Mathematical Morphology via Category Theory
aa r X i v : . [ m a t h . C T ] S e p Mathematical Morphology via
Category Theory
Hossein Memarzadeh Sharifipour and Bardia Yousefi , Department of Computer Science, Laval University, Qubec, CA
[email protected] Department of Electrical and Computer Engineering, Laval University, Qubec, CA Address:
University of Pennsylvania, Philadelphia PA 19104
Abstract.
Mathematical morphology contributes many profitable toolsto image processing area. Some of these methods considered to be basicbut the most important fundamental of data processing in many variousapplications. In this paper, we modify the fundamental of morphologicaloperations such as dilation and erosion making use of limit and co-limitpreserving functors within (
Category Theory ). Adopting the well-knownmatrix representation of images, the category of matrix, called
Mat , canbe represented as an image. With enriching
Mat over various semiringssuch as
Boolean and ( max, +) semirings, one can arrive at classical defi-nition of binary and gray-scale images using the categorical tensor prod-uct in
Mat . With dilation operation in hand, the erosion can be reachedusing the famous tensor-hom adjunction. This approach enables us todefine new types of dilation and erosion between two images representedby matrices using different semirings other than
Boolean and ( max, +)semirings. The viewpoint of morphological operations from category the-ory can also shed light to the claimed concept that mathematical mor-phology is a model for linear logic.
Keywords:
Mathematical morphology · Closed monoidal categories · Day convolution. Enriched categories, category of semirings
Mathematical morphology is a structure-based analysis of images constructedon set theory concepts. Two main induced transformations in mathematicalmorphology are dilation and erosion established initially by translations, unionsand intersections on subsets of euclidean spaces. These transformations wereextended to complete lattices later. In a more general form, dilation and erosionare Galois connections over mappings between complete lattices.According to [9], every operator on a complete lattice which preserves finitesupremum( ∨ ) is regarded as a dilation and each infimum preserving operator canbe inspected as an erosion. This is the most universal definition of dilation anderosion appeared in literature up to our knowledge. Looking toward categorytheory viewpoint, definition of the dilation and erosion can be generalized to leftand right adjoint functors that preserve co-limits and limits respectively. H.M. Sharifipour & B. Yousefi
The first section stares at mathematical morphology and category theorybriefly. Most of the consisting material for category theory can be found in[1,15,2,13] .
An image on a computer springs up from quantization of both image space andintensities into discrete values. It is merely a 2D rectangular array of integervalues. It is widely accepted to record intensity as an integer number over theinterval [0 , , f defined on spatial variables x, y . Intuitively, f ( x, y ) defines the intensity value of the pixel at point ( x, y ). The followingdefinitions correspond to binary, gray-scaled and color images. Definition 1.
A binary image is a rectangular matrix with all elements values0 or 1.
Definition 2.
A gray-scaled image is a rectangular matrix with values rangingwithin [0 , . Definition 3.
A color image is a 2D image which has vector of 3 values at eachspatial point or pixel.
Dilation and erosion constitute the basic operations which construct the back-bone for clarifying other widely used operations such as opening, closing, hit-missand a few others. They have their roots in set theory and boil from a set the-oretical view point. These transformations are defined on elements of two setscalled the source and structuring element respectively. The structuring elementis generally much smaller sized comparing to the image that it acts on. It func-tions as a pattern probing the source image, targeting at finding its structure.First we define dilation for binary images as follows [9].
Definition 4. If A, B are two sets determining the source image and the struc-turing element respectively, dilation of A and B is defined by [8] and [18] as A ⊕ B = { a + b | a ∈ A, b ∈ B } , (1)This operation is also called the Minkowski sum. The pertinence of dilation inimage analysis area varies from image expanding to filling holes. Erosion is thedual of dilation defined by: Definition 5.
