1. Mathematical Morphology A Geometric Approach to Image Processing and Analysis John Goutsias Department of Electrical and Computer Engineering Image Analysis and Communications Laboratory The Johns Hopkins University Baltimore, MD 21218

2. 2 Question What is Mathematical Morphology ?

3. 3 A Commercial Answer  Mathematical Morphology is FAST !!  Mathematical Morphology is CHEAP !!

4. 4 An (imprecise) Mathematical Answer A mathematical tool for investigating geometric structure in binary and grayscale images.

5. 5 Shape Processing and Analysis  Visual perception requires transformation of images so as to make explicit particular shape information.  Goal: Distinguish meaningful shape information from irrelevant one.  The vast majority of shape processing and analysis techniques are based on designing a shape operator which satisfies desirable properties.

6. 6 Example  Image analysis consists of obtaining measurements characteristic to images under consideration.  Geometric measurements (e.g., object location, orientation, area, length of perimeter) Grayscale Images Binary Images

7. 7 Morphological Shape Operators  Objects are opaque and shape information is not additive !!  Shapes are usually combined by means of:  Set Union (overlapping objects):  Set Intersection (occluded objects):

8. 8 Morphological Shape Operators  Shape operators should distribute over set-unions and set-intersections (a type of “ linearity ” ) ! Morphological Dilation Morphological Erosion

9. 9 Morphological Operators  Erosions and dilations are the most elementary operators of mathematical morphology.  More complicated morphological operators can be designed by means of combining erosions and dilations.

10. 10 Question What is Mathematical Morphology ?

11. 11 A (precise) Mathematical Answer Algebra Complete Lattices Operators Erosions-Dilations Mathematical Morphology Topology Hit-or-Miss Geometry Convexity - Connectivity Distance Applications Image Processing and Analysis A mathematical tool that studies operators on complete lattices

12. 12 Some History  George Matheron (1975) Random Sets and Integral Geometry, John Wiley.  Jean Serra (1982) Image Analysis and Mathematical Morphology, Academic Press.  Petros Maragos (1985) A Unified Theory of Translations-Invariant Systems with Applications to Morphological Analysis and Coding of Images, Doctoral Thesis, Georgia Tech.

13. 13 Translation Invariant Operators

14. 14 Morphological Erosion “ LINEARITY ” TRANSLATION INVARIANCE

15. 15 Morphological Erosion

16. 16 Morphological Erosion Pablo Picasso, Pass with the Cape, 1960 Structuring Element

17. 17 Morphological Dilation “ LINEARITY ” TRANSLATION INVARIANCE

18. 18 Morphological Dilation

19. 19 Morphological Dilation Pablo Picasso, Pass with the Cape, 1960 Structuring Element

20. 20 Morphological Opening

21. 21 Morphological Opening Pablo Picasso, Pass with the Cape, 1960 Structuring Element

22. 22 Morphological Opening  Is a smoothing filter !  Amount and type of smoothing is determined by the shape and size of the structuring element.  Approximates a shape from below, since

23. 23 Filtering Example Henri Matisse, Woman with Amphora and Pomegranates, 1952 ORIGINAL FILTERED DEGRADED

24. 24 An Important Result Increasing Operator Translation Invariant Operator + !!

25. 25 Main Idea  Examine the geometrical structure of an image by matching it with small patterns at various locations.  By varying the size and shape of the matching patterns, called structuring elements, one can extract useful information about the shape of the different parts of the image and their interrelations.  Results in image operators which are well suited for the analysis of the geometrical and topological structure of an image.

26. 26 Question What about gray-scale images ?

27. 27 Grayscale Erosion TRANSLATION INVARIANCE “ LINEARITY ” MINIMUM

28. 28 Grayscale Dilation “ LINEARITY ” MAXIMUM TRANSLATION INVARIANCE

29. 29 Remark Flat Erosion Flat Dilation

30. 30 Grayscale Morphology ORIGINAL EROSION DILATION OPENING

31. 31 Grayscale Opening Structuring Element

32. 32 Question Can we automatically extract the largest connected component (the woman ’ s body) in this image ? Henri Matisse, Woman with Amphora and Pomegranates, 1952

33. 33 Answer ORIGINAL MARKER EROSION MARKER This is a morphological operator that filters out connected image components of a certain size and shape CONNECTED OPERATORS !!

34. 34 An Application - Target Detection DATA MARKER OPENING MORPHOLOGICAL RECONSTRUCTION Targets

35. 35 An Application: Target Detection MORPHOLOGICAL RECONSTRUCTION MARKER CLOSING DATA

36. 36 An Application: Target Detection THRESHOLDING DATA FINAL RESULT Correctly detected targets Incorrectly detected target

37. 37 The End