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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?
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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.
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| − | '''Case 1''': <u>One prototile type</u>
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| − | <u>Given:</u>
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| − | * one prototile in an image $S with width $w_S and height $h_S on a transparent background with no borders;
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| − | : number of pixel in $S: $wh_S = $w_S * $h_S
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| − | : number of non-transparent pixel in $S: $s
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| − | * Pattern image $P with width $w_P and height $h_P
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| − | : number of pixel in $P: $wh_P = $w_P * $h_P
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| − | * command list with m 5-tuples ($S, $r, $m, $x, $y)
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| − | <u>Countable:</u>
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| − | * number of transparent pixel in $P in step t: $gap_t
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| − | * number of non-transparent pixel in $P in step t: $p_t
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| − | <u>Compute:</u>
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| − | * number of non-transparent pixel in $P in step t that are covered more than one time: $overlap_t
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| − | <u>Compositing process:</u>
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| − | * step 0: $P is still empty => $gap_0 = $wh_P and $overlap_0 = $p_0 = 0
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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
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| − | * step 2: $S is composed over $P according to the second 5-tuple
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| − | : minimum overlap if second $S is composed in a way that there is no intersection to the first $S: $gap_2 = $wh_P - 2 * $s and $overlap_2 = 0
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| − | : maximum overlap if second $S is composed over first $S: $gap_2 = $wh_P - $s and $overlap_2 = $s
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| − | : count the number of non-transparent pixel in $P: $p_2 => $gap_2 = $wh_P - $p_2 and $overlap_2 = 2 * $s - $p_2
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| − | ...
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| − | * step k: count the number of non-transparent pixel in $P: $p_k => $gap_k = $wh_P - $p_k and $overlap_k = k * $s - $p_k
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| − | ...
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| − | * step m: count the number of non-transparent pixel in $P: $p_m => <span style="color:red;">$gap_m = $wh_P - $p_m and $overlap_m = m * $s - $p_m</span>
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| − | '''Case 2''': <u>n > 1 prototile types</u>
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| − | <u>Given:</u>
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| − | * A set of n > 1 different prototile types @S = ($S_i | i = 1, …, n) with width $w_S_i and height $h_S_i each on a transparent background with no borders
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| − | : number of pixel in $S_i: $wh_S_i = $w_S_i * $h_S_i
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| − | : number of non-transparent pixel in $S_i: $s_i
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| − | * Pattern image $P with width $w_P and height $h_P
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| − | : number of pixel in $P: $wh_P = $w_P * $h_P
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| − | * command list with m 5-tuples ($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.
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| − | <u>Countable:</u>
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| − | * number of transparent pixel in $P in step t: $gap_t
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| − | * number of non-transparent pixel in $P in step t: $p_t
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| − | <u>Compute:</u>
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| − | * number of non-transparent pixel in $P in step t that are covered more than one time: $overlap_t
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| − | <u>Compositing process:</u>
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| − | * step 0: $P is still empty => $gap_0 = $wh_P and $overlap_0 = $p_0 = 0
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| − | * step 1: $S_i_1 is composed over $P according to the first 5-tuple => $gap_1 = $wh_P - $s_i_1 and $overlap_1 = 0 and $p_1 = $s_i_1
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| − | * step 2: $S_i_2 is composed over $P according to the second 5-tuple
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| − | : minimum overlap if $S_i_2 is composed in a way that there is no intersection to $S_i_1: $gap_2 = $wh_P - ($s_i_1 + $ and $overlap_2 = 0
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| − | : maximum overlap if second $S is composed over first $S: $gap_2 = $wh_P - $s and $overlap_2 = $s
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| − | : count the number of non-transparent pixel in $P: $p_2 => $gap_2 = $wh_P - $p_2 and $overlap_2 = 2 * $s - $p_2
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| − | ...
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| − | * step k: count the number of non-transparent pixel in $P: $p_k => $gap_k = $wh_P - $p_k and $overlap_k = k * $s - $p_k
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| − | ...
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| − | * step m: count the number of non-transparent pixel in $P: $p_m => <span style="color:red;">$gap_m = $wh_P - $p_m and $overlap_m = m * $s - $p_m</span>
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| − | The given command list consists of m 5-tuple which means that a selection of different elements of @S are composed over the pattern image and every image in @S is at least used one time. The composing list @C_S = ($S_i_j | j = 1, …, m) respresents one possible combination of such a composing pipeline.
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| − | The overlap after m composings is the sum of the non-transparent pixels of all elements in @C_S ($s_j, j = 1, …, m) minus the number of non-transparent pixel in $P after the m composings ($p_m): $overlap_m = Σj=1->m $s_j - $p_m.
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