OPTIMAL GLOBAL THRESHOLD ESTIMATION USING STATISTICAL CHANGE-POINT DETECTION

Signal & Image Processing : An International Journal (SIPIJ) Vol.8, No.4, August 2017
DOI : 10.5121/sipij.2017.8402 15
OPTIMAL GLOBAL THRESHOLD ESTIMATION
USING STATISTICAL CHANGE-POINT
DETECTION
Rohit Kamal Chatterjee1
and Avijit Kar2
1
Department of Computer Science & Engineering,
Birla Institute of Technology, Mesra, Ranchi, India.
2
Department of Computer Science & Engineering,
Jadavpur University, Kolkata, India.
ABSTRACT
Aim of this paper is reformulation of global image thresholding problem as a well-founded statistical
method known as change-point detection (CPD) problem. Our proposed CPD thresholding algorithm does
not assume any prior statistical distribution of background and object grey levels. Further, this method is
less influenced by an outlier due to our judicious derivation of a robust criterion function depending on
Kullback-Leibler (KL) divergence measure. Experimental result shows efficacy of proposed method
compared to other popular methods available for global image thresholding. In this paper we also propose
a performance criterion for comparison of thresholding algorithms. This performance criteria does not
depend on any ground truth image. We have used this performance criterion to compare the results of
proposed thresholding algorithm with most cited global thresholding algorithms in the literature.
KEYWORDS
Global image thresholding, Change-point detection, Kullback-Leibler divergence, robust statistical
measure, thresholding performance criteria.
1. INTRODUCTION
A grey-level digital image is a two dimensional signal LI →→→→ΖΖΖΖ××××ΖΖΖΖ: , where L={ li∈ and
i=1,2,…,M} is the set of M grey-levels. The problem of automatic thresholding is to estimate an
optimal threshold t0 which segments the image into two meaningful sets, viz. background
B={bb(x,y)=1| I(x,y)<t0 } and foreground F={bf(x,y)=1| I(x,y)≥ t0} or the opposite. The function
I(x,y) can take any random value li∈L; so, sampling distribution of grey levels becomes an
important deciding factor for t0. In many image processing applications, automating the process
of optimal thresholding is extremely important for low-level segmentation or even final
segmentation of object and background.
In general, automatic thresholding algorithms are divided into two groups, viz. global and local
methods. Global methods estimate a single threshold for the entire image; local methods find an
adaptive threshold for each pixel depending on the characteristics of its neighborhood. Global
methods are used if the image is considered as a mixture of two or more statistical distributions.
In this paper, we address the global thresholding methods guided by the image histogram. Most of
the cases global thresholding methods try to estimate the threshold (t0) iteratively by optimizing a
criterion function [1]. Some other methods attempt to estimate optimal t0 depending on histogram
shape [2, 3], image attribute such as topology [5] or some clustering techniques [4, 8, 20].
Comprehensive surveys discussing various aspects of thresholding methods can be found in the
references [1, 6, 7].

