Enhanced helix filters

The autocorrelation of the helix derivative filter H is the negative of the finite-difference representation of the Laplacian operator $\nabla^{2}$,

$$\bold{H^{'}} \bold{H} = \bold{R} = - \nabla^{2}$$ (1)

The coefficients of the causal helix filter H can be found by spectral factorization.

The helix low-cut filter H/D is designed by doing two spectral factorizations, one for the numerator of H, and another for the denominator of D. It is expressed by  

$$\frac{k^{2}} {k^{2}+k_{0}^{2}} \, \approx \, \frac {-\nabl... ...k_{0}^{2}} \, \approx \, \frac {\overline{H}H} {\overline{D}D}$$ (2)

where k is the frequency , k₀ affects the cut-off frequency, $\overline{H}$ and $\overline{D}$ are the conjugate anti-casual filters of H and D.

Both the filters do not remove the zero frequency completely, degrading the contrast and details of the roughened image. A way to solve this problem as suggested by Claerbout, is to rescale all the coefficients of H with nonzero lags by a. If s is the sum of all the coefficients with nonzero lag (which are all negative), a is expressed by  

$$a = 1 + \left( \frac{1}{\vert s\vert}-1 \right) \rho$$ (3)

$\rho=0$ denotes the original unscaled filter, while $\rho=1$guarantees the filter really removes the zero frequency component.

Now I have the enhanced helix derivative with adjustable parameters n_a and $\rho$, the enhanced helix low-cut filter with n_a, k₀ and $\rho$. Here n_a is the half length of the helix filter.

Compared with the conventional helix filters, the enhanced filters have a new adjustable parameter, $\rho$.