The erosion of a set A with a structuring element B is: [18,9] A ⊖ B = { x ∈ Z | for every b ∈ B, there exists an a ∈ A such that x = a − b } (2) athematical Morphology via Category Theory 3
Erosion of two sets
A, B can also be defined as A ⊖ B = { h ∈ Z | B h ⊆ A } (3) where B h = { b + h | b ∈ B } is the translating of B along the vector h and reflectionof the set B with respect to origin is defined like ˘ B = { x ∈ Z | for some b ∈ B, x = − b } (4)duality of dilation and erosion means that erosion can be written in terms of thedilation: ( A ⊖ B ) c = A c ⊕ ˘ B (5)where ˘ B has been defined before (3.13). In other words, dilating the foregroundis the same as eroding the background, but the structuring element reflectsbetween the two. Likewise, eroding the foreground is the same as dilating thebackground.Dilation and erosion are defined for gray scale images in a different way.Dilation and erosion of f where f : F → Z , F ⊆ Z is a function that maps( x, y ) ∈ Z to gray scale value of pixel at ( x, y ) with structuring element B isdenoted as ( f ⊕ B )( x, y ) = max ( s,t ) ∈ B { f ( x − s, y − t ) } (6)( f ΘB )( x, y ) = min ( s,t ) ∈ B { f ( x + s, y + t ) } (7)The following more general definition of dilation and erosion can be found in [9]. Definition 6.
Let L be a complete lattice and E , E be arbitrary sets. Theoperator δ : L E −→ L E is a dilation if and only if for every x ∈ E and y ∈ E there exists a δ y,x : L −→ L such that for F ∈ L E and y ∈ E , δ ( F )( y ) = _ x ∈ E δ y,x ( F ( x )) The erosion ǫ : L E −→ L E is given by: ǫ ( F )( x ) = ^ y ∈ E ǫ y,x ( F ( y )) Definition 7.
Let ( A, ≤ ) , ( B, ≤ ) be two partially ordered sets with two map-pings, F : A −→ B and U : B −→ A . A monotone Galois connection between F, U is for all x ∈ A and y ∈ B : F ( x ) ≤ y ⇔ x ≤ U ( y ) F is referred as the left adjoint and U as an right adjoint. Monotone Galois con-nections are entitled as adjunctions in literature [7,19] likewise. The other varietyof Galois connections called antitone Galois connections emerges in literature asfollows. H.M. Sharifipour & B. Yousefi
Definition 8.
Let ( A, ≤ ) , ( B, ≤ ) be two partially ordered sets. Let F : A −→ B and U : B −→ A . F, U are an antitone Galois connection if for all x ∈ A and y ∈ B : y ≤ F ( x ) ⇔ x ≤ U ( y ) Theorem 1. δ ( F ) ≤ F ⇔ ǫ ( F ) ≤ F This theorem confirms that dilation and erosion engage in a monotonic Galoisconnection.
Category theory is an effort for generalizing and simplifying many properties ofmathematical systems by denoting them with objects and arrows. Each arrow f : A → B represents a function from an object A to another object B . Acategory is small if its set of objects and arrows are small.A contravariant functor F : A OP → B maps every object A ∈ A to F ( A ) ∈ B and there exist a mapping A ( A, A ′ ) → B ( F A, F A ′ ) for each mapping f : A → A ′ ∈ A . Reminding that A ( A, A ′ ) is an arrow between the objects, these dataare subject to following two conditions: – any two morphism f ∈ A ( A, A ′ ) and g ∈ A ( A ′ , A ′′ ) can be decomposed by F ( g ◦ f ) = F ( f ) ◦ F ( g ). – For any object A ∈ A , F (1 A ) = 1 F A .A contravariant functor reverses the direction of arrows. For example f : A → B gets F ( f ) : f ( B ) → f ( A ).Conversely, a covariant functor F : C → D preserves the direction of arrows.Everything is the same as the contravariant functor except : F ( f ◦ g ) = F ( f ) ◦ F ( g ) for arrows f, g in C . Natural transformationsDefinition 9.
Let A and B be categories and F, G two functors
F, G : A → B .A natural transformation between
F, G is an arrow α x : F ( x ) → G ( x ) for anyobject X ∈ A such that for any arrow f : X → Y ∈ A , the diagram depicted infigure 1 commutes, Natural transformations resemble functor isomorphisms. (Co)-limitsDefinition 10.
Given a functor F : A → B , a cone of F is an object O ∈ B together with a family of arrows φ x : O → F ( x ) for each x ∈ A such that foreach arrow f : x → y in A we have F f ◦ φ x = φ y according to figure 2. athematical Morphology via Category Theory 5 F ( X ) F ( Y ) G ( X ) G ( Y ) F ( f ) α X G ( f ) α Y Fig. 1: Natural transformations OF ( X ) F ( Y ) φ x φ y F ( f )Fig. 2: Diagram of a cone Definition 11.
A limit of a functor F : A → B , is a universal cone ( L, φ x ) such that for every other cone ( N, ψ x ) of F there is a unique arrow u : N → L such that ψ x = φ x ◦ u for every X in B . Definition 12.
