PseudoSemiOverlap FunctionsBased Fuzzy Rough Sets Applied to Image Edge Extraction ()
1. Introduction
Zadeh introduced fuzzy sets in 1965 [1], and Pawlak explored rough sets in 1982 [2]. In 1990, Dubois and Prade combined fuzzy sets and rough sets using the fuzzy operators min and max to create fuzzy rough sets [3]. Since then, numerous scholars have explored the theory of fuzzy rough sets and their practical applications in depth. In 2002, Radzikowska et al. employed a broader method for fuzzily rough sets and introduced a fuzzy rough set that relies on Tnorm and fuzzy implication [4]. Subsequently, Qiao [5] and Wen et al. [6] formulated the (IO, O)fuzzy rough sets. Zhang et al. [7] introduced (I, O)fuzzy rough sets by substituting the IO with a broader I in the (IO, O)fuzzy rough sets. Wu et al. [8] proposed a novel form of (I, T)fuzzy rough sets, relying on the general fuzzy binary relation. Mieszkowicz Rolka et al. [9] and Zhan et al. [10] presented the theories of variable precision fuzzy rough sets and coveringbased multigranulation fuzzy rough sets, respectively. These theories have been widely used in digital image processing [11] [12], attribution reduction [13] [14], webpage classification [15], tumor detection [16], big data analysis [17], and other applications [18] [19].
With the advancement of fuzzy rough sets based on various operators, fuzzy rough sets based on clustering functions with overlap function as an important representative have performed well in image edge extraction [20][23] and decisionmaking application [24][26]. Along with the rapid development of overlap functions as a class of clustering functions, scholars have proposed more extensive clustering functions. For example, Zhang et al. [26] removed the symmetry in the overlap function, proposed the pseudooverlap function, and discussed its applications in decisionmaking and image processing. In 2022, Zhang [27] updated the concept of overlap functions by removing the right continuity. Therefore, semioverlap functions were proposed as new aggregation functions. Subsequently, a novel classification algorithm based on semioverlap functions was discovered and successfully applied. In addition to clustering functions, other proposed functions include quasioverlap functions [28], intervalvalued pseudooverlap functions [29], and general overlap functions [30]. Simultaneously, many scholars have combined the clustering function with fuzzy rough sets and proposed new fuzzy rough sets. Zhang et al. [31] proposed a fuzzy rough set comprising overlap functions and fuzzy implication and applied it to image edge extraction and attribute reduction. A link between a group of approximate operators in (I, O)fuzzy rough sets and a group of fuzzy dilation and erosion operators is present in image edge extraction applications [7]. Thus, the IOFCM image edge extraction algorithm was introduced and effectively implemented. However, for practical applications, due to the strict requirement of continuity aspects of the overlap functions, both left and right continuity must be satisfied. Hence, the flexibility of the algorithm is low, and its practical applications are limited.
Therefore, this paper conducts research by considering the broad range of applications of fuzzy rough sets, as well as the successful utilization of fuzzy rough sets based on clustering functions in image edge extraction. First, the symmetry of semioverlap functions was removed, and the pseudosemioverlap functions with their two construction methods were proposed. Second, the (I, O)fuzzy rough set was extended to the (I, PSO)fuzzy rough set, and the overlap function was replaced by a pseudosemioverlap function. Further, the theory and properties related to the (I, PSO)fuzzy rough set, along with the looser constraints of the PSO operator, were explored for wider application in image edge extraction. Compared to existing applications of fuzzy rough sets in image edge extraction, (I, PSO)fuzzy rough sets are superior in the following aspects: 1) The pseudosemioverlap function is an important aggregation function that can effectively distinguish the foreground and background of an image. Compared to existing clustering functions [32], the pseudosemioverlap function has more relaxed requirements for continuity and does not need symmetry. Therefore, (I, PSO)fuzzy rough sets have broader applications, better practical adaptability, and a higher theoretical conversion rate. 2) The upper and lower approximation operators in the (I, PSO)fuzzy rough set correlate with the fuzzy dilation and fuzzy erosion operators, respectively, in fuzzy mathematical morphology. Therefore, a new set of morphological operators with higher flexibility, IPSOFMM operators, is proposed, and the relevant properties in fuzzy rough sets and fuzzy mathematical morphology are studied. 3) The FCMIPSO image edge extraction algorithm obtained via a combination of the fuzzy Cmeans algorithm and the IPSOFMM operators exhibits superior image edge extraction results compared to those obtained using the Canny operator, Laplacian operator, Prewitt operator, Roberts operator, and Sobel operator. In other words, the FCMIPSO algorithm provides improved image edge information with a minimum noise introduction rate.
The rest of this paper is organized as follows: Section 2 presents the fundamental concepts. Section 3 begins with the definition of a pseudooverlap function and elaborates on two methods for constructing this function. Subsequently, (I, PSO)fuzzy rough sets are defined, and the related theories and properties are systematically described. Section 4 introduces a new set of fuzzy mathematical morphological operators, IPSOFMM operators, and delves into their properties. In Section 5, the FCMIPSO image edge extraction algorithm is proposed, and its performance is assessed using five gray images. The experimental results demonstrate the exceptional performance of the FCMIPSO algorithm over existing classical algorithms. An overview of the study is presented in Figure 1. The concluding remarks, along with subsequent future studies, are summarized in Section 6.
Figure 1. Outline of the study.
2. Fundamental Definitions
Definition 1 ([20]). A bivariate function $f:{\left[0,\text{1}\right]}^{\text{2}}\to \left[0,\text{1}\right]$
, for any $m,n\in \left[0,1\right]$
,
(1) f (m, n) = 0 iff mn = 0;
(2) f (m, n) = 1 iff mn = 1;
(3) f (m, n) = f (n, m);
(4) f is increasing;
(5) f is continuous.
A binary function is termed an overlap function (denoted as O) if it conforms to conditions (1)(5).
Definition 2 ([24]). A binary function $f:{\left[0,\text{1}\right]}^{\text{2}}\to \left[0,\text{1}\right]$
, for any $m,n\in \left[0,1\right]$
,
(1) f (m, n) = 0 iff mn = 0;
(2) f (m, n) = 1 iff mn = 1;
(3) f (m, n) = f (n, m);
(4) f is increasing;
(5) f is leftcontinuous.
A binary function is termed a semioverlap function (denoted as SO) if it conforms to conditions (1)(5).
Definition 3 ([33]) A binary function $I:{\left[0,1\right]}^{2}\to \left[0,1\right]$
, for any $l,m,n\in \left[0,1\right]$
,
(I1) I (1, 1) = I (0, 0) = 1;
(I2) I (1, 0) = 0;
(I3) If $m\le n$
, then $I\left(m,l\right)\ge I\left(n,l\right)$
;
(I4) If $n\le l$
, then $I\left(m,n\right)\le I\left(m,l\right)$
.
If the above conditions are satisfied, the binary function is called a fuzzy implication (denoted as I).
