Our eyes, and our sense in general, are most sensitive to gradients. May be that is the reason when we are asked to draw a picture from memory, we tend to draw the sketch involving primarily the edges. It is also the reason why it is difficult for a person from California to get adjusted to the winter in Canada.
Anyway we are ready to talk about gradient (
u_x, u_y) of an image
u. We already have the code
Dfx_plus for computing the partial derivative in the
x-direction.
Image gradient
Now let's compute the gradient and store it in matrices Gx and Gy. Here is the trivial code to do that.
%==========================================================================
function [Gx, Gy]=gradient_plus(u)
% Author: Prashant Athavale
% Date: 1122/2011
% Please acknowledge my name if you use this code, thank you.
% This function computes the gradient f the image u.
% It uses forward approximation of the derivatives and thus the name
% gradient_plus
%==========================================================================
if strcmp(class(u),'uint8')
u=double(u);
end
Gx=Dfx_plus(u);
Gy=Dfy_plus(u);
end
%==========================================================================
Magnitude of the image gradient
Let us now write another code for computing the magnitude of the gradient.
%==========================================================================
function Gm=grad_plus_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_plus(u);
Gm=sqrt(Gx.*Gx+Gy.*Gy);
end
%==========================================================================
Remark 1: Pointwise operation
Note in the above code you see the line Gx.*Gx. Here the .* gives point wise multiplication of the entries in the matrix Gx. This is useful as one more time we could avoid a for loop, making things faster.
Let's implement the following in the command window.
>> u=imread('barbara.png');
>> Gm=grad_plus_magnitude(u);
>> figure; imshow(Gm/255+0.5);
>> edges=(Gm>15);
>> figure; imshow(edges);
Remark 2: Logical assignments
Note the line edges=(Gm>15). Here is another instance we avoided a for loop. What it does is, it creates a matrix edges of class logical. The value edges(i, j)=1 if Gm(i, j) > 15. Write a code using a nested for loop to do this and see which one is faster.
Results
Following are the images I got for Gm and edges as a result of the above code. This is how a gradient looks like. I don't know about you, but I was very happy when I saw a gradient of an image for the first time.
 |
| Magnitude of the gradient in the image Barbara |
 |
| Edges, defined as the points where gradient is more than 15 |
Partial derivatives with backward difference scheme
The partial derivatives with backward difference scheme are given by
u_x:=(u(x, y)-u(x-h, y))/h and u_y:=(u(x, y)-u(x, y-h))/h .
I recommend that you code these as a practice.
