In this article we defined mathematical morphology image processing with set operations. First, we defined Minkowski set operations and proved their properties. Next, we defined basic image processing, dilation and erosion proving basic fact about them [5], [8].Yamazaki Hiroshi - Shinshu University, Nagano, JapanByliński Czesław - University of Białystok, PolandWasaki Katsumi - Shinshu University, Nagano, JapanCzesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Yuzhong Ding and Xiquan Liang. Preliminaries to mathematical morphology and its properties. Formalized Mathematics, 13(2):221-225, 2005.Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitary space. Formalized Mathematics, 11(1):23-28, 2003.H. J. A. M. Heijimans. Morphological Image Operators. Academic Press, 1994.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.P. Soille. Morphological Image Analysis: Principles and Applications. Springer, 2003.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990

Beata Padlewska

Czesław Byliński

H. Heijimans

Hiroshi Yamazaki

Katsumi Wasaki

Krzysztof Hryniewiecki

Noboru Endou

P. Soille

Wojciech Trybulec

Yuzhong Ding

Zinaida Trybulec

Formalized Mathematics

ArticleFormalized Mathematics. 20(1): 61-63 (2012)journal articl

Yamazaki, Hiroshi

Czesław, Byliński

Wasaki, Katsumi

Shinshu University Institutional Repository

FORMALIZED MATHEMATICSVol. 20, No. 1, Pages 61–63, 2012DOI: 10.2478/v10037-012-0008-y versita.com/fm/Morphology for Image Processing. Part I1Hiroshi YamazakiShinshu UniversityNagano, JapanCzesław BylińskiUniversity of BiałystokPolandKatsumi WasakiShinshu UniversityNagano, JapanSummary. In this article we defined mathematical morphology image pro-cessing with set operations. First, we defined Minkowski set operations and pro-ved their properties. Next, we defined basic image processing, dilation and erosionproving basic fact about them [5], [8].MML identifier: MORPH 01, version: 7.1 .0 4.1 0.11The terminology and notation used in this paper have been introduced in thefollowing papers: [10], [7], [1], [2], [6], [9], [4], and [3].1. Minkowski Set OperationsLet E be a non empty RLS structure. A binary image of E is a subset of E.In the sequel E denotes a real linear space and A denotes a binary image of E.Let E be a real linear space and let A, B be binary images of E. The functorA	B yielding a binary image of E is defined as follows:(Def. 1) A	B = {z ∈ E:∧b : element of E (b ∈ B ⇒ z − b ∈ A)}.Let a be a real number, let E be a real linear space, and let A be a subsetof E. We introduce a ·A as a synonym of aA. The following propositions aretrue:(1) Let E be a real linear space and A, B be subsets of E. If B = ∅, thenA⊕B = B and B ⊕A = B and A	B = the carrier of E.(2) For every real linear space E and for all subsets A, B of E such thatA 6= ∅ and B = ∅ holds B 	A = B.1The authors wants to thank Prof. Yasunari Shidama for his kind support during the courseof this work.61c© 2012 University of BiałystokCC-BY-SA License ver. 3.0 or laterISSN 1426–2630(p), 1898-9934(e)2 2 7 34 - 10.2478/v10037-012-0008-yDownloaded from De Gruyter Online at 09/14/2016 07:08:03AMvia Shinshu U Lib62 hiroshi yamazaki et al.(3) Let E be a real linear space and A, B be subsets of E. If B = the carrierof E and A 6= ∅, then A⊕B = B and B ⊕A = B.(4) For every real linear space E and for all subsets A, B of E such thatB = the carrier of E holds B 	A = B.(5) A⊕B =⋃{b+A; b ranges over elements of E: b ∈ B}.Let E be a non empty RLS structure. A binary image family of E is a familyof subsets of the carrier of E.We follow the rules: F , G are binary image families of E and A, B, C arenon empty binary images of E. We now state four propositions:(6) A	B =⋂{b+A; b ranges over elements of E: b ∈ B}.