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Research article

Gas-Net: A deep neural network for gastric tumor semantic segmentation

  • Currently, the gastric cancer is the source of the high mortality rate where it is diagnoses from the stomach and esophagus tests. To this end, the whole of studies in the analysis of cancer are built on AI (artificial intelligence) to develop the analysis accuracy and decrease the danger of death. Mostly, deep learning methods in images processing has made remarkable advancement. In this paper, we present a method for detection, recognition and segmentation of gastric cancer in endoscopic images. To this end, we propose a deep learning method named GAS-Net to detect and recognize gastric cancer from endoscopic images. Our method comprises at the beginning a preprocessing step for images to make all images in the same standard. After that, the GAS-Net method is based an entire architecture to form the network. A union between two loss functions is applied in order to adjust the pixel distribution of normal/abnormal areas. GAS-Net achieved excellent results in recognizing lesions on two datasets annotated by a team of expert from several disciplines (Dataset1, is a dataset of stomach cancer images of anonymous patients that was approved from a private medical-hospital clinic, Dataset2, is a publicly available and open dataset named HyperKvasir ‎[1]). The final results were hopeful and proved the efficiency of the proposal. Moreover, the accuracy of classification in the test phase was 94.06%. This proposal offers a specific mode to detect, recognize and classify gastric tumors.

    Citation: Lamia Fatiha KAZI TANI, Mohammed Yassine KAZI TANI, Benamar KADRI. Gas-Net: A deep neural network for gastric tumor semantic segmentation[J]. AIMS Bioengineering, 2022, 9(3): 266-282. doi: 10.3934/bioeng.2022018

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  • Currently, the gastric cancer is the source of the high mortality rate where it is diagnoses from the stomach and esophagus tests. To this end, the whole of studies in the analysis of cancer are built on AI (artificial intelligence) to develop the analysis accuracy and decrease the danger of death. Mostly, deep learning methods in images processing has made remarkable advancement. In this paper, we present a method for detection, recognition and segmentation of gastric cancer in endoscopic images. To this end, we propose a deep learning method named GAS-Net to detect and recognize gastric cancer from endoscopic images. Our method comprises at the beginning a preprocessing step for images to make all images in the same standard. After that, the GAS-Net method is based an entire architecture to form the network. A union between two loss functions is applied in order to adjust the pixel distribution of normal/abnormal areas. GAS-Net achieved excellent results in recognizing lesions on two datasets annotated by a team of expert from several disciplines (Dataset1, is a dataset of stomach cancer images of anonymous patients that was approved from a private medical-hospital clinic, Dataset2, is a publicly available and open dataset named HyperKvasir ‎[1]). The final results were hopeful and proved the efficiency of the proposal. Moreover, the accuracy of classification in the test phase was 94.06%. This proposal offers a specific mode to detect, recognize and classify gastric tumors.



    Henkin and Skolem introduced Hilbert algebras in the fifties for investigations in intuitionistic and other non-classical logics. Diego [4] proved that Hilbert algebras form a variety which is locally finite. Bandaru et al. introduced the notion of GE-algebras which is a generalization of Hilbert algebras, and investigated several properties (see [1,2,7,8,9]). The notion of interior operator is introduced by Vorster [12] in an arbitrary category, and it is used in [3] to study the notions of connectedness and disconnectedness in topology. Interior algebras are a certain type of algebraic structure that encodes the idea of the topological interior of a set, and are a generalization of topological spaces defined by means of topological interior operators. Rachůnek and Svoboda [6] studied interior operators on bounded residuated lattices, and Svrcek [11] studied multiplicative interior operators on GMV-algebras. Lee et al. [5] applied the interior operator theory to GE-algebras, and they introduced the concepts of (commutative, transitive, left exchangeable, belligerent, antisymmetric) interior GE-algebras and bordered interior GE-algebras, and investigated their relations and properties. Later, Song et al. [10] introduced the notions of an interior GE-filter, a weak interior GE-filter and a belligerent interior GE-filter, and investigate their relations and properties. They provided relations between a belligerent interior GE-filter and an interior GE-filter and conditions for an interior GE-filter to be a belligerent interior GE-filter is considered. Given a subset and an element, they established an interior GE-filter, and they considered conditions for a subset to be a belligerent interior GE-filter. They studied the extensibility of the belligerent interior GE-filter and established relationships between weak interior GE-filter and belligerent interior GE-filter of type 1, type 2 and type 3. Rezaei et al. [7] studied prominent GE-filters in GE-algebras. The purpose of this paper is to study by applying interior operator theory to prominent GE-filters in GE-algebras. We introduce the concept of a prominent interior GE-filter, and investigate their properties. We discuss the relationship between a prominent GE-filter and a prominent interior GE-filter and the relationship between an interior GE-filter and a prominent interior GE-filter. We find and provide examples where any interior GE-filter is not a prominent interior GE-filter and any prominent GE-filter is not a prominent interior GE-filter. We provide conditions for an interior GE-filter to be a prominent interior GE-filter. We provide conditions under which an internal GE-filter larger than a given internal GE filter can become a prominent internal GE-filter, and give an example describing it. We also introduce the concept of a prominent interior GE-filter of type 1 and type 2, and investigate their properties. We discuss the relationship between a prominent interior GE-filter and a prominent interior GE-filter of type 1. We give examples to show that A and B are independent of each other, where A and B are:

    (1) { A: prominent interior GE-filter of type 1. B: prominent interior GE-filter of type 2.

