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Reparameterization changes the t value for each point p on the bitmap curve, thus changing the place on the spline which corresponds to p. Given these new t values, Limn will then fit a new spline (see the previous section) to the bitmap, one which presumably matches it more closely.
Reparameterization is almost always a win. Only if the initial fit (see section 10.1.5.1 Initializing t) was truly terrible will reparameterization be a waste of time, and be omitted in favor of immediate subdivision (see the next section).
Limn sets the default threshold for not reparameterizing to be 30 "square pixels" (this number is compared to the least-squares error; see the previous section). This is usually only exceeded in cases such as that of an outline of `o', where one spline cannot possibly fit the entire more-or-less oval outline. You can change the threshold with the option `-reparameterize-threshold'.
If the error is less than `reparameterize-threshold', Limn reparameterizes and refits the curve until the difference in the error from the last iteration is less than some percentage (10 by default; you can change this with the option `-reparameterize-improve').
After Limn has given up reparameterization (either because the initial fit was worse than `reparameterize-threshold', or because the error did not change by more than `reparameterize-improve'), the final error is compared to another threshold, 2.0 by default. (You can specify this with the option `-error-threshold'.) If the error is larger, Limn subdivides the bitmap curve (see the next section) and fits each piece separately. Otherwise, Limn saves the fitted spline and goes on to the next piece of the pixel outline.