An Approach Towards Image Edge Detection Based …sussner/EUSFLAT_Slides.pdfof fuzzy set inclusion. F I(X), the class of IV-fuzzy setsover X, with a partial order that extends fuzzy

An Approach Towards Image Edge Detection Based onInterval-Valued Fuzzy Mathematical Morphology and

Admissible Orders

Peter SussnerLisbeth Corbacho

University of CampinasDepartment of Applied Mathematics

Campinas, Brazil

The 11th Conference of the European Society for Fuzzy Logic andTechnology

(University of Campinas) September 2019 EUSFLAT 2019 1 / 26

Introduction

Introduction

Usually, (gray-scale) image processing is based on the incorrectassumption that the pixel values are certain.

The inherent uncertainty due to image capture of a gray-scale (GS)image can be modelled in terms of an interval-valued (IV) image.

Here we use IV-fuzzy mathematical morphology (MM) for detectingthe edges of an IV image .


Introduction

The Use of IV-Fuzzy MM for Edge Detection

A GS image gives rise to another GS image called the morphologicalgradient image.

A GS morphological gradient image yields a binary edge image afterthinning and binarization.

Using IV-FMM, one can generate an IV morphological gradient imagefrom an IV image.

Conventional thinning and binarization methods cannot be applieddirectly to an IV morphological gradient image.

Therefore, we employ three different approaches, two of which use atotal ordering scheme called admissible order, for obtaining a GSmorphological gradient image from a IV image.

These approaches yield IV image edge detectors.


Introduction

Some Comments on MM and Lattice Theory

Complete lattices provide for the theoretical framework of MM;

Examples complete lattices include

P(X ), i.e., the powerset of X 6= ∅, with the partial order “⊆”;F(X ), the class of fuzzy sets over the universe X , with the partial orderof fuzzy set inclusion.FI(X ), the class of IV-fuzzy sets over X , with a partial order thatextends fuzzy set inclusion.

Let X be a point set. In different approaches towards MM, an imageis given by:

A subset of X in binary MM;A fuzzy set over X in fuzzy MM;An IV-fuzzy set over X in IV-fuzzy MM.


Introduction

Organization of this talk

1 Introduction

2 Some Lattice-Theoretical Concepts

3 L-fuzzy Mathematical morphology

4 Image Edge Detection using IV-FMM and Admissible Orders

5 Computational Experiments


Introduction


1 Introduction






Introduction


1 Introduction






Introduction


1 Introduction






Introduction


1 Introduction






Some Lattice-Theoretical Concepts

Lattices

A partially ordered set or poset is a set L 6= ∅ together with a partialorder relation ≤. Thus, a poset is a pair (L,≤). If ≤ arises clearlyfrom the context, one writes L instead of (L,≤).

For a, b ∈ L, where L is a poset, one defines the closed interval[a, b] = {x ∈ L | a ≤ x ≤ b}.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, ∀x , y ∈ L, x ≤ y or y ≤ x , then L is totally ordered or a chain .If 0L :=

∧L and 1L :=

∨L exist in L, then L is bounded .

If∧Y and

∨Y exist in L for every Y ⊆ L, then L is complete.

[0, 1]2 is a complete chain with each of the following total orders:

(x1, x2) �lex1 (y1, y2) ⇔ x1 < y1 or (x1 = y1 and x2 ≤ y2);(x1, x2) �lex2 (y1, y2) ⇔ x2 < y2 or (x2 = y2 and x1 ≤ y1);



Some Types of Lattices

If L is a lattice and X 6= ∅, then the following are lattices as well:

(Ln,≤) with (x1, . . . , xn) ≤ (y1, . . . , yn) ⇔ xi ≤ yi , ∀i = 1, . . . , n.

(Lm×n,≤) with S ≤ T ⇔ sij ≤ tij , ∀i = 1, . . . ,m, ∀j = 1, . . . , n.

LX = {f : X → L} with the following partial order:

f ≤ g ⇔ f (x) ≤ g(x) ∀x ∈ X . (1)

The class of all graphs of functions in LX , denoted FL(X ), is latticewith the partial order given as follows. If A = {(x , µA(x)) | x ∈ X}and B = {(x , µB(x)) | x ∈ X}, then A ≤ B ⇔ µA ≤ µB .

IL = {[x , x ] ⊆ L | x ≤ x} with [x , x ] ≤2 [y , y ] ⇔ x ≤ y and x ≤ y .

If L is bounded or complete, then the lattices above are also bounded orcomplete, resp..



Complete Lattice Isomorphisms

Definition:

A bijection φ : L→M, where L,M are complete lattices, is called acomplete lattice isomorphism if the following property holds ∀ x , y ∈ L:

x ≤ y ⇔ φ(x) ≤ φ(y). (2)

In this case, L and M are said to be isomorphic and one writes L 'M.