    (2) { A: prominent interior GE-filter. B: prominent interior GE-filter of type 2.

    (3) { A: interior GE-filter. B: prominent interior GE-filter of type 1.

    (4) { A: interior GE-filter. B: prominent interior GE-filter of type 2.

    Definition 2.1. [1] By a GE-algebra we mean a non-empty set X with a constant 1 and a binary operation satisfying the following axioms:

    (GE1) uu=1,

    (GE2) 1u=u,

    (GE3) u(vw)=u(v(uw))

    for all u,v,wX.

    In a GE-algebra X, a binary relation "" is defined by

    (x,yX)(xyxy=1). (2.1)

    Definition 2.2. [1,2,8] A GE-algebra X is said to be transitive if it satisfies:

    (x,y,zX)(xy(zx)(zy)). (2.2)

    Proposition 2.3. [1] Every GE-algebra X satisfies the following items:

    (uX)(u1=1). (2.3)
    (u,vX)(u(uv)=uv). (2.4)
    (u,vX)(uvu). (2.5)
    (u,v,wX)(u(vw)v(uw)). (2.6)
    (uX)(1uu=1). (2.7)
    (u,vX)(u(vu)u). (2.8)
    (u,vX)(u(uv)v). (2.9)
    (u,v,wX)(uvwvuw). (2.10)

    If X is transitive, then

    (u,v,wX)(uvwuwv,vwuw). (2.11)
    (u,v,wX)(uv(vw)(uw)). (2.12)

    Lemma 2.4. [1] In a GE-algebra X, the following facts are equivalent each other.

    (x,y,zX)(xy(zx)(zy)). (2.13)
    (x,y,zX)(xy(yz)(xz)). (2.14)

    Definition 2.5. [1] A subset F of a GE-algebra X is called a GE-filter of X if it satisfies:

    1F, (2.15)
    (x,yX)(xyF,xFyF). (2.16)

    Lemma 2.6. [1] In a GE-algebra X, every filter F of X satisfies:

    (x,yX)(xy,xFyF). (2.17)

    Definition 2.7. [7] A subset F of a GE-algebra X is called a prominent GE-filter of X if it satisfies (2.15) and

    (x,y,zX)(x(yz)F,xF((zy)y)zF). (2.18)

    Note that every prominent GE-filter is a GE-filter in a GE-algebra (see [7]).

    Definition 2.8. [5] By an interior GE-algebra we mean a pair (X,f) in which X is a GE-algebra and f:XX is a mapping such that

    (xX)(xf(x)), (2.19)
    (xX)((ff)(x)=f(x)), (2.20)
    (x,yX)(xyf(x)f(y)). (2.21)

    Definition 2.9. [10] Let (X,f) be an interior GE-algebra. A GE-filter F of X is said to be interior if it satisfies:

    (xX)(f(x)FxF). (2.22)

    Definition 3.1. Let (X,f) be an interior GE-algebra. Then a subset F of X is called a prominent interior GE-filter in (X,f) if F is a prominent GE-filter of X which satisfies the condition (2.22).

    Example 3.2. Let X={1,2,3,4,5} be a set with the Cayley table which is given in Table 1.

    Table 1.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 1 4 4
    3 1 1 1 5 5
    4 1 2 3 1 1
    5 1 2 2 1 1

     | Show Table
    DownLoad: CSV

    Then X is a GE-algebra. If we define a mapping f on X as follows:

    f:XX,x{1if x{1,4,5},2if x{2,3},

    then (X,f) is an interior GE-algebra and F={1,4,5} is a prominent interior GE-filter in (X,f).

    It is clear that every prominent interior GE-filter is a prominent GE-filter. But any prominent GE-filter may not be a prominent interior GE-filter in an interior GE-algebra as seen in the following example.

    Example 3.3. Let X={1,2,3,4,5} be a set with the Cayley table which is given in Table 2,

    Table 2.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 3 4 1
    3 1 2 1 4 5
    4 1 2 3 1 5
    5 1 1 3 4 1

     | Show Table
    DownLoad: CSV

    and define a mapping f on X as follows:

    f:XX,x{1if x{1,2,3,5},4if x=4.

    Then (X,f) is an interior GE-algebra and F:={1} is a prominent GE-filter of X. But it is not a prominent interior GE-filter in (X,f) since f(2)=1F but 2F.

    We discuss relationship between interior GE-filter and prominent interior GE-filter.

    Theorem 3.4. In an interior GE-algebra, every prominent interior GE-filter is an interior GE-filter.