Definition 4 ([28]) Assume O is an overlap function, and I is a fuzzy implication. Consider the fuzzy approximation space (U, R), where U is the domain and R is a fuzzy binary relation on U. To define a pair of fuzzy sets on U, a fuzzy set B in U (i.e., $B\in F\left(U\right)$
) can be considered: for any $m\in U$
,
$\overline{R}\left(B\right)\left(m\right)=\underset{n\in U}{\mathrm{sup}}O\left(R\left(m,n\right),B\left(n\right)\right)\text{,}$(1)
$\underset{\_}{R}\left(B\right)\left(m\right)=\underset{n\in U}{\mathrm{inf}}I\left(R\left(m,n\right),B\left(n\right)\right).$
(2)
where $\overline{R}\left(B\right)$
presents the fuzzy upper approximation and $\underset{\_}{R}\left(B\right)$
represents the fuzzy lower approximation in (I, O)fuzzy rough sets of B.
Definition 5 ([7]) Let B and C be fuzzy subsets of ${R}^{2}$
. Assume O is an overlap function, and I is a fuzzy implication. The expressions for the fuzzy dilation D_{O} (B, C) and fuzzy erosion E_{I} (B, C) of a gray image B by a gray structuring element C are as follows ($d\left(C\right)=\left\{mC\left(m\right)\ne 0\right\}\subseteq {R}^{\text{2}}$
) for $\forall m\in {R}^{2}$
:
${D}_{O}\left(B,C\right)\left(m\right)=\underset{n\in d\left(C\right)}{\mathrm{sup}}O\left(C\left(n\right),B\left(m+n\right)\right)\text{,}$
(3)
${E}_{I}\left(B,C\right)\left(m\right)=\underset{n\in d\left(C\right)}{\mathrm{inf}}I\left(C\left(n\right),B\left(m+n\right)\right).$
(4)
3. PseudoSemiOverlap Functions and (I, PSO)Fuzzy Rough Sets
This section proposes pseudosemioverlap functions and defines essential characteristics of (I, PSO)fuzzy rough sets.
Definition 6. A bivariate function $PSO:{\left[0,\text{1}\right]}^{\text{2}}\to \left[0,\text{1}\right]$
is named a pseudosemioverlap function (denoted as PSO) when it fulfills the following conditions:
(PSO_{1}) For any $m,n\in \left[0,1\right]$
, if mn = 0, then PSO (m, n) = 0;
(PSO_{2}) If m = n = 1, then PSO (m, n) = 1;
(PSO_{3}) PSO is increasing;
(PSO_{4}) PSO is leftcontinuous.
Example 1. A mapping $PSO:{\left[0,\text{1}\right]}^{\text{2}}\to \left[0,\text{1}\right]$
defined for any $m,n\in \left[0,1\right]$
, as (As shown in Figure 2).
(a) (b)
Figure 2. (a) Distribution of intervals of (2) in Example 1; (b) Visualization of the proposed function in Example 1.
$PSO\left(m,n\right)=\{\begin{array}{ll}\frac{mn}{7},\hfill & \text{if}\text{\hspace{0.17em}}\left(m,n\right)\in {A}_{1}\hfill \\ \frac{2mn}{7},\hfill & \text{if}\text{\hspace{0.17em}}\left(m,n\right)\in {A}_{2}\hfill \\ \frac{21mn+9}{30},\hfill & \text{if}\text{\hspace{0.17em}}\left(m,n\right)\in {A}_{3}\hfill \\ \frac{19mn+11}{30},\hfill & \text{if}\text{\hspace{0.17em}}\left(m,n\right)\in {A}_{4}\hfill \\ \frac{mn}{3},\hfill & \text{if}\text{\hspace{0.17em}}\left(m,n\right)\in {A}_{5}\hfill \\ \frac{mn}{2},\hfill & \text{if}\text{\hspace{0.17em}}\left(m,n\right)\in {A}_{6}\hfill \end{array}$
(5)
is a pseudosemioverlap function, where ${A}_{1}=\left\{\left(m,n\right)0\le m\le 0.4,m\le n\le 0.4\right\}$
, ${A}_{2}=\left\{\left(m,n\right)0<m\le 0.4,0\le n<m\right\}$
, ${A}_{3}=\left\{\left(m,n\right)0.4<m\le 1,m\le n\le 1\right\}$
, ${A}_{4}=\left\{\left(m,n\right)0.4<m\le 1,0.4\le n<m\right\}$
, ${A}_{\text{5}}=\left\{\left(m,n\right)0.\text{4}<m\le \text{1},0\le n<0.\text{4}\right\}$
, ${A}_{\text{6}}=\left\{\left(m,n\right)0\le m\le 0.\text{4},0.\text{4}<n\le \text{1}\right\}$
.
The specific intervals are distributed as follows:
Theorem 1. Assume that $PSO:{\left[0,\text{1}\right]}^{\text{2}}\to \left[0,\text{1}\right]$
is a pseudosemioverlap function. If PSO is commutative, then it is a semioverlap function.
Proof. The proof follows from Definitions 1 and 5.
Theorem 2. A bivariate function $PSO:{\left[0,\text{1}\right]}^{\text{2}}\to \left[0,\text{1}\right]$
is a pseudosemioverlap function if and only if two operators f and g exist on [0, 1] with
$PSO\left(m,n\right)=\frac{f\left(m,n\right)}{f\left(m,n\right)+g\left(m,n\right)}.$
(6)
Note: $f\left(m,n\right)+g\left(m,n\right)\ne 0$
.
Where
(1) f is increasing and g is decreasing;
(2) If mn = 0, then f (m, n) = 0;
(3) If m = n = 1, then g (m, n) = 0;
(4) Both f and g satisfy continuity.
Proof. ($\Leftarrow $
) By (2), for mn = 0, f (m, n) = 0. Then PSO (m, n) = 0, i.e., the binary function PSO satisfies (PSO_{1}).
By (3), for m = n = 1, g (m, n) = 0. Then PSO (m, n) = 1, i.e., the binary function PSO satisfies (PSO_{2}).
By (1), if ${m}_{\text{1}}\le {m}_{\text{2}}$
, for any $n\in \left[0,\text{1}\right]$
, then $f\left({m}_{1},n\right)g\left({m}_{2},n\right)\le f\left({m}_{2},n\right)g\left({m}_{1},n\right)$
. Next, by adding a nonnegative number f (m_{1}, n) f (m_{2}, n) to both sides of the equation simultaneously, $f\left({m}_{1},n\right)\left(f\left({m}_{2},n\right)+g\left({m}_{2},n\right)\right)\le f\left({m}_{2},n\right)\left(f\left({m}_{1},n\right)+g\left({m}_{1},n\right)\right)$
can be obtained, i.e., $PSO\left({m}_{\text{1}},n\right)\le PSO\left({m}_{\text{2}},n\right)$
. Following the same logic, if ${n}_{\text{1}}\le {n}_{\text{2}}$
, then $PSO\left(m,{n}_{1}\right)\le PSO\left(m,{n}_{1}\right)$
can be obtained. Therefore, the binary function PSO satisfies (PSO_{3}).
By (4), it is straightforward to note that the binary function PSO is continuous, i.e., the binary function PSO satisfies (PSO_{4}).
