Rank-ordered filter for edge enhancement of cellular images using interval type II fuzzy set

Abstract.

An edge-enhancement technique using an interval type II fuzzy set that uses rank-ordered filter to enhance the edges of cellular images is proposed. When cellular images from any laboratory are digitized, scanned, and stored, some kind of degradation occurs, and directly using a rank-ordered filter may not produce clear edges. These images contain uncertainties, present in edges or boundaries of the image. Fuzzy sets that take into account these uncertainties may be a good tool to process these images. However, a fuzzy set sometimes does not produce better results. We used an interval type II fuzzy set, which considers the uncertainty in a different way. It considers the membership function in the fuzzy set as “fuzzy,” so the membership values lie within an interval range. A type II fuzzy set has upper and lower membership levels, and with the two levels, a new membership function is computed using Hamacher t-conorm. A new fuzzy image is formed. A rank-ordered filter is applied to the image to obtain an edge-enhanced image. The proposed method is compared with the existing methods visually and quantitatively using entropic method. Entropy of the proposed method is higher (0.4418) than the morphology method (0.2275), crisp method (0.3599), and Sobel method (0.2669), implying that the proposed method is better.

1. Introduction

Fuzzy set theory considers vagueness in the form of membership function. But the membership function is not always properly defined. It may be Gaussian, exponential, triangular, or any other, and it depends on user choice. Using different types of membership function, different results are obtained. Zadeh¹ in 1975 introduced another degree of freedom to ordinary (type I) fuzzy set that considers the membership function as “fuzzy,” and this is called type II fuzzy set. As it considers the membership function to be fuzzy, so the membership function of Type II fuzzy set lies in an interval range. Recently, type II fuzzy set has been used in many areas of image processing and pattern recognition, especially in cellular images.

Edge detection is an essential step for cellular image analysis and gives us important information for image interpretation. Edge pixels are defined as locations in an image where there are changes in intensity gray levels. Sometimes, edges are not properly visible due to uneven illumination, detector noise, and imaging system aberration, which introduce blurring in cell images, and in that case, boundary detection becomes very difficult.

There are various types of edge detection operators such as Prewitt, Sobel, Robert, Canny, and also many other edge detection methods. These methods look for an abrupt change in intensity gray level that indicates the boundary between two regions. The advantage of these methods is their simplicity and fast computation. Most of the edge detection techniques either have fixed result such as edge thickness or some parameters must be selected such as threshold and $\sigma$ for better result. There are also many other fuzzy methods to detect the edges of images. However, each edge detector has its advantages and disadvantages. In all of these edge detectors, either true edges are missed or false edges (nonexistent edges) are detected or disconnected edges are produced. Russo² suggested a fuzzy method for edge detection. Chaira³ suggested an intuitionistic fuzzy edge enhancement technique using rank-ordered filter. Xu⁴ suggested an edge enhancement technique for cellular images using a rank-ordered filter followed by edge detection using Canny’s method.⁵ In their method, edges are enhanced using a median filter, but in some edge images, edges are so bright that the boundaries are not clear. Khamy et al.⁶ suggested a modified fuzzy Sobel-based edge detector and then fuzzy reasoning is used to find the edges. Fuzzy edges can also be obtained using fuzzy mathematical morphology.⁷ In fuzzy mathematical morphology, maximum and minimum operators are replaced by s and t norms. Then supremum and infimum operators are replaced by argmax and argmin. Edge image is obtained by taking the difference between dilated and eroded images. Barrenechea et al.⁸ suggested an interval-valued fuzzy relation to construct fuzzy edges. Bustince et al.⁹ constructed edges using interval-valued fuzzy sets and applied it on general images and not on cellular images.

Ensafi and Tizhoosh¹⁰ suggested type II fuzzy image thresholding on general images. Bhutada et al.¹¹ used wavelet and curvelet transform and the features are applied separately and adaptively in different regions. Their work is based on the removal of fuzzy edges in the homogeneous region. Chen and Chen¹² proposed convex curve membership function and then smoothed the image using median filter. Melin et al.¹³ suggested an edge-detection method on nonmedical images using interval type 2 fuzzy logic, where an interval type 2 fuzzy inference system was used. Chaira¹⁴ also suggested a contrast enhancement using interval type II fuzzy set. In this method, overall contrast of the image is enhanced to make the objects more prominent with respect to background. Dark regions become darker and bright regions become brighter.

