Copyright | (c) Alexey Kuleshevich 2018-2022 |
---|---|
License | BSD3 |
Maintainer | Alexey Kuleshevich <lehins@yandex.ru> |
Stability | experimental |
Portability | non-portable |
Safe Haskell | Safe-Inferred |
Language | Haskell2010 |
Synopsis
- data Stencil ix e a
- makeStencil :: Index ix => Sz ix -> ix -> ((ix -> e) -> a) -> Stencil ix e a
- getStencilSize :: Stencil ix e a -> Sz ix
- getStencilCenter :: Stencil ix e a -> ix
- data Padding ix e = Padding {
- paddingFromOrigin :: !(Sz ix)
- paddingFromBottom :: !(Sz ix)
- paddingWithElement :: !(Border e)
- noPadding :: Index ix => Padding ix e
- samePadding :: Index ix => Stencil ix e a -> Border e -> Padding ix e
- mapStencil :: (Index ix, Manifest r e) => Border e -> Stencil ix e a -> Array r ix e -> Array DW ix a
- applyStencil :: (Index ix, Manifest r e) => Padding ix e -> Stencil ix e a -> Array r ix e -> Array DW ix a
- idStencil :: Index ix => Stencil ix e e
- sumStencil :: (Num e, Index ix) => Sz ix -> Stencil ix e e
- productStencil :: (Num e, Index ix) => Sz ix -> Stencil ix e e
- avgStencil :: (Fractional e, Index ix) => Sz ix -> Stencil ix e e
- maxStencil :: (Bounded e, Ord e, Index ix) => Sz ix -> Stencil ix e e
- minStencil :: (Bounded e, Ord e, Index ix) => Sz ix -> Stencil ix e e
- foldlStencil :: Index ix => (a -> e -> a) -> a -> Sz ix -> Stencil ix e a
- foldrStencil :: Index ix => (e -> a -> a) -> a -> Sz ix -> Stencil ix e a
- foldStencil :: (Monoid e, Index ix) => Sz ix -> Stencil ix e e
- dimapStencil :: (c -> d) -> (a -> b) -> Stencil ix d a -> Stencil ix c b
- lmapStencil :: (c -> d) -> Stencil ix d a -> Stencil ix c a
- rmapStencil :: (a -> b) -> Stencil ix e a -> Stencil ix e b
- makeConvolutionStencil :: (Index ix, Num e) => Sz ix -> ix -> ((ix -> e -> e -> e) -> e -> e) -> Stencil ix e e
- makeConvolutionStencilFromKernel :: (Manifest r e, Index ix, Num e) => Array r ix e -> Stencil ix e e
- makeCorrelationStencil :: (Index ix, Num e) => Sz ix -> ix -> ((ix -> e -> e -> e) -> e -> e) -> Stencil ix e e
- makeCorrelationStencilFromKernel :: (Manifest r e, Index ix, Num e) => Array r ix e -> Stencil ix e e
Stencil
Stencil is abstract description of how to handle elements in the neighborhood of
every array cell in order to compute a value for the cells in the new array. Use
makeStencil
and makeConvolutionStencil
in order
to create a stencil.
Instances
:: Index ix | |
=> Sz ix | Size of the stencil |
-> ix | Center of the stencil |
-> ((ix -> e) -> a) | Stencil function that receives a "get" function as it's argument that can retrieve values of cells in the source array with respect to the center of the stencil. Stencil function must return a value that will be assigned to the cell in the result array. Offset supplied to the "get" function cannot go outside the boundaries of the stencil, otherwise an error will be raised during stencil creation. |
-> Stencil ix e a |
Construct a stencil from a function, which describes how to calculate the value at a point while having access to neighboring elements with a function that accepts idices relative to the center of stencil. Trying to index outside the stencil box will result in a runtime error upon stencil creation.
Note - Once correctness of stencil is verified then switching to makeUnsafeStencil
is recommended in order to get the most performance out of the Stencil
Example
Below is an example of creating a Stencil
, which, when mapped over a
2-dimensional array, will compute an average of all elements in a 3x3 square
for each element in that array.
