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					Chapter 10, Part II
   An approach incorporating
    ◦ The previous three approaches:
      detection of discontinuities,
      thresholding, and region growing.
    ◦ knowledge-based constraints.
       Viewing images as 3-D topography
    ◦     Two spatial coordinates plus the gray-level.

       Three types of points are considered:
    1. regional minimum
    2. catchment basin or watershed.
    3. divide lines or watershed lines.

       The objective:
    ◦     According to the concepts and find the
          watershed lines.
   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.
Illustration of Morphological
Illustration of Morphological Watershed
       The watershed lines form a connected path,
        thus giving continuous boundaries between
       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.
   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 C[n-1].
   The two components merge at the flooding
    step n and let the connected component be
    denoted as q.
       Dilate each region by the structure element,
        consider the following two conditions:
    1) The dilation has to be constrained to q.
    2) 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.
       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.
Illustration of Dam Construction
   Definitions:
    ◦   Let M1, M2 ….MR be sets denoting the coordinates
        of the points in the regional minima of an image

    ◦   Let C(Mi) be a set denoting the coordinates of the
        points in the catchment basin associated with
        regional minimum Mi .
◦ Let T[n] represent the set of coordinates (s, t) for
  which g(s,t)<n. i.e., T[n]={(g,s)|g(s,t)<n}

◦ The topology will be flooded in integer flood
  increments from n=min+1 to n=max +1 where min
  and max are the minimum and maximum value of
◦   Let Cn(Mi) denote the set of coordinates of
    points in the catchment basin associated with Mi
    that are flooded at stage n.

◦   Cn(Mi) can be viewed as a binary image given by
    Cn(Mi) = C(Mi)ÇT[n]
       Cn(Mi)=1 at location (x,y) if (x,y)Î C(Mi) AND (x,y)Î
        T[n], otherwise Cn(Mi)=0

◦   Let C[n] denote the union of the flooded
    catchment basins portion at stage n as:
◦   Then C[max+1] is the union of all catchment
    basins as:

◦   C[n-1] is subset of C[n].
◦   C[n] is a subset of T[n].
◦   The algorithm for finding the watershed lines is
    initialized with C[min+1]=T[min+1].
◦   Let Q[n] denote the set of connected components
    in T[n].
   Obtain C[n] from C[n-1] as follows:
    ◦   For each q Î Q[n], consider p = qÇC[n-1]. If
        (a) p is empty : a new minimum is encountered.
        Add q into C[n-1] to form C[n].

        (b) p contains one connected component: q lies
        within the catchment basin of some regional
        minimum. Add q into C[n-1] to form C[n]

        (c) p contains more than one component:
        Flooding would cause water level in these
        catchment basin to merge; therefore, a dam
        must built within q by dilating of p.
               Example 10.18

The segmentation boundaries have the important property of
being connected paths.
   The problem of watershed segmentation:
    ◦   Due to noise and other irregularities of the
   Solutions:
    ◦   Pre-procesing.
    ◦   Various merging algorithms can be seen in
        other references.
    ◦   Markers: determined according to predefined
        criteria. For example,
              A region surrounded by points of higher “altitude”
              Points in the regions form a connected component.
              Points in the connected region with the same gray-
               level value.
Illustration of Oversegmentation
Watershed Algorithm Using The Markers
   Segmentation using internal and external
   Simplify the problem as partition each
    region into a single object and its
   Marker selection can be based on gray-
    level value and connectivity, and more
    complex description involving size, shape,
    location, texture content, and so on.

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