lick, et al 
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Jalal Amini 
  
YT, t= (XBT, BT, NBT ©) 
There are several sequence of structuring elements ir! that are useful in practice. Many of them are given by 
a permissible rotation of a structuring element in the appropriate digital raster (e.g. hexagonal, square). The 3*3 
matrices will be shown in the first two rotations, from which the other rotations can easily be derived. The 
composite structuring element will be expressed by one matrix only. A value of one in the matrix means that this 
element belongs to T, (it is a subset of objects in the hit or miss transformations), and a value zero belongs to 
T, and is a subset of the background. An asterisk * * ' in the matrix denotes an element that is not used in the 
matching process, i.e. its value is not significant. 
Thinning and thickening sequential transformations converge at one point. The number of iterations needed 
depends on the objects in the image and the structuring element used. If two successive images in the sequence 
are identical the thinning (or thickening) can be stopped. 
4.1.4 Sequential Thinning By Structuring Element L.This sequential thinning is quite important as it serves as 
the homotopic substitute of the skeleton. The final thinned image consists only lines of one pixel width and 
isolated points. The structuring element L is given by 
000 *0* 000 "00 
4 
Li=l*1" 2-110)... p=[*1*] 7 =l1 10]. 
111 11* r11 *1* 
Assume that the homotopic substitute of the skeleton by element L^ has been found. The skeleton is usefully 
jagged due to sharp points on the outline of the object. It is possible to smooth the skelton by sequential 
thinning by structuring element E. Using n iterations , points are removed gradually depending on the number of 
iterations from free-end lines and isolated points. If thinning by element E is performed until the image does not 
change, then only closed contours remain. The structuring element E is given by 
* k 3k *(Q* * ] * 0 * * 
E.—|010hZ;=|0 1+;- Z=|o10|,*-|o10|.. 
*0* *0* 000 000 
4.2 Gray Scale Morphological Operations 
The binary morphological operations of dilation, erosion, opening and closing can be extended to gray scale 
imagery (Sternberg, 1982; Haralick, etal,1987). For such images, the minimum and maximu m values are found within 
neighborhoods represented by the structuring element. Let F and T be the domain of the gray scale image f and 
the gray scale structuring element f, respectively. The gray scale dilation and erosion can be computed by 
Dilation : (f @t)(x,y)= MAX ,_., mer imme f(x - m y — n) + t(m,n)} a 
Erosion : CfOn(x, y) = min, vemerunnielid (x +m, y 5 n) = tn, n)} 
The adaptations can be better understood by considering dilation and erosion of an image intensity profile (figure 
1.). A three pixel wide and four intensity unit high rectangle slides along the baseline of an image profile. The new 
intensity value of the center pixel is determined according to the following rules. 1) For dilation, if any pixel of the 
rectangle fits at or under the image intensity profile, the center pixel of the rectangle is given the maximum 
intensity of the pixel and its two neighbors in the original image; otherwise the pixel is set to zero intensity. 2) For 
erosion, if the whole rectangle fits at or under the image intensity profile, the center pixel is given the minimum 
intensity of the pixel and its two neighbors in the original image; otherwise the pixel is set to zero intensity. 
Q t RO 
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International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 39