Image Sharpening using second order derivative –(Laplacian)

Prerequisite: Read EdgeDetection- fundamentals

The derivative operator Laplacian for an Image is defined as

For X-direction,

For Y-direction,

By substituting, Equations in Fig.B and Fig.C in Fig.A, we obtain the following equation

The equation  represented in terms of Mask:

0	1	0
1	-4	1
0	1	0

When the diagonals also considered then the equation becomes,

The Mask representation of the above equation,

1	1	1
1	-8	1
1	1	1

Now let’s discuss further how image sharpening is done using Laplacian.

Equation:

Where f(x,y)  is the input image

              g(x,y) is the sharpened image and

               c= -1 for the above mentioned filter masks.(fig.D and fig.E)

MATLAB CODE:

%Input Image

A=imread('coins.png');

figure,imshow(A);

%Preallocate the matrices with zeros

I1=A;

I=zeros(size(A));

I2=zeros(size(A));

%Filter Masks

F1=[0 1 0;1 -4 1; 0 1 0];

F2=[1 1 1;1 -8 1; 1 1 1];

%Padarray with zeros

A=padarray(A,[1,1]);

A=double(A);

%Implementation of the equation in Fig.D

for i=1:size(A,1)-2

    for j=1:size(A,2)-2

       

        I(i,j)=sum(sum(F1.*A(i:i+2,j:j+2)));

       

    end

end

I=uint8(I);

figure,imshow(I);title('Filtered Image');

The Laplacian derivative equation has produced grayish edge lines and other areas are made dark(background)

%Sharpenend Image

%Refer Equation in Fig.F

B=I1-I;

figure,imshow(B);title('Sharpened Image');

The Filter Image is combined with the Original input image thus the background is preserved and the sharpened image is obtained .

For a Filter Mask that includes Diagonal,

1	1	1
1	-8	1
1	1	1

Filtered Image:

Sharpened Image:

Edge Detection-Fundamentals

The derivatives of a digital function are defined in terms of differences.

The above statement made me to analyze about derivatives and how it is used for edge detection.  The first time when I came across the edge detection operation [Example: edge(Image,’sobel’)], I wondered how it worked. 

Consider a single dimensional array,

A =

5

4

3

2

2

2

2

8

8

8

6

6

5

4

0

MATLAB CODE:

x=1:15;

y=[5 4 3 2 2 2 2 8 8 8 6 6 5 4 0];

figure,

plot(x,y,'-o','LineWidth',3,'MarkerEdgeColor','k','Color','y');

title('Input Array');

First-order Derivative for one dimensional function f(x):

MATLAB  CODE:

x1=1:14;

y1=diff(y,1);

figure,

plot(x1,y1,'-o','LineWidth',3,'MarkerEdgeColor','k','Color','r');

-1

-1

-1

0

0

0

6

0

0

-2

0

-1

-1

-4

NOTE: The contiguous values are zero. Since the values are nonzero for non-contiguous values, the result will be thick edges.

The first-order derivative produces thicker edges.

Second-order Derivative for one dimensional function f(x):

MATLAB CODE:

x2=1:13;

y2=diff(y,2);

figure,

plot(x2,y2,'-o','LineWidth',3,'MarkerEdgeColor','k','Color','g');

0

0

1

0

0

6

-6

0

-2

2

-1

0

-3

The Second-order derivative gives finer result compared to first-order derivative. It gives fine detailed thin lines and isolated points.  Let’s see how the second-order derivative used for Image sharpening (Laplacian) in my upcoming post.

Convert HSI Image to RGB Image

CONVERT HSI IMAGE TO RGB IMAGE:

Refer How to convert the RGB image to HSI Image

MATLAB CODE:

 figure,imshow(HSI);title('HSI Image');  
 %Obtain the Hue, Saturation and Intensity components  
 H1=HSI(:,:,1);  
 S1=HSI(:,:,2);  
 I1=HSI(:,:,3);  
    
   

 %Multiply Hue by 360 to represent in the range [0 360]  
 H1=H1*360;                                               
    


 %Preallocate the R,G and B components  
 R1=zeros(size(H1));  
 G1=zeros(size(H1));  
 B1=zeros(size(H1));  
 RGB1=zeros([size(H1),3]);  
    


 %RG Sector(0<=H<120)  
 %When H is in the above sector, the RGB components equations are  


    
 B1(H1<120)=I1(H1<120).*(1-S1(H1<120));  
 R1(H1<120)=I1(H1<120).*(1+((S1(H1<120).*cosd(H1(H1<120)))./cosd(60-H1(H1<120))));  
 G1(H1<120)=3.*I1(H1<120)-(R1(H1<120)+B1(H1<120));  


    
 %GB Sector(120<=H<240)  
 %When H is in the above sector, the RGB components equations are  


    
 %Subtract 120 from Hue  
 H2=H1-120;  


    
 R1(H1>=120&H1<240)=I1(H1>=120&H1<240).*(1-S1(H1>=120&H1<240));  
 G1(H1>=120&H1<240)=I1(H1>=120&H1<240).*(1+((S1(H1>=120&H1<240).*cosd(H2(H1>=120&H1<240)))./cosd(60-H2(H1>=120&H1<240))));  
 B1(H1>=120&H1<240)=3.*I1(H1>=120&H1<240)-(R1(H1>=120&H1<240)+G1(H1>=120&H1<240));  


    
 %BR Sector(240<=H<=360)  
 %When H is in the above sector, the RGB components equations are  


    
 %Subtract 240 from Hue  
 H2=H1-240;  


    
 G1(H1>=240&H1<=360)=I1(H1>=240&H1<=360).*(1-S1(H1>=240&H1<=360));  
 B1(H1>=240&H1<=360)=I1(H1>=240&H1<=360).*(1+((S1(H1>=240&H1<=360).*cosd(H2(H1>=240&H1<=360)))./cosd(60-H2(H1>=240&H1<=360))));  
 R1(H1>=240&H1<=360)=3.*I1(H1>=240&H1<=360)-(G1(H1>=240&H1<=360)+B1(H1>=240&H1<=360));  


    
 %Form RGB Image  
 RGB1(:,:,1)=R1;  
 RGB1(:,:,2)=G1;  
 RGB1(:,:,3)=B1;  


    
 %Represent the image in the range [0 255]  
 RGB1=im2uint8(RGB1);  
 figure,imshow(RGB1);title('RGB Image');  
    
   

  

Explanation