($\Rightarrow $
) It is known that PSO satisfies (PSO_{1})(PSO_{4}), and suppose that f (m, n) = PSO (m, n) and g (m, n) = 1 − PSO (m, n). Then, PSO (m, n) can be defined by f (m, n), g (m, n). Furthermore, note that conditions (1)(4) are satisfied.
Theorem 3. Assume $PS{O}_{\text{1}},PS{O}_{\text{2}},\cdots ,PS{O}_{m}$
be a pseudosemioverlap function and ${r}_{\text{1}},{r}_{\text{2}},\cdots ,{r}_{m}$
be nonnegative weights with ${\sum}_{j=1}^{m}{r}_{j}=1$
. Then $PSO\left(u,v\right)={\displaystyle {\sum}_{j=1}^{m}{r}_{j}}PS{O}_{j}\left(u,v\right)$
is also a pseudosemioverlap function.
Proof. (PSO_{1})(PSO_{3}) are easy proved. So, we prove PSO satisfies (PSO_{4}). If PSO is leftcontinuous, then for any $u\in \left[0,1\right]$
and for any $\left\{{v}_{i}i\in I\right\}\subseteq \left[0,1\right]$
, it follows that $PSO\left(u,\mathrm{sup}\left\{{v}_{i}i\in I\right\}\right)=\mathrm{sup}\left\{PSO\left(u,{v}_{i}\right)i\in I\right\}$
. Hence, we can get
$\begin{array}{c}PSO\left(u,\underset{i\in I}{\mathrm{sup}}{v}_{i}\right)={\displaystyle \sum _{j=1}^{m}{r}_{j}}PS{O}_{j}\left(u,\underset{i\in I}{\mathrm{sup}}{v}_{i}\right)={\displaystyle \sum _{j=1}^{m}{r}_{j}}\left(\underset{i\in I}{\mathrm{sup}}PS{O}_{j}\left(u,{v}_{i}\right)\right)\\ =\underset{i\in I}{\mathrm{sup}}{\displaystyle \sum _{j=1}^{m}{r}_{j}}PS{O}_{j}\left(u,{v}_{i}\right)=\underset{i\in I}{\mathrm{sup}}PS{O}_{j}\left(u,{v}_{i}\right)\end{array}$
Proposition 1. Assume ${\beta}_{1},{\beta}_{2},{\beta}_{3}:\left[0,\text{1}\right]\to \left[0,\text{1}\right]$
are continuous and increasing operators. For any $i\in \left[\text{1},\text{2},\text{3}\right]$
, iff m = 0, and ${\beta}_{i}\left(m\right)=1$
iff m = 1. Assuming that PSO is a binary pseudosemioverlap function, $PS{O}^{{\beta}_{1},{\beta}_{2},{\beta}_{3}}$
is defined as follows:
$PS{O}^{{\beta}_{1},}{{}^{{\beta}_{2},}}^{{\beta}_{3}}\left(m,n\right)={\beta}_{1}(PSO\left({\beta}_{2}\left(m\right),{\beta}_{3}\left(n\right)\right).$
(7)
Proof. It is easy to show that $PS{O}^{{\beta}_{1},{\beta}_{2},{\beta}_{3}}$
satisfies (PSO_{3}) and (PSO_{4}). If m = 0, ${\beta}_{2}\left(m\right)=0$
, and consequently, $PSO\left({\beta}_{2}\left(m\right),{\beta}_{3}\left(n\right)\right)=0$
. According to the known conditions, $PS{O}^{{\beta}_{1},{\beta}_{2},{\beta}_{3}}\left(m,n\right)=0$
can be easily obtained. Moreover, when n = 0, $PS{O}^{{\beta}_{1},{\beta}_{2},{\beta}_{3}}\left(m,n\right)=0$
can be obtained. Thus, $PS{O}^{{\beta}_{1},{\beta}_{2},{\beta}_{3}}$
satisfies condition (PSO_{1}). If m = 1, ${\beta}_{1}\left(m\right)={\beta}_{2}\left(m\right)=0$
can be determined. Then, based on the known conditions, $PSO\left({\beta}_{2}\left(m\right),{\beta}_{3}\left(n\right)\right)=1$
, and $PS{O}^{{\beta}_{1},{\beta}_{2},{\beta}_{3}}$
satisfies condition (PSO_{2}).
Definition 7. Assume that (U, R) is a fuzzy approximation space, where R represents the fuzzy binary relation on U. PSO represents a pseudosemioverlap function, while I represent a fuzzy implication. Given the fuzzy set B defined on the domain set U (i.e., $B\in F\left(U\right)$
), the following equation shows a couple of fuzzy sets in U for any $m\in U$
.
${\overline{R}}_{PSO}\left(B\right)\left(m\right)=\underset{n\in U}{\mathrm{sup}}PSO\left(R\left(m,n\right),B\left(n\right)\right),$
(8)
${\underset{\_}{R}}_{I}\left(B\right)\left(m\right)=\underset{n\in U}{\mathrm{inf}}I\left(R\left(m,n\right),B\left(n\right)\right).$
(9)
Here, ${\overline{R}}_{PSO}\left(B\right)$
and ${\underset{\_}{R}}_{I}\left(B\right)$
are known as the (I, PSO)fuzzy upper approximation and lower approximation of B, respectively.
Example 2. Assuming U = {m_{1}, m_{2}, m_{3}, m_{4}, m_{5}}, fuzzy set B = {0.4/m_{1}, 0.5/m_{2}, 0.7/m_{3}, 0.8/m_{4}, 0.6/m_{5}}. Table 1 lists the fuzzy relations R in the domain U.
Table 1. Fuzzy relation R in the domain U.
