Color image segmentation is a primitive operation in many image processing and computer vision applications. Accordingly, there are numerous segmentation approaches in the literature, something which can be misleading for a researcher who is looking for a practical algorithm. While many researchers are still using the tools which belong to the old color space paradigm, there are evidences in the research established in eighties that proper descriptor of color vectors should act locally in the color domain. In this paper, we use these results to propose a new color image segmentation method. The proposed method searches for the principal colors, defined as the intersections of the cylindrical representations of homogeneous blocks of the given image. As such, rather than using the noisy individual pixels, which may contain many outliers, the proposed method uses the linear representation of homogeneous blocks of the image. The paper includes comprehensive mathematical discussion of the proposed method and experimental results to show the efficiency of the proposed algorithm.
In 1988, Klinker, Shafer, and Kanade presented
a novel approach to measure the highlights in color
images [1]. In that work, they developed a proper
model for the reflected light from an arbitrary point of a
dielectric object. In 1990, they applied their approach to color
image understanding [2]. However, more than a decade
passed since the idea was successfully incorporated into a practical
algorithm. In 2003, without paying too much attention to the
theoretical aspects, Cheng and Hsia used the
principal component analysis (PCA) for color image
processing [3]. Then, in 2004, Nikolaev and
Nikolayev started the work again from the theory and proved
that the PCA is a proper tool for color image
processing [4]. The next necessary step had been
introduced in 1991, when Turk and Pentland proposed
their eigenface method [5], in a completely different
context. There, they developed a novel idea which connected the
eigenproblems in the color domain and the spatial domain together. Although,
there is this rich theoretical background for the linear local
models of color, it is quite common to see research
procedures which are based on the old color space paradigm, even
published in 2005. For a more comprehensive discussion of this topic refer to [6].
In different works, the authors have applied the PCA, and then the
fuzzy PCA (FPCA), to many different color image processing tasks.
The essential idea of those works, which was borrowed from the
abovementioned literature, is that homogeneous swatches in color
images of nature produce thin cylinders in the RGB space. In this
framework, fuzzy clustering is a proper tool for finding these
structures [7]. This theory opens plenty of
opportunities for efficient color image processing tasks. Interested
readers can refer to [8,9] for two
applications of this theory. Also, see [6] for more
examples.
In the PCA framework, each color swatch is represented by a line,
which passes through its expectation vector and is parallel to its
principal direction. Although, it is proved that the lines
representing similar materials lie on a unique
plane [1], the theory neglects the (possibly virtual)
intersection of these lines. A virtual intersection of a set of
lines is a point very close to which almost all of them pass. In
this paper, we empirically show that the lines representing similar
materials do intersect and show that this intersection leads to a
new technique for color image segmentation. We call this
intersection point the principal color of the underlying material.
A review of the color image processing literature shows the huge amount of research dedicated to the basic primitives such as color image segmentation. This results in a vast variety of approaches with different motivations and tools. The reader is referred to [10,11] and the references therein for more details. Although, there are many available segmentation algorithms, no numerical criterion is standardized to give a reasonable comparison between the results of different approaches. In fact, the only globallyaccepted, and referred to, factors are the elapsed time and the visual properness of the results. Furthermore, the utilization of many techniques includes many parameters which must be tuned by an expert. Failing to do the required tuning, may result in total or partial collapse of the method (see [12] as an example). In this paper, we compare the performance of the proposed method with that of an available approach which is based on the same theory [7] and show that the new method outperforms the previous one. For the comparison of that method with the available literature refer to [7].
The rest of this paper is organized as follows. Section II introduces the concept of principal colors and gives experimental cues for their existence. Then, it describes the proposed method to use principal colors to perform color image segmentation. Section III holds the experimental results and discussions and finally Section IV concludes the paper.
Here, we propose an investigation, through which the existence of
the principal colors could be empirically perceived. We emphasize
that this empirical result has to be confirmed by theoretical
investigation. We believe that the increasing number of research
projects devoted to processing color images will pave the way to the
formal approval of this theory, something which is outside the scope
of this paper.
Figure 1 shows three sets of color swatches representing skin, sky, and leaf materials. Each set contains 30 images, each a 64×64 JPEG file with quality of 100. The skin samples are extracted from erotic images downloaded from free resources on the Internet. The sky and leaf samples are captured by a Cannon A60 digital camera in different times of the day and in different locations. For each category, the lines representing each single swatch are drawn in a common RGB space. Figure 2 shows these three sets of lines. Note that for each material, all the lines pass through a small region in the color space. We call this (virtual) intersection of the lines, the principal color of that material. Figure 3 shows the principal colors of skin, sky, and leaf extracted by the proposed method. Section IIA gives the mathematical details about the process which finds this point. Section IIB develops the idea further to include a set of fuzzy (or weighted) lines. Section IID incorporates the principal color theory with the general clustering algorithm described in Section IIC to propose a new color image segmentation method.
(a)
(b)
(c)
Figure 1: Randomly selected samples representing three different materials. (a) Skin. (b) Sky. (c) Leaf.
(a)
(b)
(c)
Figure 2: Linear representation of the samples shown in
Figure 1. (a) Skin. (b) Sky. (c) Leaf.
(a)
(b)
(c)
Figure 3: Principal color of the samples shown in
Figure 1 extracted by the proposed method. (a)
Skin. (b) Sky. (c) Leaf.
Assume that n lines of l_{1},¼,l_{n} in \mathbbR^{m} are given. Also, assume that l_{i} is indicated by the two vectors [(h)\vec]_{i} and [(v)\vec]_{i}. As such, [(h)\vec]_{i} is a vector starting from the origin (with its head on l_{i}) and [(v)\vec]_{i} is the direction of l_{i}, satisfying [(v)\vec]_{i}=1. Hence, the points on l_{i} occupy the set L_{i} Ì \mathbbR^{m}, which is defined as,
L_{i}=
ì í
î
®
h
i
+l
®
v
i
l Î \mathbbR
ü ý
þ
.
(1)
We assume that [(h)\vec]_{i} is perpendicular to [(v)\vec]_{i} ([(h)\vec]_{i}^{T}[(v)\vec]_{i}=0). If this condition is not met, [(h)\vec]_{i} is replaced with,
®
h
* i
=
®
h
i

