Robust Stereo Matching Using Adaptive Normalized Cross-Correlation

Abstract

A majority of the existing stereo matching algorithms assume that the corresponding color values are similar to each other. However, it is not so in practice as image color values are often affected by various radiometric factors such as illumination direction, illuminant color, and imaging device changes. For this reason, the raw color recorded by a camera should not be relied on completely, and the assumption of color consistency does not hold good between stereo images in real scenes. Therefore, the performance of most conventional stereo matching algorithms can be severely degraded under the radiometric variations. In this paper, we present a new stereo matching measure that is insensitive to radiometric variations between left and right images. Unlike most stereo matching measures, we use the color formation model explicitly in our framework and propose a new measure, called the Adaptive Normalized Cross-Correlation (ANCC), for a robust and accurate correspondence measure. The advantage of our method is that it is robust to lighting geometry, illuminant color, and camera parameter changes between left and right images, and does not suffer from the fattening effect unlike conventional Normalized Cross-Correlation (NCC). Experimental results show that our method outperforms other state-of-the-art stereo methods under severely different radiometric conditions between stereo images.

Full text

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Robust Stereo Matching Using Adaptive
Normalized Cross-Correlation
Yong Seok Heo, Student Member, IEEE, Kyoung Mu Lee, Member, IEEE, and
Sang Uk Lee, Fellow, IEEE
Abstract—A majority of the existing stereo matching algorithms assume that the corresponding color values are similar to each other.
However, it is not so in practice as image color values are often affected by various radiometric factors such as illumination direction,
illuminant color, and imaging device changes. For this reason, the raw color recorded by a camera should not be relied on completely,
and the assumption of color consistency does not hold good between stereo images in real scenes. Therefore, the performance of
most conventional stereo matching algorithms can be severely degraded under the radiometric variations. In this paper, we present a
new stereo matching measure that is insensitive to radiometric variations between left and right images. Unlike most stereo matching
measures, we use the color formation model explicitly in our framework and propose a new measure, called the Adaptive Normalized
Cross-Correlation (ANCC), for a robust and accurate correspondence measure. The advantage of our method is that it is robust to
lighting geometry, illuminant color, and camera parameter changes between left and right images, and does not suffer from the
fattening effect unlike conventional Normalized Cross-Correlation (NCC). Experimental results show that our method outperforms
other state-of-the-art stereo methods under severely different radiometric conditions between stereo images.
Index Terms—Stereo matching, color, radiometric variation, illumination, camera exposure, gamma correction.
Ç
1
1.1
S
from different point of views or at different times. However,
finding the accurate correspondence is not an easy task;
there exist a number of difficulties, such as occluded
regions, textureless regions, and object boundaries. This
issue has been an important area of research in the past
several decades, and considerable progress has been made
with respect to the problem surrounding stereo matching
algorithms. Efforts toward this end have resulted in
numerous stereo algorithms that perform relatively well
for the images in the Middlebury database [1], [2].
These algorithms are based on a common assumption
that corresponding pixels have similar color values, an
assumption we refer to as color consistency. As a conse-
quence, a majority of these utilize a data cost which
includes simple L1 or L2 difference of intensities or color
values of corresponding pixels. However, it should be
noted that these methods do not hold good for stereo
images which do not have similar corresponding color
values. Nonetheless, a few studies have been performed in
order to solve this problem.
INTRODUCTION
Motivation
TEREOmatching aims to obtain 3D information by finding
the correct correspondence between images captured
In a real scenario, various factors prevent two corre-
sponding pixels from having the same color value. One
major preventing factor is a radiometric change, which
includes lighting geometry, illuminant color, and camera device
changes between stereo images. Different color values are
obtained when the same scene is viewed under a different
lighting geometry; the reason for this is that the intensity at
each point is determined by the angle between the direction
of the incident light and the direction of the surface normal
in a Lambertian model. After fixation of the lighting
geometry, the object when viewed under different illumi-
nant colors also produces different colors because there is a
change in the spectral distribution of the reflected light from
the object. Furthermore, color changes can also be induced
by using a camera device or setting changes such as
exposure variations because there are options to vary the
total amount of photon that is incident to the camera.
Common situations such as these can be a reason for
practical problems in stereo images such as aerial images.
