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