1. Digital Image Processing, Spring 2006 1 ECES 682 Digital Image Processing Week 8 Oleh Tretiak ECE Department Drexel University

2. Digital Image Processing, Spring 20062 Announcements Final exam on Monday, June 12 No class on May 29

3. Digital Image Processing, Spring 20063 Nonlinear Filtering g(x,y) = F[f(N(x,y))]  N - neighborhood  3x3 neighborhood, binary images -> 512 different transformations  Mystified by Wolfram (cellular automata)  Some simple functions obviously useful  Get rid of salt, pepper noise  Detect edges

4. Digital Image Processing, Spring 20064 Morphological Image Processing Boolean algebra Dilation and erosion Opening and closing Hit-or-miss Basic algorithms Extension to gray-scale

5. Digital Image Processing, Spring 20065 Review: Boolean Algebra A Boolean algebra is a set with at least two elements, three operations (and , or , not ‘) and two special elements (0, 1) that have the following properties.  AB is an element of the set. This function is defined for all elements A and B in the set. It is symmetric (AB = BA)  A  B has the same properties  A’ is defined for all elements in the set.  AA’=0, AA’=1  The operations  and + are distributive.  A(BC)=(AB)(AC)  A(BC)=(AB)(AC)  0 and 1 are identities, in the following sense  0A=A  1A=A

6. Digital Image Processing, Spring 20066 Examples of Boolean Algebra Switching algebra  S = {0, 1} Finite Boolean algebras  Example: S = {(0, 0), (0, 1), (1, 0), (1, 1)}  (a 1, a 2 )’ = (a’ 1, a’ 2 )  (0, 1)(1, 0) = (0, 0) Set unions/intersections  Union is like   Intersection is like   Empty set is like 0  There is no 1 (universal set)

7. Digital Image Processing, Spring 20067 Boolean Algebra of Binary Pictures

8. Digital Image Processing, Spring 20068 Continuous and Discrete Morphology There are morphology theories of continuous and discrete spaces Example of continuous space  Real line Example of discrete space  Integers We will talk about the morphology of discrete spaces

9. Digital Image Processing, Spring 20069 Additional Operations Elements of set: points  Points are integers (1-D discrete space)  Points are 2-D vectors with integer components (2-D discrete space) Operations  Addition (vector addition)  Reflection (multiply by -1)  Integer multiplication A set of points can be translated or reflected  S+x = x+S (new set consists of all points of S, translated by x)  S^ is the set reflected through the origin

10. Digital Image Processing, Spring 200610 Basic Morphological Operations  Dilation  A+B = {x| x = y+z, y in A, z in B}  Equivalent definition  {x, (x+B^)A is not empty} Erosion  A-B = {x| x+B is a subset of A}

11. Digital Image Processing, Spring 200611 More Examples

12. Digital Image Processing, Spring 200612

13. Digital Image Processing, Spring 200613

14. Digital Image Processing, Spring 200614 Combination of Dilation and Contraction

15. Digital Image Processing, Spring 200615 Morphological Opening A opened by B

16. Digital Image Processing, Spring 200616 Morphological Closing A closed by B

17. Digital Image Processing, Spring 200617 Equations and Inequalities

18. Digital Image Processing, Spring 200618 Example

19. Digital Image Processing, Spring 200619 Combination of Opening and Closing

20. Digital Image Processing, Spring 200620 Hit-or-Miss Given: points on plane Template: Set of one points (foreground) and set of zero points (background) Example foreground: B 1 = D, B 2 = D Find: Points x for which B 1 +x are 1, B 2 +x are 0 Solution:

21. Digital Image Processing, Spring 200621 Example

22. Digital Image Processing, Spring 200622 Boundary Extraction

23. Digital Image Processing, Spring 200623 Region Filling Start with point in region A. Keep expanding by dilation, using points in region A only.

24. Digital Image Processing, Spring 200624 Extraction of Connected Components Start with point on object. Keep adding points

25. Digital Image Processing, Spring 200625 Skeleton Morphological skeleton Connected skeleton

26. Digital Image Processing, Spring 200626 Morphological Skeleton

27. Digital Image Processing, Spring 200627 Connected Skeleton

28. Digital Image Processing, Spring 200628 Morphological Skeleton Start with structuring element, B Generate a sequence of elements B k =kB, B 0 =0 Construction

29. Digital Image Processing, Spring 200629 Distance Function (Transform) Useful for morphology, skeletons, alignment MATLAB has a subfunction

30. Digital Image Processing, Spring 200630 Grey-Scale Morphology

31. Digital Image Processing, Spring 200631 Erosion

32. Digital Image Processing, Spring 200632 Example

33. Digital Image Processing, Spring 200633 Opening and Closing