æ è
®
v
t i
®
h
i
ö ø
®
v
i
.
(2)
The closest point to the n lines, l_{1},¼,l_{n}, is the point [(x)\vec] Î \mathbbR^{m} which minimizes the following objective function,
D =
n å
i=1
d
æ è
®
x
,l_{i}
ö ø
2
,
(3)
where, d([(x)\vec],l_{i}) is the distance from the point [(x)\vec] to the line l_{i}. Linear algebra shows that,
Now, we will prove that the n×n matrix on the left hand side of (9), which we call U, is always nonsingular, given that not all [(v)\vec]_{i}s are parallel.
A theory in linear algebra states that, "if I+E is singular then
E ³ 1, where · is a subordinate matrix norm" [14].
We will first prove this theorem. Assume that I+E is singular.
Hence, there exists a nonzero vector [(x)\vec] satisfying
(I+E)[(x)\vec]=[(0)\vec].
Thus, [(x)\vec]=E[(x)\vec] or equivalently,
E ³
E
®
x


®
x

=1.
(10)
Thus, to prove that U in (9) is nonsingular, it
suffices to show that,

n å
i=1
®
v
i
®
v
T i
 < n.
(11)
We prove (11) by showing that for any vector
[(y)\vec] satisfying [(y)\vec]=1,
we have,

n å
i=1
®
v
i
®
v
T i
®
y
 < n,
(12)
First, we know that for any two arbitrary m×1 vectors
[(u)\vec] and [(v)\vec],