But in the proposed method, only edges are highlighted. Edges are the sharp discontinuities or rapid changes in intensity gray levels in the image that correspond to high-frequency components. Median filter in image processing is used to remove impulse noise as it allows a great deal of high spatial frequency detail to pass. So, median filtering is used in the proposed edge-enhancement method. On enhancing the edges, physicians can have some knowledge on any deformities present in the images, and then these edge-enhanced images will be used for edge detection using any edge-detecting techniques.

However, it is very difficult to find the edges in cellular images as these biological images do not have high enough contrast. When cellular images from any laboratory are digitized, scanned, and stored, some degradation occurs at the output. So these images are not clear. In cellular imaging, usually a large number of nuclei exist in a tissue sample. Many edge-detection schemes have been developed by researchers. However, challenges remain on correctly and accurately detecting edges of nuclei. The reason is that cellular images mostly have uneven brightness and it is difficult to separate nuclei from background. Considering the difficulties in cellular images, an idea to generate enhanced edges in cellular images using interval type II fuzzy set that uses rank-ordered filter is proposed. Image is considered to be a type II fuzzy image, and using Hamacher t-conorm, a new membership function is constructed using upper and lower levels of the type II membership function to obtain a new image. Then a $3\times 3$ median filter is applied on the new image. The resulting image is an edge-enhanced image.

The rest of the paper is organized as follows. Section 2 overviews the introduction to type II fuzzy set. Section 3 briefly outlines fuzzy t-norm and t-conorm. Section 4 details the proposed method. Section 5 exhibits and discusses the results, and finally, conclusion is drawn in Sec. 6.

2. Introduction to Type II Fuzzy Set

The main problem in type I fuzzy set is that the membership function defined is not always certain. Membership values are defined by human knowledge, and these are difficult to agree with accurate membership values. This is the reason why different fuzzy techniques suggested by different authors produce different results. Membership function of type II fuzzy set is considered fuzzy as against the membership function in type I fuzzy set. It accounts for the uncertainty in shape or location or other parameters by considering another degree of freedom to type I fuzzy set for better representation of uncertainty.

If type I membership function is blurred by shifting the points in the membership function curve (triangular, Gaussian, etc.) either to left or right, then there will be no longer single-valued membership function. Type II fuzzy set has grades of membership that are themselves fuzzy. Membership grade in type II fuzzy set can be any subset that lies in [0, 1], and this is the primary membership. To each primary membership, there is a secondary membership that also lies between [0, 1], which defines the possibility of primary membership.¹⁵ It means that to each input $x$ in type I fuzzy set (one-dimensional), there is a primary membership and the uncertainty in the primary membership is reflected by a bounded interval $[\text{Lower}(x),\text{Upper}(x)]$. Uncertainty is represented by a region called footprint of uncertainty shown in Fig. 1.¹¹

Fig. 1.

(a) Type I membership function and (b) interval of type II fuzzy membership function (shaded region) is the footprint of uncertainty.

Type II fuzzy set may be written as

$$A_{\text{type II}}=\{x,{\stackrel{͡}{\mu }}_A(x)|x\in X\},$$

where ${\stackrel{͡}{\mu }}_A(x)$ is a type-2 membership function.

Type II fuzzy set can be described in terms of lower and upper membership function. These are written as

$$\begin{gathered} {\mu }^{\text{upper}}={[\mu (x)]}^{\alpha } \\ {\mu }^{\text{lower}}={[\mu (x)]}^{1/\alpha }, \end{gathered}$$ (1)

where $0<\alpha <1$ and ${\mu }^{\text{upper}}$ and ${\mu }^{\text{lower}}$ are the upper and lower membership functions, respectively. So, a more practical form of representing type II fuzzy set is written as

$$\begin{gathered} A_{\text{type II}}=\{x,{\mu }_U(x),{\mu }_L(x)|x\in X\}, \\ {\mu }_L(x)<\mu (x)<{\mu }_U(x),\mu \in [\mathrm{0,1}]. \end{gathered}$$

3. Fuzzy t-Norm and t-Conorm

A fuzzy set is an extension of an ordinary set theory. Operators such as union, intersection, and complement in ordinary set theory can be extended to fuzzy theory. In fuzzy set theory, these operators are called t-norms (T) and t-conorms ($T^{*}$). At present, there are many fuzzy t-norms and t-conorms suggested by Yager,¹⁶ Hamacher,¹⁷ Dombi,¹⁸ Weber,¹⁹ and so on. These operators either contain min or max term or do not contain min or max term, i.e., purely algebraic. In this work, algebraic operator is used as this operator does not contain min or max term and reveals the actual value.

