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Chapters:
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<a href="Introduction.html">1: Introduction</a><br>
<a href="SimpleExample.html">2: Simple example</a><br>
<a href="InvokingGri.html">3: Invocation</a><br>
<a href="GettingMoreControl.html">4: Finer Control</a><br>
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<h3>9.3.11: `<font color="#82140F"><code>filter</code></font>'</h3>
<!-- latex: \index{filter} -->
<ul>
<p>
<li> `<font color="#82140F"><code>filter column x|y|z|u|v|weight recursively a[0] a[1] ... b[0] b[1] ...</code></font>'
Filter indicated column, using a two-pass recursive filter. The first
pass runs from the start to the end, while the second pass runs from the
end to the start; in this way, the phase shift inherent in this type of
filter is removed entirely. The coefficients are used in the following
formula (demonstrated on the `<font color="#82140F"><code>x</code></font>' column):
<p>
<TABLE SUMMARY="Example" BORDER="0" BGCOLOR="#efefef" WIDTH="100%">
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<PRE>
<font color="#82140F">
x_new[i] = b[0] * x[i] \
+ b[1] * x[i-1] \
+ b[2] * x[i-2] \
+ ... \
- a[1] * x_new[i-1] \
- a[2] * x_new[i-2] \
- ...
</font></PRE>
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<p>
Thus, for example, setting `<font color="#82140F"><code>a[i]</code></font>' = 0 results in a simple
backwards-looking moving-average filter applied in two passes. The real
power of this type of filter, however, comes when non-zero `<font color="#82140F"><code>a[i]</code></font>'
coefficients are given, thus adding recursion (i.e., `<font color="#82140F"><code>x_new[i]</code></font>'
depends on `<font color="#82140F"><code>x_new[i-...]</code></font>'). See any standard reference on digital
filters for an explanation. You might find that the Matlab command
`<font color="#82140F"><code>butter</code></font>' an easy way to design filter coefficients. Here are some
examples:
<p>
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<PRE>
<font color="#82140F">
# Filter x column with simple 2-point moving
# average. (This slurs into a 3-point moving
# average, in effect, since the filter is run
# forwards and then backwards.)
filter column x recursively 0 0 0.5 0.5
<p>
# Use filter designed with the Matlab
# command butter(2,0.1), which creates a
# 2nd order lowpass butterworth filter
# with a cutoff frequency of 0.1
# (in units which have a frequency
# of 1 corresponding to one-half the
# sampling rate).
filter column x recursively \
1 -1.561 0.6414 \
0.0201 0.0402 0.0201
</font></PRE>
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<p>
<li> `<font color="#82140F"><code>filter grid rows|columns recursively a[0] a[1] ... b[0] b[1] ...</code></font>'
<p>
Apply recursive filter (see `<font color="#82140F"><code>filter column ... recursively</code></font>' for
meaning of this filter operation) to the individual rows or columns of
the grid data. For example, the command
`<font color="#82140F"><code>filter grid columns recursively 0 0 0.5 0.5</code></font>'
applies a 2-point moving average filter across the columns,
smoothing the grid in the x-direction.
<p>
<li> `<font color="#82140F"><code>filter image highpass</code></font>'
Remove low-wavenumber components from image (ie, sharpen edges). Do
this by subtracting a Laplacian smoothed version of the image.
<p>
<li> `<font color="#82140F"><code>filter image lowpass</code></font>'
Remove high-wavenumber components from image (ie, smooth shapes). Do
this by Laplacian smoothing.
</ul>
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<b>See also</b> see <a href="Smooth.html#Smooth">Smooth</a>.
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