Gray Level Image Processing

Contours Detection

Method

Contour detection is identifying large variations in color between pixels close. The aim is to make appear the edges of objects present in an image. Indeed, the differences between an object and a background are their quantifications. Example, a black object on white background. To detect the contours of the object we observe the difference between neighboring pixels of the image. If the difference is large then we have detected an edge.
rotation d'une image
In the code when using the variable named 'distance' it means the removal of pixels included in the masks used.

Gradient

A simple technique is to look at the gradient that exists in the image in vertical and in horizontal.
$g r a d i e n t (i, j) = \sqrt{{(p (i, j) - p (i, j + 1))}^{2} + {(p (i, j) - p (i + 1, j))}^{2}}$

for i = 1:ligne-distance for j = 1:colonne-distance img2(i,j) = ((img(i,j)-img(i,j+distance))^2 + (img(i,j)-img(i+distance,j))^2)^0.5; end end

Gradient 2

Another possible gradient.

g r a d i e n t (i, j) = | p (i, j) - p (i, j + 1) | + | p (i, j) - p (i + 1, j) |

for i = 1:ligne-distance for j = 1:colonne-distance img2(i,j) = abs(img(i,j)-img(i,j+distance)) + abs(img(i,j)-img(i+distance,j)); end end

Roberts

Roberts filter is a gradient of size 2*2.

[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]

and

[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]

The gradients are made ??on the two possible diagonals of the mask. Edge detection will be the color variations on the diagonals. We have the following equation:

g r a d i e n t (i, j) = \frac{| p (i, j) - p (i + 1, j + 1) | + | p (i + 1, j) - p (i, j + 1) |}{2}

for i = 1:ligne-distance for j = 1:colonne-distance img2(i,j) = (abs((img(i,j)-img(i+distance,j+distance)) + (img(i+distance,j)-img(i,j+distance))))/2; end end

Prewitt

This operator do the convolution between txo mask 3*3.

[\begin{matrix} - 1 & 0 & 1 \\ - 1 & 0 & 1 \\ - 1 & 0 & 1 \end{matrix}]

and

[\begin{matrix} - 1 & - 1 & - 1 \\ 0 & 0 & 0 \\ 1 & 1 & 1 \end{matrix}]

This filter combines edge detection with a mean filter, which allow to be much less dependent to the noise than Roberts filter.

for i = 1:ligne-(distance+1) for j = 1:colonne-(distance+1) img2(i,j) = abs( (-img(i,j)-img(i+1,j)-img(i+2,j)+img(i,j+distance)+img(i+1,j+distance)+img(i+2,j+distance))+... (-img(i,j)-img(i,j+1)-img(i,j+2)+img(i+distance,j)+img(i+distance,j+1)+img(i+distance,j+2))); end end

Sobel

The Sobel filter is very similar to Prewitt, except: The filter applied is Gaussian, the noise is better filtered than Prewitt masks.

[\begin{matrix} - 1 & - 1 & 1 \\ - 2 & 0 & 2 \\ - 1 & 0 & 1 \end{matrix}]

and

[\begin{matrix} - 1 & - 2 & - 1 \\ 0 & 0 & 0 \\ 1 & 2 & 1 \end{matrix}]

for i = 1:ligne-(distance+1) for j = 1:colonne-(distance+1) img2(i,j) = abs( (-img(i,j)-2*img(i+1,j)-img(i+2,j)+img(i,j+distance)+2*img(i+1,j+distance)+img(i+2,j+distance))+... (-img(i,j)-img(i,j+1)-2*img(i,j+2)+img(i+distance,j)+2*img(i+distance,j+1)+img(i+distance,j+2))); end end

Laplacian

The estimate of the Laplacian of an image is made in the same way by convolution of the image with a mask. The Laplacian is approximated by finite differences: $▽^{2} = {1, - 2, 1}$