Definitions suggested by Roychowdhury²⁰ and Weber¹⁹ are written as follows:

$T^{*}:[\mathrm{0,1}]\times [\mathrm{0,1}]\to [\mathrm{0,1}]$ is a t-conorm iff it satisfies the following properties where $x,y,z\in [\mathrm{0,1}]$:

• 1.Boundary condition: $T^{*}(\mathrm{0,0})=0$ and $T^{*}(\mathrm{0,1})=T^{*}(\mathrm{1,0})=T^{*}(\mathrm{1,1})=1$
• 2.Commutativity: $T^{*}(x,y)=T^{*}(y,x)$
• 3.Monotonicity: $T^{*}(x,y)\le T^{*}(x,z)$ if $y\le z$
• 4.Associativity: $T^{*}[T^{*}(x,y),z]=T^{*}[x,T^{*}(y,z)]$
• 5.Zero-identity: $T^{*}(x,0)=x$

  T-conorm, $T^{*}$, is called Archimedean iff

• 6.$T^{*}$ is continuous
• 7.$T^{*}(x,x)>x$ for all $x\in (\mathrm{0,1})$

$T:[\mathrm{0,1}]\times [\mathrm{0,1}]\to [\mathrm{0,1}]$ is a t-norm iff it satisfies the following properties where $x,y,z\in [\mathrm{0,1}]$:

• 1.Boundary condition: $T(\mathrm{1,1})=1$ and $T(\mathrm{0,1})=T(\mathrm{1,0})=T(\mathrm{0,0})=0$
• 2.Associativity: $T[T(x,y),z]=T[x,T(y,z)]$
• 3.Commutativity: $T(x,y)=T(y,x)$
• 4.Monotonicity: $T(x,y)\le T(x,z)$ if $y\le z$
• 5.One-identity: $T(x,1)=x$

  A T-norm, $T$, is called Archimedian iff

• 6.$T$ is continuous
• 7.$T(x,x)<x$ for all $x\in (\mathrm{0,1})$

One can use any triangular (t) operators to generate a new membership function. In this work, Hamacher t-norm, which is an algebraic t-norm and t-conorm, is used to generate a new membership function.

T-norm: $H_{\gamma }(x,y)=x.y/\gamma +(1-\gamma ).(x+y-x·y)$, $\gamma >0$, with decreasing generator

$$f_{\gamma }(x)=\frac{1}{\gamma }\ln \frac{\gamma +(1-\gamma ).x}{x}\mathrm{and}{f}_{\gamma }^{-1}(y)=\frac{\gamma .e^{-\gamma ·y}}{1-(1-\gamma ).e^{-\gamma ·y}}.$$

This t-norm is one-identity.

$$H_{\gamma }(x,1)=\frac{x}{\gamma +(1-\gamma ).(x+1-x)}=\frac{x}{\gamma +(1-\gamma )}=x$$

$H_{\gamma }^{*}(x,y)=H_{\gamma }^{*}(y,x)$ - commutativity. Likewise, monotonicity can also be verified.

T-conorm: $H_{\gamma }^{*}(x,y)=x+y-x.y-(1-\gamma )x.y/1-(1-\gamma ).x.y$, with increasing generator

$$\begin{gathered} g_{\gamma }(x)=\frac{1}{\gamma }\ln \frac{\gamma +(1-\gamma ).(1-x)}{1-x}, \\ {g}_{\gamma }^{-1}(y)=\frac{1-e^{-\gamma .y}}{1-(1-\gamma ).e^{-\gamma .y}}. \end{gathered}$$

This is zero-identity as

$$H_{\gamma }^{*}(x,0)=\frac{x}{1}=x$$

As in t-norm, t-conorm shows commutative property and monotonicity.

