Sobel Vs Gaussian Derivative

Question

What would be the most significant difference when using either a sobel filter or a gaussian-derivative filter, apart from the obvious difference of the size of the 2 filters?

Answer 1

The Sobel kernels can also be thought of as $3 × 3$ approximations to first-derivative-of-Gaussian kernels.

The Sobel filter is nothing but jointly applying the following:

• Smoothing in the perpendicular direction of the derivative such as $h_s=[1,2,1]^T$

• Simple central difference in the derivative direction such as $h_d=[1,0,-1]$

Then $$h_{sobel} = h_s h_d$$

The smoothing factor is an approximate triangle shaped filter. A Gaussian is naturally a better replacement. In fact, if larger sizes of Sobel is desired, people first smooth the image with a Gaussian filter, then apply the Sobel kernel as is. That's because:

$$ \frac{\partial}{\partial x} {(I*G)} = I*\frac{\partial}{\partial x} {(G)} $$

This approximation enables the Sobel operator to be computed quite quickly, while Gaussian derivatives might be more accurate.