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<html>
<title>Error</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</h1>
<p>Error calculations are used to determine whether a transformation is "good"
or not, and in particular, whether one transformation is better or worse than
another. A smaller error indicates a better transformation, from the perspective
of the alignment algorithm.
<p>ALE defines an error value for each pixel of the alignment reference image, and also
comprehensive error, which 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 result of this step is 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 for more details.
<h3>Per-Pixel Error</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 alignment reference
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 for pixel <i>(i, j)</i> in the alignment reference
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 (unless a different interpolant
is specified using <code>--afilter</code>). If not specified by the --metric
option, <i>metric_exponent</i> defaults to 2.
<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 per-pixel error, 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</h3>
<p>There are two varieties of comprehensive error used by ALE: <i>exhaustive</i>
error (enabled by <code>--no-mc</code>) and <a
href="mc/"><i>Monte Carlo</i></a> error (enabled by <code>--mc</code>).
Whereas the exhaustive approach evaluates the error for each pixel in the
alignment reference image, the <a href="mc/">Monte Carlo</a> approach evaluates
only a subset of pixels.
<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 alignment reference 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</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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H2 {clear: left}
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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>
<p><b>[ <a href="../">Up</a> ]</b></p>
<h1>Monte Carlo Error</h1>
<p>Use of Monte Carlo error 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 error.
<h2>Motivation</h2>
<p>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.
<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>
<p>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 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>
<p>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. (The results are outlined in a <a
href="ratios/">table</a>.) 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.
<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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<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}
TD {padding-left: 0; padding-right: 0; border-width: 0}
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<BODY TEXT="#000000" BGCOLOR="#FFFFFF" LINK="#0000EF" VLINK="#55188A" ALINK="#FF0000">
<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>
7119814098<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