Assuming that L is a complete lattice, we have

1 FL(X ) ' LX , if X 6= ∅;2 FL(X ) ' Ln, if X = {1, . . . , n};3 FL(X ) ' Lm×n, if X = {1, . . . ,m} × {1, . . . , n}.



Admissible Orders

Definition:

Let I denote I[0,1]. A partial order � on I is said to be admissible if

1 � is a total order;

2 If [x , x ] ≤2 [y , y ], then [x , x ] � [y , y ].

Examples of admissible orders:

[x , x ] �lex1 [y , y ] ⇔ (x , x) �lex1 (y , y) (lexicographic-1),

[x , x ] �lex2 [y , y ] ⇔ (x , x) �lex2 (y , y) (lexicographic-2).

[x , x ] �α,β [y , y ] ⇔ (Kα(x , x),Kβ(x , x)) �lex1 (Kα(y , y),Kβ(y , y)),where α 6= β ∈ [0, 1] and Kα(x , x) = x + α(x − x), ∀[x , x ] ∈ I.

Let �α+ denote �α,1 for α ∈ [0, 1[ and �α− denote �α,0 for α ∈]0, 1].

�α,β equals �α+ for every β > α ∈ [0, 1[.

�α,β equals �α− for every β < α ∈]0, 1].


L-fuzzy Mathematical morphology

Definition:

An increasing mapping C : L× L→ L is an L-fuzzy conjunction ifC(0L, 0L) = C(0L, 1L) = C(1L, 0L) = 0L and C(1L, 1L) = 1L. If C is alsocommutative, associative and satisfies C(1L, x) = x , then C is called anL-fuzzy triangular norm or t-norm.

Examples:

For L = [0, 1]

TLK (x , y) = 0 ∨ (x + y − 1),

TnM(x , y) =

{0 if x + y ≤ 1,x ∧ y otherwise.

For L = I, the pessimistic conjunction with representative C :

CpC (x, y) = [C (x , y),C (x , y) ∨ C (x , y)]

where C is a fuzzy conjunction and x = [x , x ], y = [y , y ] ∈ I



L-fuzzy Implication

Definition

A binary operator I : L× L→ L is an L-fuzzy implication if I(·, z) isdecreasing for every z ∈ L, I(z , ·) is increasing for every z ∈ L and ifI(0L, 0L) = I(0L, 1L) = I(1L, 1L) = 1L and I(1L, 0L) = 0L. An L-fuzzyimplication I is called a L-fuzzy border implicator if I(1L, x) = x for everyx ∈ L.

Examples:

For L = [0, 1]

IFD(x , y) =

{1 if x ≤ y ,(1− x) ∨ y x > y .

IKD(x , y) = (1− x) ∨ y .

L = I, the optimistic implication with representative I ,

IoI (x, y) = [I (x , y) ∧ I (x , y), I (x , y)]

where I is a fuzzy implications and x = [x , x ], y = [y , y ] ∈ I(University of Campinas) September 2019 EUSFLAT 2019 11 / 26


L-fuzzy Erosion and Dilation

Definition:

Let X ⊆ Zd and C and I be resp. an L-fuzzy conjunction and implication.The L-fuzzy erosion and the L-fuzzy dilation of an image A ∈ FL(X ) by astructuring element (SE) S ∈ FL(X ) are resp. defined as follows:

EI(A, S)(x) =∧

y∈X∩Dx

I(S(y − x),A(y)), where Dx = {y | y − x ∈ X},

DC(A,S)(x) =∨

y∈X∩Dx

C(S(x − y),A(y)), where Dx = {y | x − y ∈ X},



Morphological Gradient

Proposition

Let A, S ∈ FL(X ), where X ⊆ Zd contains 0. If S(0) = 1L, C is anL-fuzzy t-norm, and I is an L-fuzzy border implicator, then we have:

EI(A,S) ≤ A ≤ DC(A,S). (3)

Remark: A morphological gradient image is given by DC(A,S)− EI(A,S)for some notion of a difference operator on FL(X ).


Image Edge Detection using IV-FMM and Admissible Orders

Modelling the Uncertainty Regarding the Pixel Values

The pixel values of a digital image are inherently uncertain due to thefollowing potential errors:

Quantization or Tonal Error: Captured value must be rounded up ordown to obtain an allowed value;Discretization or Spatial Error: Arises when mapping values of acontinuous domain to pixel values on a finite grid.

The resulting uncertainty can be modelled by mapping a digital imageO with values in {0, ..., 255} to an interval-valued image OIV , where

OIV (x) = [0 ∨∧

y∈N(x)

O(y)− 1, 255 ∧∨

y∈N(x)

O(x) + 1]. (4)

here N(x) stands for a 3× 3 neighborhood of x .