    Proof. It is straightforward because every prominent GE-filter is a GE-filter in a GE-algebra.

    In the next example, we can see that any interior GE-filter is not a prominent interior GE-filter in general.

    Example 3.5. Let X={1,2,3,4,5} be a set with the Cayley table which is given in Table 3.

    Table 3.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 1 4 4
    3 1 2 1 4 4
    4 1 1 3 1 1
    5 1 1 1 1 1

     | Show Table
    DownLoad: CSV

    Then X is a GE-algebra. If we define a mapping f on X as follows:

    f:XX,x{1if x=1,2if x{2,4,5},3if x=3,

    then (X,f) is an interior GE-algebra and F={1} is an interior GE-filter in (X,f). But it is not a prominent interior GE-filter in (X,f) since 1(23)=1F but ((32)2)3=3F.

    Proposition 3.6. Every prominent interior GE-filter F in an interior GE-algebra (X,f) satisfies:

    (x,yX)(f(xy)F((yx)x)yF). (3.1)

    Proof. Let F be a prominent interior GE-filter in (X,f). Let x,yX be such that f(xy)F. Then xyF by (2.22), and so 1(xy)=xyF by (GE2). Since 1F, it follows from (2.18) that ((yx)x)yF.

    Corollary 3.7. Every prominent interior GE-filter F in an interior GE-algebra (X,f) satisfies:

    (x,yX)(xyF((yx)x)yF). (3.2)

    Proof. Let F be a prominent interior GE-filter in (X,f). Then F is an interior GE-filter in (X,f) by Theorem 3.4. Let x,yX be such that xyF. Since xyf(xy) by (2.19), it follows from Lemma 2.6 that f(xy)F. Hence ((yx)x)yF by Proposition 3.6.

    Corollary 3.8. Every prominent interior GE-filter F in an interior GE-algebra (X,f) satisfies:

    (x,yX)(xyFf(((yx)x)y)F).

    Proof. Straightforward.

    Corollary 3.9. Every prominent interior GE-filter F in an interior GE-algebra (X,f) satisfies:

    (x,yX)(f(xy)Ff(((yx)x)y)F).

    Proof. Straightforward.

    In the following example, we can see that any interior GE-filter F in an interior GE-algebra (X,f) does not satisfy the conditions (3.1) and (3.2).

    Example 3.10. Consider the interior GE-algebra (X,f) in Example 3.4. The interior GE-filter F:={1} does not satisfy conditions (3.1) and (3.2) since f(23)=f(1)=1F and 23=1F but ((32)2)3=3F.

    We provide conditions for an interior GE-filter to be a prominent interior GE-filter.

    Theorem 3.11. If an interior GE-filter F in an interior GE-algebra (X,f) satisfies the condition (3.1), then F is a prominent interior GE-filter in (X,f).

    Proof. Let F be an interior GE-filter in (X,f) that satisfies the condition (3.1). Let x,y,zX be such that x(yz)F and xF. Then yzF. Since yzf(yz) by (2.19), it follows from Lemma 2.6 that f(yz)F. Hence ((zy)y)zF by (3.1), and therefore F is a prominent interior GE-filter in (X,f).

    Lemma 3.12. [10] In an interior GE-algebra, the intersection of interior GE-filters is also an interior GE-filter.

    Theorem 3.13. In an interior GE-algebra, the intersection of prominent interior GE-filters is also a prominent interior GE-filter.

    Proof. Let {FiiΛ} be a set of prominent interior GE-filters in an interior GE-algebra (X,f) where Λ is an index set. Then {FiiΛ} is a set of interior GE-filters in (X,f), and so {FiiΛ} is an interior GE-filter in (X,f) by Lemma 3.12. Let x,yX be such that f(xy){FiiΛ}. Then f(xy)Fi for all iΛ. It follows from Proposition 3.6 that ((yx)x)yFi for all iΛ. Hence ((yx)x)y{FiiΛ} and therefore {FiiΛ} is a prominent interior GE-filter in (X,f) by Theorem 3.11.

    Theorem 3.14. If an interior GE-filter F in an interior GE-algebra (X,f) satisfies the condition (3.2), then F is a prominent interior GE-filter in (X,f).

    Proof. Let F be an interior GE-filter in (X,f) that satisfies the condition (3.2). Let x,y,zX be such that x(yz)F and xF. Then yzF and thus ((zy)y)zF. Therefore F is a prominent interior GE-filter in (X,f).

    Given an interior GE-filter F in an interior GE-algebra (X,f), we consider an interior GE-filter G which is greater than F in (X,f), that is, we take two interior GE-filters F and G such that FG in an interior GE-algebra (X,f). We are now trying to find the condition that G can be a prominent interior GE-filter in (X,f).

    Theorem 3.15. Let (X,f) be an interior GE-algebra in which X is transitive. Let F and G be interior GE-filters in (X,f). If FG and F is a prominent interior GE-filter in (X,f), then G is also a prominent interior GE-filter in (X,f).

