Guassian Derivatives with orientations

Question

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.

Answer 1

Based on Schuyler Smith's presentation on the paper, I think they calculated signed Gaussian derivative "gradients" at 4 orientations 0°, 45°, 90°, 135°, and then got from that 8 unsigned gradients at orientations 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°.

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.