16
Many of these classical and recent schemes perform remarkably well for images with matching
underlying assumptions but fail to yield desired results otherwise. Some of the explicit or implicit
reasons for their failure could be: (i) assumption of some standard distribution (e.g. Gaussian)
[19], in reality though, foreground and background classes can have arbitrary asymmetric
distributions, (ii) use of non-robust measures for computing criterion functions which get
influenced by outliers. Further, the effectiveness of these algorithms greatly decreases when the
areas under the two classes are highly unbalanced. Some of the methods depend on user specified
constant (e.g. Renyi or Tsallis entropy based methods) [17, 18], greatly compromising their
performance without its appropriate value.
This paper proposes an algorithm for addressing these drawbacks using a statistical technique
known as change-point detection (CPD). For the last few decades, models of change-point
detection are successfully applied by researchers in statistics and control theory for detecting
abrupt changes in the statistical behavior of an observed signal or time series [9]. The general
principle of change-point detection considers an observed sequence of independent random
variables {Yk}1…n with a probability density function (pdf) pθ(y) depending on a parameter θ. If
any change occurs in the sequence then it is assumed that parameter θ takes a value θ0 before any
change and at some unknown time t0 alters to θ1 (≠θ0). The main problem of statistical change-
point detection is to decide the change in parameter and also the time of change. The theory of
CPD is used in this paper to decide the global threshold in an image depending on the change in
the histogram.
Further, in section 4 of this paper, we propose a new performance index for the evaluation of
thresholding algorithms. It depends on the structural difference between the shapes of background
and foreground. The advantage of this performance index is that it does not depend on any ground
truth image. We use this performance index to compare different thresholding algorithms
including ours.
Rest of the paper is organized as follows: Section 2 provides a short introduction to the problem
of statistical change-point detection, section 3 formulates and derives the global thresholding as a
change-point detection problem, section 4 describes our proposal for thresholding performance
criteria, section 5 presents the experimental results and compares the results with various often
cited global thresholding algorithms, and finally section 6 summarizes main ideas in this paper.
2. THE CHANGE-POINT DETECTION (CPD) PROBLEM
The Change-point detection (CPD) problem can be classified into two broad categories: real-time
or online and retrospective or offline change-point detection. The first targets applications where
the instantaneous response is desired such as robot control; on the other hand, retrospective or
offline change-point detection is used when longer reaction periods are allowed e.g. image
processing problems [10]. The later technique is likely to give more accurate detection since the
entire sample distribution is accessible. Since the image and the corresponding histogram are
available to us, we concentrate on offline change-point detection in this paper. We also assume
that there is only one change point throughout the given observations {yk}1…n. When required, this
assumption can easily be relaxed and extended to multiple change point detection that can be
applied in multi-level threshold detection problems.
2.1. Problem Statement
When taking an offline point of view about the observations y1, y2…, yn with corresponding
probability distribution functions F1, F2, …, Fn, belong to a common parametric family F(φ),
where φ∈ Rp
, p>0. Then the change point problem is to test the null hypothesis (H0) about the
population parameter φj, j = 1,2, …, n:

17
njiforH j ≤≤≤≤≤≤≤≤==== 00 : θθθθφφφφ
versus an alternative hypothesis (1)
{
1,
,1
0
1
:
kjfor
njkforjH
≤≤≤≤≤≤≤≤
≤≤≤≤<<<<
====
θθθθ
θθθθ
φφφφ
where θ0 ≠ θ1 and k is an unknown time of change.
These hypotheses together disclose the characteristics of change point inference, determining if
any change point exists in the process and estimating the time of change t0 = k. The likelihood
ratio corresponding to the hypotheses H0 and H1 is given by
where and are pdfs before and after the change occurs and is the overall probability
density. When the only unknown parameter is t0, its maximum likelihood estimate (MLE) is given
by the following statistic
2.2. Offline Estimation of the Change Time
When the problem is to estimate the change time (t0) in the sequence of observations {yj}1...n and
if we assume the existence of a change point with the same presumption as in the last section.
Therefore, considering equation (2) and (3) and the fact that is a constant for a given data,
the corresponding MLE estimate is
where is a maximum log-likelihood estimate of t0. Rewriting equation (4) as
As remains constant for a given observation, estimation of is simplified as
Therefore, the MLE of the change time t0 is the value which maximizes the sum of log-likelihood
ratio corresponding to all k possible values given by equation (6).
3. CHANGE-POINT DETECTION FORMULATION OF GLOBAL
THRESHOLDING
3.1. Assumptions
Let (χ, βχ, Pθ)θ∈∈∈∈Θ be the statistical space of discrete grey-levels associated with a random variable
Y:ℤℤℤℤxℤℤℤℤ→ℤℤℤℤ, where βχ is the σ-field of Borel subsets A ⊂⊂⊂⊂ χ and {Pθ}θ∈∈∈∈Θ is a family of probability
distributions defined on the measurable space (χ, βχ) with parameter space Θ, an open subset of

