Some Approaches towards Lattice Computing in Mathematical

Some Approaches towards Lattice Computing inMathematical Morphology and Computational

Intelligence

Peter Sussner

Mathematical Imaging and Computational Intelligence GroupDepartment of Applied Mathematics, IMECC

University of Campinas

The 11th International FLINS Conference on Decision Making andSoft Computing

Peter Sussner (Unicamp) LC in MM and Comp. Intelligence FLINS 2014 1 / 68

Introduction

Lattice Theory (LT)

Some Remarks on LT

Origins:

Boolean algebra (mid 19th century);Dedekind’s investigations on number theory (late 19th and early20th century);Major branch of abstract algebra since 1940 (Birkhoff’s book);Linked to fuzzy set theory by Goguen in 1967;Provides framework for MM as shown by Serra in 1988.

LT has found applications in many areas such as:

mathematical morphology (MM);fuzzy set theory;computational intelligence;automated decision making;formal concept analysis.


Introduction

Lattice Computing

Graña’s Original Definition:

The collection of computational intelligence tools and techniques that

make use of lattice operators inf and sup for the construction ofthe computational algorithms

or exploit lattice theory for language representation and reasoning.

Kaburlasos’ Extended Definition:

An evolving collection of tools and math. modeling methodologies with

the capacity to process lattice ordered data per se including logicvalues, numbers, sets, symbols, graphs, etc.


Introduction

Organization of this talk

1 Introduction

2 Basic Concepts of Lattice Theory and MM on Complete Lattices

3 L-Fuzzy MM

4 Lattice Fuzzy Transforms

5 General Concepts of MNNs

6 Some Examples of MNNs and Related LC Models

7 Conclusions and Perspectives for the Future


Introduction


1 Introduction


3 L-Fuzzy MM






Introduction


1 Introduction


3 L-Fuzzy MM






Introduction


1 Introduction


3 L-Fuzzy MM






Introduction


1 Introduction


3 L-Fuzzy MM






Introduction


1 Introduction


3 L-Fuzzy MM






Introduction


1 Introduction


3 L-Fuzzy MM






Basic Concepts of LT and MM on Complete Lattices

Complete Lattices

Recall that a partially ordered set or poset is a set L 6= ∅ togetherwith a partial order relation ≤. Thus, a poset is a pair (L,≤). If ≤arises clearly from the context, one writes L instead of (L,≤).A poset L is called a lattice if the infimum and the supremum of Y,denoted resp. by

∧

Y and∨

Y , exist in L ∀ finite Y (6= ∅) ⊆ L.If 0L =

∧

L and 1L =∨

L exist in L, then L is bounded.If∧

Y and∨

Y exists in L for every Y ⊆ L, then L is complete.

Examples of complete lattices include R±∞ = R ∪ {+∞,−∞},Z±∞ = Z ∪ {+∞,−∞}, {0,1}, [0,1], I = {[a,b] ⊆ [0,1]} with thepartial ordering [a,b] ≤ [c,d ] ⇔ a ≤ c and b ≤ d , and P(X), theclass of all subsets of X, with the partial ordering given by ⊆.

If L is a complete lattice, then LX = {f : X → L} for an arbitrary set

X and Ln also yield complete lattices.



L-Fuzzy Sets

Definition and Notations:

An L-fuzzy set A consists of a universe X together with a membershipfunction µA : X → L. If L is a complete lattice, then the class of L-fuzzysets over the universe X, denoted using the symbol FL(X), alsorepresents a complete lattice.If A ∈ FL(X), then we simply write A(x) instead of µA(x).

L-Fuzzy Sets for special choices of a complete lattice L:

F[0,1](X) = F(X), the class of fuzzy sets over X;

FI(X) equals the class of interval-valued fuzzy sets over X;

L = P([0,1]) yields the class of interval type-2 fuzzy sets;

L = F([0,1]) yields the class of type-2 fuzzy sets (T2-FSs).



Homomorphisms and Isomorphisms

Given complete lattices L,M, ϕ : L → M is called a completelattice homomorphism if:

ϕ(∧

X ) =∧

ϕ(X ) and ϕ(∨

X ) =∨

ϕ(X )∀X ⊆ L.

If ϕ is also bijective, then ϕ is called a (complete) latticeisomorphism. In this case, L and M are said to be isomorphic andwe write L ≃ M.

For any complete lattice L, we have FL(X) ≃ LX and if |X | = n,

then FL(X) ≃ Ln.



Two Basic Operators of MM on Complete Lattices

Complete lattices yield a mathematical framework for MM. From nowon, the symbols L and M denote complete lattices.

Consider operators ε, δ : L → M.

ε is called an (algebraic) erosion if

ε(

∧

Y)

=∧

y∈Y

ε(y) , ∀Y ⊆ L .

δ is called an (algebraic) dilation if

δ(

∨

Y)

=∨

y∈Y

δ(y) , ∀Y ⊆ L .



