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<title>Alignment</title>
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<a href="http://auricle.dyndns.org/ALE/">
<big>ALE</big>
<br>
Image Processing Software
<br>
<br>
<small>Deblurring, Anti-aliasing, and Superresolution.</small></a>
<br><br>
<big>
Local Operation
</big>
<hr>
localhost<br>
5393119533<br>
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<p><b>[ <a href="../">Up</a> | <a href="error/">Error Function</a> ]</b></p>
<h1>Alignment</h1>

ALE aligns each supplemental frame, in sequence, with the merged rendering
representing all previous frames.  This page outlines the three supported
transformation classes, the algorithm used for alignment, and the properties of
the alignment algorithm.  Following this is a discussion of practical use of
alignment options, including alignment classes and alignment in the case of
extended renderings.

<h2>Transformations</h2>

<p>ALE offers three classes of transformations: </p>

<table border>
<tr><td>Translations</td><td>introduced in version 0.0.0</td>
<tr><td>Euclidean transformations (excluding reflections)</td><td>introduced in version 0.1.0</td>
<tr><td>Projective transformations</td><td>introduced in version 0.2.0</td>
</table>

<!-- <p>ALE uses a forward transformation to map from a supplemental frame to the
accumulated image, and an inverse transformation to map from the accumulated
image to the supplemental frame.  Alignment operates on the parameters of the
forward transformation. -->

<h2>Algorithm</h2>

<p>Alignment proceeds by a deterministic search, beginning with an initial
transformation and modifying this transformation through a series of
perturbations.

<p>The initial transformation may be loaded from a file or selected by default.
The default initial transformation is either the identity transformation or
(when the <code>--follow</code> option is specified) the most recently merged
frame's final alignment.  (Note that changes in release 0.5.0 are not reflected
here; these changes affect the interaction of the --follow and --trans-load
flags.)

<p>Once the initial transformation is determined, an initial perturbation
amount is selected, and represents the step size by which each of the
transformation parameters are changed.  In translational or Euclidean
alignment, the perturbation amount is applied to translation -- in units of
pixels on the two image axes -- and rotation -- in units of degrees about the
image center.  (In version 0.4.8 and later, an additional configurable upper
bound constrains rotational perturbation separately, preventing, e.g., a
360 degree perturbation of rotation.)  In the case of projective alignment, the
perturbation amount is applied to the position of the corners of the projected
quadrilateral in units of pixels, where the projection is from the boundary
of the supplemental image into the coordinate system of the accumulated image.

<p>If possible, transformation parameters are changed to decrease the <a
href="error/">error</a> between the two images being
aligned.  The perturbation amount is halved whenever it is determined that no
parameter change of this size improves the alignment of the images.  A lower
bound on the perturbation amount determines when the alignment is complete.

<p>The order in which parameters are considered for change is specified in the
source code, and has the following property: No modified parameter is
considered for further change until all other parameters have been considered.
A consequence of this property is that parameters are always considered in a
fixed (round robin) order.

<p>When multiple levels of detail are used, the error may be calculated on
images with a reduced level of detail.  ALE versions 0.1.1 through 0.4.7 use a
level of detail twice as fine as the perturbation amount for perturbation
amounts larger than two, and full detail otherwise.  Later versions default to
this behavior, but can be configured differently.  Earlier versions do not use
reduced levels of detail.

<h2>Properties</h2>

<p>Several assumptions were made throughout the design and testing of the
algorithm outlined above.  These assumptions are outlined below.

<p>The algorithm is based on a hill-climbing approach, which requires that any
local minimum reachable from the starting point by traveling a path of
decreasing error is also a global minimum (or, in this case, the correct
alignment).  While it is possible that the algorithm outlined above succeeds in
some cases for which hill-climbing fails, it is still susceptible to entrapment
in local minima.

<p>As outlined above, depending on program options, transformation parameters
may be changed by perturbations of several units (degrees or pixels) early in
the alignment process.  As long as no change of this magnitude moves the
transformation out of the 'bowl' in which the minimum error -- and hence
correct alignment -- lies, this is not a problem.  However, it might break in
some cases where a hill-climbing approach would succeed.  (Notably, simulated
annealing suffers from a similar problem, and it seems likely that a case could
be constructed in such a way that the algorithm outlined above -- like
simulated annealing -- could, contrarily, succeed where hill-climbing fails.)

<p>Finally, the use of reduced level-of-detail relies on a high signal-to-noise
ratio at low frequencies.  Fortunately, this assumption seems to generally
hold, but camera defects or radio interference could violate the assumption,
possibly resulting in misalignment.

<h2>Use of Alignment Classes</h2>

<p>ALE is likely to be most useful when corresponding regions of different
frames can be aligned by one of the available alignment classes.

<p>As described by Steve Mann in his work on <a
href="http://wearcam.org/orbits/">Video Orbits</a>, the projective
transformation offers particular versatility for camera imaging of (ideal
Lambertian) flat scenes.  In this case, any change in camera position and
orientation can be corrected as long as points always have a defined projection
onto the rendering plane (for which ALE uses the base of the pyramid
<b>R<sub>1</sub></b>).

<p>In camera imaging of scenes with depth, correction for orientation is almost
the same as for flat scenes, since, if focus and lens distortion is ignored, a
scene with depth is indistinguishable from a flat scene from the perspective of
a camera whose position is fixed.

<p>For sequences of camera images with small changes in position or
orientation, the projective transformations for alignment may closely
approximate Euclidean transformations; in this case, using Euclidean
transformations may achieve similar results and may require less time for
alignment, since there are fewer parameters to tweak (three parameters
instead of eight).

<p>In the case of flatbed scanners that preserve the relative height and width
of scans, any change in the position or orientation of flat objects can be
corrected using the Euclidean alignment class.</p>

<p>If a flatbed scanner does not preserve relative height and width, but does
preserve straight lines, then any change in the position or orientation of flat
objects can be corrected with the projective alignment class.

<p>However, even if a transformation is within the alignment class used, the
alignment algorithm may still be unable to approximate it.

<h2>Alignment in the case of Extended Renderings</h2>

<p>By using the --extend flag, ALE can be used to create image mosaics spanning
a spatial region larger than that represented by any single image in the frame
sequence.  In these cases, if adjacent frames in the sequence tend to be more
closely aligned with each other than they are with the original frame, it may
be helpful to also use the --follow flag as a hint to the alignment algorithm.
</p>


<br><br>
<small>
</small>

<hr>
<i>Copyright 2002, 2003 <a href="mailto:dhilvert@auricle.dyndns.org">David Hilvert</a></i>
<p>Verbatim copying and distribution of this entire article is permitted in any
medium, provided this notice is preserved.
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