Boundary Extraction:

If A is an image and structuring element is B then Boundary Extraction can be given as,

Boundary (A) = A - (A Ɵ B)

It means subtracting the erode image of A from the original Image. Let A =

If B=

Then AƟB would be same as A except one pixel, A Ɵ B =

Now Boundary (A) =

Hence, it would give a pixel difference.

Region Filling:

It fills the hollow portion of the image.

$X_k = (X_{k-1} \otimes B) ∩ A^C k = 1,2,3,….$

To start the procedure of region filling,we start with a pixel p and assign a value 1 to it.

Then apply the above formula to it,$X_0 = p$ and B is the structuring element.

Consider the example A=

We start with pixel p inside the boundary.Let p=1

Therefore $X_0 = p$

Now according to formula, $Xk=(X_k-1 \otimes B)∩ A^C k = 1,2,3,….$ Therefore, X_1=(X_0 \otimes B)∩A^C i.e. $X_1$ is $X_0$ dilated by B which is then intersected by the complement of A Similarly,

$X_2 = (X_1 \otimes B) ∩ A^C$

We continue to use the formula till $X_k = X_k-1.$

$X_1 = (X_0 \otimes B) ∩ A^C$

Similarly,k=1,2,3…..,we get the final output

$X_5 = (X_4 \otimes B)∩ A^C$

Since,$X_5 = X_4 (X_k = X_{k-1}.)$we stop the algorithm here.

Therefore,$X_k = X_4$

The final step is finding the union of image A and $X_4$

$A U X_k = A ∩ X_4$