Robust Similarity Measures for Stereo Correspondence
S Srinivas Kumar, Non-member
B N Chatterji, Fellow
Normalized cross correlation (NCC), Sum of squared differences (SSD), and Sum of absolute differences (SAD) are the
linear correlation measures generally used for stereo correspondence. These measures fail to establish the correspondence
under non- ideal conditions such as specular reflection, occlusion etc. In this paper, the similarity measures for stereo
correspondence based on rank correlation techniques are considered. The performance of various rank correlation
techniques for stereo matching is compared in terms of discriminatory power, sensitiveness and minimum false matches.
Experiments conducted on real stereo images suggest the superiority of Kendall’s rank correlation over the other measures
under non-ideal conditions.
Keywords: Stereo matching; Similarity measure; Rank correlation; Normalized cross correlation (NCC)
INTRODUCTION
Stereovision is an attractive passive range sensing technique for
depth perception, a central problem in computer vision. The
process of analyzing real images acquired from two different
views to extract the depth information is called Stereovision.
Image acquisition, camera modelling, feature acquisition, image
matching, depth determination and interpolation are the various steps involved in the stereopsis . The crux of the problem
lies in establishing the correspondence between the stereo images. The correspondence needs to be established among the
homologous features that are the projections of the same physical identity in each view. The search space in finding the
correspondence of pixels in the left and right images is reduced
by acquiring the stereo images with an appropriate image
geometry to satisfy the epipolar constraint . Hence, the scene
point is projected in the left and right images in the same scan
line.
1
1
Stereo correspondence techniques are broadly classified into
three techniques: Feature-based − , Pixel-based − and Areabased − . Feature-based techniques use symbolic features derived from the intensity images such as edge pixels, edge segments, corner pixels etc. The disadvantage of this technique is
that it requires complicated pre-processing to acquire features
and post-processing such as interpolation to obtain the full
resolution disparity. The feature detection may not be reliable
due to noise, occlusion etc. Hence, feature-based algorithms for
stereo correspondence are less preferred.
2
8
4
5
7
11
The pixel-based techniques use dense low level features and
intensity values. These techniques are very sensitive to noise
and hence require pre-processing for noise reduction. Area
based, ie, window-based techniques use intensities of pixels
within a window. Since the intensity value at every pixel is used,
a dense disparity estimate can be obtained. Area based algorithms available in the current literature, are based on the
assumption, (either stated explicitly or implicitly), that the
disparities within a neighbourhood of a pixel in the stereo
S S Kumar is with JNTU College of Engineering, Kakinada 533 003 and
Prof B N Chatterji is with Indian Institute of Technology, Kharagpur
721 302.
This paper was received on May 13, 2002. Written discussion on the paper will be
received till January 31, 2005.
44
images are constant. Hence, the intensities within a considered
window around a pixel can be used to find the corresponding
pixel in the other image. The simplest measure is correlation.
In a correlation-based framework, correspondence for a pixel
in the reference image is obtained by searching in a pre-defined
region of the second image. The search space is reduced due to
epi-polar constraint. An algorithm to illustrate this technique
to establish a correspondence between pixel in the left image
and the pixel in the right image is as follows:
Step 1: The matching is restricted to
(x ′, y ′)| y ′ ≤ y ± dmax
x−ξ≤x′≤x+ξ
where ξ =3 are scan lines, dmax, the maximum disparity
estimated; (x, y) are the co- ordinates of the pixel in left image;
(x ′, y ′ ), co-ordinates of pixel in right image.
Step 2: For each pixel at (x, y) in the left image, the corresponding pixel at (x ′, y ′ ) in the right image is to be determined. A
stationary window of size N × N around (x, y) and a moving
window of size N × N around the estimated corresponding
pixel (x ′, y ′) are to be defined. Suppose, [A] and [B] are
matrices of size N × N with intensity values around (x, y) and
(x ′, y ′) respectively amd Aij and Bij represent the elements in
A and B.
Step 3: Compute the value of similarity measure such as NCC
or SSD or SAD etc., between [A] and [B].
Steps 2 and 3 are repeated when the window in the right image
moves within the estimated disparity. The matching pixel
(x ′, y ′ ) in the right image corresponding to (x, y) in left image
is the center pixel of the moving window for which the image
agreement measureisoptimum.
Normalized cross correlation (NCC), sum of squared differences (SSD) and sum of absolute differences (SAD) are the linear
correlation measures generally used in area-based techniques for
stereo correspondence.
Suppose, X and Y represent the intensities in two windows, ie,
there exists N tuples (X , Y ) ... (Xn, Yn), depending on the
1
1
IE(I) Journal-CP
size of the window used then the normalized cross correlation
(NCC) is given by
n
∑
i=
R =
__
__
(Xi − X) (Yi − Y)
1
n
n
__
__ 
 (X − X )
(Y
−
∑ i Y ) 
∑ i

