i. Opening and closing are two important morphological operations.

ii. Opening generally smoothest the contour of an object, breaks narrow isthmuses, and eliminates thin protrusions.

iii. Closing also tends to smooth sections of contours but, as opposed to opening, it generally fuses narrow breaks and long thin gulfs, eliminates small holes, and fills gaps in the contour.

$A\circ B=(A\Theta B)\oplus B$

iv. The opening of set A by structuring element B, denoted A o B, is defined as

$A ¥ B$

Thus, the opening A by B is the erosion of A by B, followed by a dilation ofthe result by B.

v. Similarly, the closing of set A by structuring element B, denoted by, isdefined as

$A ¥ B =(A\oplus B) \Theta B$

which says that the closing of A by B is simply the dilation of A by B, followed by the erosion of the result by B.

vi. The opening operation has a simple geometric interpretation (Fig.1). Suppose that we view the structuring element B as a (flat) “rolling ball.” The boundary of A • B is then established by the points in B that reach the farthest into the boundary of A as B is rolled around the inside of this boundary. This geometric fitting property of the opening operation leads to a settheoretic formulation, which states that the opening of A by B is obtained by taking the union of all translates of B that fit into A.

Figure 1 (a) Structuring element B “rolling” along the inner boundary of A (the dot indicates the origin of B), (b) Structuring element, the heavy line is the outer boundary of the opening. (c) Complete opening (shaded). We did not shade A in (a) for clarity.

vii. Closing has a similar geometric interpretation, except that now we roll B on the outside of the boundary (Fig. 2).

Figure 2 (a) Structuring element B “rolling” on the outer boundary of A (b) The heavy line is the outer boundary of the closing. (c) Complete closing(shaded). We did not shade A in (a) for clarity.

viii. This is all about the opening and closing of a digital image, we can say that opening and closing operations are duals of each other.