    Proof. Assume that F is a prominent interior GE-filter in (X,f). Then it is an interior GE-filter in (X,f) by Theorem 3.4. Let x,yX be such that f(xy)G. Then xyG by (2.22), and so 1=(xy)(xy)x((xy)y) by (GE1) and (2.6). Since 1F, it follows from Lemma 2.6 that x((xy)y)F. Hence ((((xy)y)x)x)((xy)y)FG by Corollary 3.7. Since

    ((((xy)y)x)x)((xy)y)(xy)(((((xy)y)x)x)y)

    by (2.6), we have (xy)(((((xy)y)x)x)y)G by Lemma 2.6. Hence

    ((((xy)y)x)x)yG.

    Since y(xy)y, it follows from (2.11) that

    ((((xy)y)x)x)y((yx)x)y.

    Thus ((yx)x)yG by Lemma 2.6. Therefore G is a prominent interior GE-filter in (X,f). by Theorem 3.11.

    The following example describes Theorem 3.15.

    Example 3.16. Let X={1,2,3,4,5} be a set with the Cayley table which is given in Table 4,

    Table 4.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 1 5 5
    3 1 1 1 5 5
    4 1 3 3 1 1
    5 1 3 3 1 1

     | Show Table
    DownLoad: CSV

    and define a mapping f on X as follows:

    f:XX,x{1if x=1,3if x{2,3},5if x{4,5}.

    Then (X,f) is an interior GE-algebra in which X is transitive, and F:={1} and G:={1,4,5} are interior GE-filters in (X,f) with FG. Also we can observe that F and G are prominent interior GE-filters in (X,f).

    In Theorem 3.15, if F is an interior GE-filter which is not prominent, then G is also not a prominent interior GE-filter in (X,f) as shown in the next example.

    Example 3.17. Let X={1,2,3,4,5} be a set with the Cayley table which is given in Table 5,

    Table 5.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 1 4 1
    3 1 5 1 4 5
    4 1 1 1 1 1
    5 1 1 1 4 1

     | Show Table
    DownLoad: CSV

    and define a mapping f on X as follows:

    f:XX,x{1if x=1,3if x=3,4if x=4,2if x{2,5}.

    Then (X,f) is an interior GE-algebra in which X is transitive, and F:={1} and G:={1,3} are interior GE-filters in (X,f) with FG. We can observe that F and G are not prominent interior GE-filters in (X,f) since 23=1F but ((32)2)3=(52)3=13=3F, and 42=1G but ((24)4)2=(44)2=12=2G.

    In Theorem 3.15, if X is not transitive, then Theorem 3.15 is false as seen in the following example.

    Example 3.18. Let X={1,2,3,4,5,6} be a set with the Cayley table which is given in Table 6.

    Table 6.  Cayley table for the binary operation "".
    1 2 3 4 5 6
    1 1 2 3 4 5 6
    2 1 1 1 6 6 6
    3 1 1 1 5 5 5
    4 1 1 3 1 1 1
    5 1 2 3 2 1 1
    6 1 2 3 2 1 1

     | Show Table
    DownLoad: CSV

    If we define a mapping f on X as follows:

    f:XX,x{1if x=1,4if x=4,5if x=5,6if x=6,2if x{2,3},

    then (X,f) is an interior GE-algebra in which X is not transitive. Let F:={1} and G:={1,5,6}. Then F is a prominent interior GE-filter in (X,f) and G is an interior GE-filter in (X,f) with FG. But G is not prominent interior GE-filter since 5(34)=55=1G and 5G but ((43)3)4=(33)4=14=4G.

    Definition 3.19. Let (X,f) be an interior GE-algebra and let F be a subset of X which satisfies (2.15). Then F is called:

    A prominent interior GE-filter of type 1 in (X,f) if it satisfies:

    (x,y,zX)(x(yf(z))F,f(x)F((f(z)y)y)f(z)F). (3.3)

    A prominent interior GE-filter of type 2 in (X,f) if it satisfies:

    (x,y,zX)(x(yf(z))F,f(x)F((zf(y))f(y))zF). (3.4)

    Example 3.20. (1). Let X={1,2,3,4,5} be a set with the Cayley table which is given in Table 7,

    Table 7.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 1 1 1
    3 1 2 1 2 2
    4 1 1 1 1 1
    5 1 1 1 1 1

     | Show Table
    DownLoad: CSV

    and define a mapping f on X as follows:

    f:XX,x{1if x{1,3}2if x=2,4if x=4,5if x=5.

    Then (X,f) is an interior GE-algebra and F:={1,3} is a prominent interior GE-filter of type 1 in (X,f).

    (2). Let X={1,2,3,4,5} be a set with the Cayley table which is given in Table 8,

    Table 8.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 1 1 1
    3 1 1 1 4 1
    4 1 1 1 1 5
    5 1 1 3 4 1

     | Show Table
    DownLoad: CSV

    and define a mapping f on X as follows:

    f:XX,x{1if x=1,2if x{2,3,4,5}.

    Then (X,f) is an interior GE-algebra and F:={1,3} is a prominent interior GE-filter of type 2 in (X,f).