i =
i=

__
__ 
where X and Y represent the sample means of the corresponding windows. The absolute value of normalized cross correlation lies in between -1 and 1, and a value of 1 indicates perfect
matching of windows.
2
1
Figure 1 Specular reflection causes changes in intensity values in stereo
images
2
1
Other distance metrics or similarity measures used to determine
the similarity between two images are the sum of squared
differences (SSD) or sum of absolute differences (SAD). Suppose
X and Y represent the intensities in two windows, ie, there exists
N tuples (X Y1) ... (Xn, Yn), depending on the size of the
window used. The quantity
Figure 2 Occlusion causes the pixels in the stereo images to differ
1,
n
∑
SSD =
i=
(Xi − Yi)
2
1
measures the squared euclidean distance between X and Y. A
value close to zero indicates a strong correlation.
The measure, sum of absolute difference is as follows:
n
SAD =
∑
i=
|Xi − Yi|
1
The value of this measure decreases as the similarity of intensity
values in the windows increase.
The above measures fail to establish the correspondence in the
presence of occlusion, specular reflection etc., in the stereo
images. Noise in the images also tends to corrupt the image
agreement measure. The different phenomena that affect the
window-based matching algorithms with the use of linear correlation measures are disscussed in the next section.
LIMITATIONS IN USING LINEAR CORRELATION
MEASURES FOR STEREO MATCHING
Window-based, ie, area based algorithms use the intensity values
of pixels in the stereo images. Linear correlation measures
cannot be used in stereo images due to phenomenas such as
specular reflection, depth discontinuity, occlusion and projective distortion in the stereo images.
Specular Reflection
The intensity value of pixels in the corresponding windows of
stereo images may differ due to different sensor outputs I of two
cameras mainly due to varying camera parameters, illumination
etc, The sensor output I is related to image irradiance E as:
I = gE
1
⁄Y
+ m
where ’g’ is camera gain; ’m’, bias factor and; γ accounts for
the contrast. Camera gain and bias factor account for linear
Vol 85, November 2004
Figure 3 Depth discontinuity showing different surface locations in left
and right images
variation in sensor output of the two cameras. The intenstiy
values of stereo images are linearly related even if the gain and
bias factor of the two cameras differ. Linear correlation measures are applicable to such stereo images. However, different
image contrasts lead to non-linear relation between the intensity values of stereo images and linear correlation measures fail
to establish correspondence in such stereo images. Hence,
specular reflection is a limitation to apply linear correlation
measures for stereo correspondence. The effect of specular
reflection is illustrated in Figure 1.
Occlusion
Due to different camera positions, the scene projected in the
stereo image pair may not be same. Thus, all the pixels in the
left window are not visible in the corresponding right window
and vice-versa. Hence, The pixels in the occluded region are to
be considered as insignificant in establishing the correspondence. Linear correlation measures are based on absolute values
of intensities of all pixels in the left and right windows. These
measures give equal importance to all pixel intensities in the
windows. Hence, linear correlation measures fail to establish
the correspondence, if the corresponding windows are with
occlusion zones. This phenomenon is illustrated in Figure 2.
Depth Discontinuity
If the center pixel of the window is located on a depth discontinuity, the windows represent different surface locations. The
presence of depth discontinutities also causes occlusion due to
which scene points are visible in only one of the two images.
This phenomenon is illustrated in Figure 3.
Projective Distortion
Projective distortion results in windows being different which
