doc/html-original/ALE/download/ale-0.7.x-tech/chain/index.html

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  29 <a href="http://auricle.dyndns.org/ALE/">
  30 <big>ALE</big>
  31 <br>
  32 Image Processing Software
  33 <br>
  34 <br>
  35 <small>Deblurring, Anti-aliasing, and Superresolution.</small></a>
  36 <br><br>
  37 <big>
  38 Local Operation
  39 </big>
  40 <hr>
  41 localhost<br>
  42 5393119533<br>
  43 </table>
  44
  45
  46
  47
  48 <p><b>[ <a href="../">Up</a> ]</b></p>
  49 <h1>ALE Rendering Chains</h1>
  50
  51 <p>In cases of spatially non-uniform resolution, rendering chains can maintain low
  52 aliasing in poorly-resolved regions while preserving detail in well-resolved
  53 regions.  Each chain is based on a sequence of rendering invariants, each
  54 allowing first, last, average, minimum, or maximum pixel values to be rendered.
  55 For a given invariant, exclusion regions are honored by default, but can
  56 optionally be ignored.  Finally, for a given invariant, resolution can be
  57 limited to the minimum of the input and output images, to prevent aliasing, or
  58 can use the full resolution of the output image, to prevent loss of fine
  59 details.</p>
  60
  61 <h2>Parameters</h2>
  62
  63 Parameters for rendering chain or rendering invariant <b>r</b> are as follows:</b>
  64
  65 <blockquote>
  66 <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>
  67
  68 <br><br>
  69
  70 <table>
  71 <tr><th align=left>Parameter<th><pre>   </pre><th align=left>Description</tr>
  72 <tr><td>n</td><td></td><td> the number of images contributing to output</tr>
  73 <tr><td>n'</td><td></td><td> the current image index</tr>
  74 <tr><td>(i, j)</td><td></td><td> the output pixel position</tr>
  75 <tr><td>(i', j')</td><td></td><td> the pixel position in the current image</tr>
  76 <tr><td>E<sub>x</sub></td><td></td><td> linear exposure adjustment for image x</tr>
  77 <tr><td>G</td><td></td><td> gamma correction</tr>
  78 <tr><td>P<sub>x</sub></td><td></td><td> projective transformation for image x</tr>
  79 <tr><td>B<sub>x</sub></td><td></td><td> barrel distortion for image x</tr>
  80 <tr><td>d<sub>x</sub></td><td></td><td> image x</tr>
  81 </table>
  82
  83 </blockquote>
  84
  85 Parameters for scaled sampling filter with exclusion (SSFE) <b>e</b> are as follows:
  86
  87 <blockquote>
  88 <b>e(n', i, j, i', j', E, G, P, B, d)</b>
  89
  90 <br><br>
  91
  92 <table>
  93 <tr><th align=left>Parameter<th><pre>   </pre><th align=left>Description</tr>
  94 <tr><td>n'</td><td></td><td> the current image index</tr>
  95 <tr><td>(i, j)</td><td></td><td> the output pixel position</tr>
  96 <tr><td>(i', j')</td><td></td><td> the input pixel position</tr>
  97 <tr><td>E</td><td></td><td> linear exposure adjustment</tr>
  98 <tr><td>G</td><td></td><td> gamma correction</tr>
  99 <tr><td>P</td><td></td><td> projective transformation</tr>
 100 <tr><td>B</td><td></td><td> barrel distortion</tr>
 101 <tr><td>d</td><td></td><td> image</tr>
 102 </table>
 103
 104 </blockquote>
 105
 106
 107
 108 Parameters for scaled sampling filter (SSF) <b>s</b> are as
 109 follows:</b>
 110
 111 <blockquote>
 112 <b>s(i, j, i', j', P, B, k)</b>
 113
 114 <br><br>
 115
 116 <table>
 117 <tr><th align=left>Parameter<th><pre>   </pre><th align=left>Description</tr>
 118 <tr><td>(i, j)</td><td></td><td> the output pixel position</tr>
 119 <tr><td>(i', j')</td><td></td><td> the input pixel position</tr>
 120 <tr><td>P</td><td></td><td> projective transformation</tr>
 121 <tr><td>B</td><td></td><td> barrel distortion</tr>
 122 <tr><td>k</td><td></td><td> certainty values</tr>
 123 </table>
 124
 125 </blockquote>
 126
 127 Parameters for sampling filter <b>f</b> are as follows:</b>
 128
 129 <blockquote>
 130 <b>f(p)</b>
 131
 132 <br><br>
 133
 134 <table>
 135 <tr><th align=left>Parameter<th><pre>   </pre><th align=left>Description</tr>
 136 <tr><td>p</td><td></td><td>position offset p = (i, j)</tr>
 137 </table>
 138
 139 </blockquote>
 140
 141 <h2>Chains</h2>
 142
 143 <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>:
 144
 145 <blockquote>
 146 <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>) = &sum;<sub>n'&isin;0..n</sub> &sum;<sub>(i', j')&isin;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>
 147 </blockquote>
 148
 149
 150 If <b>w<sub>t</sub></b> is the weight threshold (ALE option 'wt'), and there exists a smallest <b>x</b> such that:
 151
 152 <blockquote>
 153 <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>) &ge; w<sub>t</sub></b>
 154 </blockquote>
 155
 156 Then <b>c</b> gives:
 157
 158 <blockquote>
 159 <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>
 160 </blockquote>
 161
 162 Otherwise, <b>c</b> gives:
 163
 164 <blockquote>
 165 <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>
 166 </blockquote>
 167
 168 If both of the expressions above are undefined, then <b>c</b> gives:
 169
 170 <blockquote>
 171 <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>
 172 </blockquote>
 173
 174
 175 <h2>Invariants</h2>
 176
 177 <p>There are five types of rendering invariants, all of which are parameterized
 178 with a scaled sampling filter with exclusion, denoted here by the symbol <b>e</b>.  In particular, an
 179 invariant can be of initial, final, maximal, minimal, or average type.
 180
 181 <h3>Initial</h3>
 182
 183 <p>If an invariant <b>v</b> is of initial type, then choose the smallest <b>m</b> such that
 184 the following expression is non-zero:
 185
 186 <blockquote>
 187 <b>&sum;<sub>(i',j')&isin;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>
 188 </blockquote>
 189
 190 If such an <b>m</b> can be chosen, then, using the C trinary if-else operator to express condition:
 191
 192 <blockquote>
 193 <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>
 194 </blockquote>
 195
 196 Otherwise:
 197
 198 <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>
 199 </blockquote>
 200
 201 <h3>Final</h3>
 202
 203 <p>If an invariant <b>v</b> is of final type, then choose the largest <b>m</b> such that
 204 the following expression is non-zero:
 205
 206 <blockquote>
 207 <b>&sum;<sub>(i',j')&isin;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>
 208 </blockquote>
 209
 210 If such an <b>m</b> can be chosen, then, using the C trinary if-else operator to express condition:
 211
 212 <blockquote>
 213 <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>
 214 </blockquote>
 215
 216 Otherwise:
 217
 218 <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>
 219 </blockquote>
 220
 221 <h3>Minimal</h3>
 222
 223 <p>If an invariant <b>v</b> is of minimal type, then choose <b>m</b> such that the following
 224 expression is defined and minimal:
 225
 226 <blockquote>
 227 <b>&sum;<sub>(i',j')&isin;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>) / &sum;<sub>(i',j')&isin;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>
 228 </blockquote>
 229
 230 If such an <b>m</b> can be chosen, then, using the C trinary if-else operator to express condition:
 231
 232 <blockquote>
 233 <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>
 234 </blockquote>
 235
 236 Otherwise:
 237
 238 <blockquote>
 239 <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>
 240 </blockquote>
 241
 242 <h3>Maximal</h3>
 243
 244 <p>If an invariant <b>v</b> is of maximal type, then choose <b>m</b> such that the following
 245 expression is defined and maximal:
 246
 247 <blockquote>
 248 <b>&sum;<sub>(i',j')&isin;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>) / &sum;<sub>(i',j')&isin;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>
 249 </blockquote>
 250
 251 If such an <b>m</b> can be chosen, then, using the C trinary if-else operator to express condition:
 252
 253 <blockquote>
 254 <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>
 255 </blockquote>
 256
 257 Otherwise:
 258
 259 <blockquote>
 260 <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>
 261 </blockquote>
 262
 263 <h3>Average</h3>
 264
 265 <p>If an invariant <b>v</b> is of average type, then:
 266
 267 <blockquote>
 268 <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>
 269 </blockquote>
 270
 271 <h2>Scaled Sampling Filter with Exclusion (SSFE)</h2>
 272
 273 <p>A scaled sampling filter with exclusion <b>e</b> is parameterized with a scaled sampling filter <b>s</b>, and
 274 can be of two types: it can either honor exclusion regions or not.  Define
 275 <b>is_exclude(n', i, j)</b> to be false if point <b>(i, j)</b> is not excluded
 276 for frame <b>n'</b>, or if exclusion regions are not being honored.  Then,
 277 using the C trinary if-else operator to express condition:
 278
 279 <blockquote>
 280 <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, &kappa;G<sup>-1</sup>E<sup>-1</sup>d)</b>
 281 </blockquote>
 282
 283 <p>Where <b>&kappa;</b> is the operator for <a href="../certainty/">certainty</a>.