R 
m_{1} 
m_{2} 
m_{3} 
m_{4} 
m_{5} 
m_{1} 
1 
0.6 
0.7 
0.7 
0.6 
m_{2} 
0.6 
1 
0.4 
0.6 
0.8 
m_{3} 
0.7 
0.4 
1 
0.6 
0.7 
m_{4} 
0.7 
0.6 
0.6 
1 
0.6 
m_{5} 
0.6 
0.8 
0.7 
0.6 
1 
By Definition 7, the upper and lower approximations of the fuzzy set B in the approximation space (U, R) are deduced as follows (these relevant functions are used, including PSO_{1} and I_{1}):
Note:
${I}_{\text{1}}\left(m,n\right)=\text{min}\left(\text{1},\text{1}m+n\right),$
(10)
$PS{O}_{1}\left(m,n\right)=\{\begin{array}{ll}{m}^{2},\hfill & {m}^{2}\le {n}^{3},\hfill \\ n,\hfill & {m}^{2}>{n}^{3}.\hfill \end{array}$
(11)
${\overline{R}}_{PSO}\left(B\right)\left({m}_{1}\right)=\text{sup}\left\{0.40,\text{\hspace{0.17em}}0.50,\text{\hspace{0.17em}}0.70,\text{\hspace{0.17em}}0.49,\text{\hspace{0.17em}}0.60\right\}=0.7;$
${\overline{R}}_{PSO}\left(B\right)\left({m}_{2}\right)=\text{sup}\left\{0.40,\text{\hspace{0.17em}}0.50,\text{\hspace{0.17em}}0.16,\text{\hspace{0.17em}}0.36,\text{\hspace{0.17em}}0.60\right\}=0.6;$
${\overline{R}}_{PSO}\left(B\right)\left({m}_{3}\right)=\text{sup}\left\{0.40,\text{\hspace{0.17em}}0.50,\text{\hspace{0.17em}}0.70,\text{\hspace{0.17em}}0.36,\text{\hspace{0.17em}}0.60\right\}=0.7;$
${\overline{R}}_{PSO}\left(B\right)\left({m}_{4}\right)=\text{sup}\left\{0.40,\text{\hspace{0.17em}}0.50,\text{\hspace{0.17em}}0.70,\text{\hspace{0.17em}}0.80,\text{\hspace{0.17em}}0.60\right\}=0.8;$
${\overline{R}}_{PSO}\left(B\right)\left({m}_{5}\right)=\text{sup}\left\{0.40,\text{\hspace{0.17em}}0.50,\text{\hspace{0.17em}}0.70,\text{\hspace{0.17em}}0.36,\text{\hspace{0.17em}}0.60\right\}=0.7;$
${\underset{\_}{R}}_{I}\left(B\right)\left({m}_{1}\right)=\mathrm{inf}\left\{0.40,\text{\hspace{0.17em}}0.90,\text{\hspace{0.17em}}1.00,\text{\hspace{0.17em}}1.00,\text{\hspace{0.17em}}1.00\right\}=0.4;$
${\underset{\_}{R}}_{I}\left(B\right)\left({m}_{2}\right)=\mathrm{inf}\left\{0.80,\text{\hspace{0.17em}}0.50,\text{\hspace{0.17em}}1.00,\text{\hspace{0.17em}}1.00,\text{\hspace{0.17em}}0.80\right\}=0.5;$
${\underset{\_}{R}}_{I}\left(B\right)\left({m}_{3}\right)=\mathrm{inf}\left\{0.70,\text{\hspace{0.17em}}1.00,\text{\hspace{0.17em}}0.70,\text{\hspace{0.17em}}1.00,\text{\hspace{0.17em}}0.90\right\}=0.7;$
${\underset{\_}{R}}_{I}\left(B\right)\left({m}_{4}\right)=\mathrm{inf}\left\{0.70,\text{\hspace{0.17em}}0.90,\text{\hspace{0.17em}}1.00,\text{\hspace{0.17em}}0.80,\text{\hspace{0.17em}}1.00\right\}=0.7;$
${\underset{\_}{R}}_{I}\left(B\right)\left({m}_{5}\right)=\mathrm{inf}\left\{0.80,\text{\hspace{0.17em}}0.70,\text{\hspace{0.17em}}1.00,\text{\hspace{0.17em}}1.00,\text{\hspace{0.17em}}0.60\right\}=\mathrm{0.6.}$
Subsequently, the upper and lower approximation sets of B in the approximate space are as follows:
${\overline{R}}_{PSO}\left(B\right)=\left\{0.7/{m}_{1},\text{\hspace{0.17em}}0.6/{m}_{2},\text{\hspace{0.17em}}0.7/{m}_{3},\text{\hspace{0.17em}}0.8/{m}_{4}\text{,}\text{\hspace{0.17em}}0.7/{m}_{5}\right\};$
${\underset{\_}{R}}_{I}\left(B\right)=\left\{0.4/{m}_{1},\text{\hspace{0.17em}}0.5/{m}_{2},\text{\hspace{0.17em}}0.7/{m}_{3},\text{\hspace{0.17em}}0.7/{m}_{4}\text{,}\text{\hspace{0.17em}}0.6/{m}_{5}\right\}.$
The example illustrates the calculation process of (I, PSO)fuzzy rough sets, and then the properties of (I, PSO)fuzzy rough sets are demonstrated.
Theorem 4. Assuming PSO as a pseudosemioverlap function, R as a fuzzy reflexive relation, and I as a fuzzy implication. For (I, PSO)fuzzy rough sets, the following conditions apply for ${\overline{R}}_{PSO}\left(B\right)$
and ${\underset{\_}{R}}_{I}\left(B\right)$
:
(1) ${\overline{R}}_{PSO}(\varnothing )=\varnothing $
;
(2) ${\underset{\_}{R}}_{I}\left(U\right)=U$
;
(3) For any $m,n\in \left[0,\text{1}\right]$
, if $PSO\left(\text{1},m\right)\ge m$
and $I\left(\text{1},n\right)\le n$
, then${\underset{\_}{R}}_{I}\left(B\right)\subseteq B\subseteq {\overline{R}}_{PSO}\left(B\right)$
;
(4) If $B\subseteq C$
, then ${\overline{R}}_{PSO}\left(B\right)\subseteq {\overline{R}}_{PSO}\left(C\right)$
, ${\underset{\_}{R}}_{I}\left(B\right)\subseteq {\underset{\_}{R}}_{I}\left(C\right)$
.
Proof. (1) ${\overline{R}}_{PSO}(\varnothing )=\varnothing $
can be proven by Definition 7. (2) ${\underset{\_}{R}}_{I}\left(U\right)=U$
can be proven according to Definition 7.
(3) For the fuzzy set B, according to Definition 7, $\forall m\in U$
,
${\overline{R}}_{PSO}\left(B\right)\left(m\right)=\underset{n\in U}{\mathrm{sup}}PSO\left(R\left(x,n\right),B\left(n\right)\right)=\{\begin{array}{ll}{I}_{M}\left({\displaystyle \underset{l\ne n}{\cup}{\left[l\right]}_{R}},{\alpha}_{M}\right)\left(m\right),\hfill & R\left(m,n\right)=0,\hfill \\ 0,\hfill & R\left(m,n\right)\ne 0.\hfill \end{array}$
Thus, ${\overline{R}}_{PSO}\left(B\right)\supseteq B$
.
Moreover, according to Definition 7,
$\begin{array}{c}{\underset{\_}{R}}_{I}\left(B\right)\left(m\right)=\underset{n\in U}{\mathrm{inf}}I\left(R\left(m,n\right),B\left(n\right)\right)\\ \le I\left(R\left(m,m\right),B\left(m\right)\right)\\ =I\left(1,B\left(m\right)\right)\\ \le B\left(m\right)\end{array}$
Therefore, ${\underset{\_}{R}}_{I}\left(B\right)\subseteq B$
. Finally, it is proven that ${\underset{\_}{R}}_{I}\left(B\right)\subseteq B\subseteq {\overline{R}}_{SO}\left(B\right)$
.
(4) If $B\subseteq C$
, by (PSO_{3}) of Definition 6, $\forall m\in U$
, PSO (R (m, n) can be obtained. $B\left(n\right)\le PSO\left(R\left(m,n\right),C\left(n\right)\right)$
. Then
$\underset{n\in U}{\mathrm{sup}}PSO\left(R\left(m,n\right),B\left(n\right)\right)\le \underset{n\in U}{\mathrm{sup}}PSO\left(R\left(m,n\right),C\left(m\right)\right)$
Therefore, ${\overline{R}}_{PSO}\left(B\right)\subseteq {\overline{R}}_{PSO}\left(C\right)$
. By (I2), similarly, ${\underset{\_}{R}}_{I}\left(B\right)\subseteq {\underset{\_}{R}}_{I}\left(C\right)$
.