Note - Make sure to add an INLINE
pragma, otherwise performance will be terrible.
average3x3Stencil :: Fractional a => Stencil Ix2 a a average3x3Stencil = makeStencil (Sz (3 :. 3)) (1 :. 1) $ get -> ( get (-1 :. -1) + get (-1 :. 0) + get (-1 :. 1) + get ( 0 :. -1) + get ( 0 :. 0) + get ( 0 :. 1) + get ( 1 :. -1) + get ( 1 :. 0) + get ( 1 :. 1) ) / 9 {-# INLINE average3x3Stencil #-}
Since: 0.1.0
getStencilSize :: Stencil ix e a -> Sz ix Source #
Get the size of the stencil
Since: 0.4.3
getStencilCenter :: Stencil ix e a -> ix Source #
Get the index of the stencil's center
Since: 0.4.3
Padding
Padding of the source array before stencil application.
Examples
In order to see the affect of padding we can simply apply an identity stencil to an array:
>>>
import Data.Massiv.Array as A
>>>
a = computeAs P $ resize' (Sz2 2 3) (Ix1 1 ... 6)
>>>
applyStencil noPadding idStencil a
Array DW Seq (Sz (2 :. 3)) [ [ 1, 2, 3 ] , [ 4, 5, 6 ] ]>>>
applyStencil (Padding (Sz2 1 2) (Sz2 3 4) (Fill 0)) idStencil a
Array DW Seq (Sz (6 :. 9)) [ [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 1, 2, 3, 0, 0, 0, 0 ] , [ 0, 0, 4, 5, 6, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] , [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ]
It is also a nice technique to see the border resolution strategy in action:
>>>
applyStencil (Padding (Sz2 2 3) (Sz2 2 3) Wrap) idStencil a
Array DW Seq (Sz (6 :. 9)) [ [ 1, 2, 3, 1, 2, 3, 1, 2, 3 ] , [ 4, 5, 6, 4, 5, 6, 4, 5, 6 ] , [ 1, 2, 3, 1, 2, 3, 1, 2, 3 ] , [ 4, 5, 6, 4, 5, 6, 4, 5, 6 ] , [ 1, 2, 3, 1, 2, 3, 1, 2, 3 ] , [ 4, 5, 6, 4, 5, 6, 4, 5, 6 ] ]>>>
applyStencil (Padding (Sz2 2 3) (Sz2 2 3) Edge) idStencil a
Array DW Seq (Sz (6 :. 9)) [ [ 1, 1, 1, 1, 2, 3, 3, 3, 3 ] , [ 1, 1, 1, 1, 2, 3, 3, 3, 3 ] , [ 1, 1, 1, 1, 2, 3, 3, 3, 3 ] , [ 4, 4, 4, 4, 5, 6, 6, 6, 6 ] , [ 4, 4, 4, 4, 5, 6, 6, 6, 6 ] , [ 4, 4, 4, 4, 5, 6, 6, 6, 6 ] ]>>>
applyStencil (Padding (Sz2 2 3) (Sz2 2 3) Reflect) idStencil a
Array DW Seq (Sz (6 :. 9)) [ [ 6, 5, 4, 4, 5, 6, 6, 5, 4 ] , [ 3, 2, 1, 1, 2, 3, 3, 2, 1 ] , [ 3, 2, 1, 1, 2, 3, 3, 2, 1 ] , [ 6, 5, 4, 4, 5, 6, 6, 5, 4 ] , [ 6, 5, 4, 4, 5, 6, 6, 5, 4 ] , [ 3, 2, 1, 1, 2, 3, 3, 2, 1 ] ]>>>
applyStencil (Padding (Sz2 2 3) (Sz2 2 3) Continue) idStencil a
Array DW Seq (Sz (6 :. 9)) [ [ 1, 3, 2, 1, 2, 3, 2, 1, 3 ] , [ 4, 6, 5, 4, 5, 6, 5, 4, 6 ] , [ 1, 3, 2, 1, 2, 3, 2, 1, 3 ] , [ 4, 6, 5, 4, 5, 6, 5, 4, 6 ] , [ 1, 3, 2, 1, 2, 3, 2, 1, 3 ] , [ 4, 6, 5, 4, 5, 6, 5, 4, 6 ] ]
Since: 0.4.3
Padding | |
|
noPadding :: Index ix => Padding ix e Source #
Also known as "valid" padding. When stencil is applied to an array, that array will shrink, unless the stencil is of size 1.
Since: 0.4.3
samePadding :: Index ix => Stencil ix e a -> Border e -> Padding ix e Source #
Padding that matches the size of the stencil, which is known as "same" padding,
because when a stencil is applied to an array with such matching padding, the resulting
array will be of the same size as the source array. This is exactly the behavior of
mapStencil
Since: 0.4.3
Application
:: (Index ix, Manifest r e) | |
=> Border e | Border resolution technique |
-> Stencil ix e a | Stencil to map over the array |
-> Array r ix e | Source array |
-> Array DW ix a |
Map a constructed stencil over an array. Resulting array must be
compute
d in order to be useful.