Dual to limit, a co-limit of a functor F : A → B , is a universalco-cone ( L, φ x ) such that for any other co-cone ( N, ψ x ) of F , there exists aunique arrow u : L → N such that ψ x = u ◦ φ x for every X in B . Figure 3 (b)illustrates the diagram of a co-limit. A functor F : A → B is called small if B isa small category. (co)-limits over small functors are called small. Symbols lim −→ and lim ←− are used to denote lim and co-limit often in literature.A category is called (co)-complete if it contains all small (co)-limits. Symbolslim −→ and lim ←− are used to denote lim and co-limit often in literature. (Co)-ends (co)-ends are useful notions inspired from calculus. Particularly, anend resembles an infinite product whereas a co-end imitates the idea of an infinitesum or integral.(co)-ends are special (co)-limits defined on functors of the form F : C OP ×C →D . Defining (co)-wedge is essential since a (co)-end is a universal (co)-wedge.A wedge of a functor F : C OP × C → D is an object O ∈ D with an arrow ω c : O → F ( C, C ) for any object C ∈ C . The universal property of a wedgeenforces that for any arrow C ′ → C for C, C ′ ∈ C , the diagram illustrated in (a)commutes. H.M. Sharifipour & B. Yousefi
NLF ( X ) F ( Y ) u F ( f ) ψ x ψ y φ x φ y NLF ( X ) F ( Y ) u F ( f ) ψ x ψ y φ x φ y Fig. 3: a) Diagram of a limit b) Diagram of a co-limitConversely, co-ends are defined by natural co-wedges. A co-wedge for a func-tor F : C OP × C → D is an object O in D along with an arrow ω c : F ( C, C ) → O such that for any arrow t : C ′ → C in C , the diagram illustrated in (a) commutes. O F ( C, C ) F ( C ′ , C ′ ) F ( C, C ′ ) ω c ω ′ c F ( t, F ( , t ) F ( C, C ′ ) F ( C, C ) F ( C ′ , C ′ ) O F (1 , t ) F ( t , ) ω ′ c ω c Fig. 4: a)Diagram of a wedge b)Diagram of a co-wedgeThe abusing integral notation for denoting (co)-ends stems from the work ofN.Yoneda [14] while he came up with functors C OP × C → Ab . The subscriptedintegral notation R C F ( C, C ) denotes an end of a functor F : C OP × C → D whereas the superscripted integral notation R C F ( C, C ) demonstrates a Co-endfor F .A helpful property of ends which makes them so useful is their capabilityof representing natural transformations. This can be expressed by the followingtheorem, Theorem 2.
Given two functors
F, G : C → D , the set of natural transforma-tions between
F, G denoted by [ C, D ]( F, G ) equals to [ C, D ]( F, G ) = R c ∈ C D ( F ( c ) , G ( c )) .Proof. Suppose R c ∈ C D ( F ( c ) , G ( c )) includes morphisms h ( c ) : F ( c ) → G ( c ) in D .For any other morphism f : c → d in C , The following diagram commutes as aconsequence of an end properties. This is by definition a natural transformation F ⇒ G . athematical Morphology via Category Theory 7 F ( c ) F ( d ) G ( c ) G ( d ) h ( c ) h ( d ) F ( f ) G ( f )Some practical properties of ends that will be used later are: Z a ∈ A Z b ∈ B F ( a, a, b, b ) = Z b ∈ B Z a ∈ A F ( a, a, b, b ) = Z ( a,b ) ∈ A × B F ( a, a, b, b ) (8) Z A [ C, F ( A, A )] ≃ [ C, Z A F ( A, A )] (9) Z A [ F ( A, A ) , C ] ≃ [ A Z F ( A, A ) , C ] (10) Definition 13.
