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Fuzzy Mathematical Morphology and its Applications to Colour Image Processing

This paper explores the application of fuzzy mathematical morphology in colour image processing, specifically focusing on morphological erosion and dilation. It discusses the use of adjunctions, representation of colour images, and introduces various filters and operations.

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Fuzzy Mathematical Morphology and its Applications to Colour Image Processing

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  1. Fuzzy mathematical morphology and its applications to colour image processingAntony T. PopovFaculty of Mathematics and Informatics – Department of Information Technologies,St. Kliment Ohridski University of Sofia,5, J. Bourchier Blvd., 1164 Sofia, Bulgariatel/fax: +359 2 8687180e-mail: atpopov@fmi.uni-sofia.bg WSCG’07 Plzen

  2. MORPHOLOGICAL EROSION WSCG’07 Plzen

  3. MORPHOLOGICAL DILATION WSCG’07 Plzen

  4. WSCG’07 Plzen

  5. Openings and closings are IDEMPOTENT filters: Ψ2 = Ψ

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

  7. General grey-scale morphological operations: Drawback: may change the scale! WSCG’07 Plzen

  8. WSCG’07 Plzen

  9. ALGEBRAIC DILATION AND EROSION WSCG’07 Plzen

  10. FUZZY SETS≡ membership functions A = “young” B= “very young” Instead of μA(x) we could write A(x) WSCG’07 Plzen

  11. 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

  12. 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

  13. 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

  14. YCrCb RGB WSCG’07 Plzen

  15. Discretization of the CrCb unit square by equal intervals

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

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

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

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

  20. “original”, morphological gradient(χδB –χεB) , Laplacian of Gaussian filter, Sobel filter WSCG’07 Plzen

  21. QUESTIONS ???

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