Mathematical Morphology

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Mathematical Morphology

• Mathematical morphology (matematická morfologie)
• A special image analysis discipline based on
  morphological transformations of the image
  (usually a binary image).
• Developed
• In early 1980s by the group of Jean Serra
• Centre de Morphologie Mathématique (CMM)
• Fontainebleau, France
• http//www.ensmp.fr/Eng/Research/Domain/MathInfoA
  uto/CMM/CMM.html
• http//cmm.ensmp.fr/Recherche/recherche_eng.htm
• Motivation
• Binary image (obtained e.g. using thresholding
  or edge detection followed by edge linking and
  object filling) often needs further processing
  such as overlapping object separation, filling of
  holes, removal of narrow protrusions and other
  morphological (i.e. shape) changes.

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Mathematical Morphology

• Basic principles
• Such further processing is performed using one or
  a combination of several morphological
  transformations.
• The transformations work in a certain local
  neighborhood of each pixel (similarly to
  convolution) defined by so called structuring
  element (strukturní element). The structuring
  element can be square, cross-like or any other
  shape.
• There are two types of morphological
  transformations binary and grey-scale. Binary
  transformations transform binary images into
  binary images. Grey-scale transformations
  transform grey-scale images into grey-scale
  images.

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Mathematical Morphology Binary Images

• Definition
• The 2D binary images are seen as point sets,
  where each point is defined as an ordered pair
  (x-position, y-position)
• A morphological transformation ? is then given by
  the relation of the point set X (input image)
  with another smaller point set B called
  structuring element.

X (0,1),(1,1),(2,1),(2,2),(3,0),(4,0)
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Mathematical Morphology Binary Images

• Example of two simple structuring elements (SE)
• Any structuring element B is expressed with
  respect to its local origin O (struck out point).

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Mathematical Morphology Binary Images

• Dilation (dilatace)
• notation X ? B
• definition X ? B d ? E2 dxb for every x ?X
  and b ?B(alternative def. X ? B d ? E2
  dx-b for every x ?X and b ?B)
• meaning Dilation replaces zeros neighboring to
  ones by ones.
• exampleX (0,1),(1,1),(2,1),(2,2),(3,0),(4,0)
  B (0,0), (0,1)X ? B (0,1), (1,1), (2,1),
  (2,2), (3,0), (4,0), (0,2), (1,2), (2,2),
  (2,3), (3,1), (4,1)

source Sonka, Hlavac, Image Processing, Analysis
and Machine Vision
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Mathematical Morphology Binary Images

• Erosion (eroze)
• notation X ? B
• definition X ? B d ? E2 db ?X for every b
  ?B(alternative def. X ? B d ? E2 d-b ?X
  for every b ?B)
• meaning Erosion replaces ones neighboring to
  zeros by zeros.
• exampleX (0,1),(1,1),(2,1),(3,0),(3,1),(3,2),
  (3,3),(4,1)B (0,0), (0,1)X ? B
  (3,0),(3,1),(3,2)

source Sonka, Hlavac, Image Processing, Analysis
and Machine Vision
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Mathematical Morphology Binary Images

• Closing (uzavrení)
• Closing is dilation followed by erosion.
• Closing merges dense agglomerations of ones
  together, fills small holes and smoothes
  boundaries.
• Opening (otevrení)
• Opening is erosion followed by dilation.
• Opening removes single ones, thin lines and
  divides objects connected with a narrow path
  (neck).

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Mathematical Morphology Binary Images

• Original Erosion with a 3x3 mask
• Opening with a 3x3 mask Opening with a 5x5
  mask

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Mathematical Morphology Binary Images

• Original Dilation with a 3x3 mask
• Closing with a 3x3 mask Closing with a 5x5
  mask

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Mathematical Morphology Binary Images

• Opening Closing (3x3 mask) Opening
  Closing (5x5 mask)
• Closing Opening (3x3 mask) Closing
  Opening (5x5 mask)

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Mathematical Morphology Binary Images

• Shrinking (zcvrkávání)
• Shrinking is a modification of erosion object
  without holes erodes to a single pixel at or near
  its center of mass, object with holes erodes to
  a connected ring lying midway between each hole
  and its nearest outer boundary.
• Thinning (ztencování)
• Thinning is a modification of erosion object
  without holes erodes to a minimally connected
  stroke located equidistant from its nearest outer
  boundaries, object with holes erodes to
  a minimally connected ring lying midway between
  each hole and its nearest outer boundary.
• Skeletonizing (vytvárení skeletu neboli kostry)
• Skeletonizing is a modification of erosion
  object erodes to a set of points that are equally
  distant from two closest points of an object
  boundary (this set of points is also called
  medial axis skeleton). Skeleton uniquely
  describes the structure of the object.

