Diskussion:Exploring a Design Space for Patterns and Tilings Competition 2015: Unterschied zwischen den Versionen

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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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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.
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'''Case 1''': <u>One Prototile</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;
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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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: number of non-transparent pixel in $P in step t: $p_t
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* command list with m 5-tuples ($S, $r, $m, $x, $y)
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<u>Count:</u>
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* number of transparent pixel in $P in step t: $gap_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 => $gap_m = $wh_P - $p_m and $overlap_m = m * $s - $p_m
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'''Case 2''': <u>n > 1 Prototiles</u>
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is it possible to make a overlap measure that can be computed given the result pattern image and the command list without the m intermediate steps?
 
  
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Given is a pattern image $P with width $w_P and height $h_P; the number of pixel in $P is $wh_P = $w_P * $h_P and the number of non-transparent pixel in $P is $p;
 
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Gap Measure: number of transparent pixel in an image: here $gap_P;
 
  
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Case 1: Given is one prototile in an image $S (with width $w_S and height $h_S) with transparent background; the number of pixel in $S is $wh_S = $w_S * $h_S and the number of non-transparent pixel in $S is $s;
 
  
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The given command list consists of m 5-tuple which means that $S is composed m times over the pattern image with results in a composing list @C_S = ($S |  j = 1, …, m).
 
  
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The overlap after m composings is the sum of the non-transparent pixels of all elements in @C_S (which is m * $s because all elements in @C_S are the same image $S) minus the number of non-transparent pixel in $P after the m composings ($p_m): $overlap_m =  m * $s - $p_m
 
  
  
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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.  
 
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 - $p_m.  
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The overlap after m composings is the sum of the non-transparent pixels of all elements in @C_S ($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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Version vom 13. Februar 2015, 13:43 Uhr

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
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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
...
  • 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.

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