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Mathematical Morphology - Set-theoretic representation for binary shapes

Mathematical Morphology - Set-theoretic representation for binary shapes . Qigong Zheng Language and Media Processing Lab Center for Automation Research University of Maryland College Park October 31, 2000. What is the mathematical morphology ?.

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Mathematical Morphology - Set-theoretic representation for binary shapes

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  1. Mathematical Morphology - Set-theoretic representation for binary shapes Qigong Zheng Language and Media Processing Lab Center for Automation Research University of Maryland College Park October 31, 2000

  2. What is the mathematical morphology ? • An approach for processing digital image based on its shape • A mathematical tool for investigating geometric structure in image • The language of morphology is set theory

  3. Goal of morphological operations • Simplify image data, preserve essential shape characteristics and eliminate noise • Permits the underlying shape to be identified and optimally reconstructed from their distorted, noisy forms

  4. Shape Processing and Analysis • Identification of objects, object features and assembly defects correlate directly with shape • Shape is a prime carrier of information in machine vision

  5. Set Union (overlapping objects): • Set Intersection (occluded objects): Shape Operators • Shapes are usually combined by means of :

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

  7. 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

  8. Dilation B A

  9. Dilation • Let A be a Subset of and . The translation of A by x is defined as • The dilation of A by B can be computed as the union of translation of A by the elements of B

  10. Dilation B

  11. Dilation

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

  13. Properties of Dilation • Commutative • Associative • Extensivity • Dilation is increasing

  14. Extensitivity A B

  15. Properties of Dilation • Translation Invariance • Linearity • Containment • Decomposition of structuring element

  16. 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

  17. Erosion A B

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

  19. Erosion

  20. Erosion

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

  22. Properties of Erosion • Erosion is not commutative! • Extensivity • Dilation is increasing • Chain rule

  23. Properties of Erosion • Translation Invariance • Linearity • Containment • Decomposition of structuring element

  24. Duality Relationship • Dilation and Erosion transformation bear a marked similarity, in that what one does to image foreground and the other does for the image background. • , the reflection of B, is defined as • Erosion and Dilation Duality Theorem

  25. Opening and Closing • Opening and closing are iteratively applied dilation and erosion Opening Closing

  26. Opening and Closing

  27. Opening and Closing • They are idempotent. Their reapplication has not further effects to the previously transformed result

  28. Opening and Closing • Translation invariance • Antiextensivity of opening • Extensivity of closing • Duality

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

  30. Structuring Element Example of Closing

  31. 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.

  32. Morphological filtering • Noisy image will break down OCR systems Clean image Noisy image

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

  34. Summary • Mathematical morphology is an approach for processing digital image based on its shape • The language of morphology is set theory • The basic morphological operations are erosion and dilation • Morphological filtering can be developed to extract useful shape information

  35. THE END

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