1. Mathematical Morphology

2. A basic view to mathematical morphology :
An approach for processing digital image based on its shape
A mathematical tool for investigating geometric structure in image and signal
The language of morphology is set theory

3. Binary Domain

4. Morphological Operations
The primary morphological operations are dilation and erosion
More complicated morphological operators can be designed by means of combining erosions and dilations

5. Dilation
Dilation is the operation that combines two sets using vector addition of set elements.
Let A and B are subsets in 2-D space. A: image undergoing analysis, B: Structuring element, denotes dilation

6. Dilation B • • • A

7. Dilation • • • B •

8. Dilation

9. Pablo Picasso, Pass with the Cape, 1960
Example of Dilation
Structuring
Element
Pablo Picasso, Pass with the Cape, 1960

10. Properties of Dilation
Commutative
Associative
Extensivity
Dilation is increasing

11. Erosion
Erosion is the morphological dual to dilation. It combines two sets using the vector subtraction of set elements.
Let denotes the erosion of A by B

12. Erosion A • • • B

13. Erosion • • •

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37. Erosion Erosion can also be defined in terms of translation
In terms of intersection

38. Erosion • • • • •

39. Pablo Picasso, Pass with the Cape, 1960
Example of Erosion
Structuring
Element
Pablo Picasso, Pass with the Cape, 1960

40. Properties of Erosion Erosion is not commutative! Extensivity
Dilation is increasing

41. Opening and Closing Opening Closing
Opening and closing are iteratively applied dilation and erosion
Opening
Closing

42. Opening and Closing can be filters , because …
They are idempotent. Their reapplication has not further effects to the previously transformed result

43. Properties of Opening and Closing
Translation invariance
Antiextensivity of opening
Extensivity of closing
Duality

44. Pablo Picasso, Pass with the Cape, 1960
Example of Opening
Structuring
Element
Pablo Picasso, Pass with the Cape, 1960

45. Example of Closing
Structuring
Element

46. Dilation

47. Erosion

48. Closing

49. Opening

50. Morphological Filtering
Main idea
Examine the geometrical structure of an image by matching it with small patterns called structuring elements at various locations
By varying the size and shape of the matching patterns, we can extract useful information about the shape of the different parts of the image and their interrelations.

51. Morphological filtering
Noisy image will break down OCR systems
Clean image
Noisy image

52. Morphological filtering
By applying MF, we increase the OCR accuracy!
Restored image

53. GS dilation and erosion:

54. Example over audio signal