$[\begin{matrix} 0 & 0 & 0 \\ 1 & - 2 & 1 \\ 0 & 0 & 0 \end{matrix}] + [\begin{matrix} 0 & 1 & 0 \\ 0 & - 2 & 0 \\ 0 & 1 & 0 \end{matrix}] = [\begin{matrix} 0 & 1 & 0 \\ 1 & - 4 & 0 \\ 0 & 0 & 0 \end{matrix}]$

for i = 1:ligne-2 for j = 1:colonne-2 img2(i,j) = img(i+1,j) ... +img(i,j)-4*img(i+1,j+1)+img(i+2,j+2)+... +img(i+1,j+1) ; end end

Laplacien 2

Another conrour detection with diagonals.

$[\begin{matrix} 0 & 0 & 0 \\ 1 & - 2 & 1 \\ 0 & 0 & 0 \end{matrix}] + [\begin{matrix} 0 & 1 & 0 \\ 0 & - 2 & 0 \\ 0 & 1 & 0 \end{matrix}] + [\begin{matrix} 1 & 0 & 0 \\ 0 & - 2 & 0 \\ 0 & 0 & 1 \end{matrix}] + [\begin{matrix} 0 & 0 & 1 \\ 0 & - 2 & 0 \\ 1 & 0 & 0 \end{matrix}] = [\begin{matrix} 1 & 1 & 1 \\ 1 & - 8 & 1 \\ 1 & 1 & 1 \end{matrix}]$

for i = 1:ligne-2 for j = 1:colonne-2 img2(i,j) = img(i,j)+img(i+1,j)+img(i+2,j)+... +img(i,j)-8*img(i+1,j+1)+img(i+2,j+2)+... +img(i,j)+img(i+1,j+1)+img(i+2,j+2); end end

Freishen

It can also been used with diagonals.

$[\begin{matrix} - 1 & 0 & 1 \\ \sqrt{2} & 0 & \sqrt{2} \\ - 1 & 0 & 1 \end{matrix}] e t [\begin{matrix} - 1 & \sqrt{2} & - 1 \\ 0 & 0 & 0 \\ 1 & \sqrt{2} & 1 \end{matrix}]$

Canny

r=[cos(theta_v) -sin(theta_v); sin(theta_v) cos(theta_v)]; for i = 1 : n for j = 1 : n u = r * [j-(n+1)/2 i-(n+1)/2]'; h(i,j) = (exp(-u(1)^2/(2*s^2)) / (s*sqrt(2*pi)))*(-u(2) * ((exp(-u(2)^2/(2*s^2)) / (s*sqrt(2*pi))) / s^2)); end end frontvert= conv2(img,h,'same'); r=[cos(theta_h) -sin(theta_h); sin(theta_h) cos(theta_h)]; for i = 1 : n for j = 1 : n u = r * [j-(n+1)/2 i-(n+1)/2]'; h(i,j) = (exp(-u(1)^2/(2*s^2)) / (s*sqrt(2*pi)))*(-u(2) * ((exp(-u(2)^2/(2*s^2)) / (s*sqrt(2*pi))) / s^2)); end end fronthor= conv2(img,h,'same'); % normalisation contour=sqrt(frontvert.*frontvert+fronthor.*fronthor); % Thresholding alfa=0.1; contour_max=max(max(contour)); contour_min=min(min(contour)); seuil=alfa*(contour_max-contour_min)+contour_min; seuillage=contour; seuillage(seuillage<seuil) = seuil; %Ibw=max(NVI,level.*ones(size(NVI))); [n,m]=size(seuillage); X=[-1,0,+1;-1,0,+1;-1,0,+1]; Y=[-1,-1,-1;0,0,0;+1,+1,+1]; for i=1:n-2, for j=1:m-2, if seuillage(i+1,j+1) > seuil, Z=seuillage(i:i+2,j:j+2); XI=[frontvert(i+1,j+1)/contour(i+1,j+1), -frontvert(i+1,j+1)/contour(i+1,j+1)]; YI=[fronthor(i+1,j+1)/contour(i+1,j+1), -fronthor(i+1,j+1)/contour(i+1,j+1)]; ZI=interp2(X,Y,Z,XI,YI); if seuillage(i+1,j+1) >= ZI(1) && seuillage(i+1,j+1) >= ZI(2) contour_final(i,j)=contour_max; else contour_final(i,j)=contour_min; end else contour_final(i,j)=contour_min; end end end img2=contour_final;