Adjunctions and (Algebraic) Dilations and Erosions

Definition:

Consider δ : L → M and ε : M → L. The pair (ε, δ) is called anadjunction from L to M, in other words ε and δ are adjoint if

δ(x) ≤ y ⇔ x ≤ ε(y) ∀ x ∈ L , y ∈ M .

Useful Facts:

Let δ : L → M and ε : M → L.

If (ε, δ) is an adjunction, then δ is a dilation and ε is an erosion.

For every dilation δ there is a unique erosion ε such that (ε, δ) isan adjunction.

For every erosion ε there is a unique dilation δ such that (ε, δ) isan adjunction.


L-Fuzzy MM

Origins of Mathematical Morphology

Mathematical Morphology (MM) originated as a theory for theprocessing and analysis of images using structuring elements (SEs).

Aplications of MM include

1 noise removal;2 skeletonizing;3 edge detection;4 automatic target recognition;5 image segmentation;6 image restauration.


L-Fuzzy MM

Two Perspectives on MM

MM from two different (but not mutually exclusive) points of view:

MM in the geometrical or topological sense: employs SEs as wellas inclusion and intersection measures;

MM in the algebraic sense: usually defined in a complete latticesetting, recently extended to complete semilattices.

Basic Operators of MM in the geometrical or topological sense

(Morphological) erosion: yields a degree of inclusion of thetranslated SE in the image at every pixel;

(Morphological) dilation: yields the a degree of intersection of theimage with the (reflected and) translated SE at every pixel.


L-Fuzzy MM

Basic Concepts of L-Fuzzy MM

Consider an image a and an SE s ∈ FL(X), where X = Rd or X = Z

d .

sx(y) = s(y − x) ∀ y ∈ X , (translation of s by x)

s(y) = s(−y) ∀ y ∈ X . (reflection of s around the origin)

Let IncL and SecL be respectively inclusion and intersection measures.

An L-fuzzy (morphological) erosion of a ∈ FL(X) by s ∈ FL(X):

EL(a,s)(x) = IncL(sx,a) ∈ L, ∀x ∈ X;

An L-fuzzy (morphological) dilation of a ∈ FL(X) by s ∈ FL(X):

∆L(a,s)(x) = SecL(sx,a) ∈ L, ∀x ∈ X.


L-Fuzzy MM

Two Special Cases of L-Fuzzy MM

Binary MM:

Binary MM deals with images A and SEs S ∈ P(X) ≃ F{0,1}(X).

The operators of binary erosion and dilation can be defined usingcrisp inclusion and intersection measures.

Fuzzy MM:

Fuzzy MM deals with images a and SEs s such that a,s ∈ F(X).

Fuzzy erosions and dilations can be defined using fuzzy inclusionand intersection measures.


L-Fuzzy MM

Illustration of Binary Erosion

Figure :

Binary image A, SE S, and binary erosion of A by S.


L-Fuzzy MM

Illustration of Fuzzy Erosion

Figure :

Fuzzy image a, SE s, and fuzzy erosion of a by s.


L-Fuzzy MM

Some Comments on Gray-Scale MM

Images and SEs are viewed as elements of GX, where

G = R±∞

or G = Z±∞.

The usual approach towards gray-scale MM is the umbraapproach that:

can be derived from binary MM,yields algebraic erosions and dilations GX → GX for arbitrary, butfixed SEs,is closely related to fuzzy MM based on Lukasiewicz operators,and can be extended to images and SEs in GX, were G is anarbitrary complete l-group extension.


L-Fuzzy MM

Lattice-Ordered Groups

A lattice that also represents a group such that every grouptranslation x 7→ a + x + b is isotone is called an l-group.

An l-group F such that F is a conditionally complete lattice iscalled a conditionally complete l-group.

A complete lattice G such that F = G \ {∨

G,∧

G} forms anl-group is called a complete l-group extension.

Examples

1 Rn and Z

n are conditionally complete l-groups ∀n ∈ N .2 R±∞ = R ∪ {−∞,+∞} and Z±∞ = Z ∪ {−∞,+∞} represent

complete l-group extensions.


L-Fuzzy MM

Construction of L-Fuzzy Erosions and Dilations

Let I and C be resp. an L-fuzzy implication and an L-fuzzy conjunction.The operators EL,∆L : FL(X)×FL(X) → FL(X) are defined as follows:

EL(a,s)(x) = IncL(sx,a) =∧

y∈X

I(sx(y),a(y)) ∀x ∈ X ,

∆L(a,s)(x) = SecL(sx,a) =∨

y∈X

C(sx(y),a(y)) ∀x ∈ X .

We have:

EL(.,s) are (algebraic) erosions for all s ∈ FL(X) if and only ifI(s, .) are (algebraic) erosions for all s ∈ L.

∆L(.,s) are (algebraic) dilations for all s ∈ FL(X) if and only ifC(s, .) are (algebraic) dilations for all s ∈ L.


L-Fuzzy MM

L-Fuzzy Operators

An operator C : L× L → L is called a conjunction on L or anL-fuzzy conjunction if

C is increasing,C(0L, 0L) = C(0L, 1L) = C(1L, 0L) = 0L and C(1L, 1L) = 1L.

