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Interpose Detection Using Complete Lattice Approach of Mathematical Morphology

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Interpose Detection Using Complete Lattice Approach of Mathematical Morphology International Journal of Pure and Applied Mathematics Volume 117 No. 11 2017, 369-375 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu Interpose Detection Using Complete Lattice Approach of Mathematical Morphology Nisha A B1 and Sasigopalan2 1Department of Mathematics Cochin University of Science and Technology Thrikkakara, Kerala, India. [email protected] 2 School of Engineering, Cochin University of Science and Technology Kerala, India. [email protected] Abstract An interpose detecting method has a great role in the field of digital im- age processing to influence or change the structure of data in to its best approximate form. The idempotence of morphological operators are useful to establish a mathematical frame work in this method. This paper is proposing a mathematical model for the interpose detecting process . The idempotence of morphological theory is to be discussed. Lattice theory and the concept of Galois connection between basic morphological operators is to be established. This will result the proposed mathematical structure to detect and restrain interposes. Hence an apt morphological operator will lead to better interpose detector in the field of digital image processing. AMS Subject Classification: ... Key Words and Phrases: Dilation, Erosion, Complete Lattice, Galois Connection, Appendage 1 Introduction Photographs are considered to be the most authoritarian and reliable way of storing evi- dences. Now a days ordinary photographs are completely replaced by digital images. The easily available photo editing soft-wares made the interposing of digital images effortless.As a result ,it is very difficult to check the geniality of photographs by mere inspection. Be- cause of this the interpose detection has enumerated as an important research area in the field of applied mathematics.The Lattice theory helps to define with monotone functions of F and G which reduces to the distinction between the lower and upper adjoints. 369 International Journal of Pure and Applied Mathematics Special Issue 2 Basic Definitions from Lattice Theory The following definitions from Lattice theory are required for the topic under considera- tion. Definition 1. Partially ordered set [1] or a POset is a set in which a binary relation is defined, which satisfies x, y, z the following conditions 0 ≤0 ∀ (i)x x x (Reflexive) ≤ ∀ (ii)If x y and y x, then x = y (Antisymmetry) ≤ ≤ (iii)If x y and y z, then x z(Transitivity) ≤ ≤ ≤ It should be noted that the converse of any partial ordering is itself a partial ordering Definition 2. A lattice[1] is a poset L in which any two of whose elements have a supremum denoted by x y and an infimum denoted by x y ∨ ∧ A lattice Lc is complete[1] when each of its subsets has a supremum and an infimum in Lc Definition 3. Galois connection [5] is a particular correspondence between partially ordered sets.Generally there are two types of Galois connections. (i)Monotone Galois Connection[3] Let (A, ) and (B, ) wo partially ordbe tered sets.A monotone Galois connection ≤ ≤ between these POsets consists of two monotone functions F : A B and G : B A −→ −→ such that F (a) b a G(b) a A and b B ≤ ⇔ ≤ ∀ ∈ ∈ In this paper a Galois connection of two monotone functions F and G is denoted by (F, G). Here F is called the lower adjoint and G is called the upper adjoint. (ii) Antitone Galois Connection Let (A, ) and (B, ) wo partially ordbe tered sets.An antitone Galois connection ≤ ≤ between these POsets consists of two functions F : A B and G : B A such that −→ −→ b F (a) a G(b) a A and b B. ≤ ⇔ ≤ ∀ ∈ ∈ The symmetry of F and G in this connection eliminates the distinction between upper and lower adjoints. 