12.12: The `filter' Command

• `filter column x|y|z|u|v|r|theta recursively a[0] a[1] ... b[0] b[1] ...' 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 `x' column):

  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] \ - ...

  Thus, for example, setting `a[i]' = 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 `a[i]' coefficients are given, thus adding recursion (i.e., `x_new[i]' depends on `x_new[i-...]'). See any standard reference on digital filters for an explanation. You might find that the Matlab command `butter' an easy way to design filter coefficients. Here are some examples:

  # 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 # 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

• `filter grid rows|columns recursively a[0] a[1] ... b[0] b[1] ...' Apply recursive filter (see `filter column ... recursively' for meaning of this filter operation) to the individual rows or columns of the grid data. For example, `filter grid columns recursively 0 0 0.5 0.5' applies a 2-point moving average filter across the columns, smoothing the grid in the x-direction.
• `filter image highpass' Remove low-wavenumber components from image (ie, sharpen edges). Do this by subtracting a Laplacian smoothed version of the image.
• `filter image lowpass' Remove high-wavenumber components from image (ie, smooth shapes). Do this by Laplacian smoothing.