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

Question

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?

Bay H., Ess A., Tuytelaars T. Van Gool L. - Speed-Up Robust Features (SURF), page 3, column 2

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.

Answer 1

The answer boils down to 2 issues with the practical approximations of the Gaussian Kernel:

1. Though the Gaussian Kernel is radially symmetric its discrete approximation has a rectangle support. Unless this support will have infinite length a rotation by any angle different from a multiplication of 90 degrees will yield a shape which has to modified to fit a rectangle. Either choice (Cropping or extending) will yield different results as the weights will be different from the original approximation. One must pay attention that the longer the support of the filter the less this issue is pronounced.
2. If I remember correctly, the SURF algorithm uses Box Blur based approximation of the Gaussian Kernel. This approximation, though can be very close visually, doesn't retain the radial symmetry of the Gaussian Blur hence even with large length ha different results when rotated.

I hope this clarifies the issue.