    Theorem 3.21. In an interior GE-algebra, every prominent interior GE-filter is of type 1.

    Proof. Let F be a prominent interior GE-filter in an interior GE-algebra (X,f). Let x,y,zX be such that x(yf(z))F and f(x)F. Then xF by (2.22). It follows from (2.18) that ((f(z)y)y)f(z)F. Hence F is a prominent interior GE-filter of type 1 in (X,f).

    The following example shows that the converse of Theorem 3.21 may not be true.

    Example 3.22. Let X={1,2,3,4,5} be a set with the Cayley table which is given in Table 9,

    Table 9.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 1 1 1
    3 1 1 1 1 5
    4 1 1 3 1 1
    5 1 1 1 1 1

     | Show Table
    DownLoad: CSV

    and define a mapping f on X as follows:

    f:XX,x{1if x=1,2if x{2,3},5if x{4,5}.

    Then (X,f) is an interior GE-algebra and F:={1} is a prominent interior GE-filter of type 1 in (X,f). But it is not a prominent interior GE-filter in (X,f) since 1(34)=1F but (43)3)4=4F.

    The following example shows that prominent interior GE-filter and prominent interior GE-filter of type 2 are independent of each other, i.e., prominent interior GE-filter is not prominent interior GE-filter of type 2 and neither is the inverse.

    Example 3.23. (1). Let X={1,2,3,4,5} be a set with the Cayley table which is given in the following Table 10,

    Table 10.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 1 1 1
    3 1 5 1 1 5
    4 1 1 1 1 1
    5 1 3 3 1 1

     | Show Table
    DownLoad: CSV

    and define a mapping f on X as follows:

    f:XX,x{1if x=1,4if x{3,4}5if x{2,5}.

    Then (X,f) is an interior GE-algebra and F:={1} F is a prominent interior GE-filter in (X,f). But it is not a prominent interior GE-filter of type 2 since 1(5f(2))=55=1F and f(1)=1F but ((2f(5))f(5))2=((25)5)2=(15)2=52=3F.

    (2). Let X={1,2,3,4,5} be a set with the Cayley table which is given in the following Table 11,

    Table 11.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 1 1 1
    3 1 2 1 1 1
    4 1 1 1 1 1
    5 1 1 1 1 1

     | Show Table
    DownLoad: CSV

    and define a mapping f on X as follows:

    f:XX,x{1if x=1,5if x{2,3,4,5}.

    Then (X,f) is an interior GE-algebra and F:={1} is a prominent interior GE-filter of type2 in (X,f). But it is not a prominent interior GE-filter in (X,f) since 1(23)=11=1F and 1F but ((32)2)3=(22)3=13=3F.

    The following example shows that prominent interior GE-filter of type 1 and prominent interior GE-filter of type 2 are independent of each other.

    Example 3.24. (1). Let X={1,2,3,4,5} be a set with the Cayley table which is given in the following Table 12,

    Table 12.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 1 5 5
    3 1 1 1 1 1
    4 1 1 1 1 1
    5 1 1 1 1 1

     | Show Table
    DownLoad: CSV

    and define a mapping f on X as follows:

    f:XX,x{1if x=1,3if x{2,3},5if x{4,5}.

    Then (X,f) is an interior GE-algebra and F:={1,2,4} is a prominent interior GE-filter of type 1 in (X,f). But it is not a prominent interior GE-filter of type 2 since 1(5f(2))=1(53)=11=1F and f(1)=1F but ((2f(5))f(5))2=((25)5)2=(55)2=12=2F.

    (2). Let X={1,2,3,4,5} be a set with the Cayley table which is given in the following Table 13,

    Table 13.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 4 4 5
    3 1 1 1 1 1
    4 1 2 2 1 5
    5 1 1 1 1 1

     | Show Table
    DownLoad: CSV

    and define a mapping f on X as follows:

    f:XX,x{1if x=1,2if x=2,4if x=4,3if x{3,5}.

    Then (X,f) is an interior GE-algebra and F:={1} is a prominent interior GE-filter of type 2 in (X,f). But it is not a prominent interior GE-filter of type 1 in (X,f) since 1(5f(2))=1(52)=11=1F and f(1)F but ((f(2)5)5)f(2)=((25)5)2=(55)2=12=2F.

    The following example shows that interior GE-filter and prominent interior GE-filter of type 1 are independent of each other.

    Example 3.25. (1). Let X={1,2,3,4,5} be a set with the Cayley table which is given in the following Table 14,

    Table 14.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 5 5 5
    3 1 1 1 1 1
    4 1 1 1 1 1
    5 1 1 1 1 1

     | Show Table
    DownLoad: CSV

    and define a mapping f on X as follows:

    f:XX,x{1if x=1,2if x=2,5if x{3,4,5}.

    Then (X,f) is an interior GE-algebra and F:={1} is an interior GE-filter in (X,f). But F is not prominent interior GE-filter of type 1 since 1(5f(2))=1(52)=11=1F and f(1)=1F but ((f(2)5)5)2=((25)5)2=(55)2=12=2F.

