Helix derivative filter

First I apply the enhanced helix derivative filter to some familiar images and check the effects of adjustable parameters n_a and $\rho$.

Figure 1 shows the views of the Sea of Galilee created by helix derivative filters with different n_a and $\rho$.The top two plots (with $\rho=0$) indicate that as the half filter length n_a increases, the contrast of the image increases. This means the longer filter keeps less low frequency components, indicating a lower zero-frequency response. The vertical plots (with same n_a) indicate that as $\rho$ increases, the contrast increases also, as expected from the definition of the enhanced helix filter equation.

To compare the properties of the filters with different n_a other than the zero-frequency response, I set $\rho=1$ in the bottom two plots, so that they have the same zero-frequency response. The plots are very similar, and the difference between them is very weak.

 

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Figure 1 The Views of the Sea of Galilee roughened with helix derivative filters. Each two horizontal plots have the same $\rho$; each two vertical plots have the same n_a. As n_a or $\rho$ increases, the zero-frequency response decreases.

As the examples above show, the zero-frequency response of the enhanced helix derivative filter decreases as n_a or $\rho$ increases. Thus filters with different n_a may create the same results by adjusting $\rho$, which provides the possibility of using a short filter instead of a long filter in image processing and reduce the computational cost greatly. By adjusting $\rho$, the enhanced short filter can reach a low level of zero-frequency response, which is done conventionally by using a long filter.

Since the short filter is equivalent to the long filter when adjusting $\rho$, now $\rho$ plays the key role in image processing for the enhanced helix derivative filter, and n_a is not an important parameter. When the high frequency component is too weak compared with the low frequency, a large $\rho$ $\left( \approx 1 \right)$ should be chosen in order resolve details in the image. Otherwise, a small $\rho$ would be suitable. Based on the balance of the computation cost and response symmetry, I recommend the use of $n_a \approx 8$.

From the quantitative analysis of spectra of the enhanced helix derivative filter (see the appendix A), I can find $\rho$ according to empirical formula  

$$\rho = 1 - \frac{I_0 n_a}{0.44}$$ (4)

where I₀ is the factor of the zero-frequency component to be preserved.

For the view of the Sea of Galilee, the high frequency component is very weak, so I chose I₀=0.002, n_a=8, and $\rho=0.96$. Figure 2 shows the preferred and gradient roughened results.

 

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Figure 2 The views of the Sea of Galilee. The left is with the gradient operator d/dx; the right is with the helix derivative filter. The helix filter enhances the details both along the vertical and horizontal direction.