can be observed from the changing texture frequency. This
phenomenon also introduces the outliers in one of the windows
due to occlusion. This phenomenon can be explained as
45
window of size 3 × 3 be assumed as
R
Figure 4 Projective distortion caused by slanting surface
S
10
30
70
10
30
70
20
50
80
20
50
80
40
60
100
40
60
100
The ranks of these intensity values are given as
R
Figure 5 Projective distortion
follows: The projective distortion caused by slanting surface is
depicted in Figure 4. The region of terrain BC is projected in
camera ’Q’ as b′ − c′ but not in ’P’. The intensity values in the
region b′ − c′ in image Q are considered as outliers. Effects of
projective distortion changing the texture frequency is illustrated in Figure 5.
Image Noise
Stereo matching in the presence of noise is computationally a
challenging task. Linear correlation measures are successful,
only if the stereo images are affected by Gaussian noise. However, often this assumption is invalid in real problems. Hence,
the similarity metrics that are successful for real noise distributions are to be considered for stereo matching.
In summary, similarity measures used for stereo matching
should be (i) insensitve to outliers due to occlusion to a high
degree (ii) independent of camera gain, bias factor and contrast
(iii) report small number of false matchings in the presence of
projective distortion and depth discontinuity (iv) provide
matching successfully in the presence of real noise distributions.
REVIEW OF SIMILARITY MEASURES BASED ON
RANK CORRELATION TECHNIQUES
Linear correlation measures are invariant under positive linear
transformations of intensity values in the corresponding windows in stereo images. However, it is not invariant under all
transformations of intensities in the corresponding left and
right windows for which the order of magnitude is preserved.
In order to be distribution-free, inferences must usually be done
by relative magnitudes as opposed to absolute magnitudes of
the intensities. Hence, applying rank correlation as similarity
measures solves stereo correspondence problem. Well-known
similarity measures known as Spearman’s and Kendall’s rank
correlation techniques are considered for stereo correspondence problem and compared with ordinal measures proposed
by D N Bhat et al . The performance of these measures is
compared in terms of sensitiveness, discriminatory power, and
minimum false matches.
8
3
Motivation
Suppose the intensity values in the reference window and search
46
S
1
3
7
1
3
7
2
5
8
2
5
8
4
6
9
4
6
9
Suppose the intensity value ’100’ in the search window is
affected due to different phenomena and changed to ’255’, even
then the rank matrices do not change. This shows the advantages of rank correlation techniques over linear correlation
techniques.
If it is assumed that the left camera and the right camera gain
are ’g’ and bias factor of both cameras is zero.
Then, the intensity values in the left window are given by
I = ge
1
⁄ γ2
1
⁄ γ2
and that in the right window are given by
I = g e . The intensity values in each window are non-linearly related to the contrast ’γ’. Hence, linear correlation techniques fail to establish correspondence due to specular
reflection. However, the rank matrices of these intensity values
are not affected due to different contrast. This is due to the
reason that the ranks are dependent on relative magnitudes of
intensity values. Hence, the rank correlation techniques are
successful in the presence of specular reflection.
1
2
Spearman’s Rank Correlation Coefficient (ρ)
It is a measure of association, which requires that both the
variables be measured in an ordinal scale, so that the sample
values may be ranked in two ordered series.
A random sample of n pairs (X , Y ), (X , Y ) ... (Xn, Yn) is
taken and the simple correlation coefficient is defined as:
1
n
∑
R =
i=
1
2
2
__
__
(Xi − X) (Yi − Y)
1
n
__
 (X − X)
∑ i
i = 1