Note that the pixel values of O and OIV can be normalized so as toobtain a fuzzy and an IV-fuzzy image, resp..



1. Generating a GS Gradient from an IV Gradient Image

We employed the IV-fuzzy t-norm CpTnM

, the IV-fuzzy border implicator IoIKD and

the interval difference x−IV y = [x , x ]−IV [y , y ] = [x − y , (x − y) ∨ (x − y)].



Examples:

Gray-scale edges obtained by the convex combination of the upper andlower bound images of an IV morphological gradient:

Figure: (a) α = 0.25 (b) α = 0.45 (c) α = 0.5 (d) α = 1.



2. Applying an Admissible Order to the IV-Fuzzy Imagebefore Computing the Morph. Gradient Image

Here the fuzzy dilation and erosion was computed using TnM and IKD .(University of Campinas) September 2019 EUSFLAT 2019 17 / 26


Examples:

Gray-scale morphological gradients obtained by varying the admissibleorder on IV-image:

Figure: (a) �0.0008− (b) �0.47− (c) �0.86− (d) �1−.



3. Applying an Admissible Order After Computing theIV-Fuzzy Erosion and Dilation

We used the IV-fuzzy t-norm CpTnM

, the IV-fuzzy implication IoIKD and the usual

difference to compute the morphological gradient.(University of Campinas) September 2019 EUSFLAT 2019 19 / 26


Examples:

Gray-scale morphological gradients obtained by varying the admissibleorder

Figure: (a) �0.0008− (b) �0.4− (c) �0.78− (d) �1−.



A Visual Comparison of the Edge Detection Methods

Figure: a) Ground truth. b) IVF morph. gradient, convex comb. for α = 0.5. c)IVF image, �0.5−, morph. gradient. d) IVF Erosion and Dilation, �0.5−, - e)Canny edge detector


Computational Experiments


We considered the first 25 images of the public image dataset of theUniversity of South Florida, including their respective ground truthimages.

We applied the followed steps to generate a binary edge image:

.

To measure the perfomance of the edge detectors, we evaluatedPratt’s figure of merit (FoM). Note that 0 ≤ FoM ≤ 1.

FoM =1

max{|(DE )|, |(GT )|}∑x∈DE

1

1 + ad2.

|(DE )| and |(GT )| denote resp. the numbers of detected and ground truth

edge pixels, a is a scaling constant, and d is the separation of an actual edge

pixel to a “true” edge pixel. Here a = 1 and d is the Euclidean distance.



Results

Average FoM over 25 images for the Canny method applied to the originalimage in comparison with the average FoMs achieved by three differentmethods that were applied to IV-fuzzy images OIV for α ∈ {0, 0.02, . . . , 1}.

Figure:



Concluding remarks

We proposed three edge detectors for IV-fuzzy images based onIV-FMM, convex combinations and admissible orders.

In our experiments, we binarized the resulting morphological gradientand computed the FoM.

Our method of choice consists of applying an admissible order to theIV-fuzzy erosion and dilation, followed by taking the usual difference.

This method produced higher FoM values than

convex combinations of the upper and lower bounds of the IV-fuzzymorph. gradient of OIV ;the gray-scale morph. gradients of Kα−(OIV ), Kα+(OIV );The classical Canny method.

Recall that we used an IV-fuzzy image OIV that contains lessinformation than the original image O.



References

C. Lopez-Molina, C. Marco-Detchart, J. Cerron, H. Bustince, B. De Baets,Gradient extraction operators for discrete interval-valued data, in: 16th IFSAWorld Congress; 9th Conference of the European Society for Fuzzy Logic andTechnology, Vol. 89, Atlantis Press, 2015, pp. 836–843.

H. Bustince, J. Fernandez, A. Kolesarova, R. Mesiar, Generation of linearorders for intervals by means of aggregation functions, Fuzzy Sets andSystems 220 (2013) 69–77.

M. Gonzalez-Hidalgo, S. Massanet, A fuzzy mathematical morphology basedon discrete t-norms: fundamentals and applications to image processing, SoftComputing 18 (11) (2014) 2297–2311.

P. Sussner, M. Nachtegael, T. Melange, G. Deschrijver, E. Esmi, E. Kerre,Interval-valued and intuitionistic fuzzy mathematical morphologies as specialcases of L-fuzzy mathematical morphology, Journal of Mathematical Imagingand Vision 43 (1) (2012) 50–71.



Acknowledgments

Thanks for your interest! Any questions?


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An Approach Towards Image Edge Detection Based …sussner/EUSFLAT_Slides.pdfof fuzzy set inclusion. F I(X), the class of IV-fuzzy setsover X, with a partial order that extends fuzzy