If C is in addition commutative and associative with C(x ,1L) = x∀x ∈ L, then C is called a triangular norm or simply t-norm on L

(L-fuzzy disjunctions and s-norms are defined similarly).

An operator I : L× L → L is called an implication on L or L-fuzzyimplication if

I is decreasing in the first argument,I is increasing in the second argument,I(0L, 0L) = I(0L, 1L) = I(1L, 1L) = 1L, and I(1L, 0L) = 0L.

An involutive bijection L → L that reverses the partial ordering iscalled a negation on L.


L-Fuzzy MM

Interval-Valued Fuzzy (IV-Fuzzy) Operators

Some examples of IV-fuzzy operators, i.e., I-fuzzy operators:

The pessimistic conjunction CpC with representative C:

CpC(x,y) = [C(x1, y1),C(x1, y2) ∨ C(x2, y1)].

The symbol T pW denotes Cp

C if C = TW (Lukasiewicz t-norm) and∆

pW denotes the I-fuzzy dilation based on T p

W .

The optimistic implication IoI with representative I:

IoI (x,y) = [I(x1, y1) ∧ I(x2, y2), I(x1, y2)].

The symbol IoW denotes Io

I if I = IW (Lukasiewicz implication) andEo

W denotes the I-fuzzy erosion based on IoW .

The standard negation NI

S on I is given byNI

S(x) = [1 − x2,1 − x1]∀x = [x1, x2] ∈ I.


L-Fuzzy MM

Properties and Applications of EoW and ∆

pW

Properties:

For every s ∈ FI(X), we have:

EoW (.,s) and ∆p

W (., s) are adjoint.

EoW (., s) and ∆p

W (., s) are dual w.r.t. the standard negation NS onFI(X), where NS(a)(x) = NI

S(a(x)) ∀x ∈ I, that is:

∆pW (a, s) = NS(E

oW (NS(a), s)∀ a ∈ FI(X).

Morphological Gradient:

Given an SE s ∈ FI(X), an IV morphological gradient image ofa ∈ FI(X) is given by:

EoW (a,s)(x)−∆p

W (a, s)(x)∀ x ∈ I,

where x − y = [x1 − y2, (x1 − y1) ∨ (x2 − y2)] ∀ x, y ∈ I.Peter Sussner (Unicamp) LC in MM and Comp. Intelligence FLINS 2014 21 / 68

L-Fuzzy MM

An Application in Image Segmentation

The Watershed Transform:A region-based approach to image segmentation;

we chose F. Meyer’s flooding algorithm for our experiments;

pre- or postprocessing techniques are usually employed to avoidoversegmentation;

to this end, we used the 8-connected disk with radius 1 as astructuring element in gradient and filtering techniques.

Figure : Illustration of Watershed Algorithm.


L-Fuzzy MM

Tomographic Image Reconstruction

Experiments Using the Shepp-Logan Phantom:

Consider a discretized version of the famous Shepp-Logan phantom(on a 256 × 256 grid) as well as the reconstructions produced by thefollowing algorithms in a noiseless setting using 600 uniform views and400 equally spaced rays within each view:

filtered backprojection (FBP) with Ramlak filter,

filter of the backprojections (FOB) with Ramlak filter,

Tretiak & Metz reconstruction with attenuation parameter 0.1.


L-Fuzzy MM

Visualization of Image Reconstruction Algorithms

Original Shepp-Logan Phantom (Top Left) and Reconstructions:

Figure : Original Shepp-Logan phantom and reconstructions produced by thePeter Sussner (Unicamp) LC in MM and Comp. Intelligence FLINS 2014 24 / 68

L-Fuzzy MM

Conventional Morphological Gradient Images

Morphological Gradients of Shepp-Logan Phantom and ReconstructedImages after Applying the h-Minima Transform (with h = 0.07):


L-Fuzzy MM

Application of Watershed Transform

Watershed Transforms of Original and Reconstructed Images:

Figure :Peter Sussner (Unicamp) LC in MM and Comp. Intelligence FLINS 2014 26 / 68

L-Fuzzy MM

IV Representation of the Reconstructed Images

Lower and Upper Bounds given by the Pixelwise Minimum andMaximum of the Reconstructed Images:


L-Fuzzy MM

IV Morphological Gradient and Resulting Watershed

IV Morphological Gradient (Top Row), Mean of Upper and LowerBounds, and Watershed Transform (after h-Minima Transform):


Lattice FTs

Fuzzy Tansforms (FTs)

Introductory Remarks:

FTs come in pairs consisting of a direct and an inverse transform:

The direct transform maps functions residing in an original space tofunctions in a transformed space.The inverse transform maps functions in the transformed spaceback into the original space.

3 versions of FTs: one linear and 2 lattice-based version.

Lattice FTs were defined as discrete transforms fromF(P) → F(K) (direct transform) or from F(K) → F(P) (inversetransform) for finite universes P and K.

