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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 Watershed Illustration of Morphological Watershed 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. 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 g(x,y). ◦ 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 g(x,y). ◦ 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: Oversegmentation ◦ Due to noise and other irregularities of the gradient. 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 markers. Simplify the problem as partition each region into a single object and its background. 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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posted: | 7/28/2013 |

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