Having said that, one should not completely trust the raw
color recorded by a camera for purposes of matching, and it
remains that the color consistency assumption is no longer
valid for stereo images in real scenes. Under radiometric
variations such as these, the performance of most stereo
matching algorithms can be severely degraded. To prove
this point with an example, we show in Fig. 1 the case in
which a conventional method such as the Sum of Absolute
Difference (SAD) with Graph-cuts (GC) method fails, while
the proposed method is more robust under severe radio-
metric changes between the left and right images.
However, there exists a natural color-constancy process
in the human visual system that can compute colors
regardless of radiometric variations and also estimate the
reflectance of the object under any illumination condition
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 33,NO. 4, APRIL 2011 807
. The authors are with the Department of Electrical Engineering and
Computer Science, Automation and Systems Research Institute, Seoul
National University, 599 Gwanak-ro, Gwanak-gu, Seoul 151-744, Korea.
E-mail: hys@diehard.snu.ac.kr, kyoungmu@snu.ac.kr,
sanguk@ipl.snu.ac.kr.
Manuscript received 31 Dec. 2008; revised 11 Nov. 2009; accepted 27 Apr.
2010; published online 20 July 2010.
Recommended for acceptance by W. Fo ¨rstner.
For information on obtaining reprints of this article, please send e-mail to:
tpami@computer.org, and reference IEEECS Log Number
TPAMI-2008-12-0891.
Digital Object Identifier no. 10.1109/TPAMI.2010.136.
0162-8828/11/$26.00 ? 2011 IEEE Published by the IEEE Computer Society

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[3]. In practice, however, almost all existing stereo matching
algorithms do not take into consideration this color-
constancy process. Therefore, there is a pressing need to
include this process into the matching mechanism.
In this paper, we present a new stereo matching measure
that is robust in handling various radiometric changes,
including local radiometric variations caused by varying
lighting geometry, as well as global radiometric variations
that are brought about by changes in illuminant color and
camera parameter [4]. It should be noted that this paper is
based on the assumption of the Lambertian world. Under
the assumption, the color formation process was explicitly
modeled, unlike other methods. Then, we have extracted
the invariant color information from the color model and
thereby propose, in this paper, a new matching measure,
called Adaptive Normalized Cross-Correlation (ANCC), which
is robust to various radiometric changes.
1.2
Hirschmu ¨ller and Scharstein carried out an evaluation of
various cost functions for stereo matching on radiometri-
cally different images caused by factors such as light
configuration, camera exposure, gamma correction varia-
tions, and noise, etc [5]. Under various radiometric
conditions, they compared Birchfield and Tomasi data cost
(BT) [6], BT with Laplacian of Gaussian (LoG) filtering [7],
BT with Mean filtering, BT with Rank transform [8],
Normalized Cross-Correlation (NCC), and Hierarchical
Mutual Information (HMI) [9] with correlation-based
method, as well as the semiglobal and global method.
Although BT cost is a very popular data cost which is
insensitive to camera sampling, the major drawback of it is
that it cannot handle radiometrically different images as it
employs the use of linearly interpolated function of
intensity values. Although the LoG filter calculates sec-
ond-order derivatives that are insensitive to outliers [7], BT
with LoG filtering itself is insufficient for use in stereo
matching. BT with Rank transform [8] possesses a robust
property to radiometric variations because the Rank trans-
form is based on the principle that the rank between pixel
intensities does not vary after radiometric variations.
Although it is effective in handling global radiometric
changes, it is weak with respect to local radiometric
variations. NCC is a highly popular as well as traditional
measure for matching contrast-varying images [10], [11]. It
measures only the cosine of an angle between matching
Related Works
vectors because normalization renders the matching vectors
to have zero mean and one standard deviation. Therefore,
NCC is suitable only for matching affine-transformed
intensity or color values and it also suffers from the
fattening effect that object boundaries are not reconstructed
correctly similar to the Sum of Absolute Difference (SAD)
and the Sum of Squared Difference (SSD). Mutual Informa-
tion (MI) has been used as a similarity measure in computer
vision [9], [12], [13], [14]. For stereo matching, Kim et al. [14]
proposed a pixelwise data cost in the MAP-MRF framework
based on mutual information. Their work was improved by
Hirschmu ¨ller [9] and extended to handle occlusion and
speed up the computation by using the Hierarchical Mutual
Information (HMI). The advantage of the MI-based cost is
that it enables finding the correspondence between globally
transformed image pairs by utilizing the joint histogram of
intensities between two input images. However, local
radiometric variations due to lighting geometry changes
cannot be handled by MI-based cost. Because of the just-
discussed reasons, Hirschmu ¨ller and Scharstein [5] con-
cluded that all compared costs previously described fail to
achieve success with respect to strong local radiometric
changes caused by the light configuration changes. Besides,
Wang et al. [15] presented a new invariant measure called
light transport constancy (LTC) based on a rank constraint
for non-Lambertian surfaces. Their method required at least
two stereo image pairs with different illumination condi-
tions to be available for making use of rank constraint.