Generalized versions consist of direct transforms FL(P) → FL(K)and inverse transforms FL(K) → FL(P), where P and K arearbitrary and (L,∨,∧, ⋆,→) is a complete residuated lattice.


Lattice FTs

Residuated Lattices

A residuated lattice (RL) is an algebra (L,∨,∧, ⋆,→) such that1 (L,∨,∧) is a bounded lattice.2 (L, ⋆) is a commutative monoid whose identity element is 1L.3 The operation → is a residuation operation with respect to ⋆, i.e.,

for all x , y , z ∈ L we have:

z ⋆ x ≤ y ⇔ x ≤ z → y , ∀x , y , z ∈ L.

In other words, (z ⋆ ., z → .) is an adjunction for every z ∈ L.

If L is complete, then we speak of a complete residuated lattice.

In any residuated lattice, the operator ⋆ is an L-fuzzy t-norm.

Note that L-fuzzy MM can be conducted in a complete RL.


Lattice FTs

Definitions of Lattice FTs

Generalized Versions of Lattice FTs:

If (L,∨,∧, ⋆,→) is a complete RL, P and K are arbitrary universes, andAk ∈ FL(P) are basic functions s.t. ∀p ∈ P ∃ k ∈ K with Ak (p) 6= 0L,then F↑,F↓ : F(P) → F(K) (direct transforms) andf↑, f↓ : F(K) → F(P) (inverse transforms) are defined as follows:

F ↑(k) = [F↑(f )](k) =∨

p∈P

Ak (p) ⋆ f (p) , ∀f ∈ F(P) , ∀k ∈ K,

f ↑(p) = [f↑(F )](p) =∧

k∈K

Ak(p) → F (k) , ∀F ∈ F(K) , ∀p ∈ P,

F ↓(k) = [F↓(f )](k) =∧

p∈P

Ak (p) → f (p) , ∀f ∈ F(P) , ∀k ∈ K,

f ↓(p) = [f↓(F )](p) =∨

k∈K

Ak (p) ⋆ F (k) , ∀F ∈ F(K) , ∀p ∈ P.


Lattice FTs

Lattice FTs and MM on Complete Lattices

Theorem:

Let Ak ∈ FL(P), where k ∈ K. We have:

1 F↑ : FL(P) → FL(K) represents an (algebraic) dilation.2 f↑ : FL(K) → FL(P) represents an (algebraic) erosion.3 F↓ : FL(P) → FL(K) represents an (algebraic) erosion.4 f↓ : FL(K) → FL(P) represents an (algebraic) dilation.

Moreover, the pairs (F↑, f↑) and (F↓, f↓) represent adjunctions.


Lattice FTs

Relationship of Lattice FTs to L-Fuzzy MM

Theorem:

Given a complete RL (L,∨,∧, ⋆,→), let F↑,F↓, f↑, f↓ : FL(P) → FL(P)with Ak ∈ FL(P) for k in an arbitrary group P with identity element 0.Let s denote the SE in FL(P) that satisfies s(p) = A0(p) for all p ∈ P.If Ak+q(p + q) = Ak(p) ∀k ,p,q ∈ P, then F↑,F↓, f↑, and f↓ can bewritten as follows in terms of L-fuzzy erosions or L-fuzzy dilations:

F↑(f ) = ∆L(f , s),

f↑(F ) = EL(F , s),

F↓(f ) = EL(f , s),

f↓(F ) = ∆L(F , s).


General Concepts of MNNs

Morphological Neural Networks (MNNs)

MNNs can be seen as approaches towards lattice computing orcomputational intelligence based on lattice theory.

MNNs perform morphological operations in the lattice-algebraic orgeometrical/topological sense.

The aggregation functions of morphological neurons are given byoperations of MM.

Most MNN models have strong theoretical foundations in MM oncomplete lattices (or in MM on complete inf-semilattices).

MNNs have been used for a variety of applications such aspattern recognition, image and signal processing, computervision, approximate reasoning, and prediction.



Max Product, Min Product, and Conjugate

Let G be a complete l-group extension and F = G \ {∨

G,∧

G}. LetA ∈ F

m×n and B ∈ Gn×p.

M = A ∨� B - max product of A and B: mij =∨n

k=1(aik + bkj).

W = A ∧� B - min product of A and B: wij =∧n

k=1(aik + bkj)

A∗ - conjugate of A: A∗ = −AT

An (algebraic) erosion is givenby

εA : Gn −→ G

m

x 7−→ A ∧� x

An (algebraic) dilation is givenby

δA : Gn −→ G

m

x 7−→ A ∨� x



Max-C Product and Min-D Product

Let A ∈ Lm×n and B ∈ L

n×p.

Consider a conjunction C and a disjunction D on L. Recall thatC,D are increasing operators L

2 → L s.t.:C(0L, 0L) = C(0L, 1L) = C(1L, 0L) = D(0L, 0L) = 0L;C(1L, 1L) = D(0L, 1L) = D(1L, 0L) = D(1L, 1L) = 1L;

M = A ◦ B - max-C product of A and B: mij =∨n

k=1 C(aik ,bkj).

