Chapter 10, Part II

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Digital Image Processing

• Chapter 10, Part II

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10.5 Segmentation by Morphological Watersheds

• An approach incorporating
• The previous three approaches detection of
  discontinuities, thresholding, and region
  growing.
• knowledge-based constraints.

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10.5.1 Basic Concept

• Viewing images as 3-D topography
• Two spatial coordinates plus the gray-level.
• Three types of points are considered
• regional minimum
• catchment basin or watershed.
• divide lines or watershed lines.
• The objective
• According to the concepts and find the watershed
  lines.

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Basic Idea of the Algorithm

• Find the regional minima in the image.
• Flood the entire topography from the regional
  minima at an uniform rate.
• When the rising water in distinct watershed is
  close to merge, a dam is built to prevent the
  merging.
• The flooding eventually stops when only the tops
  of the dams are visible above the water line.
• These dam boundaries correspond to the divide
  lines of the watersheds.

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Illustration of Morphological Watershed
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Illustration of Morphological Watershed
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• The watershed lines form a connected path, thus
  giving continuous boundaries between regions.
• Regions characterized by small variations in gray
  levels have small gradient values.
• In practice, we often see watershed segmentation
  applied to the gradient of an image.

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10.5.2 Dam Construction

• Let M1 and M2 denote the set of coordinates of
  points in two regional minima.
• Cn-1(M1) and Cn-1(M2) the set of coordinates of
  points in the catchment basin associated with M1
  and M2 at stage n-1.
• Let the union of the two sets be Cn-1.
• The two components merge at the flooding step n
  and let the connected component be denoted as q.

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Procedures of Dam Construction

• Dilate each region by the structure element,
  consider the following two conditions
• The dilation has to be constrained to q.
• The dilation cannot be performed on points that
  would cause sets being dilated to merge.
• In Fig. 10.45, during the second dilation, some
  points satisfy Condition (2) but do not satisfy
  (1).
• resulting in broken perimeter.

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Procedures of Dam Construction

• The only points in q that satisfy the two
  conditions under consideration describe the
  one-pixel-thick connected path shown by
  cross-hatched points.
• The path constitutes the desired separation dam
  at stage n of flooding.

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Illustration of Dam Construction
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10.5.3 Watershed Segmentation Algorithm

• Definitions
• Let M1, M2