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: WSCG’07 Plzen

MORPHOLOGICAL EROSION WSCG’07 Plzen

MORPHOLOGICAL DILATION WSCG’07 Plzen

Openings and closings are IDEMPOTENT filters: Ψ 2 = Ψ

Original, closing, dilation, erosion and opening Grey-scale operations by a flat structuring element WSCG’07 Plzen

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) WSCG’07 Plzen

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 WSCG’07 Plzen

COLOUR IMAGES = 3D SPACE (No natural ordering of the points in this space exists ) RGBHSV Problems: When S=0 H is undefined H is measured as an angle, i.e. 0 = 360 WSCG’07 Plzen

YCrCb RGB WSCG’07 Plzen

Discretization of the CrCb unit square by equal intervals

U FOR A COLOUR IMAGE X IN YCrCb REPRESENTATION define WSCG’07 Plzen

Fuzzy dilation – erosion adjunction for colour images Thus we obtain idempotent opening and closing filters! WSCG’07 Plzen

Original, dilation, erosion; opening and closing (3 by 3 flat SE) WSCG’07 Plzen

original, dilation, erosion; opening, closing and closing through L*a*b* (5 by 5 flat SE) WSCG’07 Plzen

“original”, morphological gradient (χδ B –χε B ), Laplacian of Gaussian filter, Sobel filter WSCG’07 Plzen

QUESTIONS ???