3 Operators in Mathematical Morphology based on Complete lattices Consider a complete lattice L with a partial order relation ’ ’ supremum ’ ’ and infimum c ≤ ∨ ’ ’. Let G be the set of all mappings on L . ∧ c G = g/g : L L { c −→ c} We can stretch out the Complete lattice structure of Lc to G by defining a partial order relation on G as follows Definition 4. Let g , g G 1 2 ∈ g g g (X) g (X) X L 1 ≤ 2 ⇐⇒ 1 ≤ 2 ∀ ∈ c .This implies G is a complete lattice with supremum and infimum.The supremum and infimum in G are defined as follows, Let X L and D G ∈ c ⊆ ( D)(X) = gi(X) ∨ gi∨D ∈ 370 International Journal of Pure and Applied Mathematics Special Issue ( D)(X) = gi(X) ∧ gi∧D ∈ Acording to Jean Serra[2] Dilation and Erosion can be treated as operators Definition 5. Let ~ G, then (i)~ is called a dilation if ~( T ) = ~(X), T Lc ∈ ∨ X∨T ∀ ⊆ ∈ (ii)~ is called an erosion if ~( T ) = ~(X), T Lc ∧ X∧T ∀ ⊆ ∈ Proposition 3.1 Let Lc be a complete lattice , G = g/g : Lc Lc and ~ G.Then the following { −→ } ∈ statements are equivalent (i) ~ is increasing (ii) ~( T ) ~(X), T Lc ∨ ≥X ∨T ∀ ⊆ ∈ (iii) ~( T ) ~(X), T Lc ∧ ≤X ∧T ∀ ⊆ Proof: ∈ (i) = (ii) ⇒ Assume (i),~ is increasing Then X Y ~(X) ~(Y ) X, Y Lc ≤ ⇒ ≤ ∀ ∈ Let T L , and T = φ ⊆ c 6 T X ~( T ) ~(X) X T ( ~ is increasing) ∨ ≥ ⇒ ∨ ≥ ∀ ∈ ∵ ~( T ) ~(X) ⇒ ∨ ≥x ∨T Now to prove (ii) = (∈i) ⇒ Let Y, Z L with Y Z ∈ c ≥ Take T = Y, Z { } Then Y Z = Y = T ∨ ∨ By (ii) ~( T ) ~(X) ∨ ≥X ∨T ∈ ~(Y ) ~(Y ) ~(Z) ⇒ ≥ ∨ ~(Y ) ~(Z) ⇒ ≥ ~ is increasing ⇒ Similarly by duality(i) = (iii)&(i) = (iii). ⇒ ⇒ 4 Appendage Let Lc be a complete lattice , G be the set of all operators from Lc Lc and ~d, ~e −→ ∈ G,then (~d, ~e) is an appendage if ~d(X) Y X ~e(Y ), X, Y Lc. ≤ ⇔ ≤ ∀ ∈ Here Lc is taken as a complete lattice,therefore by definition of complete lattices Lc is a POset.Hence appendage is nothing but a Galois connection between complete lattices. Proposition 4.1 Let Lc be a complete lattice , G be the set of all operators from Lc Lc and ~d, ~e G.If −→ ∈ (~d, ~e) is an appendage then ~d is a dilation and ~e is an erosion Proof: Suppose (~d, ~e) is an appendage , ~d, ~e G ~d, ~e : Lc Lc . ∈ ⇒ → Let T L ,Y L ⊆ c ∈ c Let T L be a non-empty subse of L ,Y L ⊆ C C ∈ C 371 International Journal of Pure and Applied Mathematics Special Issue ~d(X) Y ~d(X) Y (by the definition of ) x∨T ≤ ⇔ ≤ ∨ ∈ X ~e(Y )( (~e, ~d)is an appendage) X T ⇔ ≤ ∵ ∀ ∈ T ~e(Y ) ⇔∨ ≤ ~d(T ) Y ⇔ ≤ By taking Y = ~d(X) and Y = ~d( T ) X∨T ∨ ∈ Hence ~d is a dilation By duality ~e is an erosion Remark: If (~e, ~d) is an appendage ,~e is an erosion and ~d is a dilation.By the definition of dilation, ~d satisfies (ii) of proposition 2 and similarly by the defenition of erosion, ~e satisfies (ii) of proposition 2.Therefore by the equivalence conditions both dilation and erosion are increasing. ∴ (~e, ~d) is a monotone Galois connection between complete lattices. Theorem 4.1 There exists closing and opening operators which follows monotone Galois connection between complete lattices such that they are idempotent Proof:The proof of the theorem follows the following proposition Proposition 4.2 If ~e, ~d G,the following conditions are equivalent ∈ (i)(~e, ~d) is an appendage (ii)~e~d I and I ~e~d, where I is the identity operator ≤ ≤ Also the above two conditions imply (iii)~d = ~d~e~d and ~e = ~e~d~e (iv)~e~d and ~d~e are idempotent Proof:- (i) = (ii) ⇒ Suppose (~e, ~d) is an appendage. ~d(X) Y X ~e(Y ), X, Y Lc ≤ ⇔ ≤ ∀ ∈ Y Lc, ~e(Y ) ~e(Y ) ~d[~e(Y )] Y = I(Y ) ∀ ∈ ≤ ⇔ ≤ ~d~e(Y ) I(Y ) ⇔ ≤ ~d~e I ⇔ ≤ X Lc, ~d(X) ~d(X) ~e[~d(X)] X = I(X) ∀ ∈ ≥ ⇔ ≥ ~e~d(X) I(X) ⇔ ≥ ~e~d I ⇔ ≥ Also to prove (ii) = (i) ⇒ LetX ~e(Y )andY ~d(X) ≤ ≥ X ~e(Y ) ~d(X) ~d[~e(Y )]( ~d is monotone) ≤ ⇒ ≤ ∵ ~d(X) ~d~e(Y ) ⇒ ≤ ~d(X) I(Y ) ⇒ ≤ ~d(X) Y (1) ⇒ ≤ −→ Y ~d(X) ~e(Y ) ~e[~d(X)] ≥ ⇒ ≥ ~e(Y ) ~e~d(X) ⇒ ≥ ~e(Y ) ~e~d(X) ⇒ ≥ ~e(Y ) I(X) ⇒ ≥ ~e(Y ) X (2) ⇒ ≥ −→ 372 International Journal of Pure and Applied Mathematics Special Issue F rom (1)and(2)X ~e(Y ) ~d(X) Y (~e, ~d)is an adjunction ≤ ⇔ ≤ ⇒ Now to prove (ii) = (iii) ⇒ ~d~e I ~d~e~d ~d (3) ≤ ⇒ ≤ −→ I ~e~d ~d ~d~e~d (4) From (3)and(4) ≤ ⇒ ≤ −→ ~d~e~d = ~d, Similarly ~e~d~e = ~e Also (~e~d)(~e~d) = ~e(~d~e~d) = ~e~d (~d~e)(~d~e) = (~d~e~d)~e = ~d~e ∵ ~e~d and ~d~e are idempotent 5 Interposed Area in Image Processing Grey scale images are considered in three dimensions,such as x,y and z axis.Pixel posi- tions arerepresenting in thex, y axis and intensity of each pixel is represented in the z-axis. The variation intensity values represent interposed area in images. In Mathematical Mor- phology ,dilation followed by erosion is called closing and erosion followed by dilation is called opening .By proposition 4.2 closing and opening are idempotent which means that once we apply closing or opening operation on an image no further change will betide by their repeated appliation . Inposing the dilation and erosion on the input image using reconstructed mathematical operators of the interposed area, the closing operators on the original imageI restore the resulting imageIC for further use (C) (C) Result 5.1 LetIM be the interposed version of the restored image.Closing of IM (C,C) gives IM a new image.The change in this image decides the interposed area .
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