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Morphology

Morphology. Morphology deals with form and structure Mathematical morphology is a tool for extracting image components useful in: representation and description of region shape (e.g. boundaries) pre- or post-processing (filtering, thinning, etc.) Based on set theory. Morphology.

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Morphology

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  1. Morphology • Morphology deals with form and structure • Mathematical morphology is a tool for extracting image components useful in: • representation and description of region shape (e.g. boundaries) • pre- or post-processing (filtering, thinning, etc.) • Based on set theory

  2. Morphology • Sets represent objects in images • Sets in binary images  (x,y) • Sets in gray scale images  (x,y,g) • Some morphological operations: Dilation & Erosion Opening & Closing Hit-or-Miss Transform Basic Algorithms

  3. Basic Concepts of Set Theory • A is a set in , a=(a1,a2) an element of A, aA • If not, then aA • : null (empty) set • Typical set specification: C={w|w=-d, for d  D} • A subset of B: AB • Union of A and B: C=AB • Intersection of A and B: D=AB • Disjoint sets: AB=  • Complement of A: • Difference of A and B: A-B={w|w  A, w  B}= • Reflection of B: • Translation of A by z=(z1,z2):

  4. Morphological Image Processing

  5. Morphological Image Processing

  6. Morphological Image Processing

  7. Dilation & Erosion • Basic definitions: • A,B: sets in Z2 with components a=(a1,a2) and b=(b1,b2) • Translation of A by x=(x1,x2), denoted by (A)x is defined as: (A)x = {c| c=a+x, for a∈A}

  8. Dilation & Erosion • More definitions: Reflection of B: = {x|x=-b, for b∈B} Complement of A: Ac = {x|xA} Difference of A & B: A-B = {x|x∈A, x  B} = A∩Bc

  9. Dilation & Erosion • Dilation: • : empty set; A,B: sets in Z2 • Dilation of A by B:

  10. Dilation & Erosion • Dilation: • Obtaining the reflection of B about its origin and then shifting this reflection by x • The dilation of A by B then is the set of all x displacements such that and A overlap by at least one nonzero element…

  11. Dilation & Erosion • Dilation: B is the structuring element in dilation.

  12. Morphological Image Processing

  13. Morphological Image Processing

  14. Dilation & Erosion • Erosion: i.e. the erosion of A by B is the set of all points x such that B, translated by x, is contained in A. In general:

  15. Morphological Image Processing

  16. Morphological Image Processing

  17. Opening & Closing • In essence, dilation expands an image and erosion shrinks it. • Opening: • generally smoothes the contour of an image, breaks isthmuses, eliminates protrusions. • Closing: • smoothes sections of contours, but it generally fuses breaks, holes, gaps, etc.

  18. Opening & Closing • Opening of A by structuring element B: • Closing:

  19. Morphological Image Processing

  20. Morphological Image Processing

  21. Morphological Image Processing

  22. Morphological Image Processing

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