Ogale and Aloimonos [16] presented a contrast-robust
stereo matching algorithm for local matching using multi-
ple frequency channels. Negahdaripour [17] proposed a
general linear brightness constraint for handling radio-
metric variations between images. The principle behind his
model was that the linear transformation of the brightness
between image patches was assumed, and the linear
transformation parameters and the optical flow vectors
were simultaneously estimated. Zhang et al. [18] presented
a unified MRF framework for the estimation of illumination
variation and disparity map simultaneously. Their frame-
work sought to define Illumination Ratio Map (IRM), and
assumed disparity and illumination smoothness.
A major drawback of most of the above-mentioned
methods is that these utilize only raw intensity (or color)
information, which can be sensitive to severe radiometric
variations for stereo matching due to the fact that raw
intensity depends only on the direction of the light and the
surface normal and does not take into consideration the
808IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 33,NO. 4, APRIL 2011
Fig. 1. Comparison of the proposed method with a conventional stereo method (SAD+GC) for the illumination-varying stereo images. (a)-(b) The left
and right Aloe images with varying illumination. (c) Disparity map of SAD+GC for the input stereo image pair in (a) and (b). (d) Disparity map of the
proposed method for the input stereo image pair in (a) and (b).

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color of surface and light and information regarding the
camera parameters. Although the intensity information is
applied widely in global transformations such as exposure
variations, it still remains sensitive to local variations
caused by changes in lighting geometry. Taking all of these
factors into consideration, in this paper we intend to show
that it is more efficient and robust to use invariant color
information rather than the simple raw intensity (or color)
information for handling various local and global radio-
metric variations.
2STEREO ENERGY FORMULATION
In this paper, we define stereo matching as a minimization
problem of the following energy in the MAP-MRF frame-
work [19]:
EðfÞ ¼ EdataðfÞ þ EsmoothðfÞ;
EdataðfÞ ¼
p
X
where NðpÞ is the neighborhood pixels of p and DpðfpÞ is the
data cost, which is a measure of the dissimilarity between
pixel p in the left image and pixel p þ fpin the right image.
Vpqðfp;fqÞ is the smoothness cost that favors the piecewise
smooth objects. Taking these costs together, we can arrive at
the optimal disparities by minimizing the total energy in (1).
From the findings of the study by Meltzer et al. [20], it can
be inferred that the globally optimal disparity map that was
even obtained by the powerful tree-reweighted message
passing (TRW) [21] was not perfect because of the incorrect
modeling of the energy functional. In order to find out
correspondence, the most fundamental part is a matching
cost (data cost) computation. This acted as a motivation for
us to concentrate on the modeling of a more accurate,
radiometric-insensitive data cost for the MAP-MRF frame-
work in real scenarios. In order to model radiometric-
insensitive data cost, the color formation process has to be
taken into consideration in an explicit manner and proper
color-insensitive matching information has been extracted.
X
DpðfpÞ;
X
EsmoothðfÞ ¼
p
q2NðpÞ
Vpqðfp;fqÞ;
ð1Þ
3COLOR-INVARIANT INFORMATION
Color-constancy algorithms [3], [22], [23], [24], [25], [26],
[27] attempt to separate the illumination and the reflectance
components on images similar to the human visual system.
The first attempt made toward this end was the Land’s
retinex algorithm [22] with the objective of modeling and
computing the human color constancy. It is worth mention-
ing here that retinex algorithms can calculate only the
lightness sensations and not the physical reflectance in a
given image and can compensate for nonuniform lighting in
an effective manner [3], [22], [23], [24]. Retinex algorithms,
however, are highly sensitive as well as dependent on color
variations of the adjacent objects in the images [25].