Theorem 5. Suppose PSO is a pseudosemioverlap function, I is a fuzzy implication, and R_{1} and R_{2} represent a couple of fuzzy binary relations on U. If ${R}_{\text{1}}\subseteq {R}_{\text{2}}$
, in this case,
(1) ${\overline{{R}_{1}}}_{PSO}\left(A\right)\subseteq {\overline{{R}_{2}}}_{PSO}\left(A\right)$
;
(2) ${\underset{\_}{{R}_{2}}}_{I}\left(A\right)\subseteq {\underset{\_}{{R}_{1}}}_{I}\left(A\right)$
.
Note: ${\overline{{R}_{1}}}_{PSO}\left(A\right)$
and ${\overline{{R}_{2}}}_{PSO}\left(A\right)$
represent the fuzzy sets A based on the (I, PSO)fuzzy rough set upperapproximation operators of R_{1} and R_{2}, respectively; ${\underset{\_}{{R}_{2}}}_{I}\left(A\right)$
and ${\underset{\_}{{R}_{1}}}_{I}\left(A\right)$
represent the fuzzy sets A based on (I, PSO)fuzzy rough set lowerapproximation operators of R_{1} and R_{2}, respectively.
Proof. (1) If ${R}_{\text{1}}\subseteq {R}_{\text{2}}$
, then for any $x,y\in U$
; according to (PSO_{3}) in Definition 6, the following expression can be written:
$PSO\left({R}_{\text{1}}\left(x,y\right),A\left(y\right)\right)\le PSO\left({R}_{\text{2}}\left(x,y\right),B\left(y\right)\right)$
.
Then,
$\underset{y\in U}{\mathrm{sup}}PSO\left({R}_{1}\left(x,y\right),A\left(y\right)\right)\le \underset{y\in U}{\mathrm{sup}}PSO\left({R}_{2}\left(x,y\right),A\left(y\right)\right).$
Therefore, ${\overline{{R}_{1}}}_{PSO}\left(A\right)\subseteq {\overline{{R}_{2}}}_{PSO}\left(A\right)$
.
(2) By combining the proof strategies of (1) and (I_{2}) of Definition 3, it can be proven that ${\underset{\_}{{R}_{2}}}_{I}\left(A\right)\subseteq {\underset{\_}{{R}_{1}}}_{I}\left(A\right)$
.
Theorem 6. Suppose PSO is a pseudosemioverlap function, and I is a fuzzy implication, where C and D are fuzzy sets in the domain U. Consequently, the following can be inferred:
(1) ${\overline{R}}_{PSO}\left(C\cup D\right)={\overline{R}}_{PSO}\left(C\right)\cup {\overline{R}}_{PSO}\left(D\right)$
;
(2) ${\underset{\_}{R}}_{I}\left(C\cup D\right)\supseteq {\underset{\_}{R}}_{I}\left(C\right)\cup {\underset{\_}{R}}_{I}\left(D\right)$
;
(3) ${\overline{R}}_{PSO}\left(C\cap D\right)\subseteq {\overline{R}}_{PSO}\left(C\right)\cap {\overline{R}}_{PSO}\left(D\right)$
;
(4) ${\underset{\_}{R}}_{I}\left(C\cap D\right)={\underset{\_}{R}}_{I}\left(C\right)\cap {\underset{\_}{R}}_{I}\left(D\right)$
.
Proof. The definition of (I, PSO)fuzzy rough set is obtained directly from conditions (1) and (4). Proofs for (2) and (3) are given below.
(2) From Definition 7, $\forall m\in U$
,
$\begin{array}{c}{\underset{\_}{R}}_{I}\left(C\cup D\right)\left(m\right)=\underset{n\in U}{\mathrm{inf}}I\left(R\left(m,n\right),\left(C\cup D\right)\left(n\right)\right)=\underset{n\in U}{\mathrm{inf}}I\left(R\left(m,n\right),C\left(n\right)\vee D\left(n\right)\right)\\ =\underset{n\in U}{\mathrm{inf}}\left(I\left(R\left(m,n\right),C\left(n\right)\right)\vee I\left(R\left(m,n\right),D\left(n\right)\right)\right)\\ \ge \underset{n\in U}{\mathrm{inf}}I\left(R\left(m,n\right),C\left(n\right)\right)\vee \underset{n\in U}{\mathrm{inf}}I\left(R\left(m,n\right),D\left(n\right)\right)\\ ={\underset{\_}{R}}_{I}\left(C\right)\cup {\underset{\_}{R}}_{I}\left(D\right)\left(m\right)\end{array}$
Hence, ${\underset{\_}{R}}_{I}\left(C\cup D\right)\supseteq {\underset{\_}{R}}_{I}\left(C\right)\cup {\underset{\_}{R}}_{I}\left(D\right)$
.
(3) By Definition 7, $\forall m\in U$
,
$\begin{array}{c}{\overline{R}}_{PSO}\left(C\cap D\right)\left(m\right)=\underset{n\in U}{\mathrm{sup}}PSO\left(R\left(m,n\right),\left(C\cap D\right)\left(n\right)\right)\\ =\underset{n\in U}{\mathrm{sup}}PSO\left(R\left(m,n\right),C\left(n\right)\wedge D\left(n\right)\right)\\ =\underset{n\in U}{\mathrm{sup}}\left(PSO\left(R\left(m,n\right),C\left(n\right)\right)\wedge PSO\left(R\left(m,n\right),D\left(n\right)\right)\right)\\ \le \underset{n\in U}{\mathrm{sup}}PSO\left(R\left(m,n\right),C\left(n\right)\right)\wedge \underset{n\in U}{\mathrm{sup}}PSO\left(R\left(m,n\right),D\left(n\right)\right)\\ ={\overline{R}}_{PSO}\left(C\right)\cap {\overline{R}}_{PSO}\left(D\right)\left(m\right)\end{array}$
Hence, ${\overline{R}}_{PSO}\left(C\cap D\right)\subseteq {\overline{R}}_{PSO}\left(C\right)\cap {\overline{R}}_{PSO}\left(D\right)$
.
Proposition 2. Let (M, N, R) be a fuzzy approximation space, PSO be a pseudosemioverlap function, I be a fuzzy implication, and R be a fuzzy relation from M to N. For any $\alpha \in \left[0,1\right]$
, the following statement holds:
(1) ${\overline{R}}_{PSO}\left({\alpha}_{M}\right)=PS{O}_{N}\left({\displaystyle \underset{m\in M}{\cup}{\left[m\right]}_{R}},{\alpha}_{N}\right)$
;
(2) ${\underset{\_}{R}}_{I}\left({\alpha}_{N}\right)={I}_{M}\left({\displaystyle \underset{n\in N}{\cup}{\left[n\right]}_{R}},{\alpha}_{M}\right)$
;
(3) ${\overline{R}}_{PSO}\left({\alpha}_{N}\right)=PS{O}_{M}\left({\left[n\right]}_{R},{\alpha}_{M}\right)\left(\forall n\in N\right)$
;
(4) If $\forall a\in \left[0,\text{1}\right]$
, then $I\left(a,0\right)=0$
. The following statements hold: $\forall n\in N$
,
Note: For $\forall m\in M$
, $n\in N$
, ${\left[n\right]}_{R}\left(m\right)=R\left(m,n\right)$
exists; the value of the fuzzy set N in the context of $\alpha $
is a set of constant ${\alpha}_{N}$
; the value of the fuzzy set M in the context of${\alpha}_{M}$
is a set of constant $\alpha $
.