2
n
__ 
(
Y
−
∑ i Y) 

i=1

2
The X observations and Y observations are ranked separately
using the same ranking scheme. The data then consists of n- sets
of paired ranks from which ρ can be calculated. The resulting
coefficient is known as the Spearman’s Coefficient of rank
correlation. It measures the degree of correspondence between
rankings instead of the actual sample values. A simple form of
IE(I) Journal-CP
the Spearman coefficient of rank correlation as
n
6
ρ = 1 −
∑
i=
D2i
1
n (n − 1)
2
where Di = Ri − Si and Ri = rank (Xi), Si = rank (Yi)
A possibility that may commonly arise is the occurrence of tied
observations. Therefore, it is necessary to correct the sum of
squares taking ties into account. The correction factor T is given
as
 1

− 1
aij = 

 0

if these pairs are concordant
if these pairs are discordant
If these pairs are neither concordant nor dis−
cordant because of a tie in either component
The Kendall’s Rank correlation coefficient is defined as
n
n
Aij
n (n − 1)
∑ ∑
τ =
i=
1
j=
1
When two or more observations on either X or Y variables are
tied, the effect is to change the denominator of the formula for
τ. In the case of ties, τ becomes
(t − t)
12
3
T =
The values assumed by Aij are
where ’t’ is the number of observations tied at a given rank.
When the sum of squares is corrected for ties, ρ is given as
∑
ρ =
∑y −∑
2√

∑ x ∑ y
x +
2
2
2
2
∑
n − n
− Tx
12
∑
y =
n − n
− Ty
12
1
j=
Aij
1


√
[n (n − 1) − ∑ u (u − 1)] [n (n − 1) − ∑ v (v − 1)]

where u, number of ties in X data set; v, number of ties in Y
data set.
Ordinal Measures (κ)
3
2
i=

2
3
2
n
∑ ∑
τ=
d
where,
x =
n
Tx, is correction factor for data set X; Ty, is the correction factor
for data set Y.
Kendall’s Rank Correlation Coefficient ( τ)
Ordinal measure proposed by D N Bhat et al, is another
measure of association based on relative ordering of intensity
values (ranks) in windows rather than the intensity values of
the pixels directly. Ordinal measures are defined by using the
distance between two rank permutations. The ordinal measure
is invariant to pixel photometric variations or to shot noise
since the rank of the pixel remains the same within certain
limits in the intensity values.
Let it be assumed that I is a window in the left image and I , a
window in the right image. For a set of intensity values
(I i, I i), let πi1 be the rank of I i among the I data and πi be the
rank of I i among the I data. A composition permutation s can
be defined as follows:
1
The Kendall’s rank correlation coefficient is suitable as a measure of correlation with the same sort of data for which ρ is
useful, ie, both the variables should be measured on an ordinal
scale.
A random sample of n pairs (X , Y ), (X , Y ), ..., (Xn, Yn) is
taken and are ranked either in the ascending or descending order
(R , S ), (R , S ), ..., (Rn, Sn), where Ri = rank (Xi) and
Si = rank (Yi). The order of the ranks is rearranged such that
the ranks Ri appear in the natural order, ie, (1, 2, 3, ..., n). Now,
the numbers of ranks of Si that are in the correct natural order
with respect to each other are to be determined.
1
1
1
2
1
2
The indicator variable Aij is defined as
Aij = sgn (Xj − Xi) sgn (Yj − Yi)
− 1 if u < 0

where sgn (u) =  0 if u = 0
 1 if u > 0

Vol 85, November 2004
2
2
1
2
2
1
2
1
2
2
si = πk, k = (π − )i
1
2
1
where π − 1 denotes the inverse permutation of π . The inverse
permutation is defined as follows: If πi = j, then (π − ) j = i.
Informally, s is the ranking of I with respect to that of I . Under
perfect positive correlation, s should be identically equal to the
identity permutation given by u = (1, 2, ..., n).
1
1
1
1
2
1
1
The deviation dmi for i = 1, 2, ..., n is defined as:
i
dmi
= i −
∑
j=
i
=
∑
j=
J (s j ≤ i)
1
J (s j > i)
1
47
where J(B), is an indicator function of event B ie, J(B) is 1 when
B is true and 0 otherwise. The vector of dmi is termed as the
distance vector. Each component of the distance vector indicates the number of predecessing elements in s that are out of
position. If I and I are perfectly correlated, then
dm = (0, 0, ..., 0). The maximum value that any component of
the distance vector can take is n ⁄ 2 , which must occur in the
case of perfect negative correlation.
1
2

