Stencil: Difference between revisions
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{{Built-in|Stencil|⌺}} is a primitive [[dyadic operator]] that applies its left [[operand]] to (possibly overlapping) rectangular views of the right argument [[array]]. The [[shape]] and movement of the rectangular views are dictated by the right operand. It was introduced to [[Dyalog APL]] in version 16.0 and is also known as tessellation, moving window, [[wikipedia:stencil code|stencil code]] or cut. The operator can be used for computations involving the immediate neighbours of items in an array and has applications in image processing (particularly [[wikipedia:convolution (computer science)|convolution]]), artificial [[neural networks]] and, most famously, [[wikipedia:cellular automaton|cellular automata]]. The operator derives from a subset (specifically, case 3) of the functionality of [[J]]'s [[Cut]] operator (<source lang=j inline>;.3</ | {{Built-in|Stencil|⌺}} is a primitive [[dyadic operator]] that applies its left [[operand]] to (possibly overlapping) rectangular views of the right argument [[array]]. The [[shape]] and movement of the rectangular views are dictated by the right operand. It was introduced to [[Dyalog APL]] in version 16.0 and is also known as tessellation, moving window, [[wikipedia:stencil code|stencil code]] or cut. The operator can be used for computations involving the immediate neighbours of items in an array and has applications in image processing (particularly [[wikipedia:convolution (computer science)|convolution]]), artificial [[neural networks]] and, most famously, [[wikipedia:cellular automaton|cellular automata]]. The operator derives from a subset (specifically, case 3) of the functionality of [[J]]'s [[Cut]] operator (<source lang=j inline>;.3</syntaxhighlight>),<ref>[[Eugene McDonnell]]: [https://www.jsoftware.com/papers/eem/life1.htm Life: Nasty, Brutish, and Short]. [[APL88]]. </ref> which in turn originates in the 3-cut mentioned in [[A Dictionary of APL]]<ref>[[Ken Iverson]]: [https://www.jsoftware.com/papers/APLDictionary.htm A Dictionary of APL]. [[APL Quote Quad]], Volume 18, Number 1, 1987-09.</ref> | ||
== Description == | == Description == | ||
For a call <source lang=apl inline>f⌺s</ | For a call <source lang=apl inline>f⌺s</syntaxhighlight> the right operand <source lang=apl inline>s</syntaxhighlight> can in general be a two-row [[matrix]] of strictly positive integers, where the first row describes the dimensions of the rectangles that will be passed to <source lang=apl inline>f</syntaxhighlight> and the second row describes the movement along the different [[axis|axes]]. When using stencil <source lang=apl inline>(f⌺s)Y</syntaxhighlight> the right operand <source lang=apl inline>s</syntaxhighlight> in general has <source lang=apl inline>≢⍴Y</syntaxhighlight> columns. If it has fewer, the rectangles are cut out of the [[major cell]]s of the <source lang=apl inline>(≢⍴Y) - ≢1⌷s</syntaxhighlight> axis of <source lang=apl inline>Y</syntaxhighlight>. If <source lang=apl inline>s</syntaxhighlight> is a vector or scalar it describes only the size of the rectangles and the movement defaults to 1. Rectangles are centred on [[element]]s of <source lang=apl inline>Y</syntaxhighlight> whose [[index|indices]] differ by the movements in <source lang=apl inline>s</syntaxhighlight> (defaulting to 1), starting with the first element in [[ravel order]]. For a matrix, this is the top left. For even window sizes, the centring is instead on the space between elements or [[cell]]s. Along the edges of the [[argument]] array, the windows are thus subject to be padded with [[fill element]]s. | ||
The number of fill elements in along each axis is given as a vector left argument on each call of <source lang=apl inline>f</ | The number of fill elements in along each axis is given as a vector left argument on each call of <source lang=apl inline>f</syntaxhighlight> for the respective [[subarray]]. This is designed such that [[Drop]] (<source lang=apl inline>↓</syntaxhighlight>) can take the left and right arguments to remove padding: | ||
<source lang=apl> | <source lang=apl> | ||
({⊂⍵}⌺3 3)3 4⍴⍳12 | ({⊂⍵}⌺3 3)3 4⍴⍳12 | ||
| Line 41: | Line 41: | ||
│9 10│9 10 11│10 11 12│11 12│ | │9 10│9 10 11│10 11 12│11 12│ | ||
└────┴───────┴────────┴─────┘ | └────┴───────┴────────┴─────┘ | ||
</ | </syntaxhighlight> | ||
== Examples == | == Examples == | ||
| Line 61: | Line 61: | ||