Proof. (1) By Definition 7, $\forall m\in M$
,
$\begin{array}{c}{\overline{R}}_{PSO}\left({\alpha}_{M}\right)\left(n\right)=\underset{m\in M}{\text{Sup}}PSO\left(R\left(n,m\right),{\alpha}_{X}\left(m\right)\right)=\underset{m\in M}{\text{Sup}}PSO\left(R\left(n,m\right),\alpha \right)\\ =PSO\left(\underset{m\in M}{\text{Sup}}R\left(n,m\right),\alpha \right)=PSO\left({\displaystyle \underset{m\in M}{\cup}{\left[m\right]}_{R}},{\alpha}_{N}\right)\left(n\right)\end{array}$
Thus, ${\overline{R}}_{PSO}\left({\alpha}_{M}\right)=PS{O}_{N}\left({\displaystyle \underset{m\in M}{\cup}{\left[m\right]}_{R}},{\alpha}_{N}\right)$
.
(2) By Definition 7, $\forall m\in M$
, $n\in N$
,
$\begin{array}{c}{\underset{\_}{R}}_{I}\left({\alpha}_{N}\right)\left(m\right)=\underset{n\in N}{\mathrm{inf}}I\left(R\left(m,n\right),{\alpha}_{N}\left(n\right)\right)=\underset{n\in N}{\mathrm{inf}}I\left(R\left(m,n\right),\alpha \right)\\ =I\left(\underset{n\in N}{\text{Sup}}R\left(m,n\right),\alpha \right)=I\left({\displaystyle \underset{n\in N}{\cup}{\left[n\right]}_{R}},{\alpha}_{M}\right)\left(m\right)\end{array}$
Hence, ${\underset{\_}{R}}_{I}\left({\alpha}_{N}\right)={I}_{M}\left({\displaystyle \underset{n\in N}{\cup}{\left[n\right]}_{R}},{\alpha}_{M}\right)$
.
(3) ${\overline{R}}_{PSO}\left({\alpha}_{N}\right)=PS{O}_{M}\left({\left[n\right]}_{R},{\alpha}_{M}\right)$
can be directly inferred from (1).
(4) By Definition 7, $\forall m\in M$
, $n,l\in N$
,
$\begin{array}{c}{\underset{\_}{R}}_{I}\left({\alpha}_{N}\left\{n\right\}\right)\left(m\right)=\underset{l\in N}{\mathrm{inf}}I\left(R\left(m,l\right),\left({\alpha}_{N}\left\{n\right\}\right)\left(l\right)\right)\\ =\underset{l\in N}{\mathrm{inf}}I\left(R\left(m,l\right),\alpha \right)\wedge I\left(R\left(m,l\right),0\right)\end{array}$
by I (a, 0) = 0($\forall a\in \left[0,\text{1}\right]$
),
${\underset{\_}{R}}_{I}\left({\alpha}_{N}\left\{n\right\}\right)\left(m\right)=\{\begin{array}{ll}\underset{l\ne n}{\wedge}I\left(R\left(m,l\right),\alpha \right),\hfill & R\left(m,n\right)=0,\hfill \\ 0,\hfill & R\left(m,n\right)\ne 0.\hfill \end{array}$
$\begin{array}{c}{\underset{\_}{R}}_{I}\left({\alpha}_{N}\left\{n\right\}\right)\left(m\right)=\{\begin{array}{ll}I\left(\underset{l\ne n}{\vee}{\left[l\right]}_{R},\alpha \right),\hfill & R\left(m,n\right)=0,\hfill \\ 0,\hfill & R\left(m,n\right)\ne 0,\hfill \end{array}\\ =\{\begin{array}{ll}{I}_{M}\left({\displaystyle \underset{l\ne n}{\cup}{\left[l\right]}_{R}},{\alpha}_{M}\right)\left(m\right),\hfill & R\left(m,n\right)=0,\hfill \\ 0,\hfill & R\left(m,n\right)\ne 0.\hfill \end{array}\end{array}$
Hence, statement (4) is true.
4. IPSOFMM Operators
This section presents the IPSOFMM operators, an innovative set of morphological operators based on pseudosemioverlap functions and fuzzy implications. Furthermore, an innovative algorithm for image edge extraction, called FCMIPSO, was developed by integrating IPSOFMM operators and the fuzzy Cmeans algorithm.
Definition 8. Consider R as a fuzzy binary relation on ${R}^{\text{2}}$
(i.e., $R:{R}^{\text{2}}\times {R}^{\text{2}}\to \left[0,\text{1}\right]$
). The pair (${R}^{\text{2}}$
, R) forms a fuzzy approximate space. Let PSO represent a pseudosemioverlap function and I represent a fuzzy implication. B is a fuzzy subset of ${R}^{\text{2}}$
(i.e., $B:{R}^{\text{2}}\to \left[0,\text{1}\right]$
). The fuzzy dilation operator, denoted as D_{PSO} (B, R), and the fuzzy erosion operator, denoted as E_{I} (B, R), are defined below: $\forall x,y\in {R}^{\text{2}}$
,
${D}_{PSO}\left(B,R\right)\left(x\right)=\underset{y\in {\text{R}}^{\text{2}}}{\mathrm{sup}}PSO\left(R\left(x,y\right),B\left(y\right)\right),$
(12)
${E}_{I}\left(B,R\right)\left(x\right)=\underset{y\in {\text{R}}^{\text{2}}}{\mathrm{inf}}I\left(R\left(x,y\right),B\left(y\right)\right).$
(13)
Example 3. Dilation and erosion examples performed by IPSOFMM operators are presented Figure 3.
(a) (b) (c)
Figure 3. Sample of dilation and erosion image. (a) Original image of cell. (b) Fuzzy dilation of cells. (c) Fuzzy erosion of cells.
Theorem 7. Consider B as a gray image, R as a fuzzy relation, PSO as a pseudosemioverlap function, I as a fuzzy implication, D_{PSO} (B, R) as a fuzzy dilation operator, and E_{I} (B, R) as a fuzzy erosion operator in the IPSOFMM operator. Then, for any $x,y\in {R}^{\text{2}}$
, (the symbol d (R) denotes the set of all points in R).
(1) ${D}_{PSO}\left(B,R\right)\left(x\right)=0$
iff ($\forall y\in d\left(R\right)$
,$B\left(x+y\right)=0$
);
(2) $\exists y\in d\left(R\right)$
, $R\left(y\right)=\text{1}$
and $B\left(x+y\right)=0$
iff ${D}_{PSO}\left(B,R\right)\left(x\right)=\text{1}$
;
(3) $\exists y\in d\left(R\right)$
, $R\left(y\right)=\text{1}$
and $B\left(x+y\right)=0$
iff ${E}_{I}\left(B,R\right)\left(x\right)=0$
.
