1. Made to measure morphological filters
Fernand Meyer
Center of mathematical morphology
Mines-ParisTech
France

2. Non edge preserving filters
The adjunction erosion/dilation
Openings/closings
The simplest filters
Alternate sequential filters

3. Erosions and dilations of increasing sizes

4. Erosions of increasing size
Regional minima get larger
New maxima may appear. Most maxima vanish, others get larger.

5. Dilations of increasing size
New minima may appear. Most mainima vanish, others get larger.
Regional maxima get larger, some coalesce, others disappear.

6. Erosion = minimal value in a window  darker image
Dilation = maximal value in a window  brighter image
INCREASING WINDOW + STRONGER EFFECT
Increasing erosions
Increasing dilations

7. combining an erosion and a dilation in sequence, in order to get openings or closings.

8. An erosion
Darker
Followed by
A dilation
Brighter
= opening
Darker

9. openings of increasing size
Each reg. min. of a large closing contains a reg. Min. of each smaller opening
Regional maxima get larger, some coalesce, others disappear.

10. A dilation
Brighter
Followed by
An erosion
Darker
= closing
Brighter

11. openings of increasing size
Regional minima get larger, some coalesce, others disappear.
Each reg. max. of a large closing contains a reg. Max. of each smaller closing

12. Open-close, close-open, Open-close-open, close-open-close

13. An opening
Darker
Followed by
A closing
Brighter
= morphological filter
Similar grey tone

14. = morphological filter
A closing
Brighter
Followed by
An opening
Darker
= morphological filter
Similar grey tone

15. Opening followed by closing of increasing sizes
Regional minima
Regional maxima

16. Closing followed by opening of increasing sizes
Regional minima
Regional maxima

17. The two are not comparable
Opening followed by closing of increasing sizes
Closing followed by opening of increasing sizes
The two are not comparable

18. Close-open-close  open-close-open
Open close open of increasing sizes
Close open close of increasing sizes
Close-open-close  open-close-open

19. Closing followed by opening followed by closing of increasing sizes
Regional minima
Regional maxima

20. A small closing followed by a small opening = small effects
A large closing followed by a large opening = too crude
Alternate sequential filter = sequence of increasing pairs of an opening followed by a closing

21. Alternate sequential filter starting with an opening of increasing sizes
Regional minima
Regional maxima

22. Alternate sequential filter starting with a closing of increasing sizes
Regional minima
Regional maxima

23. We will now introduce edge preserving filters :
Partial conclusion
Non edge preserving filters simplify the images but the displacement of the contours is annoying in many cases and in particular if the filter precedes image segmentation
We will now introduce edge preserving filters :
Razings
Floodings
Levelings