18
ℝq
, q>0.We consider a finite population Π of all gray-level images with N elements that could be
classified into M categories or classes L={l1, ..., lM}, i.e. each sample point in the sample image
can take any random gray-level values from the set L.
3.2. Change-Point Detection Formulation
Since we are mainly interested in discrete gray-level data, we consider the multinomial
distribution model. Let ℘℘℘℘={Ei}, i=1,...,M be a partition of χ. The formula Prθ(Ei) = pi(θ), i = 1, . .
., M, defines the probability of the li
th
gray-level in the discrete statistical model. Further we
assume {y1,. . .,yN} to be a random sample from the population described by the random variable
Y, representing the gray-level of a pixel. And let , where IE is the index function.
Then we can approximate pi(θ)≈Ni/N, i=1,…, M. Estimating θ by maximum likelihood method
consists of maximizing the joint probability distribution for fixed n1, . . . , nM,
or equivalently maximizing the log-likelihood function
Therefore, referring to equation (4), problem of estimating the threshold by MLE can be stated as
where unknown parameter θ =θ0 before the change and θ= θ1 after the change. Now, equation (9)
can be expanded as
The first term within the bracket on the right side of equation (10) is a constant and the last term
is independent of j, i.e. it cannot influence the MLE. So, eliminating these terms from equation
(10) and simplifying we get
Multiplying and dividing N on right side of equation (11) we get
assuming pi(θ)≈ni/N equation (12) can be written as

19
The expression in (13) under the summation denotes Kullback-Leibler (KL) divergence between
the density and , where and denotes the pdfs above and below the
threshold location j; therefore equation (13) can be written as
Since total sum is independent of j, i.e. a constant for a given observation, a
sample image, therefore equation (14) can be rewritten as
Hence, equation (15) provides the maximum likelihood estimation of the threshold t0. Equation
(15) can be restated as the following proposition:
Proposition 1: In a mixture of distributions, the maximum likelihood estimate of change-point is
found by minimizing the Kullback-Leibler divergence of the probability mass across successive
thresholds.
In spite of this striking property, KL divergence is not a ‘metric’ since it is not symmetric. An
alternative symmetric formula by “averaging” the two KL divergences is given as [11]
An attractive property of KL divergence is its robustness i.e. KL divergence is little influenced
even when one component of mixture distribution is considerably skewed. A proof of robustness
can be found for generalized divergence measures in [11, 12].
This method can be easily extended to find multiple thresholds for several mixture distributions
by identifying multiple change-points simultaneously.
3.3. Implementation
Section Let us consider an image I:ℤxℤ L, whose pixels assume M gray-levels in the set L={l1,
l2,…, lM }. The empirical distribution of the image can be represented by a normalized histogram
p(li)=ni/N, where ni is the number of pixels in ith
gray-level and N is the total number pixels in the
image.
Now, suppose we are grouping the pixels into two classes B and F (background and object) by
thresholding at the level k. Histogram of gray-levels can be found for the classes B and F; let us
denote them as pB(li)and pF(li). Following statistics are calculated for the level k.