4. Methodology

An image (say $A$) of size $M\times M$ is initially normalized using the formula

$${\mu }^{\prime }(g)=\frac{g-g_{\min }}{g_{\max }-g_{\min }},$$ (2)

where $g$ is the gray level of the image ranging from 0 to L-1. $g_{\max }$ and $g_{\min }$ are the maximum and minimum values of the gray levels of the image. ${\mu }^{\prime }(g)$ is the membership function of the image, i.e., the degree of belongingness of the pixels in an image.

Upper and lower ranges of the interval type II fuzzy membership function are calculated using Eq. (1). Fuzzy linguistic hedges²¹ with $\alpha \le 1$ are used to generate the lower and upper membership functions. Linguistic hedge is used to modify the membership values.

Using Hamacher t-conorm, a new membership function is created, which is written as

$$U^A({\mu }^{\text{upper}},{\mu }^{\text{lower}})=\frac{{\mu }^{\text{upper}}(g_{ij})+{\mu }^{\text{lower}}(g_{ij})+(\gamma -2){\mu }^{\text{upper}}.{\mu }^{\text{lower}}}{1-(1-\gamma ){\mu }^{\text{upper}}.{\mu }^{\text{lower}}},$$ (3)

where ${\mu }^{\text{upper}}(g_{ij})$ and ${\mu }^{\text{lower}}(g_{ij})$ in Eq. (3) are the upper and lower membership functions of the interval type II fuzzy image, respectively, which are computed using Eq. (1).

$$\begin{gathered} {\mu }^{\text{lower}}(g_{ij})={[{\mu }^{\prime }(g_{ij})]}^{1/\alpha }, \\ {\mu }^{\text{upper}}(g_{ij})={[{\mu }^{\prime }(g_{ij})]}^{\alpha }. \end{gathered}$$

So, a new type II fuzzy image is formed using Hamacher t-conorm. This new fuzzy image is not a specific edge or nonedge set. It will be discussed in a succeeding section that on applying median filter on this new image, edges in the image are highlighted.

Value of $\gamma =1.4$ is chosen. Selection of gamma is done by trial and error method followed by visual and statistical analysis. After thorough experimentation, it is observed that with $\gamma =1.8$, edges are not enhanced clearly. With $\gamma =1$ or 1.4, edges are enhanced clearly. So, $\gamma =1.4$ is used.

A sample result is shown in Fig. 2. It is observed that with $0<\gamma <1.5$, enhanced edges of the image look better. With $\gamma =1.8$, i.e., $\gamma >1.5$, edges are not clearly visible.

Fig. 2.

(a) Blood cell, (b) edge image for $\gamma =1.8$, (c) edge image for $\gamma =1.4$, and (d) edge image for $\gamma =1.0$.

Next decision has to be made about the value of $\alpha$. As mentioned earlier, fuzzy linguistic hedges are used to generate the lower and upper membership functions from type 1 fuzzy membership function. In most applications, parameter $\alpha$ is usually determined heuristically, which lies in the range $0<\alpha <1$. In this experiment, the value of $\alpha =0.75$, which should lie between 0 and 1, is chosen by trial and error method, and with this value, the enhanced edge image (after median filtering) looks better.

Median filter of size $3\times 3$ is applied on the images. In image processing, median filter is used to remove any impulse noise present, which is more robust. Since the median is one of the pixels in a window, it will not create any new pixel when it crosses the edge as compared to averaging filters. So, it preserves the discontinuities in a better way.

The area of the image covered by the median filter is the image window. The median value surrounding the current pixel $A(m,n)$ of the image window is written as follows:

$$X(m,n)=\text{median}[A(m-i,n-j)],i,j\in w(m,n),$$

and $w(m,n)$ is the window. $m=\mathrm{1,2},3,\dots M$, $n=\mathrm{1,2},3,\dots N$.

Total variation in the $3\times 3$ image window with respect to the median of the window is computed as

$$Z(m,n)=\frac{1}{9}\sum _{i,j\in W}\mathrm{abs}[A(i,j)-X(m,n)].$$ (4)

$i,j$ are the pixels in the $3\times 3$ window. $W$ is the image window. The size of the median filter and the size of the image window are same. For each pixel position, $Z(m,n)$ is computed and a difference matrix is formed. The new matrix is an edge enhanced image.

