1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
<title>ALE Rendering Chains</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}
P.footer {max-width: none; width: auto; padding-left: 0}
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>
<p><b>[ <a href="../">Up</a> ]</b></p>
<h1>ALE Rendering Chains</h1>
<p>In cases of spatially non-uniform resolution, rendering chains can maintain low
aliasing in poorly-resolved regions while preserving detail in well-resolved
regions. Each chain is based on a sequence of rendering invariants, each
allowing first, last, average, minimum, or maximum pixel values to be rendered.
For a given invariant, exclusion regions are honored by default, but can
optionally be ignored. Finally, for a given invariant, resolution can be
limited to the minimum of the input and output images, to prevent aliasing, or
can use the full resolution of the output image, to prevent loss of fine
details.</p>
<h2>Parameters</h2>
Parameters for rendering chain or rendering invariant <b>r</b> are as follows:</b>
<blockquote>
<b>r(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>)</b>
<br><br>
<table>
<tr><th align=left>Parameter<th><pre> </pre><th align=left>Description</tr>
<tr><td>n</td><td></td><td> the number of images contributing to output</tr>
<tr><td>n'</td><td></td><td> the current image index</tr>
<tr><td>(i, j)</td><td></td><td> the output pixel position</tr>
<tr><td>(i', j')</td><td></td><td> the pixel position in the current image</tr>
<tr><td>E<sub>x</sub></td><td></td><td> linear exposure adjustment for image x</tr>
<tr><td>G</td><td></td><td> gamma correction</tr>
<tr><td>P<sub>x</sub></td><td></td><td> projective transformation for image x</tr>
<tr><td>B<sub>x</sub></td><td></td><td> barrel distortion for image x</tr>
<tr><td>d<sub>x</sub></td><td></td><td> image x</tr>
</table>
</blockquote>
Parameters for scaled sampling filter with exclusion (SSFE) <b>e</b> are as follows:
<blockquote>
<b>e(n', i, j, i', j', E, G, P, B, d)</b>
<br><br>
<table>
<tr><th align=left>Parameter<th><pre> </pre><th align=left>Description</tr>
<tr><td>n'</td><td></td><td> the current image index</tr>
<tr><td>(i, j)</td><td></td><td> the output pixel position</tr>
<tr><td>(i', j')</td><td></td><td> the input pixel position</tr>
<tr><td>E</td><td></td><td> linear exposure adjustment</tr>
<tr><td>G</td><td></td><td> gamma correction</tr>
<tr><td>P</td><td></td><td> projective transformation</tr>
<tr><td>B</td><td></td><td> barrel distortion</tr>
<tr><td>d</td><td></td><td> image</tr>
</table>
</blockquote>
Parameters for scaled sampling filter (SSF) <b>s</b> are as
follows:</b>
<blockquote>
<b>s(i, j, i', j', P, B, k)</b>
<br><br>
<table>
<tr><th align=left>Parameter<th><pre> </pre><th align=left>Description</tr>
<tr><td>(i, j)</td><td></td><td> the output pixel position</tr>
<tr><td>(i', j')</td><td></td><td> the input pixel position</tr>
<tr><td>P</td><td></td><td> projective transformation</tr>
<tr><td>B</td><td></td><td> barrel distortion</tr>
<tr><td>k</td><td></td><td> certainty values</tr>
</table>
</blockquote>
Parameters for sampling filter <b>f</b> are as follows:</b>
<blockquote>
<b>f(p)</b>
<br><br>
<table>
<tr><th align=left>Parameter<th><pre> </pre><th align=left>Description</tr>
<tr><td>p</td><td></td><td>position offset p = (i, j)</tr>
</table>
</blockquote>
<h2>Chains</h2>
<p>A chain <b>c</b> is based on a sequence of rendering invariants <b>v<sub>1</sub>, v<sub>2</sub>, ..., v<sub>max</sub></b>. For each <b>v<sub>x</sub></b>, define <b>w<sub>x</sub></b>:
<blockquote>
<b>w<sub>x</sub>(n, i, j, E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) = ∑<sub>n'∈0..n</sub> ∑<sub>(i', j')∈Dom[d<sub>n'</sub>]</sub> v<sub>x</sub>(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>)</b>
</blockquote>
If <b>w<sub>t</sub></b> is the weight threshold (ALE option 'wt'), and there exists a smallest <b>x</b> such that:
<blockquote>
<b>w<sub>x</sub>(n, i, j, E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) ≥ w<sub>t</sub></b>
</blockquote>
Then <b>c</b> gives:
<blockquote>
<b>c(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) = v<sub>x</sub>(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) / w<sub>x</sub>(n, i, j, E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>)</b>
</blockquote>
Otherwise, <b>c</b> gives:
<blockquote>
<b>c(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) = v<sub>max</sub>(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) / w<sub>max</sub>(n, i, j, E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>)</b>
</blockquote>
If both of the expressions above are undefined, then <b>c</b> gives:
<blockquote>
<b>c(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) = 0</b>
</blockquote>
<h2>Invariants</h2>
<p>There are five types of rendering invariants, all of which are parameterized
with a scaled sampling filter with exclusion, denoted here by the symbol <b>e</b>. In particular, an
invariant can be of initial, final, maximal, minimal, or average type.