A functor C , D along with functors F : D → C and G : C → D , F is assumed to preserve co-limits, if lim −→ i X i which exists in C for a functor X : I → C , F (lim −→ i X i ) ≃ lim −→ i F ( X i ) . Conversely, G is assumed to preserveall small limits if it forces that the limit lim ←− i X i for a functor X : I → C in C if exists, G (lim ←− i X i ) ≃ lim ←− i G ( X i ) . One of the main intentions of mathematics is to compare two models. One way toexpress that two models or objects are similar is by equality. However, equalityis too much strong in many cases. Another similarity observation in some casesis isomorphism. Isomorphism is a weaker notion comparing with equality. Twocategories C , D are isomorphic if there exist two functors R : C → D and L : D →L such that L ◦ R = id C and R ◦ L = id D . However, the notion of isomorphismis also too ambitious to expect in many cases.Adjunction weakens even the requirements needed by isomorphism for twocategories by just asking a one way natural transformation η : id D ⇒ R ◦ L andanother natural transformation expressing ǫ : L ◦ R ⇒ id C . In the language ofcategory theory η is called the unit and ǫ is called the co-unit of the adjunction.Functor L is noted as the left adjoint to functor R and R is called the rightadjoint to L .One can also express the adjunctions in terms of triangular identities [1,3]depicted by diagrams in figure 5: H.M. Sharifipour & B. Yousefi
L L ◦ R ◦ LLL ◦ η ǫ ◦ LR R ◦ L ◦ RRη ◦ R R ◦ ǫ Fig. 5: Triangle diagrams(Adjunctions)
Example.
Adjunctions in the category of preorders corresponds to functors F : L → L OP , G : L OP → L between two preorders L , L . L is the left adjoint of G iff forany p ∈ L , q ∈ L , q ≤ F ( p ) = ⇒ p ≤ G ( q )The adjunction in category of complete lattices called Galois connections plays acrucial role in mathematical morphology area. To express more, any left adjointin the category of complete lattices is a dilation and its right adjoint is an erosionconsequently.A major property of adjoints which is widely used in category theory isthat they preserve (co)-limits. A left adjoint preserves co-limits whereas a rightadjoint preserves limits. The Yoneda lemma is a major and applicative result of category theory [17].It allows us to embed any category into the category of contravariant functorsstemming from that category to the category of sets. The Yoneda lemma makesthe life easier by suggesting that one can investigate functors from a small cate-gory to the category of sets instead of investigating directly on it. In many casesthe former inspection is much more easier. athematical Morphology via
Category Theory 9
Definition 14.
Consider a functor F : A →
Set from an arbitrary category A to the category of sets, an object A ∈ A and the corresponding functor A ( A, − ) : A → Set . There exists a bijective correlation, nat ( A ( A, − ) , F ) ≃ F A
The main idea inherited in Yoneda lemma is that every information we needabout an object A ∈ C is encoded in C [ − , A ]. Yoneda lemma can be expressedby co-ends also. Let F : C OP → Set and G : C →
Set be functors. The followingformulas express the Yoneda lemma. F ≃ A ∈C Z F A × C [ − , A ] G ≃ A ∈C Z GA × C [ A, − ] Monoidal categoriesDefinition 15.
A category C is monoidal if it is equipped with a tensor product ⊗ that satisfies some conditions. Roughly speaking ⊗ : C × C → C is a functorsatisfying: – ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ) (Associativity isomorphism law) – There exist an identity object I satisfying: λ A : A ⊗ I → A and ρ A : I ⊗ A → A – The following two commutations hold: ( A ⊗ ( B ⊗ C )) ⊗ D (( A ⊗ B ) ⊗ C ) ⊗ D A ⊗ (( B ⊗ C ) ⊗ D )( A ⊗ B ) ⊗ ( C ⊗ D ) ( A ⊗ B ( ⊗ ( C ⊗ D )) α A , B , C ⊗ D α A , B ⊗ C , D α A ⊗ B,C,D A ⊗ α B,C,D α A,B,C ⊗ D ( A ⊗ I ) ⊗ B A ⊗ ( I ⊗ B ) A ⊗ Bα A,I,B λ A ⊗ B A ⊗ ρ B A monoidal category is left closed if for each object A the functor B A ⊗ B hasa right adjoint B ( A ⇒ B ). This means that the bijection Hom C ( A ⊗ B, C ) ∼ = Hom C ( A, B ⇒ C ) between the Hom-sets called currying exists.Dually, the monoidal category C is right closed if the functor B B ⊗ A admits a right adjoint. In a symmetric monoidal category the notions of leftclosed and right closed coincide. Generally, the arrows between two objects A,B in category C illustrated by C [ A, B ] are a set. The notion of enrichment extends that structure from merelya set to more fruitful structures [11,5]. For instance the set of arrows betweentwo objects may have an abelian structure meaning that it is possible to addthe arrows between two objects. A restriction imposed on a category V on whichthe arrows are enriched over it is that it should have a monoidal structure. Moreformally, Definition 16.