W = A • B - min-D product of A and B: wij =∧n

k=1 D(aik ,bkj)

If C(a, ·) is a dilation for everya ∈ L then we have an(algebraic) dilation

δFA : Ln −→ L

m

x 7−→ A ◦ x

If D(a, ·) is an erosion for everya ∈ L then we have an(algebraic) erosion

εFA : Ln −→ L

m

x 7−→ A • x



Some Types of Morphological Neurons

Additive Max and Min Neurons

Let G be a complete l-group extension and F = G \ {∨

G,∧

G}. For aninput vector x ∈ G

n and a vector of synaptic weights w ∈ Fn, the output

y ∈ G is computed as follows:

Additive max neuron: y =∨n

j=1

(

wj + xj)

= wT ∨� x;

Additive min neuron: y =∧n

j=1

(

wj + xj)

= wT ∧� x.

Max-C and Min-D Neurons:

Let C and D be resp. a fuzzy conjunction and disjunction. For an inputvector x ∈ [0,1]n and a vector of synaptic weights w ∈ [0,1]n, theoutput y ∈ [0,1] is computed as follows:

Max-C neuron: y =∨n

j=1 C(

wj , xj)

= wT ◦ x;

Min-D neuron: y =∧n

j=1 D(

wj , xj)

= wT • x.



Morphological Neurons in Complete Lattices

Observations

Additive max and min neurons as well as max-C and min-Dneurons for continous C and D yield elementary morphologicaloperators between complete lattices.

The aggregation functions of max-C and min-D neurons can alsobe viewed as L-fuzzy dilations and erosions of x by SEs(determined by w).


Examples of MNNs and Related LC Models

Morphological Associative Memories (MAMs)

Design Problem of an Associative Memory (AM):

Given a finite set of pairs or associations{(

xξ,yξ)

: ξ = 1, . . . , k}

,determine a mapping A such that - ideally - we have:

1 A(xξ) = yξ for all ξ = 1, . . . , k ;2 A(xξ) = yξ for xξ ≈ xξ.

Comments on Original MAM Models

1 The MAMs WXY and MXY are designed by using “ ∧� " and “ ∨� " toassociate xξ with yξ.

2 WXY , MXY yield functions Rn±∞ → R

m±∞. Replacing R±∞ by any

complete l-group extension G, let F = G \ {∨

G,∧

G}.



The MAMs WXY and MXY

Definitions of WXY and MXY :

For X = [x1, . . . ,xk ] ∈ Fn×k and Y = [y1, . . . ,yk ] ∈ F

m×k , let

WXY = Y ∧� X ∗ , MXY = Y ∨� X ∗ ∈ Fm×n . (1)

Given x ∈ Gn, the outputs of WXY and MXY are resp. calculated in

terms of a dilation and an erosion Gn → G

m:

y = WXY ∨� x , z = MXY ∧� x .

If X = Y , we speak of an auto-associative MAM (AMM).



Properties of AMMs

1 Unlimited absolute storage capacity;2 One-step convergence when employed with feedback;3 The fixed points known to be the lattice polynomials in xξ;4 Basins of attraction known;5 x is attracted to supremum of x in set of fixed points using WXX ;6 x is attracted to infimum of x in set of fixed points using MXX .

Alternatives1 Use modified MAMs WXX + ν or MXX + µ;2 Substitute the complete lattice (Gn,≤) with a cisl of the form

define new AM in this setting.



Some Applications of MAMs

The MAM models WXY and MXY have been applied in diverse areassuch as:

hyperspectral image analysis;

color image segmentation;

image compression;

robot vision;

face localization;

a variety of other pattern recognition problems.

Other AMs were defined in complete lattices or quantales thatcorrespond to color spaces.



L-Fuzzy Morphological Associative Memories

Max-C L-FMAMs:Let L be a complete lattice.

Given X = [x1, . . . xp] ∈ Ln×p, Y = [y1, . . . yp] ∈ L

m×p, and anL-fuzzy conjunction C, a max-C L-FMAM model W is given by

y = W(x) = W ◦ x ,

where W ∈ Lm×n, x ∈ L

n, and y ∈ Lm.

If L = [0,1], then we simply refer to an L-FMAM as an FMAM.

Examples of Max-C FMAMs:

Kosko’s max-min and max-product FAMs;

the generalized FAM of Chung & Lee;

implicative FAMs (IFAMs).


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dilative

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Adjunction-Based Learning for Max-C L-FMAMs

Synthesis of a Weight Matrix WC for a Max-C L-FMAM WC:

Let C(., x) be a dilation ∀x ∈ L.

Consider the dilation DX : Lm×n −→ Lm×p given by

DX (A) = A ◦ X ∀A ∈ Lm×n;

Let EDX : Lm×p −→ L

m×n be the unique erosion that forms anadjunction with DX ;

Define WC = EDX (Y );

The symbol WC denotes the resulting Max-C L-FMAM.


