.79
5.67
5.67
5.61
4.87
4.94
4.82
3.81
3.86
Median Dist (×102 )
2.14
2.26
2.26
2.20
2.09
2.07
2.07
1.48
1.69
Max Dist (×102 )
17.54
19.06
19.05
19.05
19.00
18.88
17.08
20.14
21.54
RMS Dist (×102 )
3.58
4.34
4.35
4.29
3.66
3.71
3.6
2.87
2.88
The tests are based on the 900 uncalibrated images.
Table XII. Comparison of ELM to Other Algorithms via the Wilcoxon Signed-Rank Test
SA
Nearest2
Nearest-10%
Nearest-50%
No-10-Max
No-50-Max
#Median
ELM (Sigmoid)
ELM (Sine)
SA Nearest2 Nearest-10% Nearest-50% No-10-Max No-50-Max #Median ELM(Sigmoid) ELM(Sine)
+
+
+
=
=
=
−
−
−
=
=
−
−
−
−
−
−
=
=
−
−
−
−
−
−
=
=
−
−
−
−
−
=
+
+
+
=
=
−
−
=
+
+
+
=
=
−
−
=
+
+
+
=
=
−
−
+
+
+
+
+
+
+
=
+
+
+
+
+
+
+
=
The labels “+” “−” and “=” have same meanings as those shown in Table IV.
summarized in Tables XI and XII. The results show that ELM(sigmoid) and ELM(sine) still outperform
all other unsupervised combinational methods. No-N-Max methods perform better than Nearest-N%
methods, and the SA method is comparable to No-N-Max methods.
6.3.3 Comparison with Supervised Combinational Methods. Tables XIII and XIV show the comparison results between the proposed ELM method and other supervised combinational methods. They
show that ELM(sigmoid) and ELM(sine) outperform LMS, the BP-based combinational method, and
#SVR(linear), and are comparable to #SVR(RBF).
6.4
Real-World Image Dataset
Finally, our proposed strategy is evaluated on a large database provided by Ciurea and Funt [2003].
The database contains 11,346 images extracted from 2 hours of digital video, which include both indoor
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Table XIII. Comparison of ELM Method to LMS, BP Neural Networks, SVR(linear) and SVR(RBF)
Method
LMS
BP
#SVR(Linear)
#SVR(RBF)
ELM(Sigmoid)
ELM(Sine)
Median Angle
2.63
2.96
2.42
2.20
2.24
2.30
Max Angle
46.14
81.10
16.22
13.71
23.46
19.03
RMS Angle
4.97
11.31
3.89
3.78
3.81
3.86
Median Dist (×102 )
1.97
2.21
1.87
1.53
1.48
1.69
Max Dist (×102 )
44.59
44.83
13.91
12.35
20.14
21.54
RMS Dist (×102 )
4.12
9.70
2.90
2.61
2.87
2.88
The tests are based on the 900 uncalibrated images.
Table XIV. Comparison of ELM to Other Algorithms via the Wilcoxon Signed-Rank Test
LMS
LMS
BP
#SVR (Linear)
#SVR (RBF)
ELM (Sigmoid)
ELM (sine)
−
+
+
+
+
BP
+
+
+
+
+
#SVR(Linear)
−
−
+
+
+
#SVR(RBF)
−
−
−
=
=
ELM(sigmoid)
−
−
−
=
ELM(sine)
−
−
−
=
=
=
The labels “+” “−” and “=” have same meanings as those shown in Table IV.
Fig. 3. (a) Examples of images from the real-world image set. (b) Cropped images for experiments.
and outdoor scenes under a wide variety of lighting conditions. Figure 3(a) gives some example images.
A matte grey sphere ball was mounted onto the video camera, appearing at the bottom-right corner
of each image. The averaged R/G/B value on the ball is used as the ground truth of the illumination
color in the scene. However, Xiong et al. [2007] pointed out that many of these 11,346 images have
very good color balance, which creates bias in testing the illumination estimation methods. Therefore,
they eliminated majority of the correctly balanced images from the dataset. The resulting dataset contains 7,661 images, which are eventually partitioned into two subsets based on geographical location.