Proof. (1) Assume $\forall y\in d\left(R\right)$
satisfies $B\left(x+y\right)=0$
. Moreover, by condition (2) in Definition 2.1, for $\exists m\in \left[0,1\right]$
, then $PSO\left(0,m\right)=PSO\left(m,0\right)=0$
; hence,
$\underset{y\in d\left(R\right)}{\mathrm{Sup}}PSO\left(R\left(x\right),B\left(x+y\right)\right)=0$
Assume ${D}_{PSO}\left(B,R\right)\left(y\right)=0$
; then,
${D}_{PSO}\left(B,R\right)\left(x\right)=\underset{y\in d\left(R\right)}{\mathrm{Sup}}PSO\left(R\left(y\right),B\left(x+y\right)\right)=0.$
Therefore, for $\forall y\in d\left(R\right)$
, $PSO\left(R\left(y\right),B\left(x+y\right)\right)=0$
can be obtained. By condition (2) in Definition 2.1, $R\left(y\right)\cdot B\left(x+y\right)=0$
, $\forall y\in d\left(R\right)$
, $R\left(y\right)\ne 0$
; hence, $B\left(x+y\right)=0$
.
(2) Assume $\exists y\in d\left(R\right)$
, $R\left(y\right)=\text{1}$
, and $B\left(x+y\right)=1$
. By (3) in Definition 2.1, $PSO\left(R\left(y\right),B\left(x+y\right)\right)=\text{1}$
. Hence,
${D}_{PSO}\left(B,\text{}R\right)\left(x\right)=\underset{y\in d\left(R\right)}{\text{Sup}}PSO\left(R\left(y\right),B\left(x+y\right)\right)=1.$
(3) This property is straightforward from the fuzzy implication and fuzzy erosion definitions.
Based on the integrated content in Sections 3 and 4, the (I, PSO)fuzzy rough sets are more extensive than the (I, O)fuzzy rough sets and have improved practical applicability while also retaining most of the characteristics of (I, O)fuzzy rough sets. Moreover, the IPSOFMM operators exhibit greater scope than the IOFMM operators while retaining the properties of fuzzy rough sets.
5. FCMIPSO Algorithm and Edge Extraction Experiment
In this section, the importance and advantages of the pseudosemioverlap functions in mathematical morphology and the field of image processing are demonstrated experimentally using the FCMIPSO algorithm.
5.1. FCMIPSO Algorithm
The core concept of the FCMIPSO algorithm can be summarized as follows. First, the fuzzy Cmeans algorithm is applied for image clustering. This step aims to separate the background of the grayscale image from its foreground. Second, the fuzzy relation R is calculated based on the prior clustering outcomes, and $\overline{R},\underset{\_}{R}$
are calculated. Third, the value of $\overline{R}\underset{\_}{R}$
is calculated to obtain the fuzzy edge image. Finally, the image is deblurred and then binarization is applied to acquire a binary edge. The detailed procedures of the FCMIPSO algorithm are outlined as follows.
Algorithm 5.1. An image edge extraction algorithm with (I, PSO)fuzzy rough sets. 
Input: gray image GI; 
Output: edge image; 
Step 1: GI←GI/255; 
Step 2: GI is subjected to clustering using the fuzzy Cmeans algorithm. BG represents the collection of all background points; Object represents the collection of all foreground points; 
Step 3: for n in GI: 
for m in GI: 
Step 4: for n in GI: 
Calculate D_{PSO} (GI)(n), E_{I} (GI)(n); 
Step 5: fuzzyI_edge←D_{PSO} (GI)E_{I} (GI); 
Step 6: grayI_edge←fuzzy_edge×255; 
for i in edge: 
if gray_edge (GI, B_{1})(i)>a: 
edge (i)←1 
else: 
edge (i)←0 
return edge; 
5.2. Experimental Step
Step 1. Choose the datasets.
Figures 4(a)(f) displays the six standard images selected for the experiments. The Lena image was used to evaluate the FCMISO algorithm.
(a) (b) (c)
(d) (e) (f)
Figure 4. Datasets. (a) Lena, (b) Cameraman, (c) Barbara, (d) Bank, (e) Cell, and (f) House.
Step 2. Clustering analysis was performed on the Lena image using the fuzzy Cmeans algorithm. (Note: The approach is similar to the image clustering method outlined in [7]).
Step 3. Image edges were detected using the Canny, Prewitt, Roberts, Laplacian, and Sobel operators.
Step 4. The FCMIPSO algorithm was employed to compute the image edges. The fuzzy relation R was calculated using B_{1} and B_{2} as follows.
${B}_{1}=\left[\begin{array}{ccc}0.7& 0.7& 0.7\\ 0.7& 0.8& 0.7\\ 0.7& 0.7& 0.7\end{array}\right],\text{}{B}_{2}=\left[\begin{array}{ccc}0.6& 0.6& 0.6\\ 0.6& 0.7& 0.6\\ 0.6& 0.6& 0.6\end{array}\right]$
(14)
5.3. Experimental Results
First, the grayscale Lena image was clustered using the FCM algorithm. The deblurred outcomes are depicted in Figure 5. (Note: Figures 5(a)(c) belong to the Object set, whereas Figures 5(d)(f) belong to the BG set).
…
(a) (b) (c) (d) (e) (f)
Figure 5. Results of applying FCM algorithm on Lena.
Second, the grayscale images in the dataset were processed using different edge detection algorithms. The output of the FCMIPSO algorithm is shown in Figure 6. The application of classical operators to process five grayscale images is illustrated in Figures 711. The operators used include Canny, Laplace, Prewitt, Roberts, and Sobel.
(a) (b) (c) (d) (e) (f)
Figure 6. Results of FCMIPSO algorithm.
(a) (b) (c) (d) (e) (f)
Figure 7. Results from Canny operator.
(a) (b) (c) (d) (e) (f)
Figure 8. Results from Laplacian operator.
(a) (b) (c) (d) (e) (f)
Figure 9. Results from Prewitt operator.
(a) (b) (c) (d) (e) (f)
Figure 10. Results from Roberts operator.
(a) (b) (c) (d) (e) (f)
Figure 11. Results from Sobel operator.
5.4. Analysis of Experimental Results
Two central problems are frequently studied when implementing image edge algorithms: first, the feasibility of extracting the edges of a foreground object from an image, and second, whether the noise level in the image is excessively high [21][23].
Regarding the first problem, Figure 6 shows the experimental results of the FCMIPSO algorithm, indicating that it can extract edge information from each gray image. For example, the edge of the building and button in image 5(b); the edge of the kerchief, tablecloths, and books in image 5(c); the edge of the small window in image 5(d); the edge of the bubbles in image 5(e); and the edge of the beams and columns in image 5(f). Furthermore, a minimal increase in noise due to ineffective background extraction was observed. Some of the classical algorithms underperformed when extracting the edges of foreground objects in images [7].