Let V be a symmetric monoidal category. A category enrichedover V called a V -category consists of: – For all objects
A, B ∈ C , the set of arrows from A to B is an object O ∈ V . – For all object
A, B, C ∈ C , there exist composition of arrows in V such that c A,B,C : C [ A, B ] × C [ A, B ] → C [ A, B ] . – For each object A ∈ C , an identity arrow exists in V such that I → C ( A, A ) . Considering the well-known notion of dilation in the category of complete lattices(
CompLat ), we may generalize it to other categories as follows:
Definition 17.
A dilation is a co-limit preserving functor whereas an erosionis a limit preserving functor.
We expressed previously that left/right adjoint functors preserve co-limits/limitsrespectively. Thus, the claim that left/right adjoints are a major source of dila-tion and erosion functors is pretty precise.Another concept that we will utilize it for defining morphological operationson matrix representation of images is the Day convolution [4,10]. Let V be acomplete and co-complete small symmetric monoidal category. Let the enrichedYonedda embedding functor to be C → [ C OP , V ]. The intuition of Day con-volution is that a monoidal structure on C brings the monoidal structure on[ C OP , V ].The enriched co-Yoneda lemma states that any V -enriched functor [ C OP , V ]is canonically isomorphic to a co-end of representables. This can be expressed athematical Morphology via Category Theory 11 as F ( c ) ≃ Z c F ( c ) ⊗ C ( − , c ). Taking two functors F, G : C OP → V , define theirmultiplication as: F ∗ G = Z c F ( c ) ⊗ C ( − , c ) ∗ Z b G ( b ) ⊗ C ( − , b ) Assuming the multiplication operation interchanges properly with the co-end,we get: F ∗ G = Z c,b F ( c ) ⊗ G ( b ) ⊗ ( C ( − , c ) ∗ C ( − , b ) )Forcing the Yoneda embedding C → [ C OP , V ] be strongly monoidal yields to: F ∗ G ≃ Z c,b F ( c ) ⊗ G ( b ) ⊗ C ( − , c ⊗ b ) . (11) Definition 18.
A semiring category S is a category with two operations of addi-tion and multiplication shown as ( ⊕ , · ) such that ( S , ⊕ , is monoidal and sym-metric, ( S , ˙ , is monoidal. Left and right distributivity of multiplication overaddition is expressed by natural isomorphisms as: A · ( B ⊕ C ) → ( A · B ) ⊕ ( A · C )( B ⊕ C ) · A → ( B · A ) ⊕ ( C · A ) Evidently, a semiring category is a ring in which the elements lack their inversesfor addition. The max-plus category may be defined by a ⊕ b = max { a, b } and a · b = a + b along with + ∞ and acting as unit objects for addition andmultiplication. Notions such as group, ring, semirings and many other varieties of algebraicstructures are categorized via Lawvere theory [12]. If T is a Lawvere theory and C is a category with finite products such as set , a functor F : T OP → C createsa full subcategory called a model of T . The model of Lawvere theories when C is category of sets is referred as T -algebras. T -algebras built on complete andco-complete category C which is in most cases category of sets are complete andcocomplete. Literally, the forgetful functors from a T -algebra to set categorycreates and preserves limits. Definition 19.
Given two matrices R ( m,n ) and S ( p,q ) , their tensor product knownalso as Kronecker product is defined by: R ⊗ S = r , S · · · r ,n S ... ... r m, S · · · r m,n S Definition 20.