Figure 3(a) includes 3,581 images, and Figure 3(b) includes 4,080. Moreover, all images are cropped to
remove the effect of the grey sphere ball on the algorithm. The size of the remaining image is 240×240.
We then use Figure 3(a) for training, Figure 3(b) for testing, and vice versa. The combined errors and
corresponding Wilcoxon sign test results are used as final results.
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B. Li et al.
Table XV. Comparison of ELM to maxRGB, Grey-World, Shade of Grey, Grey Edge, 2nd-Order Grey Edge and
SVR Performance
Method
Parameters Median Angle Max Angle RMS Angle Median Dist (×102 ) Max Dist (×102 ) RMS Dist (×102 )
maxRGB*
9.65
27.42
12.13
6.86
21.72
8.80
Grey World*
−
6.82
43.84
9.66
5.25
45.09
7.82
SoG*
p=6
6.71
37.01
8.93
4.83
27.99
6.59
Grey Edge
e1,7,4
5.83
46.39
8.08
4.39
51.07
6.19
2nd-order Grey Edge
e2,7,5
5.92
45.71
8.15
4.49
51.68
6.18
SVR(3D)*
3D
4.91
24.80
7.03
3.62
18.62
5.16
ELM(sigmoid)
L = 200
4.51
43.70
5.26
3.28
32.22
3.87
ELM(sine)
L = 70
4.52
43.21
5.10
3.31
32.30
3.67
The tests are based on the real-world images. The results of other algorithms marked by “*” are from Xiong [2007].
Table XVI. Comparison of ELM to Other Algorithms via the Wilcoxon Signed-Rank Test
maxRGB
Grey World
SoG
Grey Edge
2nd-order Grey Edge
SVR(3D)
ELM(sigmoid)
ELM(sine)
maxRGB Grey-World SoG Grey Edge 2nd-order Grey Edge SVR(3D) ELM(Sigmoid) ELM(sine)
−
−
−
−
−
−
−
+
=
−
−
−
−
−
+
=
−
−
−
−
−
+
+
+
=
−
−
−
+
+
+
=
−
−
−
+
+
+
+
+
−
−
+
+
+
+
+
+
=
+
+
+
+
+
+
=
The labels “+” “−” and “=” have same meanings as those shown in Table IV.
Table XVII. Comparison of ELM method to Simple Average, Nearest2, Nearest-N%, No-N-Max and Median
Unsupervised Fusing Methods
Method
SA
Nearest2
Nearest-10%
Nearest-50%
No-10-Max
No-50-Max
#Median
ELM(Sigmoid)
ELM(Sine)
Median Angle
6.12
6.76
6.71
6.58
6.23
6.29
6.18
4.51
4.52
Max Angle
43.34
46.78
46.81
46.79
43.44
43.89
42.64
43.70
43.21
RMS Angle
8.32
9.78
9.7
9.54
8.55
8.75
8.35
5.26
5.10
Median Dist (×102 )
4.51
5.06
4.99
4.92
4.60
4.67
4.56
3.28
3.31
Max Dist (×102 )
32.51
37.8
37.79
37.79
31.96
32.05
31.96
32.22
32.30
RMS Dist (×102 )
6.15
7.45
7.4
7.26
6.34
6.48
6.19
3.87
3.67
The tests are based on the real-world images.
6.4.1 Comparison with Single Algorithms. Tables XV and XVI show the comparison results between the proposed method and the single-color constancy solutions on the real-world image set.
ELM(sigmoid) and ELM(sine) still perform best compared with other single-color constancy algorithms.
6.4.2 Comparison with Unsupervised Combinational Methods. The experimental results of the
comparison with unsupervised combinational methods on the real-world image set are summarized
in Tables XVII and XVIII. They tell us that ELM(sigmoid) and ELM(sine) outperform all other unsupervised combinational methods.
6.4.3 Comparison with Supervised Combinational Methods. The comparison results between the
proposed ELM method and other supervised combinational methods are tabulated in Tables XIX and
XX. ELM(sigmoid) and ELM(sine) outperform the LMS and #SVR(linear) methods, and are comparable
to the #SVR(RBF) and BP methods.