Subsequently, different (I, PSO) pairs as in the Equations (15)(21), were used in the FCMIPSO algorithm to test the noise introduction rate at the edges of the Lena, Cameraman, Barbara, Bank, Cell, and House images. The results are shown in Figure 12 and Table 2 and Table 3.
Figure 12. Noise introduction rates for each algorithm.
Table 2. Noise introduction rates of the FCMIPSO algorithm (%).

Lena 
Cameraman 
Barbara 
Cell 
Bank 
House 
(PSO_{3}, I_{2}) 
1.51 
3.07 
2.78 
0.72 
4.61 
3.06 
(PSO_{3}, I_{1}) 
2.38 
3.06 
3.05 
1.24 
5.17 
2.82 
(PSO_{3}, I_{3}) 
2.17 
2.92 
3.02 
0.89 
6.21 
2.88 
(PSO_{2}, I_{4}) 
1.77 
3.05 
2.21 
0.85 
3.97 
2.28 
(PSO_{2}, I_{2}) 
1.94 
3.08 
3.1 
0.57 
2.2 
1.48 
(PSO_{2}, I_{4}) 
1.24 
3.09 
2.85 
0.84 
3.78 
2.67 
(PSO_{2}, I_{1}) 
1.72 
3.05 
2.77 
1.02 
4.07 
2.11 
(PSO_{2}, I_{3}) 
1.25 
3.03 
2.94 
0.82 
1.46 
2.29 
(PSO_{1}, I_{2}) 
2.16 
3.06 
2.64 
0.58 
2.06 
2.63 
(PSO_{1}, I_{1}) 
1.48 
3.11 
3.22 
0.7 
2.82 
1.56 
Average noise rate 
1.72 
3.05 
2.86 
0.82 
3.64 
2.38 
Table 3. Noise introduction rates of each algorithm (%).

Lena 
Cameraman 
Barbara 
Cell 
Bank 
House 
Canny operator 
3.34 
7.65 
3.07 
3.52 
8.45 
4.92 
Laplacian operator 
4.87 
12.51 
6.32 
3.06 
28.24 
23.35 
Prewitt operator 
8.77 
5.78 
10.21 
6.31 
26.67 
19.03 
Roberts operator 
6.96 
3.1 
6.14 
4.38 
21.26 
16.17 
Sobel operator 
13.23 
9.65 
8.83 
4.81 
15.52 
8.79 
FCMIPSO algorithm 
1.72 
3.05 
2.86 
0.82 
3.64 
2.38 
${I}_{1}=\{\begin{array}{ll}1,\hfill & \sqrt{x}\le y,\hfill \\ {\left(\frac{y}{\sqrt{x}}\right)}^{2},\hfill & \text{else}\text{.}\hfill \end{array}$
(15)
${I}_{2}\left(x,y\right)=\{\begin{array}{ll}1,\hfill & x\le y,\hfill \\ \frac{y}{x},\hfill & \text{else}\text{.}\hfill \end{array}$
(16)
${I}_{3}\left(x,y\right)=\{\begin{array}{ll}0,\hfill & 0\le y\le x+1,\hfill \\ x+y1,\hfill & x+1\le y\le 1.\hfill \end{array}$
(17)
${I}_{4}\left(x,y\right)=\frac{x+y}{2}.$
(18)
$PS{O}_{1}\left(x,y\right)=\{\begin{array}{ll}\frac{xy}{6},\hfill & 0\le x\le 0.2,\text{\hspace{0.17em}}x\le y\le x+0.4,\hfill \\ \frac{xy}{3},\hfill & y<x\le y+0.4,\text{\hspace{0.17em}}0\le y<0.2,\hfill \\ x{y}^{2},\hfill & 0\le x\le 0.2,\text{\hspace{0.17em}}x+0.4<y\le 1,\hfill \\ x{y}^{2},\hfill & 0.2<x\le 0.4,\text{\hspace{0.17em}}x+0.4<y\le x,\hfill \\ xy,\hfill & 0.2<x\le 0.4,\text{\hspace{0.17em}}x+0.4<y<x,\hfill \\ xy,\hfill & 0.4<x\le 1,\text{\hspace{0.17em}}0\le y<x.\hfill \end{array}$
(19)
$PS{O}_{2}\left(x,y\right)=\{\begin{array}{ll}\frac{xy}{7},\hfill & 0\le x\le 0.4,\text{\hspace{0.17em}}x\le y\le 0.4,\hfill \\ \frac{2xy}{7},\hfill & 0\le x\le 0.4,\text{\hspace{0.17em}}0\le y<x,\hfill \\ \frac{19xy+11}{30},\hfill & 0.4<x\le 1,\text{\hspace{0.17em}}x\le y\le 1,\hfill \\ \frac{21xy+9}{30},\hfill & 0.4<x\le 1,\text{\hspace{0.17em}}0.4<y<x,\hfill \\ \frac{xy}{3},\hfill & 0.4<x\le 1,\text{\hspace{0.17em}}0\le y\le 0.4,\hfill \\ \frac{xy}{2},\hfill & 0\le x\le 0.4,\text{\hspace{0.17em}}0.4\le y\le 1.\hfill \end{array}$
(20)
$PS{O}_{3}\left(x,y\right)=\{\begin{array}{ll}{x}^{2},\hfill & {x}^{2}\le {y}^{3},\hfill \\ y,\hfill & {x}^{2}>{y}^{3}.\hfill \end{array}$
(21)
Figure 12 and Table 3 indicate that the average noise introduction rate of the FCMIPSO algorithm was generally smaller than those of the other five algorithms. This is because the image is clustered using the fuzzy Cmeans algorithm before extracting the image edge, which effectively distinguishes the image background from the foreground and thus effectively reduces noise generation.
In summary, the proposed FCMIPSO algorithm minimizes noise introduction compared to other conventional algorithms while simultaneously extracting as many complete foreground edges from the image as possible.
6. Conclusion
In this paper, the pseudosemioverlap function is defined, and two construction methods for it are presented. Subsequently, the (I, PSO)fuzzy rough set is introduced, and its theoretical properties are explored. Following that, the integration of the upper and lower approximation operators within the (I, PSO)fuzzy rough set with the fuzzy mathematical morphology operators leads to the proposal of the IPSOFMM operators, with a focus on investigating its properties. Finally, the fuzzy Cmeans algorithm is combined with the IPSOFMM operator to formulate the FCMIPSO image edge extraction algorithm, subsequently applied to six grayscale images. The pseudosemioverlap function proposed in this paper requires only the properties of asymmetry and left continuity. The PSO function enhances the FCMIPSO algorithm’s ability to handle digital image data with ambiguity, noncompleteness, and irregularity, making it flexible to be used in different application environments. However, constructing the pseudosemioverlap function becomes more intricate across diverse application contexts. Therefore, future research efforts will focus on devising pseudosemioverlap functions tailored to specific application backgrounds. Followup research work could further investigate the application of the FCMIPSO algorithm in video image edge extraction in addition to the construction method of the PSO function.
Acknowledgements
This work was financially supported by the Natural Science Foundation of China (52273315).