Median-based image thresholding☆
Highlights
► We propose median-based extensions of two very popular image-thresholding methods. ► We provide theoretical interpretation of the new approaches. ► They are as simple and efficient as, and can be more robust than, their original methods.
Introduction
Image thresholding aims to partition an image into K predetermined, mutually-exclusive classes, C1, …, CK, based on K − 1 intensity thresholds. Most commonly, K = 2 and the image is partitioned into the background and the foreground. As an initial procedure for realising image segmentation, thresholding has a long history of investigation, motivated by a broad range of practical applications of image analysis and object recognition. Comprehensive overviews and comparative studies of image thresholding can be found in [15], [4], [17], [16], for example.
Many, and the most-widely used, approaches to image thresholding are based on analysis of the histogram of intensities in an image, searching for an optimal threshold t∗ to divide the histogram into two parts, C1 with intensities lower than t∗ and C2 for the remainder.
Among these approaches, two of the most popular are Otsu's method [12] and Kittler and Illingworth's minimum-error-thresholding (MET) method [8]. Otsu's method is adopted as the method for automatic image thresholding in some free and commercial software, such as GIMP (www.gimp.org) and MATLAB (The MathWorks, Inc.). The MET method is ranked as the best in a comprehensive survey of image thresholding conducted by [16].
In image thresholding, determination of an optimal threshold t∗ is often based on the estimation of measures of location and dispersion of intensities in C1 and C2. As with many other approaches, both Otsu's method and the MET method use the sample mean and the sample standard deviation to estimate location and dispersion, respectively.
It is well known that, when the distribution for class Ck is skew or heavy-tailed, or when there are outliers in the sample from Ck, the median is a more robust estimator of location than the mean. When the median is chosen for location, the mean absolute deviation from the median (denoted by MAD) is usually chosen as the estimator of dispersion.
Therefore, in order to select a t∗ that is more robust to the presence of skew and heavy-tailed distributions for Ck than those selected by Otsu's method and the MET method, we propose in section 2 two median-based approaches to image thresholding. One of them is an extension of Otsu's method and the other is an extension of the MET method; both methods are based on the use of the MAD. Like their original versions, the two new approaches remain methodologically simple and computationally efficient.
The relationship between Otsu's method and the MET method has been investigated by [9], [21], among others. [9] shows that both methods can be derived from maximisation of log-likelihoods based on mixtures of Gaussian distributions. In section 3, we present theoretical interpretation of their median-based extensions from the perspective of the maximisation of log-likelihoods for mixtures of Laplace distributions.
Some limitations of the median-based approaches are discussed in section 4 and a summary is made in section 5.
Section snippets
Methodology
Each of the N pixels in an image χ is represented by its intensity xi, i = 1,…,N. A threshold t partitions the image into two classes C1(t) and C2(t), where C1(t) = {i: 0 ≤ xi ≤ t, 1 ≤ i ≤ N} and C2(t) = {i: t < xi ≤ T, 1 ≤ i ≤ N}, in which T is the largest possible intensity, which is 255 for an 8-bit grey-level image (i.e. xi ∈ [0, T]).
The histogram for the image χ, denoted by {h(x)}, can be constructed by counting the frequencies of the intensities and dividing them by N, such that ∑ x = 0Th(x) = 1.
Relationship with Laplace mixtures
A straightforward and intuitive interpretation of Otsu's rule, as shown in Eq. (1), is that it aims to minimise the within-classes variance JO(t), a measure of dispersion, of the intensity. Correspondingly, an interpretation of the median-based extension of Otsu's method, as shown in Eq. (4), is that the extension aims to minimise the within-classes mean absolute deviation from the median JOM(t), another measure of dispersion, of the intensity.
Alternatively and insightfully, as mentioned in
Computational efficiency
The median-based approaches preserve not only the methodological simplicity and theoretical elegance, but also the computational efficiency, of their original methods.
It is a natural concern that the calculation of a median is often slower than that of a mean; the exhaustive search of the range of candidate thresholds for an optimal g∗ makes this concern more prominent, because a considerable number of medians have to be calculated. Fortunately, by sorting all pixels once in descending or
Summary
We have proposed two median-based approaches to image thresholding, to extend Otsu's method and Kittler and Illingworth's MET method. We have provided theoretical interpretation of the new approaches, based on mixtures of Laplace distributions. We have shown that the two extensions preserve the methodological simplicity and computational efficiency of their original methods, and can in general achieve more robust performance for skew and heavy-tailed data.
Acknowledgments
The NDT images are provided through the courtesy of Dr. Mehmet Sezgin. This work was partly supported by funding to J.-H.X. from the Internal Visiting Programme of the EU-funded PASCAL2 Network of Excellence. We are grateful for the referees' generous and constructive suggestions.
References (22)
- et al.
Image thresholding based on the EM algorithm and the generalized Gaussian distribution
Pattern Recognition
(2007) - et al.
Image thresholding using a novel estimation method in generalized Gaussian distribution mixture modeling
Neurocomputing
(2008) An analysis of histogram-based thresholding algorithms
CVGIP: Graphical Models and Image Processing
(1993)- et al.
A comparative study of various meta-heuristic techniques applied to the multilevel thresholding problem
Engineering Applications of Artificial Intelligence
(2010) - et al.
Minimum error thresholding
Pattern Recognition
(1986) - et al.
Maximum likelihood thresholding based on population mixture models
Pattern Recognition
(1992) - et al.
Image thresholding: some new techniques
Signal Processing
(1993) - et al.
Image thresholding using two-dimensional Tsallis–Havrda–Charvát entropy
Pattern Recognition Letters
(2006) - et al.
A survey of thresholding techniques
Computer Vision, Graphics, and Image Processing
(1988) Unified formulation of a class of image thresholding techniques
Pattern Recognition
(1996)
On minimum error thresholding and its implementations
Pattern Recognition Letters
Cited by (56)
Simplified expression and recursive algorithm of multi-threshold Tsallis entropy
2024, Expert Systems with ApplicationsHow long did crops survive from floods caused by Cyclone Idai in Mozambique detected with multi-satellite data
2022, Remote Sensing of EnvironmentDetermination of Green Spots (Trees) for Google Satellite Images Using MATLAB
2020, Procedia Computer ScienceAn improved version of Otsu's method for segmentation of weld defects on X-radiography images
2017, OptikCitation Excerpt :An improved version of Otsu's method, namely, weighted object variance (WOV), was proposed to find optimal threshold by the cumulative probability value assigned to the weight parameter on the object variance of the between-class variance (see [20]). This apart, many modifications of Otsu's method have been proposed by Xue and Titterington [18], Yang et al. [19], and Cai et al. [5] based on individual class variance. For example, median-based method of Xue and Titterington [18] considers theoretical interpretation approach based on a mixture of Laplace distributions.
A novel generalized entropy and its application in image thresholding
2017, Signal Processing
- ☆
This paper has been recommended for acceptance by Aleix Martinez.