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: