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Is it possible to make an overlap measure that can be computed given the result pattern image and the command list without the intermediate steps? <br />
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Sketch of an induction prove that this is possible for the case that only one prototile type exists and for the case that n > 1 prototile types exists. <br />
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'''Case 1''': <u>One prototile type</u><br />
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<u>Given:</u><br />
* one prototile in an image $S with width $w_S and height $h_S on a transparent background with no borders; <br />
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: number of pixel in $S: $wh_S = $w_S * $h_S<br />
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: number of non-transparent pixel in $S: $s<br />
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* Pattern image $P with width $w_P and height $h_P<br />
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: number of pixel in $P: $wh_P = $w_P * $h_P<br />
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* command list with m 5-tuples ($S, $r, $m, $x, $y)<br />
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<u>Countable:</u><br />
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* number of transparent pixel in $P in step t: $gap_t <br />
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* number of non-transparent pixel in $P in step t: $p_t <br />
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<u>Compute:</u><br />
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* number of non-transparent pixel in $P in step t that are covered more than one time: $overlap_t<br />
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<u>Compositing process:</u><br />
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* step 0: $P is still empty => $gap_0 = $wh_P and $overlap_0 = $p_0 = 0<br />
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* step 1: $S is composed over $P according to the first 5-tuple => $gap_1 = $wh_P - $s and $overlap_1 = 0 and $p_1 = $s<br />
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* step 2: $S is composed over $P according to the second 5-tuple: $overlap_2 = 2 * $s - $p_2<br />
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...<br />
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* step k: $S is composed over $P according to the k'th 5-tuple: $overlap_k = k * $s - $p_k <br />
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...<br />
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* step m: $S is composed over $P according to the m'th 5-tuple: <span style="color:red;">$overlap_m = m * $s - $p_m</span> <br />
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'''Case 2''': <u>n > 1 prototile types</u><br />
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<u>Given:</u><br />
* A set of n > 1 different prototile types @S = ($S_i | i = 1, …, n) with width $w_S_i and height $h_S_i each on a transparent background with no borders <br />
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: number of pixel in $S_i: $wh_S_i = $w_S_i * $h_S_i<br />
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: number of non-transparent pixel in $S_i: $s_i<br />
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* Pattern image $P with width $w_P and height $h_P<br />
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: number of pixel in $P: $wh_P = $w_P * $h_P<br />
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* command list with m 5-tuples ($S_i, $r, $m, $x, $y). From the command list the sequence @C_S = ($S_i_j | j = 1, …, m) of prototiles is extracted as the list of shapes that will be step by step composed. <br />
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<u>Countable:</u><br />
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* number of transparent pixel in $P in step t: $gap_t <br />
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* number of non-transparent pixel in $P in step t: $p_t <br />
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<u>Compute:</u><br />
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* number of non-transparent pixel in $P in step t that are covered more than one time: $overlap_t<br />
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<u>Compositing process:</u><br />
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* step 0: $P is still empty => $gap_0 = $wh_P and $overlap_0 = $p_0 = 0<br />
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* step 1: $S_i_1 is composed over $P according to the first 5-tuple: $p_1 = $s_i_1 and $overlap_1 = 0<br />
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* step 2: $S_i_2 is composed over $P according to the second 5-tuple: $overlap_2 = ($s_i_1 + $s_i_2) - $p_2 <br />
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...<br />
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* step k: $S_i_k is composed over $P according to the k'th 5-tuple: $overlap_k = ($s_i_1 + ... + $s_i_k) - $p_k<br />
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...<br />
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* step m: $S_i_m is composed over $P according to the m'th 5-tuple: <span style="color:red;"> <math> $overlap_m = \sum_{j=1}^{m}$s_i_j - $p_m </math> </span></div>Gbachelier