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# Error Functions

Error functions are used to determine whether a transformation is "good" or not, and in particular, whether one transformation is better or worse than another. In particular, a smaller error indicates a better transformation.

ALE defines error functions for each pixel of the accumulated image, and also a comprehensive error function that summarizes the error over all pixels.

### Pre-Alignment Exposure Registration

When exposure registration is enabled, ALE performs an exposure registration step prior to alignment. The resulting values are dependent on the initial alignment (this can be either the default initial alignment or an alignment loaded from a transformation data file). See the source code for details.

### Per-Pixel Error Functions

The per-pixel error function for pixel (i, j) in the accumulated image A, frame B, and transformation T, is:

>     p(i, j, A, B, T) = Abs( A(i, j) - B(T_inverse(i, j)) )metric_exponent

where Abs() is the absolute value function and B(T_inverse(i, j)) is determined by bilinear interpolation. If not specified by the --metric option, metric_exponent defaults to 2.0 for ALE versions 0.1.0 and later, or 1.0 for version 0.0.0.

For coordinates where B(T_inverse(i, j)) is not defined, the error is zero. (Feedback from Angelo Pesce led to clarification of this point.)

In addition to the per-pixel error function, a per-pixel maximum error estimator is calculated, as follows:

>     p_max(i, j, A, B, T) = Max( A(i, j), B(T_inverse(i, j)) )metric_exponent

For coordinates where B(T_inverse(i, j)) is not defined, the maximum error estimator is zero. (Feedback from Angelo Pesce led to clarification of this point.)

### Comprehensive Error Functions

There are two varieties of comprehensive error function used by ALE: an exhaustive error function and, in versions 0.4.3 and later, a Monte Carlo error function. Whereas the exhaustive function evaluates the error for each pixel in the accumulated image, the Monte Carlo function evaluates only a subset of pixels.

If we consider Sum[] to provide the sum over whatever subset of pixels we are sampling (including possibly the set of all pixels), then the comprehensive error function, for accumulated image A, frame B, and transformation T, is:

>     E(A, B, T) = (Sum [ p(i, j, A, B, T) ] / Sum [ p_max(i, j, A, B, T) ])(1/metric_exponent)

Copyright 2002, 2003, 2004 David Hilvert

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