®
u
T
®
v
 £
æ Ö

®
u
^{2}
®
v
^{2}
.
(13)
Note that equality occurs for [(u)\vec][(v)\vec]. Thus, in the case discussed here we have [(v)\vec]_{i}^{T}[(y)\vec] £ 1, which results in,

n å
i=1
®
v
i
®
v
T i
®
y
 £
n å
i=1

®
v
T i
®
y

®
v
i
 £
n å
i=1

®
v
i
=n.
(14)
Here, the equality happens only if all [(v)\vec]_{i}s are parallel. Thus, ignoring this very especial case, (12) is proved, proving that the solution to (9) exists and is,
®
x
=
é ë
nI
n å
i=1
®
v
i
®
v
T i
ù û
1
n å
i=1
®
h
i
.
(15)
Here, [(x)\vec] is the closest point to the n lines l_{1},¼,l_{n}.
2.2 Finding the Closest Point to a Set of Fuzzy Lines
Assume that we are looking for the closest point to the set of n
lines l_{1},¼,l_{n} in \mathbbR^{m}. Also, assume that the
importance of l_{i} is p_{i} ³ 0. This means that when composing
the objective function, we will weigh satisfaction of l_{i} by p_{i}.
Hence, as p_{i} grows, the importance of l_{i} in decisions grows
linearly, and the vice versa. As such, we are looking for
[(x)\vec] Î \mathbbR^{m} which minimizes,
~
D
=
n å
i=1
p_{i}d
æ è
®
x
,l_{i}
ö ø
2
.
(16)
Note that setting "i, p_{i}=1 converts (16)
into (3). An analysis similar to the one performed in
Section IIA reveals that [(x)\vec] is computed as,
®
x
=
é ë
n å
i=1
p_{i}I
n å
i=1
p_{i}
®
v
i
®
v
T i
ù û
1
n å
i=1
p_{i}
®
h
i
.
(17)
Again note that assuming "i, p_{i}=1 converts (17) into (15), as expected.
Assume that the fuzzy set of n vectors
[(x)\vec]_{1},¼,[(x)\vec]_{n} in \mathbbR^{m} is
given as [(X)\tilde]={([(x)\vec]_{i};p_{i})i=1,¼,n}.
Also, assume that a human observer has a perception of homogeneity
for the subspaces of this space. For example, for m=3 every human
being is able to think and argue about the homogeneity of a set of
color vectors. Also, when considering lines in \mathbbR^{3}, which
are members of \mathbbR^{6}, the meaning of homogeneity used here
is the existence of a point which is very close to almost all of
them. In many occasions, a homogeneous set may be parameterizable.
For example, research shows that a homogeneous set of color vectors
occupies a thin cylinder which will be described by two vectors.
Similarly, a set of (virtually) intersecting lines may be
parameterized by a single point. Thus, we assume that there is a
function U that extracts this parametrization for the
homogeneous fuzzy set [(X)\tilde]. As an example, if a
homogeneous set in \mathbbR^{m} is defined as a sphere, the
U function will be a fuzzy expectation operator. Also, for
the cylindrical homogeneity, U will be the
FPCA [7]. Having defined a description of a homogeneous set
as f, there might exist a function Y which measures the
distance from an arbitrary element to f. For the spherical
example, the Y function is the Euclidean distance. Also, the
Y function of the cylindrical homogeneity, is the linear
partial reconstruction error (LPRE) distance proposed
in [6].