 284
 285 <h2>Scaled Sampling Filter (SSF)</h2>
 286
 287 <p>Define <b>bayer(i, j)</b> to be a function that returns an RGB value whose
 288 channels are either zero or one, depending on whether that color is sampled at
 289 <b>(i, j)</b>.
 290
 291 <p>A scaled sampling filter <b>s</b> is parameterized with a sampling filter
 292 <b>f</b>, and can be one of two types: fine or coarse.  If it is fine, then,
 293 using <b>P</b> and <b>B</b> as functions:
 294
 295 <blockquote>
 296 <b>s(i, j, i', j', P, B, k) = bayer(i', j') * k(i', j') * f(B(P(i', j')) - (i, j))</b>
 297 </blockquote>
 298
 299 <p>If SSF <b>s</b> is coarse, then color channels are handled separately,
 300 depending on their density relative to the output image, at point <b>(i, j)</b>
 301 in the output image.  In particular, bayer patterns can cause some colors to be
 302 more dense than others.  If the local density of channel <b>h</b> is lower in
 303 each dimension than the density of channel <b>h</b> in the output image, then:
 304
 305 <blockquote>
 306 <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>
 307 </blockquote>
 308
 309 <p>Otherwise, if channel <b>h</b> is locally less dense by a factor <b>d</b> in
 310 exactly one dimension of the input image, then set <b>d_pair</b> equal to
 311 <b>(1, d)</b> or <b>(d, 1)</b>, according to the dimension, and:
 312
 313 <blockquote>
 314 <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>
 315 </blockquote>
 316
 317 <p>Otherwise, channel <b>h</b> is locally at least as dense in both dimensions of
 318 the input image as it is dense in the output image.  In this case:
 319
 320 <blockquote>
 321 <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>
 322 </blockquote>
 323
 324
 325
 326 <h2>Sampling Filter</h2>
 327
 328 <p>Sampling filters can be one of the following:</p>
 329 <table>
 330 <tr><th align=left>Type</th><th><pre>   </pre></th><th align=left>Description</th></tr>
 331 <tr><td>sinc<td><td>Sinc filter: (sin &pi;x) / (&pi;x)</td>
 332 <tr><td>lanc:&lt;x&gt;<td><td>Lanczos, diameter x.
 333 <tr><td>triangle:&lt;x&gt;<td><td>Triangle, diameter x.
 334 <tr><td>box:&lt;x&gt;<td><td>Box, diameter x.
 335 <tr><td>zero<td><td>Zero function
 336 <tr><td>&lt;f&gt;*&lt;f&gt;<td><td>Pointwise multiplication (windowing)
 337 </table>
 338
 339
 340 <br>
 341 <hr>
 342 <i>Copyright 2002, 2003, 2004 <a href="mailto:dhilvert@auricle.dyndns.org">David Hilvert</a></i>
 343 <p>Verbatim copying and distribution of this entire article is permitted in any medium, provided this notice is preserved.
 344
 345
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