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I check the effects of adjustable parameters *k*_{0}, *n*_{a} and
on roughened images. The quantitative analysis of the effects
on the filter spectrum is provided in appendix A.

When comparing the effects of *k*_{0} and *n*_{a}, I set to make
sure the filters have the same zero-frequency response. Figure
3 shows the Bay Area map created with
different *k*_{0}. As *k*_{0} increases, more low-frequency components
were removed and the detailed structure turns out to be the main
focus of the map. This indicates the cut-off frequency increases as
*k*_{0} increases. In other words, *k*_{0} governs the cut-off frequency.
When *k*_{0} remains the same, filters with different *n*_{a} can create
very similar results if the zero-frequency response is the same by
adjusting , as shown in the middle and bottom plots in Figure
4. As expected, controls the zero-frequency
response. The larger leads to higher contrast, as shown in the
top and middle plots in Figure 4.

**bay-lct-k-a16r1
**

Figure 3 Bay Area maps roughened
by helix low-cut filters with different *k*_{0}. The top is *k*_{0}=0.1,
the middle is *k*_{0}=0.3, the bottom is *k*_{0}=0.5. *n*_{a}=16, .As *k*_{0} increases, the cut-off frequency increases.

**bay-lct-ar-k3
**

Figure 4 Bay Area maps roughened
by helix low-cut filters with different filter length. The top is *n*_{a}=8
and , the middle is *n*_{a}=16 and , the bottom is *n*_{a}=32.
*k*_{0}=0.3, . *k*_{0}=0.3. As increases, the zero-frequency
response decreases.
When the zero-frequency response remains the same, the difference of
*n*_{a} does not affect results.

Among the three adjustable parameters, *n*_{a} is the least important one
because the long filter can be replaced with a short one by adjusting .*k*_{0} controls the cut-off frequency and controls the zero-frequency
response. It is hard to tell which one is more important if I use only
this information. From the quantitative analysis, I know *k*_{0} affects
zero-frequency response significantly, but does not have such
an influence on cut-off frequency. So *k*_{0} is the most important
parameter.

For the enhanced helix low-cut filter, it is very reasonable to choose
parameter *k*_{0} first, then and *n*_{a}.

When roughening the image with the helix low-cut filter, the key point is to
choose the cut-off frequency *f*_{0} or the adjustable parameter *k*_{0}.
My suggestion is that if the lowest frequency component to be preserved
is *f*_{L}, *k*_{0} should be

| |
(5) |

Therefore, the frequency far below *f*_{L} would be cut off completely, and
the component near *f*_{L} would not be affected too much.
Since the short filter is made equivalent to the long filter by adjusting
, I suggest the use of based on the balance of
computation costs and symmetric features.

With *k*_{0} and *n*_{a} determined, I can find according to Equation
(13)

| |
(6) |

If , the zero-frequency is removed completely and leads
to the highest contrast in the roughened image.
Figure 5 consists of three maps of the Bay Area.
The top portion is a topographic map of the Bay Area. The bottom plot
is the preferred result. From one slice of the Bay Area topographic map,
I know one main low frequency component is about 0.4. So I chose
*k*_{0} = 0.3 to remove the lower frequency. I chose *n*_{a} = 16. In order
to obtain the highest contrast, I chose . The middle one is the
reference plot with *k*_{0} = 0.1, *n*_{a} =16 and . I notice that
the bottom plot removes more low frequency components than the middle
one and has clearer details, as predicted by the theory.

**bay-lct-res
**

Figure 5 Bay Area maps. The top is
the topographic map; the other two are roughened with helix low-cut
filter. The middle is *k*_{0} = 0.1; and the bottom is *k*_{0} = 0.3.
*n*_{a} = 16, .

Figure 6 consists of a normal mammogram and the
roughened images.
The main low frequency component of the mammogram slice is about
0.3, so I choose *k*_{0} = 0.2 in the right plot as the preferred
result, and use *k*_{0} = 0.1 in the middle as reference.
I use to achieve the highest contrast.

**mam-lct-res
**

Figure 6 Mammogram (medical X-ray).
The left figure is the origin map; the right two are filtered with helix
low-cut filter. *n*_{a} =16, , the middle is *k*_{0}=0.1, the right
is *k*_{0}=0.2.

** Next:** Derivative versus low-cut
** Up:** Zhao: Helix filter
** Previous:** Helix derivative filter
Stanford Exploration Project

4/20/1999