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Mathematical Morphology Grey-scale Images

• Dilation (dilatace)
• Dilation replaces the central pixel with the
  maximum of its neighbors.
• Erosion (eroze)
• Erosion replaces the central pixel with the
  minimum of its neighbors.
• Closing (uzavrení)
• Closing is dilation followed by erosion.
• Closing merges dense agglomerations of local
  maxima together, fills small holes and smoothes
  boundaries.
• Opening (otevrení)
• Opening is erosion followed by dilation.
• Opening removes single local maxima, thin lines
  and divides objects connected with a narrow path
  (neck).

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Watershed Algorithm

• Developed
• 1978 by Ch. Lantuéjoul (CMM, Fontainebleau,
  France)
• 1979 by S. Beucher (CMM, Fontainebleau, France)
• Ch. Lantuéjoul, PhD thesis, CMM, 1978
• http//cmm.ensmp.fr/beucher/wtshed.html
• Idea
• Flood simulation by increasing the water level
  step by step.The grey-scale image is considered
  as a topographic surface.Creation of catchment
  basins and watershed lines.

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Watershed Algorithm

• Method
• Image smoothing (usually Gaussian filter),the
  kernel size depends on object size and also on
  noise.
• Determination of maximum intensity (MaxI)
  andminimum intensity (MinI) of the whole image.
• Determination of so-called markers, i.e. seeds of
  future objects. Can be performed e.g. using a
  minimum filter.
• Binary image (BinImg) empty image (zeros)
• For CurI MinI to MaxI doPerform region growing
  from markers (or already existing regions of
  BinImg) so that all pixels with intensities lower
  or equal to CurI are included. Do not let the
  regions merge!
• BinImg now defines object territories. The number
  of objects is equal to the number of markers.

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Watershed Algorithm

• Notes
• The algorithm can work without markers, in this
  case markers are all local minima in the image
  (in 3x3 neighborhood).
• Markers can be determined manually or
  automatically using better approaches than
  minimum filter, e.g. using distance transform
  the image is thresholded (bilevel thresholding),
  boundaries are determined and the shortest
  distance of each pixel to the closest boundary
  point is determined. Those pixels which have
  large distance to the boundary (larger than a
  pre-defined threshold) are taken as markers.
• The loop can be performed also from MaxI down to
  MinI for inverted images (bright objects and dark
  background). This approach is called
  anti-watershed.
• The algorithm is usually applied to gradient
  image (Sobel image) in order to detect object
  boundaries.

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Watershed Algorithm

• Example of the determination of markers and
  watershed
• Original grey-scale Thresholding (green)
  Watershed result
• Distance transform Segmentation
  (red)
• Markers (red) Markers (black)

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Largest Contour Segmentation (LCS)

• Developed
• 1996 by E.M.M.Manders (Delft, The Netherlands)
• Cytometry 23 15-21 (1996)
• Idea
• Iterative region growing around each local
  intensity maximum.Works on 2D as well as 3D
  images.

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Largest Contour Segmentation (LCS)

• The Method
• 1) Noise filtering (average or median),the
  kernel size depends on noise type and magnitude.
• 2) Determination of local maxima and minimal
  intensity in the image (GlobalMinI)
• 3) Calculation of centers of local maxima, i.e.
  reduction of local maxima to 1 pixel.
• 4) For each local maximum do
• 4.1) Current threshold (CurT) (Max intensity
  (LocalMaxI) GlobalMinI) / 2
• 4.2) Number of iterations (I) 0
• 4.3) Perform region growing from the center of
  the local maximum so that all neighboring pixels
  are included whose intensity values ? CurT.
• 4.4) N the number of centers inside this
  region.
• 4.5) I I 1
• 4.6) If I BitDepth, finish iteration for the
  given center and go to step 5.
• 4.4) If Ngt1, CurT (CurT previous higher
  threshold) / 2
• 4.5) If N1, CurT (CurT previous lower
  threshold) / 2
• 4.6) Go to step 4.3.
• 5) If Ngt1, Final threshold (FinT) CurT1If
  N1, Final threshold (FinT) CurT

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Largest Contour Segmentation (LCS)

• Example
• Original grey-scale Gaussian filter 7x7
  Centers of local maxima
• GlobalMinI27
• LocalMaxI207 (left top)
• LocalMaxI183 (bot. right)

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Largest Contour Segmentation (LCS)

• Example (continued for the bottom right cell)
• Input CurT105 CurT144 CurT163
  CurT153
• CurT158 CurT160 CurT161
  CurT162 FinT162