Derivative versus low-cut

Both helix derivative and low-cut filters can be used to enhance the details of images. But which one is better? It depends on the situation.

• Focus of the structure

  Both helix derivative and low-cut filters cut off the zero-frequency component, but the functions of the frequency are different. The derivative filter's response is approximately a linear function of frequency, while the low-cut filter's response has a hole below the cut-off frequency and is flat above it. This leads to the main difference between derivative and low-cut filters: the derivative filter enhances the small-scale structures more, while the low-cut filter makes medium-scale structure much clearer.

  In Figure 2, a long line structure in the middle of the sea is very clear in the plot created by $\frac{d}{dx}$ operator; in the plot created by helix derivative filter, it is too weak to be seen. The helix low-cut filter preserves this structure quite well, as shown in Figure 7.

   
  
  gal-lct-res
  Figure 7 View of the Sea of Galilee roughened with a helix low-cut filter. k₀=0.3, n_a = 16 and $\rho=0.99$.The edge of the sea fades away, but in the middle of the sea, the long line structure is persevered quite well.
  

• Computational cost

  Another difference is the cost of image processing:

  1.

  The helix low-cut filter needs to do the deconvolution with $\bold{D}$besides the convolution with $\bold{H}$;

  2.

  The worse spectral symmetry leads helix low-cut filter to the use of a long filter.

  These two points add the burden of computation for the helix low-cut filter.

So if the main interests are the small-scale structures or the high costs of computation are not affordable, I should choose the helix derivative filter. Otherwise, the helix low-cut filter is a better choice.