kernels to compute second order derivative of digital image

Question

For an image $I$, its first order derivatives can be computed using several oprators, such as $$K_{sobel} = \left[ \begin{array}{ccc} -1 &0 &1 \\ -2 &0 &2 \\ -1 &0 &1 \end{array}\right]$$ $$I_{x} = I * K_{sobel},\ I_{y} = I * K_{sobel}.T$$. And to compute the second order derivatives $(I_{xx}, I_{yy}, I_{xy})$, I can apply $K_{sobel}$ twice on $I$, however, I wonder are there any kernels which can be applied once to get the second order derivatives.

Here is what I tried, I made up 2 kernels, $$K_{xx} = \left[ \begin{array}{ccc} 0 &0 &0 \\ 2 &-4 &2 \\ 0 &0 &0 \end{array}\right]$$ $$K_{yy} = \left[ \begin{array}{ccc} 0 &2 &0 \\ 0 &-4 &0 \\ 0 &2 &0 \end{array}\right]$$ and $I_{xx} = I * K_{xx},\ I_{yy} = I * K_{yy}$, they look pretty much the same as twice-applied-sobel results, but I failed to find a good kernel for $I_{xy}$, I need your help, any idea is appreciated.

Answer 1

Convolution $(*)$ is associative, so

$I_{xx} = K_{sobel} * (K_{sobel} * I) = (K_{sobel} * K_{sobel}) * I$

meaning the convolution of a 1st order Sobel filter kernel with another 1st order Sobel filter gives the filter kernel for a second order filter. So e.g. a 2nd order horizontal Sobel filter kernel is:

$\left( \begin{array}{ccc} -1 & 0 & 1 \\ -2 & 0 & 2 \\ -1 & 0 & 1 \\ \end{array} \right) * \left( \begin{array}{ccc} -1 & 0 & 1 \\ -2 & 0 & 2 \\ -1 & 0 & 1 \\ \end{array} \right)=\left( \begin{array}{ccccc} 1 & 0 & -2 & 0 & 1 \\ 4 & 0 & -8 & 0 & 4 \\ 6 & 0 & -12 & 0 & 6 \\ 4 & 0 & -8 & 0 & 4 \\ 1 & 0 & -2 & 0 & 1 \\ \end{array} \right)$

And for $I_{xy}$:

$\left( \begin{array}{ccc} -1 & 0 & 1 \\ -2 & 0 & 2 \\ -1 & 0 & 1 \\ \end{array} \right) * \left( \begin{array}{ccc} -1 & -2 & -1 \\ 0 & 0 & 0 \\ 1 & 2 & 1 \\ \end{array} \right)=\left( \begin{array}{ccccc} 1 & 2 & 0 & -2 & -1 \\ 2 & 4 & 0 & -4 & -2 \\ 0 & 0 & 0 & 0 & 0 \\ -2 & -4 & 0 & 4 & 2 \\ -1 & -2 & 0 & 2 & 1 \\ \end{array} \right)$

If you want a 3x3 kernel, you can use the 1st order derivative filter $\left( \begin{array}{cc} -1 & 1 \\ -1 & 1 \\ \end{array} \right)$, which leads to the second order filters:

$\left( \begin{array}{cc} -1 & 1 \\ -1 & 1 \\ \end{array} \right)*\left( \begin{array}{cc} -1 & 1 \\ -1 & 1 \\ \end{array} \right)=\left( \begin{array}{ccc} 1 & -2 & 1 \\ 2 & -4 & 2 \\ 1 & -2 & 1 \\ \end{array} \right)$

and

$\left( \begin{array}{cc} -1 & -1 \\ 1 & 1 \\ \end{array} \right)*\left( \begin{array}{cc} -1 & 1 \\ -1 & 1 \\ \end{array} \right)=\left( \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 0 & 0 \\ -1 & 0 & 1 \\ \end{array} \right)$

Answer 2

What you are looking for is the so-called "Laplacian operator" (http://en.wikipedia.org/wiki/Laplace_operator and http://en.wikipedia.org/wiki/Discrete_Laplace_operator). In image processing, people often use the Laplacian of Gaussian, which is simply the difference between the two results of convolving one input image with two different Gaussian kernels. You may check details here (http://fourier.eng.hmc.edu/e161/lectures/gradient/node9.html).