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.


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.


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