CEC Multilevel Thresholding

CEC Multilevel Thresholding

Authors: M. Malik, P. Spurek

History: 2014/10/11: First version

Bugs: 0 bug(s) listed.

Plugin: CEC_multilevel_thresholding-1.0.0.jar

Installation: Download CEC_multilevel_thresholding-1.0.0.jar to the plugins folder (restart ImageJ) and there will be a new command: Plugins/Binarization/CEC thresholding

References: M. Malik, P. Spurek, J. TaborCross-Entropy based image thresholding,
Description:

CEC Multilevel Thresholding

This plugin presents a novel global thresholding algorithm for the binarization of documents and gray–scale images using Cross Entropy Clustering. Firstly, gray–level histogram is constructed and then the Gaussian densities are fitted for them. The thresholds are then determined as cross– points of the Gaussian densities. This approach automatically detects number components (only the upper limit of Gaussian densities is required).
Let us define the Cross Entropy Clustering for histograms. Firstly, let the space of all possible gray levels be:

$X = [0, 255].$
Then, the histogram is
$h: X \to \mathbb{R},$
where
$h(x) = y$
means that the color x occurs y times on the image. In the following calculations we consider only colors which exist on the image
$\bar{X} = \{x \in X: h(x) \neq 0\}.$

In such case as an estimator for mean and variancewe we use
$\mathrm{mean}(\bar{X}) = \sum \limits_{x \in \bar{X}} h(x) \cdot x$
$\var(\bar{X})= \sum \limits_{x \in \bar{X}} h(x) \cdot (x-\mathrm{mean}(\bar{X}))^2.$

Now we can introduce cost function, which we minimize:
$E(X_1,\ldots,X_k ,h) = \sum_{i = 1}^k p_i \cdot \left(-\ln(p_i) \frac{1}{2}\ln(2 \pi e) \frac{1}{2} \ln(\sigma_i^2) \right),$

where
$\sigma_i^2 = \var(X_i),$
$p_i = \frac{|X_i|}{|\bar{X}|},$
$\bar{X} = X_1 \cup \ldots \cup X_k.$

Example: The effect of CEC Multilevel Thresholding on sample gray-scale printed document for 2 thresholds (i.e. 3 clusters). Original image:

Results of thresholding:

lower cluster

middle cluster

upper cluster

The histogram along with chosen thresholds:

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Authors:	M. Malik, P. Spurek
History:	2014/10/11: First version
Bugs:	0 bug(s) listed.
Plugin:	CEC_multilevel_thresholding-1.0.0.jar
Installation:	Download CEC_multilevel_thresholding-1.0.0.jar to the plugins folder (restart ImageJ) and there will be a new command: `Plugins/Binarization/CEC thresholding`
References:	M. Malik, P. Spurek, J. TaborCross-Entropy based image thresholding,
Description:	CEC Multilevel Thresholding This plugin presents a novel global thresholding algorithm for the binarization of documents and gray–scale images using Cross Entropy Clustering. Firstly, gray–level histogram is constructed and then the Gaussian densities are fitted for them. The thresholds are then determined as cross– points of the Gaussian densities. This approach automatically detects number components (only the upper limit of Gaussian densities is required). Let us define the Cross Entropy Clustering for histograms. Firstly, let the space of all possible gray levels be: $X = [0, 255].$ Then, the histogram is $h: X \to \mathbb{R},$ where $h(x) = y$ means that the color x occurs y times on the image. In the following calculations we consider only colors which exist on the image $\bar{X} = \{x \in X: h(x) \neq 0\}.$ In such case as an estimator for mean and variancewe we use $\mathrm{mean}(\bar{X}) = \sum \limits_{x \in \bar{X}} h(x) \cdot x$ $\var(\bar{X})= \sum \limits_{x \in \bar{X}} h(x) \cdot (x-\mathrm{mean}(\bar{X}))^2.$ Now we can introduce cost function, which we minimize: $E(X_1,\ldots,X_k ,h) = \sum_{i = 1}^k p_i \cdot \left(-\ln(p_i) \frac{1}{2}\ln(2 \pi e) \frac{1}{2} \ln(\sigma_i^2) \right),$ where $\sigma_i^2 = \var(X_i),$ $p_i = \frac{\|X_i\|}{\|\bar{X}\|},$ $\bar{X} = X_1 \cup \ldots \cup X_k.$ Example: The effect of CEC Multilevel Thresholding on sample gray-scale printed document for 2 thresholds (i.e. 3 clusters). Original image: Results of thresholding: lower cluster middle cluster upper cluster The histogram along with chosen thresholds: