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<html>
<title>Error Functions</title>
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</head><body>
<table align=right valign=top width=160>
<td valign=top height=600 width=160>
<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>
</table>
<p><b>[ <a href="../">Up</a> | <a href="mc/">Monte Carlo</a> ]</b></p>
<h1>Error Functions</h1>
<p>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.
<!--
The error functions used by ALE are based on two principles:
<ol>
<li>A relatively small change in camera position or orientation should be
reflected by a small transformation.
<li>When a new frame is merged, the accumulated image should change as little
as possible.
</ol>
-->
<!--
<p>Since sufficiently large transformations can move a new frame so far that it
no longer overlaps the accumulated image, it is important that the alignment
process not mistakenly move images too far, although in principle it is
possible, as in the case of a chain-link fence, that two fairly similar images
do, in fact, represent areas that are spatially separated. This is the
motivation for the first criterion.
<p>The second criterion is based on the general idea that, if two inputs to ALE
look vaguely like a petunia, the output should look like a petunia also. Hence
it should typically be the case that the accumulated image not change much when
new frames are merged.
-->
<p>ALE defines error functions for each pixel of the accumulated image, and
also a comprehensive error function that summarizes the error over all
pixels.
<h3>Pre-Alignment Exposure Registration</h3>
<p>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.
<h3>Per-Pixel Error Functions</h3>
<!-- <p>(Alternatives to normalization exist, including the use of inner products; see
Steve Mann's paper, available on the website linked in footnote 1, for a
discussion of alignment approaches.) -->
<!--
<p>Based on principle 2, ALE calculates the difference between each accumulated
image pixel and the value with which it would be merged given a candidate
transformation. This latter value is the <i>overlapping value</i>, as
described in the section on <a href="../../merging/">merging</a>. This difference
is then raised by the error metric exponent, as specified by the --metric
option. The exponent defaults to 2.0 for ALE versions 0.1.0 and later, or 1.0
for version 0.0.0. Hence,
-->
<p>The per-pixel error function for pixel <i>(i, j)</i> in the accumulated
image <i>A</i>, frame <i>B</i>, and transformation <i>T</i>, is:
<p><blockquote>
<pre>
<i>p(i, j, A, B, T) = Abs( A(i, j) - B(T_inverse(i, j)) )<sup>metric_exponent</sup></i>
</pre>
</blockquote>
<p>where <i>Abs()</i> is the absolute value function and <i>B(T_inverse(i,
j))</i> is determined by bilinear interpolation. If not specified by the
--metric option, <i>metric_exponent</i> defaults to 2.0 for ALE versions 0.1.0
and later, or 1.0 for version 0.0.0.
<p>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.)
<p>In addition to the per-pixel error function, a per-pixel <i>maximum error
estimator</i> is calculated, as follows:
<p><blockquote>
<pre>
<i>p_max(i, j, A, B, T) = Max( A(i, j), B(T_inverse(i, j)) )<sup>metric_exponent</sup></i>
</pre>
</blockquote>
<p>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.)
<h3>Comprehensive Error Functions</h3>
There are two varieties of comprehensive error function used by ALE: an
<i>exhaustive</i> error function and, in versions 0.4.3 and later, a
<a href="mc/"><i>Monte Carlo</i></a> error function. Whereas the exhaustive function
evaluates the error for each pixel in the accumulated image, the <a href="mc/">Monte
Carlo</a> function evaluates only a subset of pixels. <!-- The latter may require
less time to compute, possibly at the expense of precision. -->
<p>If we consider <i>Sum[]</i> 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 <i>A</i>, frame
<i>B</i>, and transformation <i>T</i>, is:
<p><blockquote>
<pre>
<i>E(A, B, T) = (Sum [ p(i, j, A, B, T) ] / Sum [ p_max(i, j, A, B, T) ])<sup>(1/metric_exponent)</sup></i>
</pre>
</blockquote>
<br><br>
<hr>
<i>Copyright 2002, 2003, 2004 <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.
</body>
</html>

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<html>
<title>Monte Carlo Error Function</title>
<style type="text/css">
TABLE.ba { max-width: 678; text-align: center; padding-bottom: 15; padding-top: 5}
TABLE.inline { padding-right: 300; clear: left}
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</style>
</head><body>
<table align=right valign=top width=160>
<td valign=top height=600 width=160>
<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>
</table>
<p><b>[ <a href="../">Up</a> ]</b></p>
<h1>Monte Carlo Error Function</h1>
Monte Carlo alignment can decrease the time required to align large images,
since performing coordinate transformations and memory accesses at every pixel
can be expensive. Sections in this page describe the motivation for Monte
Carlo alignment, the ratio calculations used, the sampling algorithm, cache
behavior for the algorithm, randomization approaches, typical deviations from
the specified ratios, special handling of level-of-detail, and practical
observations regarding the use of Monte Carlo alignment.
<h2>Motivation</h2>
Performing large numbers of coordinate transformations and memory accesses in
order to determine alignment error can be computationally expensive. One
approach to mitigating this expense is to use reduced level-of-detail.
However, using reduced level-of-detail can also reduce alignment precision. In
particular, reducing the level of detail by a factor of two can make impossible
the task of precisely aligning a horizontal line one pixel high. However, if
just a few pixels from the line are sampled at full detail, exact alignment is
possible.
<h2>Ratio Calculations</h2>
<p>In Monte Carlo alignment, a ratio
<blockquote>
<i>s = (expected # of pixel
samples) / (# of total pixels in the accumulated image)</i>
</blockquote>
is specified. From this ratio, a new ratio
<blockquote>
<i>u = (expected # of unsampled pixels) / (expected # of sampled pixels)</i>
</blockquote>
is calculated. Pixels are sampled in such a manner that <i>u</i> is
approximately satisified.
<p>At this stage, the region of overlap with the new frame is not considered.
With a limited area of overlap, the number of actual samples contributing to
the final error value will typically be reduced proportionally. (Angelo Pesce
has pointed out that better approaches may be possible, wherein explicit
calculation of overlapping areas reduces the number of coordinate
transformations performed.)
<h2>Sampling Algorithm</h2>
<p>Pixels are considered in order of index, where the accumulated image pixel
at position <i>(i, j)</i> is numbered with an index <i>(i * width + j)</i>.
In order of index, we skip and sample pixels in such a manner that the
expected size of a run of consecutive skipped pixels preceding a sampled pixel
is <i>u</i>. We select the size of each run of consecutive skipped pixels as
follows:
<p>If <i>2 * u</i> is an integer, then we draw uniformly from integer values in
the interval <i>[0,2u]</i>. If it is not an integer, then we draw from integer values in
the interval <i>[0,2u + 1]</i> in such a manner that integer values in
<i>[0,2u]</i> are equally likely to be chosen. (There is only one probability
distribution of this kind that satisfies the expected value <i>u</i>. Version
0.4.3 deviates slightly from this distribution, and so a deviation in the
expected value of <i>s</i> occurs, as outlined in this <a
href="ratios/">table</a>. This problem is fixed in version 0.4.4.)
<p>(Also, see the section below on interaction with level-of-detail.)
<h2>Cache behavior</h2>
<p>Since indices are monotonically increasing in memory address, this approach
to sampling may make effective use of memory cache where other approaches (e.g.
repeated random draws from the entire index space) would not.
<h2>Randomization</h2>
ALE versions 0.4.7 and earlier do not reseed the pseudorandom number generator,
and so a new random subset is selected every time the error function is
evaluated. Hence, as more or fewer of the pixels critical to alignment are
sampled, the reported alignment can worsen or improve even in the absence of
any change in transformation.
<p>With this approach, since many transformations are inspected during the
alignment of any given frame, it is likely, especially with greater precision
of alignment, that some measured differences between transformations are due to
a difference in sample sets rather than a difference in alignment accuracy.
<p>By reseeding the pseudorandom number generator, ALE versions 0.4.8 and later
instead use a consistent set of pixels from the accumulated image when
comparing two transformations. Tests sampling 3% of pixels from a set of
320x240 frames indicate that this approach improves alignment.
<h2>Sampling characteristics</h2>
For an image with 100,000 pixels and specified
<i>s</i> in the interval <i>[0.005,0.995]</i>, ALE's sampling method results in
a ratio <i>s</i> within 0.000003 of the specified <i>s</i>. This number
improves with image size. These results are outlined in the <a href="ratios/">table</a> linked
above. However, note that s only represents an expected value, and the actual number
of sampled pixels may vary by more than the numbers given here.
<h2>Interaction with level-of-detail</h2>
<p>When reduced level-of-detail is used, the number of reduced-detail pixels
sampled is taken to be a percentage of the total number of pixels in the
full-detail image, rather than as a percentage of the total number of pixels in
the reduced-detail image. (When this fraction of pixels in the full-detail
image is more than the number of reduced-detail pixels available, all
reduced-detail pixels are used.) This may improve the likelihood of successful
alignment, but may also add overhead to the alignment process.
<h2>Use of Monte Carlo Alignment</h2>
<p> If it is not known in advance what settings will work well for a series of
frames, it may be desirable to begin by sampling a small percentage of pixels,
saving the results of alignment, and then, if the output suggests that proper
alignment is occuring, performing more precise alignment with a larger
percentage of pixels on later passes, using smaller perturbation upper bounds.
If alignment problems occur on the first pass, the percentage of pixels can be
increased and alignment performed again.</p>
<br><br>
<hr>
<i>Copyright 2002, 2003, 2004 <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.
</body>
</html>

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<html>
<head>
<title>Expected sample ratio for --mc &lt;x&gt; argument</title>
<style type="text/css">
TABLE.ba { max-width: 678; text-align: center; padding-bottom: 15; padding-top: 5}
TABLE.inline { padding-right: 300; clear: left}
TD.text_table {padding-left: 2; padding-right: 2; border-width: 1}
H2 {clear: left}
P {max-width: none; padding-right: 300; clear: left}
BLOCKQUOTE {padding-right: 400 }
LI {max-width: 640; clear: left}
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P.header {max-width: none; width: auto; padding-left: 0}
HR.main {max-width: 640; clear: left; padding-left: 0; margin-left: 0}
HR.footer {clear: both}
</style>
</head><body>
<table align=right valign=top width=160>
<td valign=top height=600 width=160>
<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>
</table>
<pre>
Expected sample ratio for specified --mc &lt;x&gt; argument in a 100,000 pixel
image. This table illustrates the effects of a bug in ALE version 0.4.3.
Column A: Specified expected sample ratio x*10^-2
Column B: Expected sample ratio in release 0.4.4
Column C: Expected sample ratio in release 0.4.3
A B C (buggy)
0.005000 0.004997 0.004997
0.010000 0.009997 0.009997
0.015000 0.014997 0.014997
0.020000 0.019997 0.019997
0.025000 0.024997 0.024997
0.030000 0.029997 0.029995
0.035000 0.034997 0.034995
0.040000 0.039997 0.039997
0.045000 0.044997 0.044991
0.050000 0.049997 0.049997
0.055000 0.054997 0.054987
0.060000 0.059997 0.059984
0.065000 0.064997 0.064984
0.070000 0.069997 0.069975
0.075000 0.074997 0.074973
0.080000 0.079997 0.079997
0.085000 0.084997 0.084957
0.090000 0.089997 0.089964
0.095000 0.094997 0.094986
0.100000 0.099997 0.099997
0.105000 0.104997 0.104983
0.110000 0.109997 0.109945
0.115000 0.114997 0.114901
0.120000 0.119997 0.119895
0.125000 0.124997 0.124997
0.130000 0.129997 0.129858
0.135000 0.134997 0.134898
0.140000 0.139997 0.139847
0.145000 0.144997 0.144862
0.150000 0.149997 0.149795
0.155000 0.154997 0.154909
0.160000 0.159997 0.159719
0.165000 0.164997 0.164867
0.170000 0.169997 0.169756
0.175000 0.174997 0.174638
0.180000 0.179997 0.179839
0.185000 0.184997 0.184730
0.190000 0.189997 0.189526
0.195000 0.194997 0.194606
0.200000 0.199997 0.199997
0.205000 0.204997 0.204556
0.210000 0.209997 0.209354
0.215000 0.214997 0.214412
0.220000 0.219997 0.219750
0.225000 0.224997 0.224681
0.230000 0.229997 0.229272
0.235000 0.234997 0.234082
0.240000 0.239997 0.239128
0.245000 0.244997 0.244426
0.250000 0.249998 0.249997
0.255000 0.254998 0.254371
0.260000 0.259998 0.258926
0.265000 0.264998 0.263675
0.270000 0.269998 0.268631
0.275000 0.274998 0.273807
0.280000 0.279998 0.279218
0.285000 0.284998 0.284881
0.290000 0.289998 0.289338
0.295000 0.294998 0.293710
0.300000 0.299998 0.298243
0.305000 0.304998 0.302947
0.310000 0.309998 0.307830
0.315000 0.314998 0.312904
0.320000 0.319998 0.318180
0.325000 0.324998 0.323669
0.330000 0.329998 0.329386
0.335000 0.334998 0.334671
0.340000 0.339998 0.338773
0.345000 0.344998 0.343003
0.350000 0.349998 0.347366
0.355000 0.354998 0.351870
0.360000 0.359998 0.356520
0.365000 0.364998 0.361324
0.370000 0.369998 0.366290
0.375000 0.374998 0.371427
0.380000 0.379998 0.376742
0.385000 0.384998 0.382247
0.390000 0.389998 0.387950
0.395000 0.394998 0.393863
0.400000 0.399998 0.399998
0.405000 0.404998 0.403795
0.410000 0.409998 0.407690
0.415000 0.414998 0.411686
0.420000 0.419998 0.415788
0.425000 0.424998 0.419998
0.430000 0.429998 0.424322
0.435000 0.434998 0.428765
0.440000 0.439998 0.433332
0.445000 0.444998 0.438026
0.450000 0.449998 0.442855
0.455000 0.454998 0.447824
0.460000 0.459998 0.452939
0.465000 0.464998 0.458207
0.470000 0.469998 0.463635
0.475000 0.474998 0.469229
0.480000 0.479998 0.474998
0.485000 0.484998 0.480951
0.490000 0.489998 0.487095
0.495000 0.494998 0.493441
0.500000 0.499998 0.499998
0.505000 0.504998 0.503365
0.510000 0.509998 0.506801
0.515000 0.514998 0.510308
0.520000 0.519998 0.513887
0.525000 0.524998 0.517542
0.530000 0.529999 0.521275
0.535000 0.534999 0.525088
0.540000 0.539999 0.528984
0.545000 0.544999 0.532966
0.550000 0.549999 0.537036
0.555000 0.554999 0.541197
0.560000 0.559999 0.545453
0.565000 0.564999 0.549807
0.570000 0.569999 0.554262
0.575000 0.574999 0.558822
0.580000 0.579999 0.563491
0.585000 0.584999 0.568272
0.590000 0.589999 0.573169
0.595000 0.594999 0.578188
0.600000 0.599999 0.583332
0.605000 0.604999 0.588606
0.610000 0.609999 0.594016
0.615000 0.614999 0.599566
0.620000 0.619999 0.605262
0.625000 0.624999 0.611110
0.630000 0.629999 0.617116
0.635000 0.634999 0.623287
0.640000 0.639999 0.629628
0.645000 0.644999 0.636149
0.650000 0.649999 0.642856
0.655000 0.654999 0.649757
0.660000 0.659999 0.656862
0.665000 0.664999 0.664178
0.670000 0.669999 0.668341
0.675000 0.674999 0.670885
0.680000 0.679999 0.673468
0.685000 0.684999 0.676091
0.690000 0.689999 0.678755
0.695000 0.694999 0.681461
0.700000 0.699999 0.684210
0.705000 0.704999 0.687002
0.710000 0.709999 0.689839
0.715000 0.714999 0.692721
0.720000 0.719999 0.695651
0.725000 0.724999 0.698629
0.730000 0.729999 0.701657
0.735000 0.734999 0.704735
0.740000 0.739999 0.707864
0.745000 0.744999 0.711047
0.750000 0.749999 0.714285
0.755000 0.754999 0.717578
0.760000 0.759999 0.720929
0.765000 0.764999 0.724339
0.770000 0.769999 0.727810
0.775000 0.774999 0.731343
0.780000 0.780000 0.734939
0.785000 0.785000 0.738601
0.790000 0.790000 0.742331
0.795000 0.795000 0.746129
0.800000 0.800000 0.749999
0.805000 0.805000 0.753943
0.810000 0.810000 0.757961
0.815000 0.815000 0.762057
0.820000 0.820000 0.766233
0.825000 0.825000 0.770491
0.830000 0.830000 0.774834
0.835000 0.835000 0.779264
0.840000 0.840000 0.783783
0.845000 0.845000 0.788395
0.850000 0.850000 0.793103
0.855000 0.855000 0.797909
0.860000 0.860000 0.802817
0.865000 0.865000 0.807829
0.870000 0.870000 0.812949
0.875000 0.875000 0.818181
0.880000 0.880000 0.823529
0.885000 0.885000 0.828996
0.890000 0.890000 0.834586
0.895000 0.895000 0.840304
0.900000 0.900000 0.846154
0.905000 0.905000 0.852140
0.910000 0.910000 0.858268
0.915000 0.915000 0.864542
0.920000 0.920000 0.870968
0.925000 0.925000 0.877551
0.930000 0.930000 0.884297
0.935000 0.935000 0.891213
0.940000 0.940000 0.898305
0.945000 0.945000 0.905579
0.950000 0.950000 0.913043
0.955000 0.955000 0.920705
0.960000 0.960000 0.928571
0.965000 0.965000 0.936652
0.970000 0.970000 0.944954
0.975000 0.975000 0.953488
0.980000 0.980000 0.962264
0.985000 0.985000 0.971292
0.990000 0.990000 0.980583
0.995000 0.995000 0.990148