# Why Does the Odd Multiple of $\frac{\pi}{4}$ on Gaussian Cause Loss in Repeatability Under Image Rotations?

I couldn't figure out below paragraph on SURF paper and hope that someone can help me to understand it. Why image rotations around odd multiples of $\frac{\pi}{4}$ lead to a loss of repeatability?

Gaussians are optimal for scale-space analysis, but in practice they have to be discretized and cropped (figure 2 left half). This leads to a loss in repeatability under image rotations around odd multiples of $\frac{\pi}{4}$. This weakness holds for Hessian-based detectors in general. Figure 3 shows the repeatability rate of two detectors based on the Hessian matrix for pure image rotation. The repeatability attains a maximum around multiples of $\frac{\pi}{2}$. This is due to the square shape of the filter.

• Because you can't rotate a square filter by $\pi$/4 and have it still be a square filter of the same size. – Aaron Jun 13 '14 at 19:14