Having proper dual functions U and Y, and a fuzzy set [(X)\tilde], an important problem is how to separate [(X)\tilde] into c homogenous clusters, where c is known. This problem is called the general clustering problem [7]. The special cases of this problem are the fuzzy Cmeans (FCM) [15] (in which the clusters are spherical) and fuzzy Cvarieties (FCV) [16] (in which the clusters are subspaces). Other examples are GathGeva [17], fuzzy elliptotypes [18], GustafsonKessel [19], and FPCAbased clustering (FPCAC) [7] algorithms. Figure 4 shows the pseudocode of an iterative solution to the general clustering problem. The interested reader is referred to [8] for the related mathematics and the proof of the convergence.
As shown in Figure 4, the algorithm first randomly initializes the clusters. Then, it iterates between computing the distance of each member to each cluster and then renewing each cluster according to the membership value of the realizations to it. The algorithm contains an important parameter of fuzziness [20], m, which is always more than unity. The general clustering algorithm enables the clustering of any kind of data, given that a proper distance function and its dual are available. In Section IID we will give a distance function which searches a set of lines for lines passing through or getting very close to a common point. The distance function is then integrated into a novel color image segmentation procedure.
Aim: Clustering data according to the given model.
Inputs:
Appropriate distance function (Y and its dual U).
Set of realizations (X={[(x)\vec]_{1},¼,[(x)\vec]_{n}}).
Number of clusters (c).
Fuzzyness (m).
Halting threshold (d).
Outputs:
Fuzzy membership values (p_{ij}).
c cluster descriptors (f_{1},¼,f_{c}).
Method:
1 k=0 and randomize f_{1},¼,f_{c}.
2 k=k+1.
3 D_{ij}=Y([(x)\vec]_{i},f_{j}).
4 p_{ij}=D_{ij}^{1/(m1)}/å_{k=1}^{c}D_{ik}^{1/(m1)}.
5 F_{kij}=p_{ij}.
6 f_{j}=U{([(x)\vec]_{1};p_{1j}^{m}),¼,([(x)\vec]_{n};p_{nj}^{m})}.
7 if k=1, then goto 2.
8 d_{k}^{2} = min_{1 £ p £ l1}E_{i,j}{(F_{pij}F_{kij})^{2}}.
9 if d_{k} > d, then goto 2, else return.
Figure 4: Pseudocode of the general clustering
algorithm [7].
2.4 Color Image Segmentation using Principal Colors
Assume that the H×W color image I is given. Also, assume
that the integer c is given as the number of homogeneous
segments that I should be split into. For now, assume that both
H and W are divisible by 2^{n}, where n is an integer larger
than 2. It is clear that, using zeropadding or resampling,
the same operation is applicable to any image. We propose to cut I
into 2^{2n}WH nonoverlapping rectangular blocks of
2^{n}×2^{n} pixels.
In [6], the authors proposed a novel homogeneity criterion for color swatches. In that work, the LPRE criterion was compared to the Euclidean and Mahalanobis distances and its superiority was proven. As such, the nonhomogeneity of the swatch i is defined as [6],
t(i)^{2}=E_{[(c)\vec] Î i}
ì í
î

