Sobel kernels

8-point neighborhood on a 2D grid

The Sobel kernel contains weights for each pixel from the 8-point neighbourhood of a tested pixel. These are 3x3 kernels.

There are 2 Sobel kernels, one for computing horizontal changes and other for computing vertical changes. Notice that a large horizontal change may indicate a vertical border, and a large vertical change may indicate a horizontal border. The x-coordinate is here defined as increasing in the "right"-direction, and the y-coordinate is defined as increasing in the "down"-direction.

The Sobel kernel for computing horizontal changes is:

${\displaystyle \mathbf {H} ={\begin{bmatrix}H_{1}&H_{2}&H_{3}\\H_{4}&H_{5}&H_{6}\\H_{7}&H_{8}&H_{9}\end{bmatrix}}={\begin{bmatrix}-1&0&+1\\-2&0&+2\\-1&0&+1\end{bmatrix}}}$

The Sobel kernel for computing vertical changes is:

${\displaystyle \mathbf {V} ={\begin{bmatrix}-1&-2&-1\\\ \ 0&\ \ 0&\ \ 0\\+1&+2&+1\end{bmatrix}}}$

Note that:

• sum of weights of kernels are zero

${\displaystyle \sum _{i=1}^{9}H_{i}=0}$

${\displaystyle \sum _{i=1}^{9}V_{i}=0}$

• One kernel is simply the other rotated by 90 degrees.^[18]
• 3 weights in each kernal are zero.

Pixel kernel

Pixel kernel A containing central pixel ${\displaystyle A_{5}}$ with its 3x3 neighbourhood:

${\displaystyle \mathbf {A} ={\begin{bmatrix}A_{1}&A_{2}&A_{3}\\A_{4}&A_{5}&A_{6}\\A_{7}&A_{8}&A_{9}\end{bmatrix}}}$

Other notations for pixel kernel:

${\displaystyle \mathbf {A} ={\begin{bmatrix}A_{1}&A_{2}&A_{3}\\A_{4}&A_{5}&A_{6}\\A_{7}&A_{8}&A_{9}\end{bmatrix}}={\begin{bmatrix}ul&um&ur\\ml&mm&mr\\ll&lm&lr\end{bmatrix}}}$

where:^[19]

unsigned char ul, // upper left
unsigned char um, // upper middle
unsigned char ur, // upper right
unsigned char ml, // middle left
unsigned char mm, // middle = central pixel
unsigned char mr, // middle right
unsigned char ll, // lower left
unsigned char lm, // lower middle
unsigned char lr, // lower right

Pixel 3x3 neighbourhood (with Y axis directed down)

In array notation it is:^[20]

${\displaystyle \mathbf {A} ={\begin{bmatrix}A[x-1][y+1]&A[x][y+1]&A[x+1][y+1]\\A[x-1][y]&A[x][y]&A[x+1][y]\\A[x-1][y-1]&A[x][y-1]&A[x+1][y-1]\end{bmatrix}}}$

In geographic notation usede in cellular aotomats it is central pixel of Moore neighbourhood.

So central (tested) pixel is:

${\displaystyle A_{5}=mm=A[x][y]\,}$

Sobel filters

Compute Sobel filters (where ${\displaystyle *}$ here denotes the 2-dimensional convolution operation not matrix multiplication). It is a sum of products of pixel and its weights:

${\displaystyle \mathbf {G} _{h}=\mathbf {H} *A=A_{1}H_{1}+A_{2}H_{2}+\cdots +A_{9}H_{9}=\sum _{r=1}^{9}A_{r}H_{r},}$

${\displaystyle \mathbf {G} _{v}=\mathbf {V} *A=A_{1}V_{1}+A_{2}V_{2}+\cdots +A_{9}V_{9}=\sum _{r=1}^{9}A_{r}V_{r},}$

Because 3 weights in each kernal are zero so there are only 6 products.^[21]

short Gh = ur + 2*mr + lr - ul - 2*ml - ll;
short Gv = ul + 2*um + ur - ll - 2*lm - lr;

Result

Result is computed (magnitude of gradient):

${\displaystyle \mathbf {G} (A_{5})={\sqrt {{\mathbf {G} _{h}}^{2}+{\mathbf {G} _{v}}^{2}}}}$

It is a color of tested pixel.

One can also approximate result by sum of 2 magnitudes:

${\displaystyle \mathbf {G} (A_{5})=\left|\mathbf {G} _{h}\right|+\left|\mathbf {G} _{v}\right|}$

which is much faster to compute.^[22]