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and let C be commutative.


Examples of Max-C L-FMAMs

The Max-CF FMAM WF and the Max-CrF IV-FMAM W r

F :

The cross-ratio uninorm CF defined below represents acommutative conjunction and C(., x) is a dilation ∀x ∈ [0,1].

CF (x , y) =

{

1, if (x , y) = (0,1) or (1,0) ,xy

(1−y)(1−x)+xy)) , otherwise .

The symbols WF and WF denote resp. WC and WC if C = CF .

Given a fuzzy conjunction C, an IV-fuzzy conjunction CrC , called

the C-representable conjunction, is defined as follows:

CrC(x,y) = [C(x1, y1),C(x2, y2)] ∀ x,y ∈ I.

CrF stands for Cr

C where C = CF .

The symbols W rF and W r

F denote resp. WC and WC if C = CrF .


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An Application to Time-Series Prediction in Industry

Problem of forecasting the average monthly streamflow of Furnas,a large hydroelectric plant in southern Brazil;

Seasonality of the streamflow suggests the use of 12 differentpredictor models;

The objective is to predict the monthly streamflow sγ from asubset of standardized past values;

Using (interval-valued) fuzzy c-means, the training data (pγ , sγ),where pγ = [sγ−3, sγ−2, sγ−1], yield the centers and standarddeviations of Gaussian membership functions xξ and yξ.

{(xξ,yξ) : ξ = 1, . . . , k} can be stored implicitly in W , whereW ∈ [0,1]n×m or W ∈ I

n×m;

The estimated value is obtained by (type-reducing and)defuzzifying yγ = W(xγ) (using the centroid method);



Some Details on the IV Fuzzy Clustering

IV fuzzy c-means clustering on the training data (pγ ,sγ), wherepγ = [sγ−3, sγ−2, sγ−1] ∈ [−5,5]3 and sγ ∈ [−5,5], yields cξ

x, cξy

and σξx, σ

ξy of IV membership functions xξ ∈ I

n and yξ ∈ Im.

We used a constant number of clusters c = 5 and “fuzzifier"parameters m = [m1,m2] = [2 − α,2 + α] for α = 0,0.1, . . . ,0.4.



Performance of IV-FMAMs on Validation Data

m1 m2 MRE RMSE MAE

2 2 22.95 181.01 223.961.9 2.1 22.80 181.17 223.351.8 2.2 22.65 183.05 223.461.7 2.3 22.31 186.23 221.971.6 2.4 22.52 187.47 223.28

Table : Prediction errors produced by W rF using the data 1931 - 1995 for

different values of m.



Performance of Several Predictors on Test Data

MRE RMSE MAEModel (%) (m3/s) (m3/s)PAR 18.08 266.13 154.44

ANFIS 20.12 262.21 166.31C-FSM 20.19 260.82 163.48A-FSM 19.08 278.42 167.77

FMAM WF 18.8 278.08 167.33IV-FMAM W r

F 18.58 276.96 165.78

Table : Comparison of the prediction errors produced by WF , W rF , an online

adaptive (first order Takagi-Sugeno) fuzzy system model (A-FSM), an offlineconstructive (first order Takagi-Sugeno) fuzzy system model (C-FSM), theadaptive network- based fuzzy inference system (ANFIS) of Jang, and aperiodic autoregressive (PAR) model using the data from 1996-2005.



Predictions Produced by the Max-CrF FMAM W r

F

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 20060

500

1000

1500

2000

2500

3000

3500

4000

Year

Str

eam

flow

(m

3 /s)

Actual ValuesPredicted Values

Figure : Streamflow prediction for the Furnas reservoir from 1996 to 2005.



Another Type of a Morphological Neuron

Let S be a fuzzy subsethood measure. Given a weight vectorw ∈ [0,1]n and input x ∈ [0,1]n, compute y ∈ [0,1] as follows:

y = S(w,x) .

We have a fuzzy erosion of x by the SE w but not an algebraic erosion.Recall that S : F(X)×F(X) → [0,1] satisfies:

1 If A ⊆ B then S(A,B) = 1;2 S(X, ∅) = 0;3 If A ⊆ B ⊆ C then S(C,A) ≤ S(B,A) and S(C,A) ≤ S(C,B).

Let T be a t-norm. A similarity measure SM is given by:

SM(A,B) = T (S(A,B),S(B,A)) ∀A,B ∈ F(X) .

F(X) can be replaced by any bounded lattice L in the def’s of S, SM.Peter Sussner (Unicamp) LC in MM and Comp. Intelligence FLINS 2014 51 / 68


Construction of Subsethood Measures

Let I be a fuzzy implication that satisfies I(a,b) = 1 for all a ≤ b and letυ : F(X) → [0,1] be a increasing function s.t. υ(∅) = 0 and υ(X) = 1.The following operators S∩ and S∪ yield subsethood measures:

(a) S∩(A,B) = I(υ(A), υ(A ∩ B)) (2)

(b) S∪(A,B) = I(υ(A ∪ B), υ(B)) (3)

Let X = {x1, . . . ,xk} and let I = IP (Goguen implication).

