Image gradient with central difference is fun

We have computed image derivatives with forward finite difference. Let us now take partial derivatives with central finite difference method.

du/dx:=(u(x+h, y)-u(x-h, y))/2h and du/dy:=(u(x, y+h)-u(x, y-h))/2h


Having written the codes one can use them to compute the image difference with central difference scheme.

Image gradient with central difference 

%==========================================================================

function [Gx, Gy]=gradient_central(u)

% Author: Prashant Athavale

% Date: 11/22/2011

% Please acknowledge my name if you use this code, thank you.

% This function computes the gradient of the image u.

% It uses central difference approximation of the derivatives 

% and thus the name gradient_central

%==========================================================================

if strcmp(class(u),'uint8')

    u=double(u);

end

Gx=Dfx_central(u);

Gy=Dfy_central(u);

end % end of the function gradient_central

%==========================================================================

%==========================================================================

function Gm=grad_central_magnitude(u)

% Author: Prashant Athavale

% Date: 11/22/2011

% Please acknowledge my name if you use this code, thank you.

% This function takes the image stored in u and gives the magnitude of the

% gradient

%==========================================================================

if strcmp(class(u),'uint8')

    u=double(u);

end

[Gx, Gy]=gradient_central(u);

Gm=sqrt(Gx.*Gx+Gy.*Gy);

end % end of the function grad_central_magnitude

%==========================================================================

Image gradient results with central difference

Here are the results of the above codes with the following commands in the command window

u=imread('barbara.png');

Gm=grad_central_magnitude(u);

ucx=Dfx_central(u); figure; imshow(ucx/255+0.4);

ucy=Dfy_central(u); figure; imshow(ucy/255+0.4);

figure; imshow(Gm/255+0.4);

Partial derivative in x-direction with central difference

Partial derivative in y-direction with central difference

Magnitude of the gradient with central difference