(7) A⊕B = {v ∈ E: (v + (−1) ·B) ∩A 6= ∅}.(8) A	B = {v ∈ E: v + (−1) ·B ⊆ A}.(9) ((The carrier of E)\A)⊕B = (the carrier of E)\A	B and ((the carrierof E) \A)	B = (the carrier of E) \A⊕B.Let E be a non empty Abelian additive loop structure and let A, B besubsets of E. Let us note that the functor A⊕B is commutative.One can prove the following propositions:(10) For every non empty add-associative additive loop structure E and forall subsets A, B, C of E holds (A+B) + C = A+ (B + C).(11) (A⊕B)⊕ C = A⊕ (B ⊕ C).(12)⋃F ⊕B =⋃{X ⊕B;X ranges over binary images of E: X ∈ F}.(13) A⊕⋃F =⋃{A⊕X;X ranges over binary images of E: X ∈ F}.(14)⋂F ⊕B ⊆⋂{X ⊕B;X ranges over binary images of E: X ∈ F}.(15) A⊕⋂F ⊆⋂{A⊕X;X ranges over binary images of E: X ∈ F}.(16) For every non empty additive loop structure E and for all subsets A, B,C of E such that B ⊆ C holds A+B ⊆ A+ C.(17) (v +A)⊕B = A⊕ (v +B) and (v +A)⊕B = v +A⊕B.(18)⋂F 	B =⋂{X 	B;X ranges over binary images of E: X ∈ F}.(19)⋂{B 	X;X ranges over binary images of E: X ∈ F} ⊆ B 	⋂F.(20)⋃{X 	B;X ranges over binary images of E: X ∈ F} ⊆⋃F 	B.(21) If F 6= ∅, then B 	⋃F =⋂{B 	X;X ranges over binary images of E:X ∈ F}.(22) If A ⊆ B, then A	 C ⊆ B 	 C.(23) If A ⊆ B, then C 	B ⊆ C 	A.(24) (v +A)	B = A	 (v +B) and (v +A)	B = v +A	B.(25) A	B 	 C = A	 (B ⊕ C). - 10.2478/v10037-012-0008-yDownloaded from De Gruyter Online at 09/14/2016 07:08:03AMvia Shinshu U LibMorphology for image processing. Part I 632. Dilation and ErosionLet E be a real linear space and let B be a binary image of E. The functordilationB yields a function from 2the carrier of E into 2the carrier of E and is definedas follows:(Def. 2) For every binary image A of E holds (dilationB)(A) = A⊕B.Let E be a real linear space and let B be a binary image of E. The functorerosionB yields a function from 2the carrier of E into 2the carrier of E and is definedby:(Def. 3) For every binary image A of E holds (erosionB)(A) = A	B.The following propositions are true:(26) (dilationB)(⋃F ) =⋃{(dilationB)(X);X ranges over binary images ofE: X ∈ F}.(27) If A ⊆ B, then (dilationC)(A) ⊆ (dilationC)(B).(28) (dilationC)(v +A) = v + (dilationC)(A).(29) (erosionB)(⋂F ) =⋂{(erosionB)(X);X ranges over binary images ofE: X ∈ F}.(30) If A ⊆ B, then (erosionC)(A) ⊆ (erosionC)(B).(31) (erosionC)(v +A) = v + (erosionC)(A).(32) (dilationC)((the carrier of E) \A) = (the carrier of E) \ (erosionC)(A)and (erosionC)((the carrier of E)\A) = (the carrier of E)\(dilationC)(A).(33) (dilationC)((dilationB)(A)) = (dilation(dilationC)(B))(A) and(erosionC)((erosionB)(A)) = (erosion(dilationC)(B))(A).References[1] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.[2] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164,1990.[3] Yuzhong Ding and Xiquan Liang. Preliminaries to mathematical morphology and itsproperties. Formalized Mathematics, 13(2):221–225, 2005.[4] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitaryspace. Formalized Mathematics, 11(1):23–28, 2003.[5] H.J.A.M. Heijimans. Morphological Image Operators. Academic Press, 1994.[6] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics,1(1):35–40, 1990.[7] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.[8] P. Soille. Morphological Image Analysis: Principles and Applications. Springer, 2003.[9] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296,1990.[10] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Received September 21, 2011 - 10.2478/v10037-012-0008-yDownloaded from De Gruyter Online at 09/14/2016 07:08:03AMvia Shinshu U Lib

