Algorithm 2 Image Edge Removal Algorithm
Part I	1:  Given a grayscale image $H\in R^{M\times N}$.
	2:  Process and get gray binary image $G_d$.
	3:  for I = 0, …, M do
	4:  Set the array of IEC to $Left[i]$, set the array of OEC to $Right[i]$.
	5:  for j = 0, …, N − 2 do
	6:   if $G_{\mathrm{d}}[i,j+k]==0(k=0,1,2)$
	7:       Get $Left[i]=j$.
	8:   end
	9:   end
	10:     for j = $Left[i]$, …, N − 4 do
	11:   if $G_{\mathrm{d}}[i,j+k]==255(k=0,\dots ,4)$
	12:       Get $Right[i]=j$.
	13:   end
	14:  end
	15:  for j = 0, …, N do
	16:     if $Left[i]\le j\le Right[i]$
	17:       Set $G_d[i,j]=255$
	18:     end
	19:   end
	20: end
	21: Output the edge-removed gray binary image $G_d$.
Part II	1:	Given a grayscale image $H\in R^{M\times N}$.
	2:	Obtain a binary image $G_{d1}$ and an image $G_{\mathrm{h}}$ with high threshold binarization. Use Sobel operator to obtain gradient binary image $G_w$.
	3:	for i = 0, …, M do
	4:	Set the array of IEC to $TopL[i]$, set the array of LEC $B\mathrm{ot}[i]$ and set the array of OEC to $Top[i]$.
	5:	for j = 0, …, N − 4 do
	7:	if $G_{\mathrm{h}}[i,j+k]==0(k=0,\cdots ,4)$
	8:	Get $TopL[i]=j$.
	9:	end
	10:	end
	11:	for j = $TopL[i]$, …, N − 4 do
	13:	if $G_{\mathrm{h}}[i,j+k]==255(k=0,\cdots ,4)$
	14:	Get $B\mathrm{ot}[i]=j$.
	15:	if $G_{d1}[i,j+k]==255(k=0,\cdots ,4)$
	16:	Get $T\mathrm{op}[i]=j$.
	17:	end
	18:	end
	19:	for j = 0, …, N do
	20:	if $0\le j\le Top[\mathrm{i}]+15$ or $\left|j-Bot[\mathrm{i}]\right|\le 20$
	21:	Set $G_w[i,j]=0$.
	22:	end
	23:	end
	24:	end
	25:	Output the edge-removed torus gradient image $G_w$.
Part III	1:	Given a grayscale end face image $H\in R^{M\times N}$.
	2:	Process and get binary image $G_{\mathrm{h}}$.
	3:	Create a white circle $C_1$ with $\left(Center\left(x,y\right),\left(Top+Bot\right),w_1\right)$ as its center, radius and line width, respectively.
	4:	forj = y, y − 1, …,4 do
	5:	Scan upward from $Center\left(x,y\right)$.
	6:	if $G_{\mathrm{h}}[x,j-k]==0(k=0,\cdots ,4)$
	7:	Get $R_2=y-j$.
	8:	end
	9:	end
	10:	Create a white circle $C_2$ with $(Center\left(x,y\right),R_2,w_1)$ as its center, radius and line width, respectively.
	11:	Use $C_1$ and $C_2$ to remove edges.
	12:	Output the edge-removed grayscale end face image.