<h3>Initial</h3>
<p>If an invariant <b>v</b> is of initial type, then choose the smallest <b>m</b> such that
the following expression is non-zero:
<blockquote>
<b>∑<sub>(i',j')∈Dom[d<sub>m</sub>]</sub> e(m, i, j, i', j', E<sub>m</sub>, G, P<sub>m</sub>, B<sub>m</sub>, d<sub>m</sub>)</b>
</blockquote>
If such an <b>m</b> can be chosen, then, using the C trinary if-else operator to express condition:
<blockquote>
<b>v(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) = (n' == m) ? e(m, i, j, i', j', E<sub>m</sub>, G, P<sub>m</sub>, B<sub>m</sub>, d<sub>m</sub>) : 0</b>
</blockquote>
Otherwise:
<blockquote> <b>v(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) = 0</b>
</blockquote>
<h3>Final</h3>
<p>If an invariant <b>v</b> is of final type, then choose the largest <b>m</b> such that
the following expression is non-zero:
<blockquote>
<b>∑<sub>(i',j')∈Dom[d<sub>m</sub>]</sub> e(m, i, j, i', j', E<sub>m</sub>, G, P<sub>m</sub>, B<sub>m</sub>, d<sub>m</sub>)</b>
</blockquote>
If such an <b>m</b> can be chosen, then, using the C trinary if-else operator to express condition:
<blockquote>
<b>v(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) = (n' == m) ? e(m, i, j, i', j', E<sub>m</sub>, G, P<sub>m</sub>, B<sub>m</sub>, d<sub>m</sub>) : 0</b>
</blockquote>
Otherwise:
<blockquote> <b>v(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) = 0</b>
</blockquote>
<h3>Minimal</h3>
<p>If an invariant <b>v</b> is of minimal type, then choose <b>m</b> such that the following
expression is defined and minimal:
<blockquote>
<b>∑<sub>(i',j')∈Dom[d<sub>m</sub>]</sub> E<sub>m</sub><sup>-1</sup>G<sup>-1</sup>d<sub>m</sub>(i',j') * e(m, i, j, i', j', E<sub>m</sub>, G, P<sub>m</sub>, B<sub>m</sub>, d<sub>m</sub>) / ∑<sub>(i',j')∈Dom[d<sub>m</sub>]</sub> e(m, i, j, i', j', E<sub>m</sub>, G, P<sub>m</sub>, B<sub>m</sub>, d<sub>m</sub>)</b>
</blockquote>
If such an <b>m</b> can be chosen, then, using the C trinary if-else operator to express condition:
<blockquote>
<b>v(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) = (n' == m) ? e(m, i, j, i', j', E<sub>m</sub>, G, P<sub>m</sub>, B<sub>m</sub>, d<sub>m</sub>) : 0</b>
</blockquote>
Otherwise:
<blockquote>
<b>v(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) = 0</b>
</blockquote>
<h3>Maximal</h3>
<p>If an invariant <b>v</b> is of maximal type, then choose <b>m</b> such that the following
expression is defined and maximal:
<blockquote>
<b>∑<sub>(i',j')∈Dom[d<sub>m</sub>]</sub> E<sub>m</sub><sup>-1</sup>G<sup>-1</sup>d<sub>m</sub>(i',j') * e(m, i, j, i', j', E<sub>m</sub>, G, P<sub>m</sub>, B<sub>m</sub>, d<sub>m</sub>) / ∑<sub>(i',j')∈Dom[d<sub>m</sub>]</sub> e(m, i, j, i', j', E<sub>m</sub>, G, P<sub>m</sub>, B<sub>m</sub>, d<sub>m</sub>)</b>
</blockquote>
If such an <b>m</b> can be chosen, then, using the C trinary if-else operator to express condition:
<blockquote>
<b>v(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) = (n' == m) ? e(m, i, j, i', j', E<sub>m</sub>, G, P<sub>m</sub>, B<sub>m</sub>, d<sub>m</sub>) : 0</b>
</blockquote>
Otherwise:
<blockquote>
<b>v(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) = 0</b>
</blockquote>
<h3>Average</h3>
<p>If an invariant <b>v</b> is of average type, then:
