# Guassian Derivatives with orientations

I am reading the paper Selective Search for Object Recognition here. In Section 3.2, they give a similarity measure between two regions of an image based on the texture of the regions with what they refer to as "fast SIFT-like measurements". On page 4, bottom right side of the page, they write:

We take Gaussian derivatives in eight orientations using

$$\sigma = 1$$ for each colour channel. For each orientation for each colour channel we extract a histogram using a bin size of $$10$$.

I understand that a derivative of Gaussian filter is the filter of size $$n \times n$$ consisting of a discrete approximation of the derivative of a bivariate gaussian function of mean $$0$$ with some standard deviation.

What do the authors mean by "with eight orientations"? Is this some kind of modification to the filter? Any insights appreciated.

If you have gradients $$\frac{\partial}{\partial x}$$ and $$\frac{\partial}{\partial y}$$ along coordinates $$x$$ and $$y$$ at orientations 0° and 90° (so that they are orthogonal, that is, perpendicular), you can calculate the gradient $$\frac{\partial}{\partial x_\phi}$$ with respect to a coordinate $$x_\phi$$ at any given orientation $$\phi$$ by:
$$\frac{\partial}{\partial x_\phi} = \cos(\phi)\frac{\partial}{\partial x} + \sin(\phi)\frac{\partial}{\partial y}.\tag{1}$$
The same way you can get the output of a Gaussian derivative filter at any orientation by taking a $$\cos(\phi)$$ and $$\sin(\phi)$$ weighted sum of the outputs of the $$x$$ and $$y$$ coordinate orientation Gaussian derivative filters.