Program

function guicontour clear all; close all; figure( 'name','détection de contour',... 'NumberTitle','off',... 'MenuBar','none',... 'color',[0.3137 0.3137 0.5098]); a(1)=axes('units','normalized',... 'position',[0.05 0.4 0.4 0.4]); a(2)=axes('units','normalized',... 'position',[0.5 0.4 0.4 0.4]); uicontrol( 'style','pushbutton',... 'string','load',... 'Position', [10 10 50 20],... 'callback',@loadimage); ctr(1)=uicontrol( 'style','popup',... 'string','gradian|gradian2|roberts|prewitt|sobel|laplacien|laplacien2|freishen|canny',... 'Min',0,'Max',8,... 'Position', [90 10 100 20],... 'callback',{@choixtxt,'typecontour'}); text(1)=uicontrol( 'style','text',... 'string','Type de contour ',... 'Position', [90 30 100 20]); ctr(2)=uicontrol( 'style','popup',... 'string','1|2|3|4|5',... 'Min',0,'Max',4,... 'Position', [200 10 50 20],... 'callback',{@choix,'distance'}); text(2)=uicontrol( 'style','text',... 'string','distance ',... 'Position', [200 30 50 20]); ctr(3)=uicontrol( 'style','popup',... 'string','non|oui',... 'Min',0,'Max',4,... 'Position', [260 10 50 20],... 'callback',{@choixseuil,'seuillage'}); text(3)=uicontrol( 'style','text',... 'string','seuillage?',... 'Position', [260 30 50 20]); ctr(4)=uicontrol( 'style','slider',... 'string','seuil',... 'Min' ,0,'Max',255, ... 'Position', [320 10 100 20],... 'Value', 0,... 'SliderStep',[1/255 10/255], ... 'callback',{@choix,'seuil'}); text(4)=uicontrol( 'style','text',... 'string','seuil',... 'Position', [320 30 100 20]); uicontrol( 'style','pushbutton',... 'string','appliquer',... 'Position', [500 10 50 20],... 'callback',@appliquer); ctr(5)=uicontrol( 'style','edit',... 'string','10',... 'Position', [200 10 50 20],... 'callback',{@choixnb,'n'}); text(5)=uicontrol( 'style','text',... 'string','distance',... 'Position', [200 30 50 20]); ctr(6)=uicontrol( 'style','edit',... 'string','1',... 'Position', [260 10 50 20],... 'callback',{@choixnb,'s'}); text(6)=uicontrol( 'style','text',... 'string','sigma',... 'Position', [320 30 50 20]); ctr(7)=uicontrol( 'style','edit',... 'string','1.5708',... 'Position', [320 10 50 20],... 'callback',{@choixnb,'theta_v'}); text(7)=uicontrol( 'style','text',... 'string','theta_v',... 'Position', [380 30 50 20]); ctr(8)=uicontrol( 'style','edit',... 'string','0',... 'Position', [380 10 50 20],... 'callback',{@choixnb,'theta_h'}); text(8)=uicontrol( 'style','text',... 'string','theta_h',... 