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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.

### Per-Pixel Error Functions

Since differences in exposure may alter the absolute values of pixels even in images that are perfectly aligned, ALE usually normalizes the images so that overall image intensity matches. This normalization step does not affect the pixel values contributing to the merged image, but it does affect the values with which the per-pixel error function is calculated. (The normalization step is skipped when --extend or --follow is specified.)

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 the overlapping value described in the merging section. 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.

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

### 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 David Hilvert

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