Notes 20110324 CIS 6320 Image Processing - Jamileh

From SnOwy - Ed's Wiki Notebook

Presentation Notes: Hit or Miss Transformation

• requirements: morphological mathematics

Contents 1 Mathematical Morphology 1.1 Application 1.2 Set Theory 1.3 Operations 1.3.1 Dilation Properties 1.4 Erosion 1.4.1 Erosion Properties 1.4.2 Erosion Application 1.5 Duality of Dilation and Erosion 1.6 Compound Operations 1.6.1 Opening 1.6.2 Opening Properties 1.6.3 Opening Applications 1.6.4 Closing 1.6.5 Properties of Closing 1.6.6 Properties of Opening and Closing 2 Hit or Miss Transform 2.1 Uses

Mathematical Morphology

• set theory, topology, geometry
• analyzes the shape and form of an object
• quantitative analysis
• set theory language for this analysis
• process digital image based on its shape
• suited more for binary images (as opposed to grey levels)
• grey level components
  • boundaries
  • skeletons
  • convex hull
• also used to remove noise and small objects (holes)
• isolates objects
• separates circles from lines
• able to remove dust marks and other artefacts from a fingerprint

Application

• many applications

Set Theory

• operations: intersect, union, complement, difference
• two geometrical examples also exist which are critical to morphology
  • reflection of a set (in the origin)
  • translation of a set (z is a vector (displacement vector, offset) -- foreach element a in A, a' = a+z)

Operations

• structuring element (Kernel)
• set of local neighbourhood with specific origin
• common structuring elements: cross, disc, circle
• hits and fits operations
  • fit: all on pixels in SE cover on pixels in the image (A)
  • {x|A_x⊆F}
  • hit: any pixel on pixel in SE covers an on pixel in image (B)
  • {x|B_x⊆F}
• dilation --
  • dilation of image F by structuring element S is given by F ⊕ S
• the S is positioned with its origin at (x, y) and new pixel value given by a rule (FIXME)
• structuring element is used as a mask to determine which pixels to turn on in the resulting image
• dilation
  • let F and S be sets in Z²
• F -- an image or shape
• S -- a structuring element
• a = (a₁, a₂) is an element in F
• b = (b₁, b₂) is an element in S
• dilation of F by S is the union of the translation of F by elements of S

Dilation Properties

• increasing, commutative, associative, distributive over set union
• increasing theorem -- F ⊆ K → F⊕S ⊆ K⊕S ...
• commutative theorem
• ...
• theorem distribution over set union

Erosion

• is the reverse of dilation, we instead fill in pixels that are fits (instead of hits)
• erosion of F ⊖ S -- foreach
• erosion of an image can vary depending on the structuring element

Erosion Properties

• increasing
• chain rule erosion (there is no associative property)
• distributive over set intersect
• does not distribute over the union -- has a different property
• no associative / no commutative

Erosion Application

• can split apart joined objects
• can shrink objects

Duality of Dilation and Erosion

• dilation and erosion are related; we need to take the complement of F though

Compound Operations

• we can combine erosion and dilation into higher level operations
• may use same or different structuring elements
• two most important compounds are
  • opening (isolates objects; removes small objects (better than erosion))
  • closing (fill holes (better than dilation))

Opening

• preserves size and shape of original object after removing small irrelevant objects
• notation: F°S = (F ⊖ S) ⊕ S
  • union of all translations of S which are fits in F
• F°S = &union;{S_b / S_b ⊆} -- FIXME
• is equal to a erosion followed by a dilation (using the same structuring element)

Opening Properties

• anti-extensive
• increasing
• idempotent (additional applications of the same operation does nothing more)
  • F°S°S = F°S

Opening Applications

• FIXME

Closing

• fills holes, but preserves shape and size
• a dilation followed by an erosion
• F⋅S
• is the complement of all the unions of all translations ...
• F⋅S = &union{} -- FIXME

Properties of Closing

• F⊆F⋅S

Properties of Opening and Closing

• (F⋅S)^c = F^c°S-hat
• where S-hat is the reflection of S across the origin
• FIXME

Hit or Miss Transform

• defined as --
  • all points that S1 hits in F AND
  • all the points that S2 hits in F complement
• the first one says S1 hits F
• the second one says S2 misses F
• notation: F⊗S where S = {S1, S2}
• requirements: S1 and S2 are arbitrary, except we require that S1 and S2 do not intersect
  • that is, the intersection of the two may not have any ones (where zero is empty)

Uses

• feature extraction
• corner extraction
• structuring elements used for locating various binary features