    (2). Let X={1,2,3,4,5} be a set with the Cayley table which is given in the following Table 15,

    Table 15.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 5 1 5
    3 1 2 1 1 1
    4 1 1 3 1 5
    5 1 1 1 1 1

     | Show Table
    DownLoad: CSV

    and define a mapping f on X as follows:

    f:XX,x{1if x{1,2,4},5if x{3,5}.

    Then (X,f) is an interior GE-algebra and F:={1,2} is a prominent interior GE-filter of type 1 in (X,f). But it is not an interior GE-filter in (X,f) since 24=1 and 2F but 4F.

    The following example shows that interior GE-filter and prominent interior GE-filter of type 2 are independent of each other.

    Example 3.26. (1). Let X={1,2,3,4,5} be a set with the Cayley table which is given in the following Table 16,

    Table 16.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 1 1 1
    3 1 2 1 1 2
    4 1 2 3 1 5
    5 1 1 1 1 1

     | Show Table
    DownLoad: CSV

    and define a mapping f on X as follows:

    f:XX,x{1if x{1,4}2if x=2,3if x=3,5if x=5.

    Then (X,f) is an interior GE-algebra and F:={1,4} is an interior GE-filter in (X,f). But F is not prominent interior GE-filter of type 2 since 4(2f(3))=4(23)=41=1F and f(4)=1F but ((3f(2))f(2))3=((32)2)3=(22)3=13=3F.

    (2). Let X={1,2,3,4,5} be a set with the Cayley table which is given in the following Table 17,

    Table 17.  Cayley table for the binary operation "".
    1 2 3 4 5
    1 1 2 3 4 5
    2 1 1 1 1 5
    3 1 1 1 1 1
    4 1 1 1 1 5
    5 1 1 1 1 1

     | Show Table
    DownLoad: CSV

    and define a mapping f on X as follows:

    f:XX,x{1if x=1,3if x{2,3,4,5}.

    Then (X,f) is an interior GE-algebra and F:={1,2,5} is a prominent interior GE-filter of type 2 in (X,f). But it is not an interior GE-filter in (X,f) since 54=1 and 5F but 4F.

    Before we conclude this paper, we raise the following question.

    Question. Let (X,f) be an interior GE-algebra. Let F and G be interior GE-filters in (X,f). If FG and F is a prominent interior GE-filter of type 1 (resp., type 2) in (X,f), then is G also a prominent interior GE-filter of type 1 (resp., type 2) in (X,f)?

    We have introduced the concept of a prominent interior GE-filter (of type 1 and type 2), and have investigated their properties. We have discussed the relationship between a prominent GE-filter and a prominent interior GE-filter and the relationship between an interior GE-filter and a prominent interior GE-filter. We have found and provide examples where any interior GE-filter is not a prominent interior GE-filter and any prominent GE-filter is not a prominent interior GE-filter. We have provided conditions for an interior GE-filter to be a prominent interior GE-filter. We have given conditions under which an internal GE-filter larger than a given internal GE filter can become a prominent internal GE-filter, and have provided an example describing it. We have considered the relationship between a prominent interior GE-filter and a prominent interior GE-filter of type 1. We have found and provide examples to verify that a prominent interior GE-filter of type 1 and a prominent interior GE-filter of type 2, a prominent interior GE-filter and a prominent interior GE-filter of type 2, an interior GE-filter and a prominent interior GE-filter of type 1, and an interior GE-filter and a prominent interior GE-filter of type 2 are independent each other. In future, we will study the prime and maximal prominent interior GE-filters and their topological properties. Moreover, based on the ideas and results obtained in this paper, we will study the interior operator theory in related algebraic systems such as MV-algebra, BL-algebra, EQ-algebra, etc. It will also be used for pseudo algebra systems and further to conduct research related to the very true operator theory.

    This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B02006812).

    The authors wish to thank the anonymous reviewers for their valuable suggestions.

    All authors declare no conflicts of interest in this paper.


    Acknowledgments



    This research was partially supported by the Ministry of higher education and scientific research who provided insight and expertise that greatly assisted the research.

    Conflict of interest



    Authors declare that they have no conflict of interest.

    Author contributions



    Lamia Fatiha KAZI TANI, Mohammed Yassine KAZI TANI and Benamar KADRI contribute to realize the presented idea. Lamia Fatiha KAZI TANI developed the theory, achieved the programs and verified the critical methods. Mohammed Yassine KAZI TANI and Benamar KADRI supervised the results of this work. All authors discussed the results and contributed to the final manuscript.

