Helix derivative filter

For the helix derivative filter, the response is nearly a linear function of $\vert\bold k\vert = \sqrt{k_x^2+k_y^2}$ and does not have the ``cut-off frequency''. The main feature here is the zero-frequency response. Figure 8 shows the zero-frequency response R₀ as the function of filter length when $\rho=0$. n_a is the half size of the filter. I find that R₀ decreases as n_a increases and $R_0 \propto n_a^{-1}$. The empirical relationship between R₀ and n_a is  

$$R_0 \approx \frac{0.44}{n_a}$$ (7)

R₀ is the sum of the helix filter's coefficients, so when $\rho \neq 0$,according to Equation (3), the zero-frequency response is  

$$R_0 \approx \frac{0.44}{n_a}(1-\rho)$$ (8)

 

drv-r0f-ar0
Figure 8 The zero-frequency response of helix derivative filter when $\rho=0$. The approximate zero-frequency response is $\frac{0.44}{n_a}$.

The zero-frequency response of the enhanced helix derivative filter is controlled by both n_a and $\rho$, I can compute the approximate value of R₀ using Equation (8).