Jalal Amini 
  
eliminating irrelevancies. Serra(1982) introduced the basis and theory of mathematical morphology. Maralick, et al 
(1987) discussed the basic mathematical morphological operations and their relations in a N-dimensional 
properties of the basic binary and multi-level morphological operations with both 1D and 2D structuring elements. 
4.1 Binary Mathematical Operations 
4.1.1 Dilation, Erosion, Opening, and Closing. Consider a discrete binary image set X in an N-D distance 
gridz~. 
Let T € Z" denote a structuring element, 7 — [-u € 7i denote the symmetric set of T with respect to the 
origin, and £ denote the empty set. 
The translation of X by a point z e Z is denoted by x , and defined by X, 2 Ixrdxe X]. Then the four 
basic binary mathematical morphological operations of X by T' are defined as follows: 
V 
Dilation X ®T ={dT, nX #£}=X, 
teT 
Erosion XOT = lar. C X= NX. 2) 
teT 
Opening X oT -(XOT)OT 
Closing XeT-(X 9 T)OT 
Dilation is used to fill small holes and fill narrow gaps in objects or expand image objects, whereas erosion shrinks 
the image objects. If we want to find the contours of objects in an image very quickly this can, for instance, be 
achieved by the subtraction from the original picture of its eroded version. Opening is used to eliminate specific 
image details smaller than the structuring element while closing connects objects that are close to each other, fill 
up small holes, and smoothes the object outline by filling up narrow gaps. Unlike dilation and erosion, opening 
and closing are invariant to translation of the structuring element. It means, if an image is eroded and then dilated 
the original image is not obtainable. 
4.1.2 Hit/miss Transform. In addition to the four morphological operations, the hit/miss transformation is also 
an important morphological operation used to detect the occurrence of an exact pattern in the image. Let T be 
composed of two subsets T, and T, ; then the hit/miss transform of X by T can be defined as the set of all points 
where T, is included in X and T, is included in x « 
X®T={x:T, cX:T,, c X*} = (XOT,)n (X°OT,) © 
where X ^ is the complement of X and T, (i — 1,2) denotes the translation of T, by x . When T,is chosen as 
the window complement of T, equation (3) can be written as(Serra, 1983) 
X OT (XOT) A(X'OGQW oT?) (4) 
where Wis the window with finite support. 
4.1.3 Thinning. Equation (4) identifies the areas which can be removed from the structure while maintaining the 
connectivity. Therefore thinning of X by T is defined as: 
XOT = X/(X OT)= Xn(X OT)‘ (5) 
where “/” is the set difference. 
The operation X ® T in equation (5) locates all occurrences of the structuring element T in X and the operation 
/ removes from X those points which have been located. 
Thinning transformations are very often used sequentially. Let denote a sequence of composite 
Tov TU 
structuring elements Ta — (T, , T,,) - Sequential thinning can then be expressed as: 
  
38 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 
The 
a pe 
mat 
com 
elen 
T, 3 
mat 
Thir 
depe 
are i 
4.1.4 
the | 
isola 
Asst 
jagg 
thinr 
itera 
chan 
43 ! 
The | 
imag 
neigt 
the g 
Dila 
Eros 
The a 
1.). 2 
inten: 
rectar 
inten: 
erosic 
intens