<blockquote>
<b>v(n, n', i, j, i', j', E<sub>0</sub>, ..., E<sub>n</sub>, G, P<sub>0</sub>, ..., P<sub>n</sub>, B<sub>0</sub>, ..., B<sub>n</sub>, d<sub>0</sub>, ..., d<sub>n</sub>) = e(n', i, j, i', j', E<sub>n'</sub>, G, P<sub>n'</sub>, B<sub>n'</sub>, d<sub>n'</sub>)</b>
</blockquote>
<h2>Scaled Sampling Filter with Exclusion (SSFE)</h2>
<p>A scaled sampling filter with exclusion <b>e</b> is parameterized with a scaled sampling filter <b>s</b>, and
can be of two types: it can either honor exclusion regions or not. Define
<b>is_exclude(n', i, j)</b> to be false if point <b>(i, j)</b> is not excluded
for frame <b>n'</b>, or if exclusion regions are not being honored. Then,
using the C trinary if-else operator to express condition:
<blockquote>
<b>e(n', i, j, i', j', E, G, P, B, d) = is_exclude(n', i, j) ? 0 : s(i, j, i', j', P, B, κG<sup>-1</sup>E<sup>-1</sup>d)</b>
</blockquote>
<p>Where <b>κ</b> is the operator for <a href="../certainty/">certainty</a>.
<h2>Scaled Sampling Filter (SSF)</h2>
<p>Define <b>bayer(i, j)</b> to be a function that returns an RGB value whose
channels are either zero or one, depending on whether that color is sampled at
<b>(i, j)</b>.
<p>A scaled sampling filter <b>s</b> is parameterized with a sampling filter
<b>f</b>, and can be one of two types: fine or coarse. If it is fine, then,
using <b>P</b> and <b>B</b> as functions:
<blockquote>
<b>s(i, j, i', j', P, B, k) = bayer(i', j') * k(i', j') * f(B(P(i', j')) - (i, j))</b>
</blockquote>
<p>If SSF <b>s</b> is coarse, then color channels are handled separately,
depending on their density relative to the output image, at point <b>(i, j)</b>
in the output image. In particular, bayer patterns can cause some colors to be
more dense than others. If the local density of channel <b>h</b> is lower in
each dimension than the density of channel <b>h</b> in the output image, then:
<blockquote>
<b>[s(i, j, i', j', P, B, k)]<sub>h</sub> = [bayer(i', j') * k(i', j') * f((i', j') - P<sup>-1</sup>(B<sup>-1</sup>(i, j)))]<sub>h</sub></b>
</blockquote>
<p>Otherwise, if channel <b>h</b> is locally less dense by a factor <b>d</b> in
exactly one dimension of the input image, then set <b>d_pair</b> equal to
<b>(1, d)</b> or <b>(d, 1)</b>, according to the dimension, and:
<blockquote>
<b>[s(i, j, i', j', P, B, k)]<sub>h</sub> = [bayer(i', j') * k(i', j') * f(d_pair * (B(P(i', j')) - (i, j)))]<sub>h</sub></b>
</blockquote>
<p>Otherwise, channel <b>h</b> is locally at least as dense in both dimensions of
the input image as it is dense in the output image. In this case:
<blockquote>
<b>[s(i, j, i', j', P, B, k)]<sub>h</sub> = [bayer(i', j') * k(i', j') * f(B(P(i', j')) - (i, j))]<sub>h</sub></b>
</blockquote>
<h2>Sampling Filter</h2>
<p>Sampling filters can be one of the following:</p>
<table>
<tr><th align=left>Type</th><th><pre> </pre></th><th align=left>Description</th></tr>
<tr><td>sinc<td><td>Sinc filter: (sin πx) / (πx)</td>
<tr><td>lanc:<x><td><td>Lanczos, diameter x.
<tr><td>triangle:<x><td><td>Triangle, diameter x.
<tr><td>box:<x><td><td>Box, diameter x.
<tr><td>zero<td><td>Zero function
<tr><td><f>*<f><td><td>Pointwise multiplication (windowing)
</table>
<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>
|