æ è
®
c

®
h
i
ö ø

®
v
T i
æ è
®
c

®
h
i
ö ø
®
v
i
^{2}
ü ý
þ
.
(18)
Here, [(h)\vec]_{i} is the expectation vector of i and [(v)\vec]_{i} is the principal direction of i. Also, [(x)\vec] is the Euclidean length of the vector [(x)\vec]. We use t(i) < q to decide whether a swatch is homogeneous or not, where q is a constant threshold. Experiments show that q = 10 is a proper choice [6].
In the proposed segmentation method, from the 2^{2n}WH blocks, those which are not homogeneous are deleted.
For the remaining blocks, the PCA representations are saved as the m lines of l_{1},¼,l_{m}, all in \mathbbR^{3}. Now, the problem is to find c proper intersections of these lines. Using the notations of the general clustering algorithm (see Section IIC) each realization is a line in \mathbbR^{3} and the cluster model is a 3D point. As such, the distance between the line l (denoted by [(h)\vec] and [(v)\vec]) and the point [(x)\vec] is defined as,
Y
æ è
l,
®
x
ö ø
=
æ è
®
x

®
h
ö ø

®
v
T
æ è
®
x

®
h
ö ø
®
v
^{2}.
(19)
In this framework, the dual function U is given by (17).
Here, the fuzzy set is [(L)\tilde]={(l_{i};p_{i})i=1 ¼,m}, where l_{i} is represented
by [(h)\vec]_{i} and [(v)\vec]_{i}.
The general clustering algorithm results in a set of c points in
\mathbbR^{3}, where each point is the principal color of the
respective cluster. Note that at the beginning of this process, the
nonhomogeneous blocks were removed. That was done because the
principal direction of those blocks are not meaningful. Therefore,
at this point, the process should assign each block to a cluster,
even those that are not homogeneous enough. Also, note that the
results of the general clustering algorithm is a set of fuzzy
membership values for each block and each cluster. Thus, using a
maximumlikelihood operator, each block is deterministically
assigned to one of the clusters. By finding the distance of each
omitted block to all clusters, the process assigns them to one of
the computed clusters. The next step is to interpolate the
assignment for all pixels. As each block is a 2^{n}×2^{n}
swatch, performing bilinear doublesampling on the resulting index
map, for n times, yields the desired segmentation. In practice,
blocks of size 2^{n1}×2^{n1} are used to find the clusters
and then 2^{n2}×2^{n2} blocks are examined to compute the
indexmap.
The proposed algorithm is developed in MATLAB 6.5, on a PIV 2600 MHz personal computer with 256 MB of RAM.
All sample images used in this paper are 512×512 color images in JPEG format with quality of 100
(see Figure 5).
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5: Some sample images. a) Peppers. b) House. c) Splash. a,b,c) Courtesy of USCSIPI, Signal and Image Processing Institute at the University of Southern California. d) Food Peppers, courtesy of freeimages.co.uk. e) Toy, courtesy of Computational Vision at Caltech, http://www.vision.caltech.edu. f) Dessert.
Figure 6 shows the results of three independent runs of the proposed method on the image Peppers. Here, m=2, c=2, n_{1}=4, and n_{2}=3 are used. This figure shows that while the proposed method always results acceptably, its output is never identical in different runs, even with the identical values of the parameters, similar to the entire category of fuzzy clustering algorithms [21]. It should be emphasized, though, that in contrast with the conventional pixelbased fuzzy clustering approaches, in which local minimum is a common unsolved problem [21], the proposed method shows less tendency to falling into local minima. A local minimum in a fuzzy clustering algorithm can be identified by a solution which does not satisfy the merits which the problem defines but still cannot be improved by the local approach utilized in the general clustering algorithm [21]. We argue that the relative immunity of the proposed method against getting trapped in local minimums is the result of two innovations in it. First, rather than working on the original realizations, which are noisy and may contain many outliers, the proposed method works on more robust statistical parameters of blocks of the image. Secondly, the proposed distance function models the physical phenomenon more appropriately, compared to the pixellevel operators used before. It is worth to emphasize that the proposed method is more repeatable, compared to the FPCAC [7], which uses a similar distance measure in the pixels level. In the experiments shown in Figure 6, the algorithm has converged in 9, 6, and 5 iterations, elapsing 16s, 14s, and 14s, respectively.
(a)
(b)
(c)
Figure 6: Results of three independent runs of the proposed method on the image Peppers.
Top) Results of segmentation. Bottom) Principal colors.
Figure 7 shows the results of the proposed method applied on the image House, while different values of m are used. Here, with c=3 and m Î {1^{1}/_{3}, 1^{2}/_{3}, 2}, the algorithm has converged in 4, 10, and 9 iterations, elapsing 13s, 16s, and 16s, respectively. More experiments show that the actual value of m has minimal effect on the results of the proposed method. Hence, the value of m=2 is used as the default value. Similarly, the values of n_{1} and n_{2} only affect the spatial precision of the results and the elapsed time. As a compromise, we always select n_{1}=4 and n_{2}=3.
