Fuzzy mathematical morphology and its applications to colour image processing

Fuzzy mathematical morphology and its applications to colour image processing

Antony T. Popov

Faculty of Mathematics and Informatics – Department of Information Technologies,

St. Kliment Ohridski University of Sofia, 5, J. Bourchier Blvd., 1164 Sofia, Bulgaria

tel/fax: +359 2 868 7180

e-mail: atpopov@fmi.uni-sofia.bg

WSCG’07 Plzen

MORPHOLOGICAL EROSION

MORPHOLOGICAL DILATION

WSCG’07 Plzen

Openings and closings are IDEMPOTENT filters: Ψ

= Ψ

Original, closing, dilation, erosion and opening

Grey-scale operations by a flat structuring element

General grey-scale morphological operations:

Drawback: may change the scale!

WSCG’07 Plzen

ALGEBRAIC DILATION AND EROSION

WSCG’07 Plzen

FUZZY SETS ≡ membership functions

A = “young”

B= “very young”

Instead of μ_A(x) we could write A(x)

) ( 1

)

( x

x

X

U





 

)) (

), (

( )

( x max

x

x

Y

X

 





)) (

), (

( )

( x min

x

x

Y

X

 





An operation c: [0,1]x[0,1]→[0,1] is a conjunctor

(a fuzzy generalization of the logical AND operation), or t-norm, if it is commutative, increasing in both arguments,

c(x,1) = x for all x, c(x,c(y,z)) = c(x(x,y),z).

An operation I: [0,1]x[0,1]→[0,1] is an implicator

if it decreases by the first and increases by the second argument, I(0,1) = I(1,1)=1 , I(1,0) = 0.

Lukasiewicz: c(x,y) = max (0,x+y-1) ; I(x,y) = min(1,y-x+1),

“classical” : c(x,y) = min(x,y) ; I(x,y) = y if y<x, and 1 otherwise.

WSCG’07 Plzen

Grey –scale images can be represented as fuzzy sets!

Say that a conjunctor and implicator form an ADJUNCTION when

C(b,y) ≤ x if and only if y ≤ I(b,x)

Having an adjunction between implicator and conjunctor, we

define

COLOUR IMAGES = 3D SPACE

(No natural ordering of the points in this space exists

)

RGB HSV

Problems: When S=0 H is undefined H is measured as an angle , i.e. 0 =

360

WSCG’07 Plzen

YCrCb

RGB

Discretization of the CrCb unit square by equal intervals

1 4 5 16 17 ……

2 3 6 15 18 …..

9 8 7 14 19 …..

10 11 12 13 20 ….

25 24 23 22 21 ….

… …. … … … …..

UFOR A COLOUR IMAGE

X

YCrCb

define

Fuzzy dilation – erosion adjunction for colour images

Thus we obtain idempotent opening and closing filters!

WSCG’07 Plzen

Original, dilation, erosion;

opening and closing

original, dilation, erosion;

opening, closing and closing through L*a*b*

(5 by 5 flat SE)

WSCG’07 Plzen

“original”, morphological gradient (χδ

–χε

) ,

Laplacian of Gaussian filter, Sobel filter

QUESTIONS ???