The algorithm of the proposed method is as follows:

• 1.Select an image.
• 2.Normalize the image gray levels to lie the values in the interval [0, 1] using Eq. (2).
• 3.Select an alpha value using Eq. (1).
• 4.Compute upper and lower membership functions.
• 5.Calculate a new membership function using Eq. (3).
• 6.Apply median filter to the image.
• 7.Edge enhanced image is obtained.

5. Results and Discussion

The algorithm is applied on several images, and the results on five images are shown. Images chosen are either very dark or very light or blurry. They are prepared in a disinfected slide. After smear preparation, these images are optically grabbed by Leica Observer (LeicaDM750, Leica Microsystems) under $100\times$ oil objective (N.A. 1.515) having 0.064 m resolution. Some input images are the microphotographs from blood smears. Some cell images are taken from trypan blue exclusion test where trypan blue is added to identify live cells. Many blurry images are also downloaded from different university websites (University of Utah, Wisconsin, Berkeley). The proposed method is compared with the edge image of Xu’s median filter based edge detection method,⁴ Sobel operator, and fuzzy morphology method.⁷

Also, a simple form of membership function may be used by taking the mean of the two membership levels. Result on a single image is shown in Fig. 3. From the result, it is observed that the edges are not enhanced properly.

Fig. 3.

(a) Cell image and (b) edge image using mean of the membership levels.

Figure 4 is a cell image where the cells are not visible. It is observed that the spatial artifacts are not visible in other methods, but these are in picture in the proposed method. It means that the proposed edge-enhancement method can highlight the edges that are not visible. Even with the morphological method, Sobel edge-operator method, and Xu’s method, edges are not visible clearly. Type II fuzzy method can show enhanced edges. This is the worst image that has been selected to show the efficacy of the proposed method.

Fig. 4.

(a) Cell image, (b) edge image using Xu’s method, (c) edge image using fuzzy morphology, (d) edge image using the proposed method, and (e) edge image using Sobel operator.

Figure 5 is an image of a blood cell where trypan blue dye is added to find viable cell present in cell suspension. It is observed that the intensity changes in the nucleus and cytoplasm are not visible in the original image. Sobel’s method produces broken edges as compared to Xu’s method, but the enhanced edges are not so clear as compared to the proposed type II fuzzy method, where the edges of nucleus and cytoplasm are clear and prominent. Intensity changes of the cells in the image are visible in the form of circular rings. Morphological edge image does not show enhanced edges.

Fig. 5.

(a) Blood cell, (b) edge image using Xu’s method, (c) edge image using fuzzy morphology, (d) edge image using the proposed method, and (e) edge image using Sobel operator.

Figure 6 shows a cell image where Xu’s edge method performs better than Sobel’s method in terms of brightness, but the edges are much more prominent and enhanced using the proposed type II fuzzy method.

Fig. 6.

(a) Cell image, (b) edge image using Xu’s method, (c) edge image using fuzzy morphology, (d) edge image using the proposed method, and (e) edge image using Sobel operator.

Figure 7 is also a cell image where inner rings in the image using the proposed method in Fig. 7(d) and Xu’s method in Fig. 7(b) are visible, but the inner rings are shown better in the proposed method. Sobel’s edge image shows some broken edges.

Fig. 7.

(a) Abnormal RBC image, (b) edge image using Xu’s method, (c) edge image using fuzzy morphology, (d) edge image using the proposed method, and (e) edge image using Sobel operator.

From the results, it is observed that fuzzy morphological edge image does not enhance all the edges of the cells. Xu’s method performs better than Sobel edge detector method and morphological edge method. But the proposed method performs better than Xu’s method with clear edges. So, it is concluded that the proposed method performs the best among the existing methods. As the edges of the cellular images are not clear, edges are required to be enhanced so that the physicians can have a clear view about the image structures.

Using $5\times 5$ median filter, edge enhancement is also performed. It is observed from Fig. 8 that the edges in the images are not properly visible. This is due to the reason that as the filter size is increased, edges are not clearly detected.

Fig. 8.

(a) Abnormal RBC image and (b) edge image using $5\times 5$ median filter.