The category of
Mat contains the set of integers as objects witharrows between two object m, n as m × n matrices with the matrix multiplicationas composition of arrows. S − mat is deducted from enriching Mat on a semiringlike S . In other words, S − mat has sets as objects and s -valued matrices m × n → S as morphisms. For instance, if S = ( { , } , ∨ , ∧ ) is the Boolean semiring, thenthe S − mat is exactly the well-known category of Rel . The V − mat category with the array multiplication, Kronecker product be-having as composition and tensor product respectively with the identity matrixconstitute a monoidal category. Let X be the discrete category containing tuplesof integers as objects with arrows X (( m, n ) , ( m, n )) = Z . Let us define F, G astwo dimensional matrices over a semiring S . Eventually, X will act for indexingthe two matrices. Day convolution of F, G can be defined by: F ∗ G ≃ Z ( m,n ) , ( p,q ) F ( m,n ) ⊗ G ( p,q ) ⊗ C ( − , ( m, n ) ⊗ ( p, q )) . (12)Defining the tensor product ( m, n ) ⊗ ( p, q ) on the discrete category X as ( m, n ) ⊗ ( p, q ) = ( m + p, n + q ) brings it a monoidal structure. Thus, 12 can be writtenas: ( F ∗ G ) ( r,s ) = M F ( m,n ) · G ( p,q ) . (13)where m + p = r ∧ n + q = s .assuming F, G as the source image and the structuring element, ( F ∗ G ) canbe defined as their dilation. The following example illustrates dilation of twobinary images.Example:Given F = and G = , Assuming F, G defined on Booleansemiring, one can derive the formula from 13 using ∨ and ∧ as addition andmultiplication respectively. So F ∗ G can be calculated like( F ∗ G )[ r, s ] = _ ( F [ m, n ] ∧ G [ p, q ]) (14)where m + p = r ∧ n + q = s .Hence, the following matrix will be induced by 14 F ∗ G = .It should be noted that this result corresponds exactly to dilating binaryimage F with the structuring element G using well-known existing formulation.However, by migrating from Boolean to max-plus semiring, formulation of gray-scaled images dilation is calculated by the following Day convolution:( F ∗ G ) ( r,s ) = max( F ( m,n ) + G ( p,q ) ) . (15) athematical Morphology via Category Theory 13
Other known morphological operation than can be derived from 13 is the fuzzydilation first appeared in [6]. For that intention we need to use the semiring with a ⊕ b = max( a, b ) and a · b = min( a, b ) denoted as the min − max to get theformula: ( F ∗ G ) ( r,s ) = max(min( F ( m,n ) , G ( p,q ) )) . (16)in which r = m + p ∧ s = n + q . We can concentrate on celebrated tensor-homadjunction for extracting the expression of erosion from dilation. Given a closedmonoidal category C , tensor-hom adjunction states that for an object A ∈ C ,tensor product − ⊗ A is the left adjoint with the internal hom functor [ A, − ].This can be expressed by: C [ A ⊗ B, C ] ≃ C [ A, [ B, C ]] (17)The morphism C [ A ⊗ B, C ] → C [ A, [ B, C ]] is nominated as currying in literature.Intuitively, currying is achieved by τ : m ⊗ n → τ ( m )( n ). The hom-tensoradjunction 17 is used to calculate the right adjoint which is equal to erosion. V [ F ∗ G, E ]= h Natural transformations representation by ends i R C [( F ∗ G ) C, EC ]= h Definition of Day convolution i R C [ A,B R ( F A ⊗ GB ⊗ θ ( A ⊗ B, C ) , EC ] ≃ h i R C R A [ F A, [ B R GB ⊗ θ ( A ⊗ B, C ) , EC ] ≃ h Commutativity of ends i R A R C [ F A, [ B R GB ⊗ θ ( A ⊗ B, C ) , EC ]] ≃ h i R A [ F A, R B,C [ θ ( A ⊗ B, C ) , [ GB, EC ]]] ≃ h i [ A, [ B, C ]]Thus the erosion of two binary matrices
F, G depicted by F ∗ ′ G can be writtenas: ( F ∗ ′ G )[ r, s ] = ^ ( F [ m, n ] ∧ G [ p, q ]) (18)where r = m − p and s = n − q . The erosion of two gray-scaled images can beextracted by using max-plus semiring,( F ∗ ′ G )[ r, s ] = min( F [ m, n ] − G [ p, q ]) (19) where r = m − p and s = n − q . Category theory provides an abstract unified framework for almost all aspects ofmathematics. This is the first research conducted on creating a unified definitionfor two fundamental morphological operations of dilation and erosion. We haveunified morphological operations appeared in contrasting situations like binary,gray-scaled and fuzzy operators into a uniform definition that can be extendedto some new variations with using different semirings.An interesting horizon for the future research on category theory and math-ematical morphology can be imagined by *-autonomous categories and mathe-matical morphology. *-autonomous categories are categorical representation oflinear logic which has been a major research area. Many models for linear logicsuch as Petri nets and game semantics has been suggested but none of themis satisfying. Mathematical morphology is claimed to be a model of linear logic[20]. The authors have shown that every derivable formula in linear logic shouldbe a model of mathematical morphology but the reverse is an open problem.Conducting research on *-autonomous which are just symmetric monoidal cate-gories with an involution object and morphological operations will help to shedthe light over relation of linear logic and mathematical morphology.
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