The illuminant in given images can be estimated using
the gamut-mapping algorithm [26] and the color-by-
correlation algorithm [27]. However, the estimation of the
illuminant is not an easy task because the color constancy
problem is ill-posed [28].
The color-invariant approaches [29], [30], [31], [32], [33],
[34], [35], [36], [37], [38], [39] find the function which is
independent of lighting conditions and imaging devices.
There exist two types of invariants, namely, the Lambertian-
based color invariant and the invariant that is designed to
deal with non-Lambertian reflectance effects such as
specular highlights.
With regard to Lambertian object, Geusebroek et al. [32]
derived invariant color features such as edges based on the
Gaussian scale-space framework. Gevers and Smeulders
[33] proposed various color-invariant features for robust
object recognition. However, these invariants suffer from
instabilities under certain conditions. Gevers and Stokman
[34] constructed a color-invariant histogram using variable
kernel density estimation to resolve these instabilities.
Finlayson et al. [35] made a suggestion in their study for
the removal of shadow from color images. Ebner [36]
suggested an algorithm to shift colors of the input image in
the direction of the gray vector according to the gray
world hypothesis.
Various studies also exist in the published literature for
finding invariants for non-Lambertian objects. Berwick and
Lee [37] suggested utilizing a chromaticity space based on
log-ratio of sensor responses for illumination pose and color
and specular invariance. Tan and Ikeuchi [38] devised a
method for the separation of the specular components from
diffuse reflection components using the iterative logarith-
mic differentiation framework. Zickler et al. [39] suggested
a specular-invariant by making use of the dichromatic
reflectance model [40], [41]. Their method is based on the
linear transformation of RGB color space (called SUV color
space) with the assumption that the source-illuminant color
is known [39].
Among these color-invariant approaches, the most
simple and commonly used methods for Lambertian sur-
face are the chromaticity normalization and the gray world
assumption [42]. Chromaticity normalization is primarily
used for the removal of the lighting geometry effects,
whereas the gray world assumption is used for the removal
of the illuminant color effects. However, neither of the two
methods can remove the dependency of both lighting
geometry and illuminant color simultaneously. A compre-
hensive normalization method strives to remove both of
them iteratively [29] and noniteratively [31].
3.1
The following equation describes an image taken by a linear
imaging device [27], [30]:
Z
where hx
krepresents the kth sensor (color channel) response
at a point x in the scene and ? is the wavelength. Eð?Þ
represents the spectral power distribution of the incident
illuminant, Sxð?Þ represents the surface reflectance at a
point x in the scene, and Qkð?Þ stands for the spectral
response of the kth sensor. As this work is based on the
assumption of the Lambertian surface, the radiometric
variations induced by non-Lambertian reflectance effects
Color Image Formation Model
hx
k¼
!
Eð?ÞSxð?ÞQkð?Þd?;
ð2Þ
HEO ET AL.: ROBUST STEREO MATCHING USING ADAPTIVE NORMALIZED CROSS-CORRELATION809

Page 4

such as specular highlights are not considered. For a camera
sensor model, we also assume the diagonal model [43], [44],
[45], from which the spectral response is approximated as
the Dirac delta function. Therefore, Qkð?Þ ¼ ?ð? ? ?kÞ,
which represents a Dirac delta function at wavelength ?k.
Now, (2) simply becomes
hx
k¼ Eð?kÞSxð?kÞ:
ð3Þ
During the image acquisition process, the device responds
in a linear fashion. However, for the compression of the
dynamic range, there is a nonlinear transformation of the
image data before the storage process. This process is called
gamma correction and it results in the raising of the value of
each RGB response to a power function of an exponent ?
value depending on the camera [46], [47], [48]. Taking all of
these factors into consideration, the color image formation
model at pixel p can be represented as follows [31]:
0
~BðpÞ
where each pixel p has its own individual brightness factor
?ðpÞ which depends on the angle between the direction of
the light and the direction of the surface normal at that
point. Changing illumination color while fixing the lighting
geometry would result in changes in the responses in three
color channels by the global scale factor a, b and c,
respectively.
RðpÞ
GðpÞ
BðpÞ
@
1
A!
~RðpÞ
~GðpÞ
0
@
1
A¼
?ðpÞaRðpÞ?
?ðpÞbGðpÞ?
?ðpÞcBðpÞ?