A measure of correlation κ (I1, I2) is defined as:
κ (I1, I2) = 1 −
2 maxni = dim
n
| |
2
1
If I and I are perfectly correlated then κ = 1. It falls to − 1
when (I I ) are perfectly negatively correlated.
1
2
1,
2
CHARACTERISTICS OF PROPOSED MEASURES
There are different issues to be considered for choosing an
appropriate similarity measure. Three characteristics of a similarity measure, ie, the ability of the measure to provide minimum number of false matches, sensitiveness and discriminatory
power are considered to compare the performance of the rank
correlation techniques.
The robustness of similarity measure determines the amount of
data inconsistency that can be withstood by the measure at the
corresponding windows before mismatches begin to occur. The
performance of a similarity measure is generally degraded in the
presence of occlusion, specular reflection, depth discontinuities
etc. in the stereo images. A reliable similarity measure is one
that provides a minimum number or no false matches even in
the presence of such discrepancies in the images.
The similarity measure value is optimum for a pair of perfectly
matched windows in the stereo images. However, in the presence of the phenomena such as occlusion, specular reflection,
depth discontinuities, projective distortion and noise the intensity data in the corresponding windows may be corrupted.
Hence, only part of the data is valid for correlation and the rest
can be regarded as outliers. The outliers tend to corrupt the
absolute value of the similarity measure of two corresponding
windows. Hence, a mismatch may result. The similarity measure should be either insensitive to these outliers or should be
detected and discarded. In this work, the former approach is
considered. The performance of the similarity measure depends
on its sensitiveness to the insignificant data in the corresponding
windows. The effect of such distortions in the images on the
similarity measure is measured in terms of the sensitiveness.
The value of the measure should not be much affected by the
presence of such insignificant data in the corresponding windows of the stereo images. The similarity measure should
capture the general relationship between the data without being
affected by unusual data.
The discriminatory power is concerned with the ability of the
measure to reject two noncorresponding windows. The factors
that affect the discriminatory power are the window size and
sensitiveness of similarity measure. If the sensitiveness of the
similarity measure is less, it may match two non-corresponding
48
windows. However, increasing the size of the window can solve
this problem. A window size of 3 × 3 involves only 9 intensity
values. Hence, the discriminatory power of the measure becomes low and mismatches result with high probability. An
increase in the window size increases the consistency of the
measure and hence the discriminatory power of the measure
increases. However, a continual increase in the window size
results in the inclusion of outliers in the window, due to
occluded regions. Hence, the performance of the measure decreases with continual increase in size of the window. Hence,
appropriate size of the window should be chosen for stereo
correspondence.
EXPERIMENTAL RESULTS
The proposed measures, ie, Spearman’s (ρ), Kendall’s (τ) and
Ordinal measure (κ) are compared, considering various factors
such as the ability of the measure to provide exact matches,
sensitiveness to outliers and discriminatory power.
The algorithms are implemented using MATLAB version 5.2.
Stereo images satisfying epipolar constraint such as Pentagon,
Shr_rub and Corridor are considered and the ability of the
measure to provide an exact match is compared. The left image
in the stereo image pair is taken as the reference image and the
right image is generated, by adding salt and pepper noise of