(a)
(b)
(c)
Figure 7: Effects of m on the results of the proposed method on
the image House. Top) Results of segmentation. Bottom)
Principal colors.
In a proper segmentation process, it is expected that increasing c should result in an increasingly better depiction of different regions in the image. Using values of c=2, c=3, and c=4, this aspect of the proposed method is investigated in Figure 8. As expected, increasing the number of clusters results in more detailed segmentation results. In these experiments, the algorithm has converged in 9, 5, and 8 iterations, elapsing 12s, 14s, and 18s, respectively. It should be emphasized that the elapsed time of the proposed algorithm depends almost linearly on the number of clusters.
(a)
(b)
(c)
Figure 8: Effects of c on the results of the proposed method on
the image Splash. Top) Results of segmentation. Bottom)
Principal colors.
Figure 9 compares the performance of the proposed method with the FPCAC [7]. This comparison is important because both methods search for cylindrical structures in the color space. Note that while the FPCAC works on individual pixels, the proposed method utilizes more robust statistically computed features. In this experiment, the FPCAC is utilized with its own default value of m=1.05 [7]. Table I lists the elapsed time and the number of iterations of each method when working on each image. Also, the value of c=4 is used for both methods in all cases.
As mentioned in Section I, there is no standard method for comparing the performance of different segmentation methods. Thus, here, we compare the results of the proposed method and the FPCAC, as shown in Figure 9, based on a heuristic approach. Comparing Figure 9a with Figure 9d, a very common problem of many segmentation methods can be seen. This way, in Figure 9a, FPCAC has mistakenly included parts of the gray background in the yellow pepper. Also, the shadow of the red pepper and other parts of the background are included in the red and green peppers, respectively. However, looking at Figure 9d, we see a proper segmentation of the image into four distinguished parts of red, yellow, and green peppers, and the background by the proposed method. Figures 9b and e show an example where the two methods perform similarly, except for the fact that FPCAC has resulted in an oversegmentation. This way, as seen in Figures 9b, FPCAC has segmented the text on the object into two different classes (because of being pixelbased). The proposed method, on the other hand, has given a smoother segmentation result. This effect is also visible in Figure 9c, where FPCAC has again resulted in many tiny segments. In contrary, the proposed method has again produced a smooth segmentation result. It is worth to mention that the proposed method is also faster than the FPCAC. The reader is referred to [6] for more examples.
In analyzing the performance of the proposed method, it is worth to mention that as the proposed algorithm is primarily concerned with spectral accuracy, details in the spatial domain might be ignored. For example, looking at Figure 9e, we observe that the proposed method has missed the details in the text on the object. This is a direct result of looking at the image in lower resolutions, as done using n_{1} and n_{2}. While this is useful in applications which look for the general geometry of the objects, in other applications which demand more precision, including coding, a pixelbased approach would yield more applicable results. Thus, we argue that the proposed method produces acceptable results when the aim is to find homogeneous regions in an image with more emphasis on the color information, and not on the spatial details.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 9: Comparison of the proposed method with the FPCAC [7]. (a), (b), and (c) Results of the FPCAC.
(d), (e), and (f) Results of the proposed method.
Table 1: Performance comparison of the proposed method and the
FPCAC [7]. [t: Elapsed time. i: Number of
iterations before convergence.]
In this paper, a new color image segmentation method is proposed
which utilizes the general clustering algorithm with an innovative
distance function. The mathematics of the proposed method is
discussed comprehensively and experimental results are presented.
Comparison of the performance of the proposed method with an
available clustering method, which searches for similar cylindrical
structures in the pixel domain, shows that the proposed method is
more stable and faster. We argue that this is mainly because of
applying the clustering task on a more robust statistical feature.
It is also observed that the proposed method decreases the
probability of local minimum entrapment. The repeatability of the
proposed segmentation method is also more than the available one.
Furthermore, while the proposed method gives a more perceptually
satisfactory segmentation, it demands less processing resources.
Acknowledgement
This work was in part supported by a grant from ITRC. Also, Arash Abadpour wishes to thank Ms. Azadeh Yadollahi for her encouragement and invaluable discussions. We also appreciate the respected anonymous referees for their constructive suggestions.
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