# The Statistic Distribution of Image Gradient?

The gradient of an image $f$ is defined as:

$\nabla f=\begin{bmatrix} \nabla f_{x} \\ \nabla f_{y} \end{bmatrix} = \begin{bmatrix} \frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y} \end{bmatrix} ,$

Its discrete calculation can be as simple as finite difference. For example

$\nabla f_{x} = \frac{f_n-f_{n-1}}{x_{n}-x_{n-1}}$ and $\nabla f_{y} = \frac{f_n-f_{n-1}}{y_{n}-y_{n-1}}.$

I can simply define the total\whole image gradient is the norm of x and y gradient component:

$||\nabla f|| = \sqrt{(\nabla f_{x})^2+(\nabla f_{y})^2}.$ Nothing fancy so far.

Now I am just wondering, what is the distribution of the image gradient in equation above? Here is an example:

In above image, the histogram of the image gradient really looks exponential to me. This is just an example, but I have seen similar shape of the histogram in many cases.

Can I claim the distribution of an image gradient follows exponential? If not, with what condition I can/cannot make this guess? Thanks a lot.