0
@
1
A;
ð4Þ
3.2
As mentioned previously, chromaticity normalization [42]
is commonly employed for the elimination of the effect of
lighting geometry that depends only on the direction of
surface normal and the direction of light in the Lambertian
model. At pixel p, if we divide each of~RðpÞ,~GðpÞ, and~BðpÞ
by the averages of them, then we can obtain the ?ð?Þ
independent color representation as follows:
0
B
On the other hand, we can acquire an illuminant color-
independent representation by the gray world assumption
[42]. By dividing each of~RðpÞ,~GðpÞ, and~BðpÞ by its channel
mean value, we can obtain the a, b, and c independent
expressions as follows:
Color Image Normalization
3?ðpÞaRðpÞ?
3?ðpÞbGðpÞ?
3?ðpÞcBðpÞ?
?ðpÞðaRðpÞ?þbGðpÞ?þcBðpÞ?Þ
?ðpÞðaRðpÞ?þbGðpÞ?þcBðpÞ?Þ
?ðpÞðaRðpÞ?þbGðpÞ?þcBðpÞ?Þ
B
@
1
C
C
A¼
3aRðpÞ?
3bGðpÞ?
3cBðpÞ?
ðaRðpÞ?þbGðpÞ?þcBðpÞ?Þ
ðaRðpÞ?þbGðpÞ?þcBðpÞ?Þ
ðaRðpÞ?þbGðpÞ?þcBðpÞ?Þ
0
B
B
@
1
C
C
A:
ð5Þ
?ðpÞaRðpÞ?
?ðpÞbGðpÞ?
?ðpÞcBðpÞ?
0
@
B
1
A !
C
?ðpÞaRðpÞ?
~ Rmean
?ðpÞbGðpÞ?
~Gmean
?ðpÞcBðpÞ?
~ Bmean
0
B
B
B
@
1
C
P
C
C
A¼
p2I~GðpÞ
N
N?ðpÞRðpÞ?
P
P
P
;
p2I?ðpÞRðpÞ?
N?ðpÞGðpÞ?
p2I?ðpÞGðpÞ?
N?ðpÞBðpÞ?
p2I?ðpÞBðpÞ?
0
B
@
B
B
B
B
1
C
A
C
C
C
C
;
~Rmean¼
P
P
p2I~RðpÞ
N
p2I~BðpÞ
N
;~Gmean¼
~Bmean¼
;
ð6Þ
where N represents the number of pixels in the image I.
As mentioned earlier, it should be noted that neither
chromaticity normalization nor gray world normalization
can remove both the dependency of lighting geometry and
illuminant color simultaneously. Finlayson et al. proposed
a comprehensive normalization method that combines the
two normalization methods in one framework iteratively
[29] and noniteratively [31]. However, because their
method requires global image information such as channel
mean value, it is not appropriate to apply the method in a
naive manner to the stereo matching problem. For instance,
in the gray world normalization process described in (6),
the left and right~Rmean values are mostly not the same
because of the view changes characteristic of stereo images.
The inference that can be drawn from this is that the true
corresponding pixel values are still not the same between
stereo images even after comprehensive normalization.
Thus, applying the stereo matching algorithm to the
prenormalized images by this method does not produce
satisfactory results as shown in our experiments.
4STEREO MATCHING USING ANCC
Let us make an assumption that ILðpÞ and IRðp þ fpÞ are
corresponding pixel values, where I 2 fR;G;Bg and ILðpÞ
is the value in the left image at pixel p and IRðp þ fpÞ is the
value in the right image at pixel p þ fp in each color
channel, respectively. NCC [10] is a well-known similarity
measure between two pixels with neighbors that is defined
by the equation
NCCðfpÞ
¼
X
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
tL2WLðpÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
tL2WLðpÞ;
tR2WRðpþfpÞ
X
s
½ILðtLÞ ??ILðpÞ? ? ½IRðtRÞ ??IRðp þ fpÞ?
0
@
0
B
1
A
C
,
jILðtLÞ ??ILðpÞj2
s
@
?