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Table XVIII. Comparison of ELM to Other Algorithms via the Wilcoxon Signed-Rank Test
SA
Nearest2
Nearest-10%
Nearest-50%
No-10-Max
No-50-Max
#Median
ELM(sigmoid)
ELM(sine)
SA Nearest2 Nearest-10% Nearest-50% No-10-Max No-50-Max #Median ELM(Sigmoid) ELM(Sine)
+
+
+
=
=
=
−
−
−
=
=
−
−
−
−
−
−
=
=
−
−
−
−
−
−
=
=
−
−
−
−
−
=
+
+
+
=
=
−
−
=
+
+
+
=
=
−
−
=
+
+
+
=
=
−
−
+
+
+
+
+
+
+
=
+
+
+
+
+
+
+
=
The labels “+” “−” and “=” have same meanings as those shown in Table IV.
Table XIX. Comparison of ELM Method to LMS, BP Neural Networks, SVR(linear), and SVR(RBF)
Method
LMS
BP
#SVR(Linear)
#SVR(RBF)
ELM(sigmoid)
ELM(sine)
Median Angle
4.86
4.50
4.79
4.48
4.51
4.52
Max Angle
47.18
44.74
44.69
43.61
43.70
43.21
RMS Angle
7.61
7.12
7.29
6.82
5.26
5.10
Median Dist (×102 )
3.58
3.32
3.53
3. 33
3.28
3.31
Max Dist (×102 )
44.56
32.48
33.43
32.17
32.22
32.30
RMS Dist (×102 )
5.65
5.18
5.35
4.99
3.87
3.67
The tests are based on the real-world images.
Table XX. Comparison of ELM to Other Algorithms via the Wilcoxon Signed-Rank
Test
LMS
LMS
BP
#SVR(Linear)
#SVR(RBF)
ELM(Sigmoid)
ELM(sine)
+
=
+
+
+
BP
−
−
=
=
=
#SVR(Linear)
=
+
+
+
+
#SVR(RBF)
−
=
−
=
=
ELM(Sigmoid)
−
=
−
=
=
ELM(sine)
−
=
−
=
=
The labels “+” “−” and “=” have same meanings as those shown in Table IV.
6.5
Discussion
In the previous experiments, we have demonstrated that the proposed combination strategy for illumination estimation gives out comparable results to many existing algorithms including single-color
constancy methods, unsupervised combinational strategies, and supervised combinational strategies.
All things considered, our proposed scheme with sigmoid activation function get better results than
others; its performance is also comparable to combinational scheme based on SVR technique.
Evaluation and comparison among different datasets show that the errors of most algorithms on the
real-world set are higher than the errors from 321 SFU set and 900 uncalibrated set. The main reason
lies in that the images in the real-world set cannot satisfy the uniform illumination assumption, while
all images in 321 SFU set and most images in 900 uncalibrated set are taken indoors, which are easy
to satisfy the prerequisite well.
The two facts embedded in these experiments are that combinational methods generally have better
performances than single algorithms and supervised combinational methods generally outperform the
unsupervised ones. The facts can help us understand the complexity of color constancy perception, and
can at least tell us that a good color constancy mechanism should take advantage of prior knowledge
and multiply clues in the scene.
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7.
•
B. Li et al.
CONCLUSION
This article describes a new combinational strategy for illumination estimation using ELM. The main
interest of this work is to integrate multiple outputs of edge-based color constancy solution with different parameter settings through ELM algorithm and produce an estimation of light color. The experiments on large real-image databases show that our proposed method is much better than most
single-color constancy algorithms and some combinational strategies. Our future work will focus on
color constancy under multi-illumination.
ACKNOWLEDGMENTS
The authors would like to thank Computational Vision Laboratory of Simon Fraser University for
providing the image datasets. We are also very appreciative to the anonymous reviewers for their very
valuable comments and suggestions. The author Bing Li also gratefully acknowledges the support of
K. C. Wong Education Foundation, Hong Kong.
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Received September 2008; revised July 2009, August 2009, November 2009; accepted December 2009
ACM Transactions on Applied Perception, Vol. 8, No. 1, Article 5, Publication date: October 2010.