'Position', [260 30 50 20]); %parametre initial setappdata(gcf,'distance',1); setappdata(gcf,'seuillage','non'); setappdata(gcf,'seuil',0); setappdata(gcf,'x',1); setappdata(gcf,'typecontour','gradian'); setappdata(gcf,'n',10); setappdata(gcf,'s',1); setappdata(gcf,'theta_v',1.5708); setappdata(gcf,'theta_h',0); set(ctr(5),'Enable','off'); set(ctr(5),'Visible','off'); set(ctr(6),'Enable','off'); set(ctr(6),'Visible','off'); set(ctr(7),'Enable','off'); set(ctr(7),'Visible','off'); set(ctr(8),'Enable','off'); set(ctr(8),'Visible','off'); set(text(5),'Enable','off'); set(text(5),'Visible','off'); set(text(6),'Enable','off'); set(text(6),'Visible','off'); set(text(7),'Enable','off'); set(text(7),'Visible','off'); set(text(8),'Enable','off'); set(text(8),'Visible','off'); function loadimage(~,~) % appeler quand appui check box [filename, pathname] = uigetfile({'*.jpg;*.tif;*.PNG;*.gif;*.bmp','All Image Files';... '*.*','All Files' },'mytitle',... 'C:\Work\myfile.jpg') x = imread(filename); if (length(size(x))>2) x=x(:,:,1);% on prend une seul plan image noir et blanc chaque plan sont egaux end k = whos('x'); if k.class == 'uint8' k=8; end setappdata(1,'k',k); setappdata(1,'x',x); axes(a(1)) imshow(x) axes(a(2)) imshow(x) end function choixtxt(hObj,~,name) % Called when user activates popup menu val = get(hObj,'Value'); %0|1=ligne, 2=colonne, 3=croix, 4=carre set(ctr(2),'Enable','on'); set(ctr(2),'Visible','on'); set(ctr(3),'Enable','on'); set(ctr(3),'Visible','on'); set(ctr(4),'Enable','on'); set(ctr(4),'Visible','on'); set(text(2),'Enable','on'); set(text(2),'Visible','on'); set(text(3),'Enable','on'); set(text(3),'Visible','on'); set(text(4),'Enable','on'); set(text(4),'Visible','on'); set(ctr(5),'Enable','on'); set(ctr(5),'Visible','off'); set(ctr(6),'Enable','off'); set(ctr(6),'Visible','off'); set(ctr(7),'Enable','off'); set(ctr(7),'Visible','off'); set(ctr(8),'Enable','off'); set(ctr(8),'Visible','off'); set(text(5),'Enable','off'); set(text(5),'Visible','off'); set(text(6),'Enable','off'); set(text(6),'Visible','off'); set(text(7),'Enable','off'); set(text(7),'Visible','off'); set(text(8),'Enable','off'); set(text(8),'Visible','off'); if ( val == 0 || val == 1 ) setappdata(1,name,'gradian'); elseif ( val == 2 ) setappdata(1,name,'gradian2'); elseif ( val == 3 ) setappdata(1,name,'roberts'); elseif ( val == 4 ) setappdata(1,name,'prewitt'); elseif ( val == 5 ) setappdata(1,name,'sobel'); elseif ( val == 6 ) setappdata(1,name,'laplacien'); set(ctr(2),'Visible','off'); set(text(2),'Visible','off'); elseif ( val == 7 ) setappdata(1,name,'laplacien2'); set(ctr(2),'Visible','off'); set(text(2),'Visible','off'); elseif ( val == 8 ) setappdata(1,name,'freishen'); elseif ( val == 9 ) setappdata(1,name,'canny'); set(ctr(5),'Enable','on'); set(ctr(5),'Visible','on'); set(ctr(6),'Enable','on'); set(ctr(6),'Visible','on'); set(ctr(7),'Enable','on'); set(ctr(7),'Visible','on'); set(ctr(8),'Enable','on'); set(ctr(8),'Visible','on'); set(text(5),'Enable','on'); set(text(5),'Visible','on'); set(text(6),'Enable','on'); set(text(6),'Visible','on'); set(text(7),'Enable','on'); set(text(7),'Visible','on'); set(text(8),'Enable','on'); set(text(8),'Visible','on'); set(ctr(2),'Enable','off'); set(ctr(2),'Visible','off'); set(ctr(3),'Enable','off'); set(ctr(3),'Visible','off'); set(ctr(4),'Enable','off'); set(ctr(4),'Visible','off'); set(text(2),'Enable','off'); set(text(2),'Visible','off'); set(text(3),'Enable','off'); set(text(3),'Visible','off'); set(text(4),'Enable','off'); set(text(4),'Visible','off'); end end function choixseuil(hObj,~,name) % Called when user activates popup menu val = get(hObj,'Value'); %0|1=ligne, 2=colonne, 3=croix, 4=carre if ( val == 0 || val == 1 ) setappdata(1,name,'non'); elseif ( val == 2 ) setappdata(1,name,'oui'); end end function choix(hObj,~,Name) % Called when user activates popup menu val = get(hObj,'Value'); %0|1=ligne, 2=colonne, 3=croix, 4=carre setappdata(1,Name,val); if(strcmp(Name,'seuil')) val=round(val); setappdata(1,Name,val); set( text(4), 'string' , strcat('seuil = ',num2str(val))); end end function choixnb(hObj,~,Name) % Called when user activates popup menu val = get(hObj,'string'); setappdata(1,Name,str2double(val)); end % function variable(hObj,~,name) % val = round(get(hObj,'Value')); % setappdata(1,name,val); % end function appliquer(hObj,~,name) typecontour = getappdata(1,'typecontour'); distance = getappdata(1,'distance'); img = double(getappdata(1,'x')); [ligne colonne]=size(img); img2=zeros(ligne,colonne); if (strcmp(typecontour,'gradian')) for i = 1:ligne-distance for j = 1:colonne-distance img2(i,j) = ((img(i,j)-img(i,j+distance))^2 + (img(i,j)-img(i+distance,j))^2)^0.5; end end elseif (strcmp(typecontour,'gradian2')) for i = 1:ligne-distance for j = 1:colonne-distance img2(i,j) = abs(img(i,j)-img(i,j+distance)) + abs(img(i,j)-img(i+distance,j)); end end elseif (strcmp(typecontour,'roberts')) for i = 1:ligne-distance for j = 1:colonne-distance img2(i,j) = (abs((img(i,j)-img(i+distance,j+distance)) + (img(i+distance,j)-img(i,j+distance))))/2; end end elseif (strcmp(typecontour,'prewitt')) for i = 1:ligne-(distance+1) for j = 1:colonne-(distance+1) img2(i,j) = abs( (-img(i,j)-img(i+1,j)-img(i+2,j)+img(i,j+distance)+img(i+1,j+distance)+img(i+2,j+distance))+... (-img(i,j)-img(i,j+1)-img(i,j+2)+img(i+distance,j)+img(i+distance,j+1)+img(i+distance,j+2))); end end elseif (strcmp(typecontour,'sobel')) for i = 1:ligne-(distance+1) for j = 1:colonne-(distance+1) img2(i,j) = abs( (-img(i,j)-2*img(i+1,j)-img(i+2,j)+img(i,j+distance)+2*img(i+1,j+distance)+img(i+2,j+distance))+... (-img(i,j)-img(i,j+1)-2*img(i,j+2)+img(i+distance,j)+2*img(i+distance,j+1)+img(i+distance,j+2))); end end elseif (strcmp(typecontour,'laplacien')) for i = 1:ligne-2 for j = 1:colonne-2 img2(i,j) = img(i+1,j) ... +img(i,j)-4*img(i+1,j+1)+img(i+2,j+2)+... +img(i+1,j+1) ; end end elseif (strcmp(typecontour,'laplacien2')) for i = 1:ligne-2 for j = 1:colonne-2 img2(i,j) = img(i,j)+img(i+1,j)+img(i+2,j)+... +img(i,j)-8*img(i+1,j+1)+img(i+2,j+2)+... +img(i,j)+img(i+1,j+1)+img(i+2,j+2); end end f=img2; for i = 1:ligne-2 for j = 1:colonne-2 img2(i,j) = f(i,j)+f(i+1,j)+f(i+2,j)+... +f(i,j)-8*f(i+1,j+1)+f(i+2,j+2)+... +f(i,j)+f(i+1,j+1)+f(i+2,j+2); end end elseif (strcmp(typecontour,'freishen')) for i = 1:ligne-distance-1 for j = 1:colonne-distance-1 img2(i,j) = abs( (1/(2+sqrt(2)))*(-img(i,j)-sqrt(2)*img(i+1,j)-img(i+2,j)+img(i,j+distance)+sqrt(2)*img(i+1,j+distance)+img(i+2,j+distance))+... (1/(2+sqrt(2)))*(-img(i,j)-img(i,j+1)-sqrt(2)*img(i,j+2)+img(i+distance,j)+sqrt(2)*img(i+distance,j+1)+img(i+distance,j+2))); end end elseif (strcmp(typecontour,'canny')) n=getappdata(1,'n') s=getappdata(1,'s') theta_v=getappdata(1,'theta_v') theta_h=getappdata(1,'theta_h') r=[cos(theta_v) -sin(theta_v); sin(theta_v) cos(theta_v)]; for i = 1 : n for j = 1 : n u = r * [j-(n+1)/2 i-(n+1)/2]'; h(i,j) = (exp(-u(1)^2/(2*s^2)) / (s*sqrt(2*pi)))*(-u(2) * ((exp(-u(2)^2/(2*s^2)) / (s*sqrt(2*pi))) / s^2)); end end frontvert= conv2(img,h,'same'); r=[cos(theta_h) -sin(theta_h); sin(theta_h) cos(theta_h)]; for i = 1 : n for j = 1 : n u = r * [j-(n+1)/2 i-(n+1)/2]'; h(i,j) = (exp(-u(1)^2/(2*s^2)) / (s*sqrt(2*pi)))*(-u(2) * ((exp(-u(2)^2/(2*s^2)) / (s*sqrt(2*pi))) / s^2)); end end fronthor= conv2(img,h,'same'); % normalisation contour=sqrt(frontvert.*frontvert+fronthor.*fronthor); % Thresholding alfa=0.1; contour_max=max(max(contour)); contour_min=min(min(contour)); seuil=alfa*(contour_max-contour_min)+contour_min; seuillage=contour; seuillage(seuillage<seuil) = seuil; %Ibw=max(NVI,level.*ones(size(NVI))); [n,m]=size(seuillage); X=[-1,0,+1;-1,0,+1;-1,0,+1]; Y=[-1,-1,-1;0,0,0;+1,+1,+1]; for i=1:n-2, for j=1:m-2, if seuillage(i+1,j+1) > seuil, Z=seuillage(i:i+2,j:j+2); XI=[frontvert(i+1,j+1)/contour(i+1,j+1), -frontvert(i+1,j+1)/contour(i+1,j+1)]; YI=[fronthor(i+1,j+1)/contour(i+1,j+1), -fronthor(i+1,j+1)/contour(i+1,j+1)]; ZI=interp2(X,Y,Z,XI,YI); if seuillage(i+1,j+1) >= ZI(1) && seuillage(i+1,j+1) >= ZI(2) contour_final(i,j)=contour_max; else contour_final(i,j)=contour_min; end else contour_final(i,j)=contour_min; end end end img2=contour_final; end seuillage = getappdata(1,'seuillage'); if (strcmp(typecontour,'canny') == 0) if strcmp(seuillage,'oui') img2 = uint8(img2); seuil = getappdata(1,'seuil'); img2(img2<seuil) = 0; img2(img2>=seuil) = 255; end end axes(a(2)) imshow(img2,[min(min(img2)) max(max(img2))]) end end