    [1] Borgli H, de Lange T, Eskeland SL, et al. (2020) HyperKvasir, a comprehensive multi-class image and video dataset for gastrointestinal endoscopy. Sci Data 7: 283. https://doi.org/10.1038/s41597-020-00622-y
    [2] Cutler J, Grannell A, Rae R (2021) Size-susceptibility of Cornu aspersum exposed to the malacopathogenic nematodes Phasmarhabditis hermaphrodita and P. californica. Biocontrol Sci Technol 31: 1149-1160. https://doi.org/10.1080/09583157.2021.1929072
    [3] Kitagawa Y, Wada N (2021) Application of AI in Endoscopic Surgical Operations. Surgery and Operating Room Innovation. Singapore: Springer 71-77. https://doi.org/10.1007/978-981-15-8979-9_8
    [4] Jia Y, Shelhamer E, Donahue J, et al. (2014) Caffe: Convolutional architecture for fast feature embedding. Proceedings of the 22nd ACM international conference on Multimedia MM14: 675-678. https://doi.org/10.1145/2647868.2654889
    [5] Hirasawa T, Aoyama K, Tanimoto T, et al. (2018) Application of artificial intelligence using a convolutional neural network for detecting gastric cancer in endoscopic images. Gastric Cancer 21: 653-660. https://doi.org/10.1007/s10120-018-0793-2
    [6] Kim JH, Yoon HJ (2020) Lesion-based convolutional neural network in diagnosis of early gastric cancer. Clin Endosc 53: 127-131. https://doi.org/10.5946/ce.2020.046
    [7] Sakai Y, Takemoto S, Hori K, et al. (2018) Automatic detection of early gastric cancer in endoscopic images using a transferring convolutional neural network. In 2018 40th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC). IEEE : 4138-4141. https://doi.org/10.1109/EMBC.2018.8513274
    [8] de Groof AJ, Struyvenberg MR, van der Putten J, et al. (2020) Deep-learning system detects neoplasia in patients with barrett's esophagus with higher accuracy than endoscopists in a multistep training and validation study with benchmarking. Gastroenterology 158: 915-929. e4. https://doi.org/10.1053/j.gastro.2019.11.030
    [9] Haque A, Lyu B (2018) Deep learning based tumor type classification using gene expression data. Proceedings of the 2018 ACM international conference on bioinformatics, computational biology, and health informatics 18: 89-96. https://doi.org/10.1145/3233547.3233588
    [10] Wang P, Xiao X, Glissen Brown JR, et al. (2018) Development and validation of a deep-learning algorithm for the detection of polyps during colonoscopy. Nat Biomed Eng 2: 741-748. https://doi.org/10.1038/s41551-018-0301-3
    [11] Arsalan M, Owais M, Mahmood T, et al. (2020) Artificial intelligence-based diagnosis of cardiac and related diseases. J Clin Med 9: 871. https://www.mdpi.com/2077-0383/9/3/871
    [12] Nopour R, Shanbehzadeh M, Kazemi-Arpanahi H (2021) Developing a clinical decision support system based on the fuzzy logic and decision tree to predict colorectal cancer. Med J Islam Repub Iran 35: 44. https://doi.org/10.47176/mjiri.35.44
    [13] Darrell T, Long J, Shelhamer E (2016) Fully convolutional networks for semantic segmentation. IEEE transactions on pattern analysis and machine intelligence 39: 640-651. https://doi.org/10.1109/TPAMI.2016.2572683
    [14] Ghomari A, Kazi Tani MY, Lablack A, et al. (2017) OVIS: ontology video surveillance indexing and retrieval system. Int J Multimed Info Retr 6: 295-316. https://doi.org/10.1007/s13735-017-0133-z
    [15] He K, Gkioxari G, Dollár P, et al. (2017) Mask r-cnn. In Proceedings of the IEEE international conference on computer vision. IEEE : 2961-2969. https://doi.org/10.48550/arXiv.1703.06870
    [16] Tani LFK, Ghomari A, Tani MYK (2019) A semi-automatic soccer video annotation system based on Ontology paradigm. 2019 10th International Conference on Information and Communication Systems. ICICS: : 88-93. https://doi.org/0.1109/IACS.2019.8809161
    [17] KAZI TANI LF Conception and Implementation of a Semi-Automatic Tool Dedicated To The Analysis And Research Of Web Video Content. Doctoral Thesis, Oran, Algeria, 2020. Available from: https://theses.univ-oran1.dz/document/TH5141.pdf
    [18] Tani LFK, Ghomari A, Tani MYK (2019) Events recognition for a semi-automatic annotation of soccer videos: a study based deep learning. The International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences 42: 135-141. https://doi.org/10.5194/isprs-archives-XLII-2-W16-135-2019
    [19] Sun JY, Lee SW, Kang MC, et al. (2018) A novel gastric ulcer differentiation system using convolutional neural networks. 2018 IEEE 31st International Symposium on Computer-Based Medical Systems (CBMS). IEEE : 351-356. https://doi.org/10.1109/CBMS.2018.00068
    [20] Martin DR, Hanson JA, Gullapalli RR, et al. (2020) A deep learning convolutional neural network can recognize common patterns of injury in gastric pathology. Arch Pathol Lab Med 144: 370-378. https://doi.org/10.5858/arpa.2019-0004-OA
