A quick divergence to the cutest Laplacian in the world.

Let $\mathbf{u}=(u_1, u_2)$ be a vector valued function. Then the divergence operator: $\mbox{div }\mathbf{u}:=\frac{du_1}{dx}+\frac{du_2}{dx}$ is a very useful thing to have on our side. Again, I will leave this little code upto you. It is really a one liner!

Note, that if $\nabla u=(u_x, u_y)$ then $\mbox{div}(\nabla u)$ gives the Laplacian $\Delta u$ of the image $u$. This could be useful.

Here is the Laplacian of Boo displayed between -50 to 50.
i.e. use the commands to display the Laplacian d :
>> m=-50; M=50; figure; imshow((d-m)/(M-m));

Laplacian of Boo (-50 to 50)

Here is the original.

Boo, the cutest dog in the world

The edge of darkness is upon us!

I assume that now we know how to find edges in a given image u. It's simple, just take the derivative!

Today, I want to find the smooth regions. Well, it's kind of silly, as now that we have edges, the rest of the image is kind of edgeless. Right... but I would like to assign a number from 0 to 1 to it ... where 1 indicates a flat region and zero indicates an edge, in other words I want to make the edges appear dark.

There are many ways to do it. One of the ways to do it is to look at the function $$g(x, y):=\frac{1}{\sqrt{1+|K*\nabla u(x, y)|}}.$$Where, K is your favourite smoothing kernel. (I will leave it to you to code this as an exercise. It is a four liner.)

Staring at this function is a refreshing activity, something that I love to do in my spare time.

Boo in the edge of darkness

Lenna in the edge of darkness

Barbara in the edge of darkness

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