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

Reflections on convolutions and animal testing

Convolution is a very tool important in image processing. There are inbuilt code to perform convolution in Matlab. But while solving many problems in image processing we assume Neumann boundary conditions. For this reason we reflected the boundary with our cute little code expand, while we performed our algorithms in the interior of the image. Finally we got rid of the extra image boundary by our shrink program. Many a times filtering is realized by performing a convolution of some kernel K with the image u.

In a discrete case, the convolution K*u is essentially the "masking process" as described in my blog about the Sobel filtering.

For example, the Laplacian operator, or the Sobel operator we have discussed before could be coded elegantly using Matlab's inbuilt filter2 command (which uses conv2).

I don't like two things about using using these two in-built commands for filtering.

1. Matlab uses "zero padding" in these in built commands, whereas we need a reflecting boundary. i.e. it adds zeros on the boundary of an image.
2. These commands can be used for only matrices.

Well, these are really trivial issues, and we can take care of them by writing our own little function say ifilter to avoid these issues.

This code performs a convolution of the image u with any 3x3 kernel h with Neumann boundary conditions. i.e. This code alone can find gradients, perform Sobel, Prewitt, Roberts filtering, or any other filtering with 3x3 window in a very nice, elegant way ! 

Of course you can write a more generalized code that is not restricted to 3x3 window. I won't do it because it is 3.00 am and I can hardly stay awake.

Let's see which filter should I chose... Let's design a filter that picks up derivatives at 45 degrees angle.

          0     1     0

h =   -1     0     1

          0    -1     0

We can normalized the above filter by dividing it by 2.

Let us do some animal testing now. I want to use an image of a Boston terrier.

Boston terrier

Here is the result of the filter, after using the following commands:

>> u=imread('bostonterrier.png');

>> h=(1/2)*[0 -1 0; -1 0 1; 0 1 0];

>> uf=ifilter(u, h);m=min(min(min(uf))); M=max(max(max(uf))); imshow((uf-m)/(M-m));

Directional derivative

               
Laplacian dog

Last time we used Laplacian, lets use ifilter to find the Laplacian of the Boson terrier.


>> h=[0 1 0; 1 -4 1; 0 1 0];
>> uf=ifilter(u, h);m=min(min(min(uf))); M=max(max(max(uf))); imshow((uf-m)/(M-m));

Laplacian dog

Prewitt operator

Here is the classical Prewitt operator implemented using ifilter. 

>> hx=(1/3)*[-1 -1 -1; 0 0 0; 1 1 1]


   -0.3333   -0.3333   -0.3333
         0         0         0
    0.3333    0.3333    0.3333


>> hy=(1/3)*[-1 0 1; -1 0 1; -1 0 1]


   -0.3333         0    0.3333
   -0.3333         0    0.3333
   -0.3333         0    0.3333


>> ux=ifilter(u, hx); uy=ifilter(u, hy); prewitt=sqrt(ux.^2+uy.^2);
>> m=min(min(min(uf))); M=max(max(max(prewitt))); imshow((prewitt-m)/(M-m));

Prewitt dog