5.1. Quantitative Analysis

Apart from the visual assessment, quantitative assessment is also required. Enhancement results are appreciated if the visual appearance of the edge enhanced images looks clear and better. By visual quality, overenhancement or any unwanted artifacts can be judged. For cellular images, quantitative analysis of quality assessment is very difficult. Since some kind of assessment is to be done, Shannon’s entropy is used in this work to compute the quality. Though the entropic method is a very old method, but at least one can have some information about the images. It measures the amount of information associated in the gray levels in an image. So, the entropies of the original image and the enhanced images are computed. Shannon’s entropy is defined as

$$E(X)=\frac{1}{M\times M\log 2}\sum _i\sum _j(-\mu (x_{ij})·\log [\mu (x_{ij})]-\{1-\mu (x_{ij})·\log [\mu (x_{ij})]\}),$$

where $\mu (x_{ij})$ is the membership value of $(i,j)$’th pixel of the image, and $M\times M$ is the size of the image.

Table 1 shows the entropy of all the images for all the methods and the average is computed. It is observed from the table that Xu’s method shows average entropy of 0.3599, Sobel edge method shows average entropy of 0.2699, morphology method shows average entropy of 0.2275, and the proposed interval type II fuzzy method shows average entropy of 0.4418. The proposed method has an entropic value higher than the other methods. Highest entropic value of an image brings out the best information content of the image. Entropy closer to the image brings out the maximum information from the image.

Table 1.

Entropy for different methods.

Images	Different methods			Sobel
	Crisp method (Xu’s method)	Morphology method	Proposed type II fuzzy method
Basophil (0.5952)	0.3544	0.2452	0.4560	0.2754
Nuclei (0.7617)	0.2906	0.1832	0.4099	0.2918
Cell 1 (0.4608)	0.2466	0.1883	0.3677	0.2083
Cell 3 (0.8364)	0.2336	0.1468	0.3662	0.2086
Cell 2 (0.6292)	0.4592	0.2389	0.5452	0.2534
Cell 4 (0.6301)	0.5479	0.2176	0.5865	0.2597
RBC (0.8512)	0.3955	0.3017	0.4365	0.3421
Nuclei 1 (0.6312)	0.5181	0.3515	0.5732	0.4008
Cell 5 (0.2450)	0.1936	0.1740	0.2352	0.1897
Average (0.6149)	0.3599	0.2275	0.4418	0.2699

So, the proposed method performs the best. Xu’s method performs better than the Sobel operator and morphological method; Sobel edge image performs better than morphological edge image.

6. Conclusion

Edge enhancement is a crucial task in cellular image processing. As cellular images are not equally illuminated, so the edges are not properly highlighted. Edges give an important clue for interpreting cellular images especially in detecting the boundaries of abnormal cells/lesions. Though there are many edge-enhancement techniques in literature, edge enhancement of cellular images is very few. In this paper, an edge-enhancement technique using interval type II fuzzy set on cellular images is proposed. Type II fuzzy set represents the uncertainty in a better way, so the results using the proposed method are observed to be better. Results are quantitatively and qualitatively compared with the existing methods, and it is observed that the proposed method using type II fuzzy method performs better. Type II fuzzy set is useful to those images where type I fuzzy set does not perform better.

This technique can also be applied in noncytology applications. It can be used not only on blood cells, but also in any boundary enhancement of structures. In noncytological cases, this algorithm can be better used in edge enhancement of tumor or any abnormal lesions where the boundaries are not clearly visible, and in that case, it is very difficult for the medical experts to visualize with bare eye. On enhancing the edges, physicians can make out some ideas for further diagnosis.

Future work is to extract the edges from the enhanced edge image. In cellular images, edges are not clear, so obtaining edges from an enhanced edge images is a crucial task. In that case, filtering may be required in removing false edges and extracting true edges.

Biography

Tamalika Chaira received her PhD from the ECE Department, Indian Institute of Technology, India. She is a research scientist at the Indian Institute of Technology, Delhi. She is the author of two books, Fuzzy Image Processing and Applications with MATLAB and Medical Image Processing—Advanced Fuzzy Set Theoretic Techniques with CRC Press. She has published numerous papers in international journals and conferences. Her research interests include image processing and medical image analysis using fuzzy set/intuitionistic fuzzy set/type II fuzzy set theory.