density 0.5 to the left image. This pair of images is considered
as a stereo pair to compare the performance of the measures in
terms of false matches. The estimated disparity is considered as
± 20 pixels. The image pairs are shown in Figures 6, 7 and 8.
2
Figure 6 Pentagon image pair (a) pentagon left image without noise
(b) Right image generated by adding salt & pepper noise
(Density 0.5) to the left image
Figure 7 Shr_rub image pair (a) shr_rub left image with out noise
(b) Right image generated by adding salt & pepper noise
(Density 0.5) to the left image
IE(I) Journal-CP
Let it be assumed that the intensity values in the reference
window and the search window of size 3x3 are as follows:
10
20
40
Figure 8 Corridor image pair (a) corridor left image with out noise
(b) Right image generated by adding salt & pepper noise
(Density 0.5) to the left image
Table 1 Percentage of false matches in the presence of salt and paper
noise with density 0.5
Stereo
Images
Pentagon
Shr_rub
Corridor
Measure
7×7
9×9
11 × 11
NCC
45
28
10
Spearman, ρ
48
35
18
Kendall’s, τ
24
10
0
Ordinal Measure, κ
56
41
21
NCC
83
76
69
Spearman, ρ
78
63
59
Kendall’s, τ
61
33
20
Ordinal Measure, κ
86
70
66
NCC
90
79
52
Spearman, ρ
84
78
47
Kendall’s, τ
69
51
22
Ordinal Measure, κ
83
80
68
A sample of 1000 points is considered and the results of matching are compared in terms of the minimum number of false
matches for different window sizes. The performance of ordinal
measure is compared with that of Spearman’s and Kendall ’s
rank correlation coefficients. However, it is observed that the
number of false matches for Kendall’s (τ) is very less compared
to the other measures in the presence of salt and pepper noise.
The results are shown in Table.1. It is noticed that the number
of false matches reduces with increasing window size as the
discriminatory power of the measure increases. A continual
increase in the window size results in the inclusion of outliers
due to occluded regions resulting in a degraded performance of
the measure. Hence, an optimal size of the window plays a vital
role in locating an exact match. In the present case, for the
images considered, the optimal size of the window is taken to
be 11 × 11. It is noted that for images with Gaussian noise, the
normalized correlation coefficient (NCC) continues to be better. The symmetry of the measures is verified by interchanging
the left and right images. It is observed that the measured value
as well as the matching pixel remains unchanged with such
interchange of the images and thus satisfies left-right consistency .
1
Vol 85, November 2004
R
30
50
60
70
80
100
10
20
40
S
30
50
60
The ranks of these intensity values are given as
R
S
1
3
7
1
3
2
5
8
2
5
4
6
9
4
6
70
80
100
7
8
9
The values of NCC, ρ, κ, τ are ‘1’. If the value of 100 in the
search window ‘S’ is changed to 255, the values of NCC, ρ, κ,
τ are 0.8367,1,1,1 respectively. If the value of 100 in the search
window ’S’ is changed to ‘0’, the values of NCC, ρ, κ, τ are
0.311, 0.4, 0.5, 0.5556 respectively. It is noted that the value of
‘τ’ is less sensitive, even if the intensity value in search window
is corrupted. Hence, Kendall’s rank correlation is preferred for
matching two corresponding windows, even if the intensity
values are corrupted due to different phenomena.
Stereo correspondence results for Shr_rub (Figure 9), and Corridor (Figure 10) stereo image pairs are presented in Tables 2-3
obtained by linear correlation measure like NCC and rank
correlation measures such as Spearman’s( ρ), Ordinal measure
(κ) and Kendall’s (τ) using window size as 11 × 11. It is noticed
Figure 9 Shr_rub stereo image pair (a) Shr_rub left image (b) Shr_rub
right image
Figure 10 Corridor stereo image pair (a) Corridor left image
(b) Corridor right image
49
Table 2 Stereo correspondence results for Shr_rub stereo image pair using NCC, Spearman’s (ρ), ordinal measure (κ) and Kendell’s (τ) using