X
tR2WRðpþfpÞ
jIRðtRÞ ??IRðp þ fpÞj2
1
A;
ð7Þ
where?ILðpÞ and?IRðp þ fpÞ are the mean values of pixels in
the window WLðpÞ centered at pixel p in the left image and
the mean values of pixels in the window WRðp þ fpÞ
centered at pixel p þ fpin the right image, respectively. It
should be mentioned here that this NCC is insensitive to the
following affine transformations of image intensity values
of two images [10]:
IL! ?LILþ ?L;IR! ?RIRþ ?R:
ð8Þ
Simply applying this NCC directly in stereo matching of
general image pairs would result in two critically important
problems: According to (4), there could be a complicated
nonlinear relationship between two corresponding pixels
between stereo images, thereby causing NCC to cease
working. Thus, applying NCC to raw stereo images in a
naive fashion does not work well because the various
radiometric changes caused by ?, a, b, c, and ? are not taken
into consideration. The second serious problem is that the
810IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,VOL. 33, NO. 4,APRIL 2011

Page 5

supporting windows in the left and right images do not
appear exactly because of the view changes. As a con-
sequence, NCC usually produces a fattening effect near the
object boundaries similar to conventional window-correla-
tion-based matching measures such as SSD and SAD.
Therefore, in this work, we seek to find remedies to these
two problems. First and foremost, the nonlinear relation-
ship that exists between corresponding pixel color values
because of various unknown radiometric changes is
transformed into a linear one by employing log-chromaticity
color space. Second, in order to reduce the fattening effect
and increase the accuracy between matching windows, we
define a modified NCC measure called the Adaptive
Normalized Cross-Correlation which utilizes adaptive weight-
ing scheme. This approach is described in detail in the
following sections.
4.1
Weprovideanoverviewofourapproachinthissection.Fig.2
depicts the overview. The principle behind our novel
approach is that the color formation model is explicitly
modeled and incorporated into a new stereo correlation
measure. To achieve this, we considered the color formation
process in an explicit manner instead of using the raw color
valueforhandlingthevariousradiometricchangesthatoccur
between stereo images. From this color model, we propose a
new data cost that is not only insensitive to the various
radiometric changes but is also able to reduce the problems
faced with window-based stereo methods such as the
fattening effect simultaneously. In order to render the color
values changed by various radiometric factors suitable for
stereo matching, we carried out a proper transformation of
the nonlinear relationship to the affine-transformed relation-
ship between the corresponding pixels that can be solved
effectively by the NCC framework. In addition, to ensure
accurate and coherent matching, we subtracted bilateral
filtered value instead of the simple window mean value for
coherent normalization around window pixels. Taking into
consideration these bilateral filtered weights again, we
Overview of Our Approach
defined a new correlation measure called ANCC, using
whichwedefinedanewdatacostforglobalenergymodeling.
4.2
If the reflectance model is Lambertian and the spectral
response function is the Dirac delta function, then the image
is formed on the basis of (4). If the lighting geometries,
illuminant colors, and camera parameters of the left and
right images are different, they can be represented by the
following equations, respectively:
0
@
RRðp þ fpÞ
GRðp þ fpÞ
BRðp þ fpÞ
Without loss of generality, only the red channel value is
considered. According to (9), the corresponding left and
right color values have a nonlinear relationship because of
different gamma values, ?Land ?R, respectively. In order to
transform this nonlinear relationship into a linear one, the
logarithms of both images have to be taken. Then, each
color value can be represented by
Log-Chromaticity Normalization
RLðpÞ
GLðpÞ
BLðpÞ
B
0
@
1
A !
C
?LðpÞaLR?L
?LðpÞbLG?L
?LðpÞcLB?L
?Rðp þ fpÞaRR?R
?Rðp þ fpÞbRG?R
?Rðp þ fpÞcRB?R
LðpÞ
LðpÞ
LðpÞ
0
@
C
B
1
A !
1
A;
C
B
Rðp þ fpÞ
Rðp þ fpÞ
Rðp þ fpÞ
0
@
B
1
A:
C
ð9Þ
R0
LðpÞ ¼ log?LðpÞ þ logaLþ ?LlogRLðpÞ;
Rðp þ fpÞ ¼ log?Rðp þ fpÞ þ logaRþ ?RlogRRðp þ fpÞ;
R0
ð10Þ
where ?ð?Þ term is dependent on each pixel position relative
to the direction of the lighting and surface normal. This
log?ð?Þ term can be eliminated by simply subtracting the
average of the transformed color values in R;G;B channels
(chromaticity normalization), which is defined by the
following equation:
HEO ET AL.: ROBUST STEREO MATCHING USING ADAPTIVE NORMALIZED CROSS-CORRELATION811
Fig. 2. Overview of the proposed algorithm.