│0 0 0│0 0 0│0 0 0│ | │0 0 0│0 0 0│0 0 0│ | ||
└─────┴─────┴─────┘ | └─────┴─────┴─────┘ | ||
</ | </syntaxhighlight> | ||
In the following example, each neighbourhood is immediately [[ravel]]led and [[sum]]med: | In the following example, each neighbourhood is immediately [[ravel]]led and [[sum]]med: | ||
<source lang=apl> | <source lang=apl> | ||
{+/,⍵}⌺3 3⊢3 3⍴⍳9 | {+/,⍵}⌺3 3⊢3 3⍴⍳9 | ||
12 21 16 27 45 33 24 39 28 | 12 21 16 27 45 33 24 39 28 | ||
</ | </syntaxhighlight> | ||
If the number of columns in the right operand to stencil does not match the rank of the right argument, the windowing is applied on the major cells: | If the number of columns in the right operand to stencil does not match the rank of the right argument, the windowing is applied on the major cells: | ||
<source lang=apl> | <source lang=apl> | ||
| Line 74: | Line 74: | ||
│7 8 9 10 11 12│19 20 21 22 23 24│31 32 33 34 35 36│43 44 45 46 47 48│55 56 57 58 59 60│ | │7 8 9 10 11 12│19 20 21 22 23 24│31 32 33 34 35 36│43 44 45 46 47 48│55 56 57 58 59 60│ | ||
└──────────────┴─────────────────┴─────────────────┴─────────────────┴─────────────────┘ | └──────────────┴─────────────────┴─────────────────┴─────────────────┴─────────────────┘ | ||
</ | </syntaxhighlight> | ||
Stencil allows for a very succinct expression of a [[dfn]] that calculates the next iteration in [[Conway's Game of Life]] (on a rectangle bound by zeros). Inspired by an algorithm from [[Arthur Whitney]] written in [[K]] and adapted to APL by Jay Foad: | Stencil allows for a very succinct expression of a [[dfn]] that calculates the next iteration in [[Conway's Game of Life]] (on a rectangle bound by zeros). Inspired by an algorithm from [[Arthur Whitney]] written in [[K]] and adapted to APL by Jay Foad: | ||
<source lang=apl> | <source lang=apl> | ||
| Line 88: | Line 88: | ||
│.....│.....│.....│.....│.....│.....│.....│...⍠.│ | │.....│.....│.....│.....│.....│.....│.....│...⍠.│ | ||
└─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘ | └─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘ | ||
</ | </syntaxhighlight> | ||
== See also == | == See also == | ||
* [[Windowed Reduce]] | * [[Windowed Reduce]] | ||
Revision as of 10:36, 11 September 2022
⌺
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Stencil (⌺) is a primitive dyadic operator that applies its left operand to (possibly overlapping) rectangular views of the right argument array. The shape and movement of the rectangular views are dictated by the right operand. It was introduced to Dyalog APL in version 16.0 and is also known as tessellation, moving window, stencil code or cut. The operator can be used for computations involving the immediate neighbours of items in an array and has applications in image processing (particularly convolution), artificial neural networks and, most famously, cellular automata. The operator derives from a subset (specifically, case 3) of the functionality of J's Cut operator (<source lang=j inline>;.3</syntaxhighlight>),[1] which in turn originates in the 3-cut mentioned in A Dictionary of APL[2]
Description
For a call <source lang=apl inline>f⌺s</syntaxhighlight> the right operand <source lang=apl inline>s</syntaxhighlight> can in general be a two-row matrix of strictly positive integers, where the first row describes the dimensions of the rectangles that will be passed to <source lang=apl inline>f</syntaxhighlight> and the second row describes the movement along the different axes. When using stencil <source lang=apl inline>(f⌺s)Y</syntaxhighlight> the right operand <source lang=apl inline>s</syntaxhighlight> in general has <source lang=apl inline>≢⍴Y</syntaxhighlight> columns. If it has fewer, the rectangles are cut out of the major cells of the <source lang=apl inline>(≢⍴Y) - ≢1⌷s</syntaxhighlight> axis of <source lang=apl inline>Y</syntaxhighlight>. If <source lang=apl inline>s</syntaxhighlight> is a vector or scalar it describes only the size of the rectangles and the movement defaults to 1. Rectangles are centred on elements of <source lang=apl inline>Y</syntaxhighlight> whose indices differ by the movements in <source lang=apl inline>s</syntaxhighlight> (defaulting to 1), starting with the first element in ravel order. For a matrix, this is the top left. For even window sizes, the centring is instead on the space between elements or cells. Along the edges of the argument array, the windows are thus subject to be padded with fill elements.