    [21] Gammulle H, Denman S, Sridharan S, et al. (2020) Two-stream deep feature modelling for automated video endoscopy data analysis. International Conference on Medical Image Computing and Computer-Assisted Intervention. Cham: Springer 742-751. https://doi.org/10.1007/978-3-030-59716-0_71
    [22] Li Z, Togo R, Ogawa T, et al. (2019) Semi-supervised learning based on tri-training for gastritis classification using gastric X-ray images. 2019 IEEE International Symposium on Circuits and Systems (ISCAS). IEEE : 1-5. https://doi.org/10.1109/ISCAS.2019.8702261
    [23] Kanai M, Togo R, Ogawa T, et al. (2019) Gastritis detection from gastric X-ray images via fine-tuning of patch-based deep convolutional neural network. 2019 IEEE International Conference on Image Processing (ICIP). IEEE : 1371-1375. https://doi.org/0.1109/ICIP.2019.8803705
    [24] Cho BJ, Bang CS, Park SW, et al. (2019) Automated classification of gastric neoplasms in endoscopic images using a convolutional neural network. Endoscopy 51: 1121-1129. https://doi.org/10.1055/a-0981-6133
    [25] Redmon J, Divvala S, Girshick R, et al. (2015) You look only once: unified real-time object detection. arXiv preprint arXiv : 1506.02640. https://doi.org/10.48550/arXiv.1506.02640
    [26] Islam M, Seenivasan L, Ming LC, et al. (2020) Learning and reasoning with the graph structure representation in robotic surgery. International Conference on Medical Image Computing and Computer-Assisted Intervention. Cham: Springer 627-636. https://doi.org/10.1007/978-3-030-59716-0_60
    [27] Tran VP, Tran TS, Lee HJ, et al. (2021) One stage detector (RetinaNet)-based crack detection for asphalt pavements considering pavement distresses and surface objects. J Civil Struct Health Monit 11: 205-222. https://doi.org/0.1007/s13349-020-00447-8
    [28] Badrinarayanan V, Cipolla R, Kendall A (2017) Segnet: A deep convolutional encoder-decoder architecture for image segmentation. IEEE transactions on pattern analysis and machine intelligence 39: 2481-2495. https://doi.org/10.1109/TPAMI.2016.2644615
    [29] Urbanos G, Martín A, Vázquez G, et al. (2021) Supervised machine learning methods and hyperspectral imaging techniques jointly applied for brain cancer classification. Sensors 21: 3827. https://doi.org/10.3390/s21113827
    [30] Urban G, Tripathi P, Alkayali T, et al. (2018) Deep learning localizes and identifies polyps in real time with 96% accuracy in screening colonoscopy. Gastroenterology 155: 1069-1078.e8. https//doi.org/10.1053/j.gastro.2018.06.037
    [31] Li L, Chen Y, Shen Z, et al. (2020) Convolutional neural network for the diagnosis of early gastric cancer based on magnifying narrow band imaging. Gastric Cancer 23: 126-132. https://doi.org/10.1007/s10120-019-00992-2
    [32] Ishihara K, Ogawa T, Haseyama M (2017) Detection of gastric cancer risk from X-ray images via patch-based convolutional neural network. 2017 IEEE International Conference on Image Processing (ICIP). IEEE : 2055-2059. https://doi.org/10.1109/ICIP.2017.8296643
    [33] Song Z, Zou S, Zhou W, et al. (2020) Clinically applicable histopathological diagnosis system for gastric cancer detection using deep learning. Nat Commun 11: 1-9. https://doi.org/10.1038/s41467-020-18147-8
    [34] Ikenoyama Y, Hirasawa T, Ishioka M, et al. (2021) Detecting early gastric cancer: Comparison between the diagnostic ability of convolutional neural networks and endoscopists. Digest Endosc 33: 141-150. https://doi.org/10.1111/den.13688
    [35] Ronneberger O, Fischer P, Brox T (2015) U-net: Convolutional networks for biomedical image segmentation. International Conference on Medical image computing and computer-assisted intervention. Cham: Springer 234-241. https://doi.org/10.1007/978-3-319-24574-4_28
    [36] Folgoc LL, Heinrich M, Lee M, et al. Attention u-net: Learning where to look for the pancreas (2018). Available from: https://arxiv.org/pdf/1804.03999.pdf%E4%BB%A3%E7%A0%81%E5%9C%B0%E5%9D%80%EF%BC%9Ahttps://github.com/ozan-oktay/Attention-Gated-Networks
    [37] Li X, Chen H, Qi X, et al. (2018) H-DenseUNet: hybrid densely connected UNet for liver and tumor segmentation from CT volumes. IEEE transactions on medical imaging 37: 2663-2674. https://doi.org/10.1109/TMI.2018.2845918
    [38] Zhou Z, Rahman Siddiquee MM, Tajbakhsh N, et al. (2018) Unet++: A nested u-net architecture for medical image segmentation. Deep learning in medical image analysis and multimodal learning for clinical decision support : 3-11. https://doi.org/10.1007/978-3-030-00889-5_1
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