:: Morphology for Image Processing, Part {I}
:: by Hiroshi Yamazaki , Czes\l aw Byli\'nski and Katsumi Wasaki
::
:: Received September 21, 2011
:: Copyright (c) 2011-2018 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies TARSKI, RLVECT_1, FUNCT_1, ARYTM_1, RELAT_1, MATHMORP, SETFAM_1,
ARYTM_3, ALGSTR_0, MORPH_01, ZFMISC_1, SUBSET_1, STRUCT_0, XBOOLE_0,
REAL_1, SUPINF_2, VALUED_2;
notations TARSKI, XBOOLE_0, ORDINAL1, SUBSET_1, SETFAM_1, FUNCT_1, FUNCT_2,
XCMPLX_0, XREAL_0, REAL_1, STRUCT_0, ALGSTR_0, RLVECT_1, RUSUB_4,
MATHMORP;
constructors SETFAM_1, REAL_1, RUSUB_4, RVSUM_1, RELSET_1, MATHMORP;
registrations SUBSET_1, RELSET_1, MEMBERED, STRUCT_0, RLVECT_1, XREAL_0;
requirements NUMERALS, SUBSET, BOOLE;
begin :: Minkowski set operations
definition let E be non empty RLSStruct;
mode binary-image of E is Subset of E;
end;
reserve E for RealLinearSpace;
reserve A, B, C for binary-image of E;
reserve a, b, v for Element of E;
definition
let E be RealLinearSpace;
let A, B be binary-image of E;
func A(-)B -> binary-image of E equals
:: MORPH_01:def 1
{ z where z is Element of E :
for b be Element of E st b in B holds z - b in A };
end;
notation
let a be Real, E be RealLinearSpace,
A be Subset of E;
synonym a * A for a (.) A;
end;
theorem :: MORPH_01:1
for E be RealLinearSpace, A, B be Subset of E st B = {}
holds A(+)B = B & B(+)A = B & A(-)B = the carrier of E;
theorem :: MORPH_01:2
for E be RealLinearSpace, A, B be Subset of E st A <> {} & B = {}
holds B(-)A = B;
theorem :: MORPH_01:3
for E be RealLinearSpace,
A, B be Subset of E st B = the carrier of E & A <> {}
holds A(+)B = B & B(+)A = B;
theorem :: MORPH_01:4
for E be RealLinearSpace,
A, B be Subset of E st B = the carrier of E
holds B(-)A = B;
theorem :: MORPH_01:5
A(+)B = union {b + A where b is Element of E: b in B};
definition
let E be non empty RLSStruct;
mode binary-image-family of E is Subset-Family of the carrier of E;
end;
reserve F, G for binary-image-family of E;
reserve A, B, C for non empty binary-image of E;
theorem :: MORPH_01:6
A(-)B = meet {b + A where b is Element of E: b in B};
theorem :: MORPH_01:7
A(+)B = {v where v is Element of E: (v + (-1)*B) /\ A <> {}};
theorem :: MORPH_01:8
A(-)B = {v where v is Element of E: v + (-1)*B c= A};
theorem :: MORPH_01:9
((the carrier of E) \ A)(+)B = (the carrier of E) \ (A(-)B)
& ((the carrier of E) \ A)(-)B = (the carrier of E) \ (A(+)B);
definition let E be non empty Abelian addLoopStr, A, B be Subset of E;
redefine func A(+)B;
commutativity;
end;
theorem :: MORPH_01:10
for E be non empty add-associative addLoopStr, A, B, C be Subset of E
holds A + B + C = A + (B + C);
theorem :: MORPH_01:11
A(+)B(+)C = A(+)(B(+)C);
theorem :: MORPH_01:12
(union F)(+)B = union {X(+)B where X is binary-image of E: X in F};
theorem :: MORPH_01:13
A(+)(union F) = union {A(+)X where X is binary-image of E: X in F};
theorem :: MORPH_01:14
(meet F)(+)B c= meet {X(+)B where X is binary-image of E: X in F };
theorem :: MORPH_01:15
A(+)(meet F) c= meet { A(+)X where X is binary-image of E: X in F};
theorem :: MORPH_01:16
for E be non empty addLoopStr, A, B, C be Subset of E
holds B c= C implies A + B c= A + C;
theorem :: MORPH_01:17
(v + A)(+)B = A(+)(v+B) & (v+A)(+)B = v+ (A(+)B);
theorem :: MORPH_01:18
(meet F)(-)B = meet { X(-)B where X is binary-image of E: X in F};
theorem :: MORPH_01:19
meet {B (-)X where X is binary-image of E: X in F} c= B(-)(meet F);
theorem :: MORPH_01:20
union {X(-)B where X is binary-image of E: X in F} c= (union F)(-)B;
theorem :: MORPH_01:21
F <> {}
implies B(-) (union F) = meet {B(-)X where X is binary-image of E: X in F};
theorem :: MORPH_01:22
A c= B implies A(-)C c= B(-)C;
theorem :: MORPH_01:23
A c= B implies C(-)B c= C(-)A;
theorem :: MORPH_01:24
(v+A)(-)B = A(-)(v+B) & (v+A)(-)B = v+(A(-)B);
theorem :: MORPH_01:25
(A(-)B)(-)C = A(-)(B(+)C);
begin :: Dilation and erosion
definition
let E be RealLinearSpace;
let B be binary-image of E;
func dilation(B)
-> Function of bool the carrier of E, bool the carrier of E means
:: MORPH_01:def 2
for A be binary-image of E holds it.A = A(+)B;
end;
definition
let E be RealLinearSpace;
let B be binary-image of E;
func erosion (B)
-> Function of bool the carrier of E, bool the carrier of E means
:: MORPH_01:def 3
for A be binary-image of E holds it.A = A(-)B;
end;
theorem :: MORPH_01:26
(dilation(B)).(union F)
= union {(dilation(B)).X where X is binary-image of E: X in F};
theorem :: MORPH_01:27
A c= B implies (dilation(C)).A c= (dilation(C)).B;
theorem :: MORPH_01:28
(dilation(C)).(v+A) = v+((dilation(C)).A);
theorem :: MORPH_01:29
(erosion(B)).(meet F)
= meet {(erosion(B)).X where X is binary-image of E: X in F};
theorem :: MORPH_01:30
A c= B implies (erosion(C)).A c= (erosion(C)).B;
theorem :: MORPH_01:31
(erosion(C)).(v+A) = v+((erosion(C)).A);
theorem :: MORPH_01:32
(dilation(C)).((the carrier of E) \ A)
= (the carrier of E) \ ((erosion(C)).A)
& (erosion(C)).( (the carrier of E) \ A)
= (the carrier of E) \ ((dilation(C)).A );
theorem :: MORPH_01:33
(dilation(C)).((dilation(B)).A) = (dilation((dilation(C)).B)).A
& (erosion(C)).((erosion(B)).A) = (erosion((dilation(C)).B)).A;