For υM given by υM(A) =∑k

i=1 µA(xi)k , S∩ and S∪ correspond resp.

to SK (Kosko’s subsethood) and SW (Willmot’s subsethood).

S∩p and S∪

p arise by using υp : F(X) → [0,1] for p ∈ (0,+∞):

υp(A) =k

∑

i=1

1 − cos(π[µA(xi)]p)

2k, ∀A ∈ F(X).



Θ-Fuzzy Associative Memories (Θ-FAMs)

Let Θξ : F(X) → [0,1] and w ∈ V ⊆ Rk , where V is closed and convex.

Figure : Topology of the Θ-FAM, based on Θξ and w, that associatesAξ ∈ F(X) with Bξ ∈ F(Y) for finite universes X and Y.



Weighted S-, dual S-, and SM-FAMs

Consider the Θ-FAM based on Θξ and w.1 If Θξ is given by Θξ( · ) = S(Aξ, · ), where ξ = 1, . . . , k , for some

subsethood measure S, then the corresponding Θ-FAM is called aweighted S-FAM;

2 If Θξ is given by Θξ( · ) = S( · ,Aξ), , where ξ = 1, . . . , k , for somesubsethood measure S, then the corresponding Θ-FAM is called aweighted dual S-FAM;

3 If Θξ is given by Θξ( · ) = SM(Aξ, · ), where ξ = 1, . . . , k , for somesimilarity measure SM, then the corresponding Θ-FAM is called aweighted SM-FAM;



Training of Θ-FAMs

Remarks on Θ-FAM Training Algorithm:

A supervised training algorithm was specifically designed forΘ-FAMs;

proven to converge in a finite number of steps (< f (w0), where f isthe proposed objective function and w0 is the initial weight vector);

proven to reach a local minimum of f in a finite number of steps;

Tunable Equivalence Measure FAMs (TE-FAMs):

Θ-FAMs can be extended to deal with pattern cues in L, where L

is an arbitrary bounded lattice.

A Θ-FAM is called TE-FAM if Θξ(.) = E ξ(.,xξ), where E ξ representparametrized equivalence measures; TE-FAMs can be trained by:

1 extracting a fundamental memory set from the training set,2 optimizing the weights and parameters.



Applications of Θ-FAMs

In our simulations, we considered weighted S-FAM, dual S-FAM,and SM-FAMs with S = SK ,SW , and S∩

p , S∪p for p ∈ P =

{0.25,0.5,0.75,1,1.5,2,2.5,3,4}, as well as SM given bySM(A,B) = S(A,B) ∧ S(B,A);

We employed pre-processing techniques to convert vectorswhose entries represent numerical or categorical attributes intofuzzy sets;

We chose initial weights w (0)j ∈ [0,1] as solutions to certain simple

optimization problems;



Some Classification Problems (KEEL Repository)

Instances Categorical Numerical ClassesFeatures Features

Appendicitis 106 0 7 2Cleveland 297 0 13 5

Crx 653 9 6 2Ecoli 336 0 7 8Glass 214 0 9 7Heart 270 0 13 2

Iris 150 0 4 3Monks 432 0 6 2

Movementlibras 360 0 90 15Pima 768 0 8 2Sonar 208 0 60 2

Spectfheart 267 0 44 2Vowel 990 0 13 11Wdbc 569 0 30 2Wine 178 0 13 3



Some Details on the Experiments

Same approach as in a recent article (IEEE TFS, Vol. 19, no. 5,Oct. 2011) on the fuzzy association rule-based classificationmethod for high-dimensional problems (FARC-HD);

Partitioning of the data into 10 folds;

Extraction of the set of Θ-FAMs that produced the lowest meanclassification error with the lowest variance during training.



Average Classification Rates in the Testing Phase

2SLAVE FH-GBML SGERD CBA CBA2 CMAR CPAR C4.5 FARC-HD Θ-FAM

Appendicitis 82.91 86 84.48 89.6 89.6 89.7 87.8 83.3 84.2 81.18Cleveland 48.82 53.51 51.59 56.9 54.9 53.9 54.9 54.5 55.2 51.17

Crx 74.06 86.6 85.03 83.6 85 85 87.3 85.3 86 82.02Ecoli 84.53 69.38 74.05 78 77.1 77.7 76.2 79.5 82.2 76.78Glass 58.05 57.99 58.49 70.8 71.3 70.3 68.9 67.4 70.2 70.49Heart 71.36 75.93 73.21 83 81.5 82.2 80.7 78.5 84.4 78.15

Iris 94.44 94 94.89 93.3 93.3 94 96 96 96 96Monks 97.26 98.18 80.65 100 100 100 100 100 99.8 98.63

Movementlibras 67.04 68.89 68.09 36.1 7.2 39.2 63.6 69.4 76.7 84.72Pima 73.71 75.26 73.37 72.7 72.5 75.1 74.5 74 75.7 67.44Sonar 71.42 68.24 71.9 75.4 77.9 78.8 75 70.5 80.2 80.69