English

Morphology for Image Processing, Part I

Yamazaki Hiroshi

Wasaki Katsumi

Morphology for Image Processing. Part I
        In this article we defined mathematical morphology image processing with set operations. First, we defined Minkowski set operations and proved their properties. Next, we defined basic image processing, dilation and erosion proving basic fact about them [5], [8].</jats:p

Crossref

Morphology for Image Processing. Part I

Byliński, Czesław

Repozytorium Uniwersytetu w Białymstoku (University of Bialystok Repository)

FORMALIZED MATHEMATICSVol. 20, No. 1, Pages 61–63, 2012DOI: 10.2478/v10037-012-0008-y versita.com/fm/Morphology for Image Processing. Part I1Hiroshi YamazakiShinshu UniversityNagano, JapanCzesław BylińskiUniversity of BiałystokPolandKatsumi WasakiShinshu UniversityNagano, JapanSummary. In this article we defined mathematical morphology image pro-cessing with set operations. First, we defined Minkowski set operations and pro-ved their properties. Next, we defined basic image processing, dilation and erosionproving basic fact about them [5], [8].MML identifier: MORPH 01, version: 7.1 .0 4.1 0.11The terminology and notation used in this paper have been introduced in thefollowing papers: [10], [7], [1], [2], [6], [9], [4], and [3].1. Minkowski Set OperationsLet E be a non empty RLS structure. A binary image of E is a subset of E.In the sequel E denotes a real linear space and A denotes a binary image of E.Let E be a real linear space and let A, B be binary images of E. The functorA	B yielding a binary image of E is defined as follows:(Def. 1) A	B = {z ∈ E: ∧b : element of E (b ∈ B ⇒ z − b ∈ A)}.Let a be a real number, let E be a real linear space, and let A be a subsetof E. We introduce a ·A as a synonym of aA. The following propositions aretrue:(1) Let E be a real linear space and A, B be subsets of E. If B = ∅, thenA⊕B = B and B ⊕A = B and A	B = the carrier of E.(2) For every real linear space E and for all subsets A, B of E such thatA 6= ∅ and B = ∅ holds B 	A = B.1The authors wants to thank Prof. Yasunari Shidama for his kind support during the courseof this work.61c© 2012 University of BiałystokCC-BY-SA License ver. 3.0 or laterISSN 1426–2630(p), 1898-9934(e)2 2 7 34Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 12/8/15 12:05 PM62 hiroshi yamazaki et al.(3) Let E be a real linear space and A, B be subsets of E. If B = the carrierof E and A 6= ∅, then A⊕B = B and B ⊕A = B.(4) For every real linear space E and for all subsets A, B of E such thatB = the carrier of E holds B 	A = B.(5) A⊕B = ⋃{b+A; b ranges over elements of E: b ∈ B}.Let E be a non empty RLS structure. A binary image family of E is a familyof subsets of the carrier of E.We follow the rules: F , G are binary image families of E and A, B, C arenon empty binary images of E. We now state four propositions:(6) A	B = ⋂{b+A; b ranges over elements of E: b ∈ B}.(7) A⊕B = {v ∈ E: (v + (−1) ·B) ∩A 6= ∅}.(8) A	B = {v ∈ E: v + (−1) ·B ⊆ A}.(9) ((The carrier of E)\A)⊕B = (the carrier of E)\A	B and ((the carrierof E) \A)	B = (the carrier of E) \A⊕B.Let E be a non empty Abelian additive loop structure and let A, B besubsets of E. Let us note that the functor A⊕B is commutative.One can prove the following propositions:(10) For every non empty add-associative additive loop structure E and forall subsets A, B, C of E holds (A+B) + C = A+ (B + C).