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 $\cos(\phi)\frac{\partial}{\partial x}$ and $\cos(\phi)\frac{\partial}{\partial y}$ 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.

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.

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 $\cos(\phi)\frac{\partial}{\partial x}$ and $\cos(\phi)\frac{\partial}{\partial y}$ 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:

$$\frac{\partial}{\partial x_\phi} = \cos(\phi)\frac{\partial}{\partial x} + \sin(\phi)\frac{\partial}{\partial y}.\tag{1}$$

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 $\cos(\phi)\frac{\partial}{\partial x}$ and $\cos(\phi)\frac{\partial}{\partial y}$ 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.