window size as 11 × 11
Left Image Co-ordinates
XL
YL
Right Image
Co-ordinates using NCC
XR
YR
Right Image Co-ordinates
using Spearman (ρ)
XR
YR
Right Image Co-ordinates
using Ordinal Measures (κ)
Right Image Co-ordinates
using Kendall (τ)
XR
YR
XR
YR
153
89
153
91*
153
73*
153
72, 73, 74, 91*
153
73*
211
200
211
194*
211
194*
211
192, 194, 195*
211
192
157
185
157
181
157
181
157
180, 181*
157
181
159
147
159
142*
159
142*
159
145, 146*
159
143
178
200
178
187*
178
189
178
187, 188, 189*
178
189
195
100
195
95*
195
104
195
84, 85, 104*
195
104*
203
147
203
166
203
166
203
166, 167*
203
166
210
210
210
204
210
204
210
201, 204, 205*
210
204
140
112
140
115*
140
112
140
98, 111, 128*
140
112
100
83
100
85
100
84*
100
85, 86*
100
85
185
110
185
130
185
127*
185
126, 130*
185
130
210
95
210
100
210
100
210
99, 100*
210
100
183
103
183
122
183
122
183
122, 123*
183
122
145
113
145
116*
145
109*
145
108, 125*
145
132*
134
159
134
156
134
156
134
153, 154, 155, 156*
134
156
* Indicate false matching pixels — do not satisfy the left-right consistency
Table 3 Stereo correspondence results for Corridor stereo image pair using NCC, Spearman’s (ρ), ordinal measure (κ) and Kendell’s (τ)
using window size as 11 × 11
Left Image Coordinates
Right Image
Co-ordinates using NCC
Right Image Co-ordinates
using Spearman (ρ)
Right Image Co-ordinates
using Ordinal Measures (κ)
Right Image Co-ordinates
using Kendall (τ)
XL
YL
XR
YR
XR
YR
XR
YR
XR
YR
10
109
10
102
10
102
10
101, 102*
10
102
35
159
35
150
35
152*
35
143, 144, 148-153,
157, 158, 164*
35
150
41
75
41
71
41
71
41
68, 71, 72, 74, 75*
41
76*
241
54
241
59*
241
50
124
53 - 60*
124
58*
29
80
29
76*
29
77
29
74, 76, 77, 78*
29
77
239
145
239
136
239
135*
239
134, 135, 136, 137*
239
136
37
140
37
135*
37
136
37
133 - 135*
37
136
45
91
45
89
45
89
45
74, 75, 79*
45
109*
181
147
181
142*
181
142*
181
142, 143*
181
143
25
102
25
96
25
96
25
94, 97*
25
96
220
140
220
131
220
130*
220
129, 130*
220
131
39
85
39
82*
39
83
39
80, 81, 85-87*
39
83
38
187
38
181
38
181
38
172, 176, 179*
38
181
65
125
65
112
65
109*
65
111, 112, 125, 126*
65
126*
29
161
29
151
29
151
29
148, 150, 151*
29
151
* Indicate false matching pixels — do not satisfy the left-right consistency
that the false matching pixels are less for Kendall’s (τ) rank
correlation measure. Ordinal measure (κ) is providing multiple
matching pixels in the search image (right image) for the each
pixel in the reference (left image) image. It is due to the lack of
discriminatory power for ordinal measure (κ). Hence, Kendall’s
(τ) is preferred for stereo correspondence due to its ability to
provide minimum false matching pixels, less sensitive to insignificant data, having good discriminatory power.
50
CONCLUSIONS
Window based algorithms using similarity measures based on
ranking of intensity values are proposed for stereo correspondence in this work. The Ordinal measures, based on ordinal
metrics, are found to be less efficient than Kendall’s rank
correlation measure in terms of minimum number of false
matches, discriminatory power and sensitiveness. The size of
the window also plays an important role in establishing correct
IE(I) Journal-CP
matches and in discriminating the other false matching pixels.
The sensitiveness to insignificant data of the ordinal measures
(κ) is observed to be higher than the other measures. Kendall’s
rank correlation coefficient exhibits a comparatively better
performance than the other measures in terms of minimum
false matches and discriminatory power for stereo correspondence. However, the computation time for stereo matching
using Kendall’s rank correlation is higher. Hence, fast algorithm to compute Kendall’ rank correlation measure is to be
developed.
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Vol 85, November 2004
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