The number of fill elements in along each axis is given as a vector left argument on each call of <source lang=apl inline>f</syntaxhighlight> for the respective subarray. This is designed such that Drop (<source lang=apl inline>↓</syntaxhighlight>) can take the left and right arguments to remove padding: <source lang=apl>
({⊂⍵}⌺3 3)3 4⍴⍳12
┌──────┬───────┬────────┬───────┐ │0 0 0 │0 0 0 │0 0 0 │0 0 0 │ │0 1 2 │1 2 3 │2 3 4 │3 4 0 │ │0 5 6 │5 6 7 │6 7 8 │7 8 0 │ ├──────┼───────┼────────┼───────┤ │0 1 2│1 2 3│ 2 3 4│ 3 4 0│ │0 5 6│5 6 7│ 6 7 8│ 7 8 0│ │0 9 10│9 10 11│10 11 12│11 12 0│ ├──────┼───────┼────────┼───────┤ │0 5 6│5 6 7│ 6 7 8│ 7 8 0│ │0 9 10│9 10 11│10 11 12│11 12 0│ │0 0 0│0 0 0│ 0 0 0│ 0 0 0│ └──────┴───────┴────────┴───────┘
({⊂⍺}⌺3 3)3 4⍴⍳12
┌────┬────┬────┬─────┐ │1 1 │1 0 │1 0 │1 ¯1 │ ├────┼────┼────┼─────┤ │0 1 │0 0 │0 0 │0 ¯1 │ ├────┼────┼────┼─────┤ │¯1 1│¯1 0│¯1 0│¯1 ¯1│ └────┴────┴────┴─────┘
({⊂⍺↓⍵}⌺3 3)3 4⍴⍳12
┌────┬───────┬────────┬─────┐ │1 2 │1 2 3 │2 3 4 │3 4 │ │5 6 │5 6 7 │6 7 8 │7 8 │ ├────┼───────┼────────┼─────┤ │1 2│1 2 3│ 2 3 4│ 3 4│ │5 6│5 6 7│ 6 7 8│ 7 8│ │9 10│9 10 11│10 11 12│11 12│ ├────┼───────┼────────┼─────┤ │5 6│5 6 7│ 6 7 8│ 7 8│ │9 10│9 10 11│10 11 12│11 12│ └────┴───────┴────────┴─────┘ </syntaxhighlight>
Examples
With a default movement in every direction of 1, the rectangles are centred on neighbouring elements of the array. Here, we simply enclose each neighbourhood: <source lang=apl>
{⊂⍵}⌺3 3⊢3 3⍴⍳9
┌─────┬─────┬─────┐ │0 0 0│0 0 0│0 0 0│ │0 1 2│1 2 3│2 3 0│ │0 4 5│4 5 6│5 6 0│ ├─────┼─────┼─────┤ │0 1 2│1 2 3│2 3 0│ │0 4 5│4 5 6│5 6 0│ │0 7 8│7 8 9│8 9 0│ ├─────┼─────┼─────┤ │0 4 5│4 5 6│5 6 0│ │0 7 8│7 8 9│8 9 0│ │0 0 0│0 0 0│0 0 0│ └─────┴─────┴─────┘ </syntaxhighlight> In the following example, each neighbourhood is immediately ravelled and summed: <source lang=apl>
{+/,⍵}⌺3 3⊢3 3⍴⍳9
12 21 16 27 45 33 24 39 28 </syntaxhighlight> If the number of columns in the right operand to stencil does not match the rank of the right argument, the windowing is applied on the major cells: <source lang=apl>
{⊂⍵}⌺(⍪2 2)⊢10 6⍴⍳100
┌──────────────┬─────────────────┬─────────────────┬─────────────────┬─────────────────┐ │1 2 3 4 5 6│13 14 15 16 17 18│25 26 27 28 29 30│37 38 39 40 41 42│49 50 51 52 53 54│ │7 8 9 10 11 12│19 20 21 22 23 24│31 32 33 34 35 36│43 44 45 46 47 48│55 56 57 58 59 60│ └──────────────┴─────────────────┴─────────────────┴─────────────────┴─────────────────┘ </syntaxhighlight> Stencil allows for a very succinct expression of a dfn that calculates the next iteration in Conway's Game of Life (on a rectangle bound by zeros). Inspired by an algorithm from Arthur Whitney written in K and adapted to APL by Jay Foad: <source lang=apl>
⎕IO←0
life ← {3=s-⍵∧4=s←{+/,⍵}⌺3 3⊢⍵}
{'.⍠'[⍵]}¨ (⍳8) {life⍣⍺⊢⍵}¨ ⊂glider
┌─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┐ │..⍠..│.⍠...│..⍠..│.....│.....│.....│.....│.....│ │⍠.⍠..│..⍠⍠.│...⍠.│.⍠.⍠.│...⍠.│..⍠..│...⍠.│.....│ │.⍠⍠..│.⍠⍠..│.⍠⍠⍠.│..⍠⍠.│.⍠.⍠.│...⍠⍠│....⍠│..⍠.⍠│ │.....│.....│.....│..⍠..│..⍠⍠.│..⍠⍠.│..⍠⍠⍠│...⍠⍠│ │.....│.....│.....│.....│.....│.....│.....│...⍠.│ └─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘ </syntaxhighlight>
See also
External links
Documentation
Publications
- Dyalog '16 presentation by Roger Hui: New Primitive Functions and Operators
- Dyalog blog posts by Roger Hui: Stencil Lives, Towards Improvements to Stencil
References
- ↑ Eugene McDonnell: Life: Nasty, Brutish, and Short. APL88.
- ↑ Ken Iverson: A Dictionary of APL. APL Quote Quad, Volume 18, Number 1, 1987-09.