(11) (A⊕B)⊕ C = A⊕ (B ⊕ C).(12)⋃F ⊕B = ⋃{X ⊕B;X ranges over binary images of E: X ∈ F}.(13) A⊕⋃F = ⋃{A⊕X;X ranges over binary images of E: X ∈ F}.(14)⋂F ⊕B ⊆ ⋂{X ⊕B;X ranges over binary images of E: X ∈ F}.(15) A⊕⋂F ⊆ ⋂{A⊕X;X ranges over binary images of E: X ∈ F}.(16) For every non empty additive loop structure E and for all subsets A, B,C of E such that B ⊆ C holds A+B ⊆ A+ C.(17) (v +A)⊕B = A⊕ (v +B) and (v +A)⊕B = v +A⊕B.(18)⋂F 	B = ⋂{X 	B;X ranges over binary images of E: X ∈ F}.(19)⋂{B 	X;X ranges over binary images of E: X ∈ F} ⊆ B 	⋂F.(20)⋃{X 	B;X ranges over binary images of E: X ∈ F} ⊆ ⋃F 	B.(21) If F 6= ∅, then B 	⋃F = ⋂{B 	X;X ranges over binary images of E:X ∈ F}.(22) If A ⊆ B, then A	 C ⊆ B 	 C.(23) If A ⊆ B, then C 	B ⊆ C 	A.(24) (v +A)	B = A	 (v +B) and (v +A)	B = v +A	B.(25) A	B 	 C = A	 (B ⊕ C).Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 12/8/15 12:05 PMMorphology for image processing. Part I 632. Dilation and ErosionLet E be a real linear space and let B be a binary image of E. The functordilationB yields a function from 2the carrier of E into 2the carrier of E and is definedas follows:(Def. 2) For every binary image A of E holds (dilationB)(A) = A⊕B.Let E be a real linear space and let B be a binary image of E. The functorerosionB yields a function from 2the carrier of E into 2the carrier of E and is definedby:(Def. 3) For every binary image A of E holds (erosionB)(A) = A	B.The following propositions are true:(26) (dilationB)(⋃F ) =⋃{(dilationB)(X);X ranges over binary images ofE: X ∈ F}.(27) If A ⊆ B, then (dilationC)(A) ⊆ (dilationC)(B).(28) (dilationC)(v +A) = v + (dilationC)(A).(29) (erosionB)(⋂F ) =⋂{(erosionB)(X);X ranges over binary images ofE: X ∈ F}.(30) If A ⊆ B, then (erosionC)(A) ⊆ (erosionC)(B).(31) (erosionC)(v +A) = v + (erosionC)(A).(32) (dilationC)((the carrier of E) \A) = (the carrier of E) \ (erosionC)(A)and (erosionC)((the carrier of E)\A) = (the carrier of E)\(dilationC)(A).(33) (dilationC)((dilationB)(A)) = (dilation(dilationC)(B))(A) and(erosionC)((erosionB)(A)) = (erosion(dilationC)(B))(A).References[1] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.[2] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164,1990.[3] Yuzhong Ding and Xiquan Liang. Preliminaries to mathematical morphology and itsproperties. Formalized Mathematics, 13(2):221–225, 2005.[4] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitaryspace. Formalized Mathematics, 11(1):23–28, 2003.[5] H.J.A.M. Heijimans. Morphological Image Operators. Academic Press, 1994.[6] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics,1(1):35–40, 1990.[7] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.[8] P. Soille. Morphological Image Analysis: Principles and Applications. Springer, 2003.[9] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296,1990.[10] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Received September 21, 2011Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 12/8/15 12:05 PM

https://soar-ir.repo.nii.ac.jp/?action=repository_action_common_download&item_id=18423&item_no=1&attribute_id=65&file_no=1

Morphology for Image Processing. Part I

Abstract

Similar works

Full text

Available Versions

Shinshu University Institutional Repository

Shinshu University Institutional Repository

Crossref

Repozytorium Uniwersytetu w Białymstoku (University of Bialystok Repository)