Table 1.

The BAE algorithm

Process	Equation
(a)Evaluate each pixel in n-label for 0 degree	$f\left(x,y\right)=\left\{\begin{array}{c}\hfill 0,I_{x+k,y}^b\mathsf{\text{if}}\mathsf{\text{b = n,}}\hfill \\ \hfill 1,I_{x+k,y}^b\mathsf{\text{if}}\mathsf{\text{b}}\ne \mathsf{\text{n,}}\hfill \\ \hfill \hfill \end{array}\right)$ Where k = 1→L L = maximum number of column -x
Evaluate each pixel for 45 degree	$f\left(x,y\right)=\left\{\begin{array}{c}\hfill 0,I_{x+m,y-k}^b\mathsf{\text{if}}\mathsf{\text{b = n,}}\hfill \\ \hfill 1,I_{x+m,y-k}^b\mathsf{\text{if}}\mathsf{\text{b}}\ne \mathsf{\text{n,}}\mathsf{\text{.}}\hfill \\ \hfill \hfill \end{array}\right)$ Where k = 1→y - 1, m = 1→L L = maximum number of column -x
Evaluate each pixel in n-label for 90 degree	$f\left(x,y\right)=\left\{\begin{array}{c}\hfill 0,I_{x,y-k}^b\mathsf{\text{if}}\mathsf{\text{b = n,}}\hfill \\ \hfill 1,I_{x,y-k}^b\mathsf{\text{if}}\mathsf{\text{b}}\ne \mathsf{\text{n,}}\hfill \\ \hfill \hfill \end{array}\right)$ Where k = 1→y - 1
Evaluate each pixel in n-label for 135 degree	$f\left(x,y\right)=\left\{\begin{array}{c}\hfill 0,I_{x-m,y-k}^b\mathsf{\text{if}}\mathsf{\text{b = n,}}\hfill \\ \hfill 1,I_{x-m,y-k}^b\mathsf{\text{if}}\mathsf{\text{b}}\ne \mathsf{\text{n,}}\hfill \\ \hfill \hfill \end{array}\right)$ Where k = 1→y - 1, m = 1→x - 1
Evaluate each pixel in n-label for 180 degree	$f\left(x,y\right)=\left\{\begin{array}{c}\hfill 0,I_{x-m,y}^b\mathsf{\text{if}}\mathsf{\text{b = n,}}\hfill \\ \hfill 1,I_{x-m,v}^b\mathsf{\text{if}}\mathsf{\text{b}}\ne \mathsf{\text{n,}}\hfill \\ \hfill \hfill \end{array}\right)$ Where k = 1→y - 1, m = 1→x - 1
Evaluate each pixel in n-label for 225 degree	$f\left(x,y\right)=\left\{\begin{array}{c}\hfill 0,I_{x-m,y+k}^b\mathsf{\text{if}}\mathsf{\text{b = n,}}\hfill \\ \hfill 1,I_{x-m,y+k}^b\mathsf{\text{if}}\mathsf{\text{b}}\ne \mathsf{\text{n,}}\hfill \\ \hfill \hfill \end{array}\right)$ Where k = 1→L, m = 1→x - 1 L = maximum number of row -x
Evaluate each pixel in n-label for 270 degree	$f\left(x,y\right)=\left\{\begin{array}{c}\hfill 0,I_{x,y+k}^b\mathsf{\text{if}}\mathsf{\text{b = n,}}\hfill \\ \hfill 1,I_{x,y+k}^b\mathsf{\text{if}}\mathsf{\text{b}}\ne \mathsf{\text{n,}}\hfill \\ \hfill \hfill \end{array}\right)$ Where k = 1→L L = maximum number of row -x
Evaluate each pixel in n-label for 315 degree	$f\left(x,y\right)=\left\{\begin{array}{c}\hfill 0,I_{x+m,y+k}^b\mathsf{\text{if}}\mathsf{\text{b = n,}}\hfill \\ \hfill 1,I_{x+m,y+k}^b\mathsf{\text{if}}\mathsf{\text{b}}\ne \mathsf{\text{n,}}\hfill \\ \hfill \hfill \end{array}\right)$ Where k = 1→L1, m = 1→L2 L1 = maximum number of row-y L2 = maximum number of column -x
(b)Stopping criteria:	[f(x, y) = 1] ∪ n = N N = maximum label
(c)Verification of bounded area for n-label	$boundedarea,if\frac{{\sum }_x^{maxx}{\sum }_y^{maxy}f\left(x,y\right)I_{x,y}^n}{\begin{array}{c}totalnumberofpixelsinn-label\\ non-boundedarea,otherwise\\ \end{array}}=1$
(d)Repeat the process	n = n+1, where n denotes the label number of pixels in image.
(e)Fill the pixels belong to bounded area with original value/background value	$I_{x,y}=I_{x,y}^n$

This table illustrates the steps in the BAE process. Step in part (a) demonstrates the labelling process in each direction. Step in part (b) explains the stopping criteria. Step in part (c) defines the recognition of bounded area, for it a noise or lost data. The entire process mentioned above is repeated in step in part (d). Last step involves the filling in the lost data or elimination of noise.