20
and finally,
The minimum value of CPD(k) for all values of k in the range [1,…, M] gives an optimal
estimate of threshold t0.
4. THRESHOLDING PERFORMANCE CRITERIA
The objective of the global thresholding algorithm is to divide the image into two binary images
generally called background and foreground (object). Most of the histogram-based thresholding
algorithms try to devise a criterion function which produces a threshold to separate the shapes and
patterns of the foreground and background as much as possible. A good thresholding algorithm
can be judged by how well it sets apart the object and the background binary images, i.e. how
much dissimilarity exists between the foreground and the background. Since the background and
foreground images are binary images dissimilarity between them can be measured by any binary
distance measures. Based on this observation, we propose a threshold evaluation criterion, which
tries to find the dissimilarity between the patterns and shapes in foreground and background.
A number of binary similarity and distance measures have been proposed in different areas, a
comprehensive survey of them can be found in Choi et al. [13]. In order to understand the
distance measure used in our work, it is helpful to refer to the following contingency table (Table
1):
Table 1. Binary contingency table
Foreground
Background
1 0
1 a b
0 c d
The cell entries in Table 1 are the number of pixel locations for which the two binary images
agree or differ. For example, cell entry ‘a’ is the total count of pixel locations where both binary
images take a value one. Hence, b + c denote the total count where foreground and background
pixels differ (Hamming distance) and a+d is the total count where they agree.
In order to extract the shapes and patterns present in the foreground (F) and background (B)
images, we use binary morphological gradient. The binary Morphological gradient is the
difference between the eroded and dilated images. Obviously, any other edge or texture detection
algorithm for binary images can be also used to extract the objects present in foreground and
background.
In this paper, we use a simple binary distance measure known as Normalised Manhattan distance
(DNM) given by
where Fg and Bg denote Binary Morphological gradients of foreground (F) and background(B)
respectively. The range of this distance measure is the interval [0, 1]. It is expected that well-
segmented image will have DNM close to 1, while in the worst case DNM =0. The advantage of this
algorithm is that it does not require any ground truth image.

21
5. EXPERIMENTAL RESULTS WITH DISCUSSION
To validate the applicability of proposed Change-Point Detection (CPD) thresholding algorithm,
we provide experimental results and compare the results with existing algorithms. The first row of
Figure 1 shows test images that are labeled from left to right as Dice, Rice, Object, Denise, Train,
and Lena respectively.
TABLE 2: Threshold evaluation criterion (DNM) for the test images (A) Dice, (B) Rice, (C) Object, (D)
Denise, (E) Train, and (F) Lena
The images have deliberately been so selected that the difference of areas between foreground
and background is hugely disproportionate. This gives us an opportunity to test the robustness of
CPD algorithm. To compare the results, we selected five most popular thresholding algorithms,
namely, Kittler-Illingworth [14], Otsu [15], Kurita [16], Sahoo [17] and Entropy [18].
In Figure 1 third row onwards show the outputs of different thresholding algorithms. The last
row shows the output of the proposed CPD thresholding algorithm. Due to substantial skewness
in the distributions of gray-levels in object or background, most of the algorithms confused
foreground with background. But results in the last row clearly show that CPD works
significantly better in all cases.
Table 2 shows optimal thresholds of five selected algorithms and the CPD algorithm using our
proposed performance criteria. It is clear that CPD performs reasonably well. For example,
consider the Denise image and Train image, Kittler-Illingworth thresholding totally fails to
distinguish the object from the background due to its assumption of Gaussian distribution for both
foreground and background [19]. Otsu’s and Kurita’s method yield almost same output due to
their common assumptions. Corresponding histograms are also reproduced in Figure 2 marked
with threshold locations of all the six algorithms above for reference. The threshold locations
show that CPD algorithm is very little influenced by the asymmetry of object of background
distributions.
6. CONCLUSIONS
In this paper we propose a novel global image thresholding algorithm based on Statistical
Change-Point detection (CPD), which is derived based on a symmetric version of Kullback-
Leibler divergence measure. The experimental results clearly show this algorithm is largely
unaffected by disproportionate dispersal of object and background scene and also very little

22
influenced by the skewness of distributions of object and background compared to other well-
known algorithms. We also propose a thresholding performance criterion using dissimilarity
between foreground and background binary images. Advantage of this performance criterion is
that it does not require any ground truth image.
Figure 1. Result of thresholding algorithms on tested images: Row-1: Original Images; Row-2: Shapes of
histograms; Row-3: Kittler; Row-4: Otsu; Row-5: Kurita; Row-6: Sahoo; Row-7: Entropy: Row-8: CPD
Threshold.

23
Figure 2: Histogram of (a) Denise and (b) Train image with threshold locations
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