Spectfheart 79.17 72.36 78.16 79.8 79.8 79.4 78.3 76.5 79.8 81.3Vowel 71.11 67.07 65.83 63.6 74.9 60.4 63.00 81.5 71.8 97.07Wdbc 92.33 92.26 90.68 94.7 95.1 94.9 95.1 95.2 95.3 96.14Wine 89.47 92.61 91.88 93.8 93.8 96.7 95.6 93.3 94.3 97.24Mean 77.05 77.22 76.15 78.09 76.93 78.49 79.79 80.33 82.12 82.60

Comparison with structural learning algorithm on vague environment (2SLAVE), fuzzy hybrid genetic based machine learning

algorithm (FH-GBML), steady-state genetic algorithm for extracting fuzzy classification rules from data (SGERD), classification

based on associations (CBA), an improved version of the CBA method (CBA2), classification based on multiple association rules

(CMAR), C4.5 decision tree, classification based on predictive association rules (CPAR), and fuzzy association rule-based

classification method for high-dimensional problems (FARC-HD).



A Vision-Based Self-Localization Problem

Figure : Landmark images that were used for building the map.



Θ-FAMs for Vision-Based Self-Localization

Comparison of the results produced by the selected Θ-FAMs bestresults previously obtained for each walk using off-line mapping byLICA, MF-ICA and MS-ICA, and endmember selection for SLAM.Walks 1 and 2 were used resp. for training and validation.

Walk 3 Walk 4 Walk 5 Walk 6 AverageDual S∩

1.5-FAM 0.78 0.72 0.78 0.77 0.76Dual S∪

1.5-FAM 0.79 0.72 0.79 0.77 0.77LICA 0.75 0.66 0.73 0.75 0.72

MF-ICA 0.62 0.54 0.65 0.53 0.58MS-ICA 0.69 0.62 0.74 0.69 0.68SLAM 0.76 0.6 0.69 0.64 0.67



Fuzzy Lattice Reasoning (FLR) Models

Fuzzy Lattice

A fuzzy lattice is a pair (L, µ) consisting of a lattice L and a functionµ : L× L → [0,1] such that µ(x , y) = 1 if and only if x ≤ y .

Inclusion Measure

For a complete lattice L, an “inclusion measure" σ is a functionL× L → [0,1] such that ∀x , y , z ∈ L:

1 σ(x , x) = 1 ∀x ∈ L;2 y ≤ z ⇒ σ(x , y) ≤ σ(x , z);3 x 6≤ y ⇒ σ(x , y) < 1,

In this case, (L, σ) is a fuzzy lattice.



Some Details on FLR Classifiers

Training Phase

Given a set of training data {(E1, c1), . . . , (EL, cL)}, where Ei areinformation granules such as fuzzy interval’s numbers or hyperboxes inR

n and ci are class labels, a clustering algorithm is perfomed resultingin a set {(E1, c1), . . . , (EM , cM )} with elements of the same type.

Recall Phase

Upon presentation of a granule F , compute σ(F , Ej) ∀j = 1, . . . ,M.Assign F to the class label cJ such that J = arg maxj σ(F , Ej).

Remarks

σ(F ,E) is a component of an erosion of E by the SE F ;

(Unlike MP/CL), FLR training depends on order of training data;

FLR models have been proposed for some general types of data.



Fuzzy Inference Systems Based on FLR

Consider the antecedent part of a fuzzy rule:



Sparse Rule Activation Using FLR

By choosing appropriate "inclusion" or subsethood measures,inputs outside the support of the antecedents can activate rules.

Parametrized versions of σ or S can be tuned.Peter Sussner (Unicamp) LC in MM and Comp. Intelligence FLINS 2014 65 / 68

Conclusions and Perspectives for the Future

Concluding Remarks

Lattice Computing, or LC for short, has been proposed forprocessing diverse types of data using lattice theory.

We reviewed some basic concepts of lattice theory and some offollowing approaches towards LC:

Mathematical morphology (MM) on complete lattices;L-fuzzy MM;Lattice fuzzy transforms;Morphological neural networks (MNNs) and related LC models.

We presented some applications of LC in:

Image segmentation (based on image reconstruction methods);Some benchmark classification problems;A vision-based self-localization problem in robotics;Time-series prediction in industry;Fuzzy inference systems in general.


Conclusions and Perspectives for the Future

Perspectives for the Future of Lattice Computing

LT is concerned with general lattice structures. Thus, LCapproaches can be applied to disparate types of data.

LT and LC allow for a top-down view on various methodologies,that are already existent or under development.

Since many classes of information granules are lattice ordered,LC approaches can be applied to granular computing.

In particular, the advent of L-fuzzy MM provides an access toMNNs for extended fuzzy sets

Extensions of fuzzy sets have proven to be very useful inrule-based systems for applications in engineering and computingwith words as well as in approximate reasoning.

Thanks for your interest!


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Some Approaches towards Lattice Computing in Mathematical