Sobel filter kernel of large size

Question

I am using a sobel filter of size 3x3 to calculate the image derivative. Looking at some articles on the internet, it seems that kernels for sobel filter for size 5x5 and 7x7 are also common, but I am not able to find their kernel values.

Could someone please let me know the kernel values for sobel filter of size 5x5 and 7x7? Also, if someone could share a method to generate the kernel values, that will be much useful.

Thanks in advance.

Answer 1

Other sources seem to give different definitions of the larger kernels. The Intel IPP library, for example, gives the 5x5 kernel as

1  2 0  -2 -1
4  8 0  -8 -4
6 12 0 -12 -6
4  8 0  -8 -4
1  2 0  -2 -1

Intuitively, this makes more sense to me because you're paying more attention to the elements closer to the centre. It also has a natural definition in terms of the 3x3 kernel which is easy to extend to generate larger kernels. That said, in my brief search I've found 3 different definitions of the 5x5 kernel - so I suspect that (as Paul says) the larger kernels are ad hoc, and so this is by no means the definitive answer.

The 3x3 kernel is the outer product of a smoothing kernel and a gradient kernel, in Matlab this is something like

sob3x3 = [ 1 2 1 ]' * [1 0 -1]

the larger kernels can be defined by convolving the 3x3 kernel with another smoothing kernel

sob5x5 = conv2( [ 1 2 1 ]' * [1 2 1], sob3x3 )

you can repeat the process to get progressively larger kernels

sob7x7 = conv2( [ 1 2 1 ]' * [1 2 1], sob5x5 )
sob9x9 = conv2( [ 1 2 1 ]' * [1 2 1], sob7x7 )
...

there are a lot of other ways of writing it, but I think this explains exactly what is happening best. Basically, you start off with a smoothing kernel in one direction and a finite differences estimate of the derivative in the other and then just apply smoothing until you get the kernel size you want.

Because it's just a series of convolutions, all the nice properties hold, (commutativity, associativity and so forth) which might be useful for your implementation. For example, you can trivially separate the 5x5 kernel into its smoothing and derivative components:

sob5x5 = conv([1 2 1],[1 2 1])' * conv([1 2 1],[-1 0 1])

Note that in order to be a "proper" derivative estimator, the 3x3 Sobel should be scaled by a factor of 1/8:

sob3x3 = 1/8 * [ 1 2 1 ]' * [1 0 -1]

and each larger kernel needs to be scaled by an additional factor of 1/16 (because the smoothing kernels are not normalised):

sob5x5 = 1/16 * conv2( [ 1 2 1 ]' * [1 2 1], sob3x3 )
sob7x7 = 1/16 * conv2( [ 1 2 1 ]' * [1 2 1], sob5x5 )
...