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# Merging

Merging uses bilinear interpolation to determine each frame's contribution to the accumulated image, and each contribution is assigned equal weight. Sections below outline the merging algorithm and its properties.

## Calculating the Overlapping Value

(This section is a very verbose explanation of bilinear interpolation.)

Determining what values from the new frame overlap pixels in the accumulated image is an important part of merging. Since transformations may involve sub-pixel alignment, and the accumulated image may be configured to be of finer resolution than the input frames, the manner in which these values are determined is not necessarily obvious.

ALE uses the following rules to determine the overlapping value:

1. If the new frame overlaps the accumulated image pixel in such a manner that the accumulated image pixel is aligned with a pixel in the new frame, then the value of this pixel is the overlapping value.
2. If the point overlapping the accumulated image pixel falls on a line between two pixels in the new frame, then the overlapping value is a linear interpolation between these pixels in the coordinate system of the new frame.
3. If the point overlapping the accumulated image pixel falls within a quadrilateral outlined by the four closest pixels in the new frame, then the overlapping value is a bilinear interpolation among these pixels in the coordinate system of the new frame.
4. If none of the above conditions apply, then there is no overlapping value for the accumulated image pixel in the new frame.

## Calculating the Merged Value

The merged value for an accumulated image pixel is a weighted average between the current accumulated image pixel value and the overlapping value. The weights are selected so that overlapping values from all frames contribute with equal weight.

## Definition of Merging

Merging a new frame replaces the existing value of each accumulated image pixel with the merged value.

## Properties

### Density and false local minima

The merging method outlined above has the property of always updating a dense set of pixels in the accumulated image, even when the accumulated image is of finer resolution than the new frame. This ensures that the area of defined pixels in the accumulated image is dense and that any uniform change in intensity is reflected uniformly over a dense area. The absence of these properties may cause alignment to fail due to entrapment in local minima, and so merging is always used internally to create the renderings used by the alignment algorithm. Except in the case of a large pixel footprint, drizzling does not share these properties.

### Convolution with the Bartlett (triangle) filter as the limiting case

Assuming predicates for translation, point sampling with simple optics, and a very large, uniform input sequence, the result of merging is equivalent to convolution of pixel data with the Bartlett, or triangle, filter. This is derived from the fact that bilinear interpolation of an image defined at discrete points is equivalent to convolution with a Barlett filter. This property is still approximately true even when the translation predicate is not satisfied, so long as large changes in scale do not occur.

Copyright 2002, 2003 David Hilvert

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