Diskussion:Exploring a Design Space for Patterns and Tilings Competition 2015
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?
Given is a pattern image $P with width $w_P and height $h_P. In the beginning (t=0) $P is empty i.e. all pixel are transparent resulting in the notation $P_0. Given is also a command list with m 5-tuples ($S, $r, $m, $x, $y). From the command list the sequence @C_S = ($S | j = 1, …, m) of prototiles is extracted as the list of shapes that will be step by step composed independant if there is only one or more prototile types given.
Case 1: One Prototile
Given:
- one prototile in an image $S with width $w_S and height $h_S on a transparent background;
- number of pixel in $S: $wh_S = $w_S * $h_S
- number of non-transparent pixel in $S: $s
- Pattern image $P with width $w_P and height $h_P
- number of pixel in $P: $wh_P = $w_P * $h_P
- number of non-transparent pixel in $P in step t: $p_t
- command list with m 5-tuples ($S, $r, $m, $x, $y)
Count:
- number of transparent pixel in $P in step t: $gap_t
Compute:
- number of non-transparent pixel in $P in step t that are covered more than one time: $overlap_t
Compositing process:
- step 0: $P is still empty => $gap_0 = $wh_P and $overlap_0 = $p_0 = 0
- 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
- step 2: $S is composed over $P according to the second 5-tuple
- 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
- maximum overlap if second $S is composed over first $S: $gap_2 = $wh_P - $s and $overlap_2 = $s
- count the number of non-transparent pixel in $P: $p_2 => $gap_2 = $wh_P - $p_2 and $overlap_2 = 2 * $s - $p_2
- ...
- 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
- ...
- step m: count the number of non-transparent pixel in $P: $p_m => $gap_m = $wh_P - $p_m and $overlap_m = m * $s - $p_m
Case 2: n > 1 Prototiles
Case 2: Given are n prototiles in images @S = ($S_i | i = 1, …, n) with transparent background; the number of pixel in $S_i is $wh_S_i = $w_S_i * $h_S_i and the number of non-transparent pixel in $S is $s_i;
The given command list consists of m 5-tuple which means that a selection of different elements of @S are composed over the pattern image and every image in @S is at least used one time. The composing list @C_S = ($S_i_j | j = 1, …, m) respresents one possible combination of such a composing pipeline.
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.