Since: 0.1.0
:: (Index ix, Manifest r e) | |
=> Padding ix e | Padding to be applied to the source array. This will dictate the resulting size of the array. No padding will cause it to shrink by the size of the stencil |
-> Stencil ix e a | Stencil to apply to the array |
-> Array r ix e | Source array |
-> Array DW ix a |
Apply a constructed stencil over an array. Resulting array must be
compute
d in order to be useful. Unlike mapStencil
, the size of
the resulting array will not necesserally be the same as the source array, which will
depend on the padding.
Since: 0.4.3
Common stencils
idStencil :: Index ix => Stencil ix e e Source #
Identity stencil that does not change the elements of the source array.
Since: 0.4.3
sumStencil :: (Num e, Index ix) => Sz ix -> Stencil ix e e Source #
Sum all elements in the stencil region
Examples
>>>
import Data.Massiv.Array as A
>>>
a = computeAs P $ iterateN (Sz2 2 5) (* 2) (1 :: Int)
>>>
a
Array P Seq (Sz (2 :. 5)) [ [ 2, 4, 8, 16, 32 ] , [ 64, 128, 256, 512, 1024 ] ]>>>
applyStencil noPadding (sumStencil (Sz2 1 2)) a
Array DW Seq (Sz (2 :. 4)) [ [ 6, 12, 24, 48 ] , [ 192, 384, 768, 1536 ] ]>>>
[2 + 4, 4 + 8, 8 + 16, 16 + 32] :: [Int]
[6,12,24,48]
Since: 0.4.3
productStencil :: (Num e, Index ix) => Sz ix -> Stencil ix e e Source #
Multiply all elements in the stencil region
Examples
>>>
import Data.Massiv.Array as A
>>>
a = computeAs P $ iterateN (Sz2 2 2) (+1) (0 :: Int)
>>>
a
Array P Seq (Sz (2 :. 2)) [ [ 1, 2 ] , [ 3, 4 ] ]>>>
applyStencil (Padding 0 2 (Fill 0)) (productStencil 2) a
Array DW Seq (Sz (3 :. 3)) [ [ 24, 0, 0 ] , [ 0, 0, 0 ] , [ 0, 0, 0 ] ]>>>
applyStencil (Padding 0 2 Reflect) (productStencil 2) a
Array DW Seq (Sz (3 :. 3)) [ [ 24, 64, 24 ] , [ 144, 256, 144 ] , [ 24, 64, 24 ] ]
Since: 0.4.3
avgStencil :: (Fractional e, Index ix) => Sz ix -> Stencil ix e e Source #
Find the average value of all elements in the stencil region
Example
>>>
import Data.Massiv.Array as A
>>>
a = computeAs P $ iterateN (Sz2 3 4) (+1) (10 :: Double)
>>>
a
Array P Seq (Sz (3 :. 4)) [ [ 11.0, 12.0, 13.0, 14.0 ] , [ 15.0, 16.0, 17.0, 18.0 ] , [ 19.0, 20.0, 21.0, 22.0 ] ]>>>
applyStencil noPadding (avgStencil (Sz2 2 3)) a
Array DW Seq (Sz (2 :. 2)) [ [ 14.0, 15.0 ] , [ 18.0, 19.0 ] ]>>>
Prelude.sum [11.0, 12.0, 13.0, 15.0, 16.0, 17.0] / 6 :: Double
14.0
Since: 0.4.3
maxStencil :: (Bounded e, Ord e, Index ix) => Sz ix -> Stencil ix e e Source #
Create a stencil centered at 0 that will extract the maximum value in the region of supplied size.
Example
Here is a sample implementation of max pooling.
>>>
import Data.Massiv.Array as A
>>>
a <- computeAs P <$> resizeM (Sz2 9 9) (Ix1 10 ..: 91)
>>>
a
Array P Seq (Sz (9 :. 9)) [ [ 10, 11, 12, 13, 14, 15, 16, 17, 18 ] , [ 19, 20, 21, 22, 23, 24, 25, 26, 27 ] , [ 28, 29, 30, 31, 32, 33, 34, 35, 36 ] , [ 37, 38, 39, 40, 41, 42, 43, 44, 45 ] , [ 46, 47, 48, 49, 50, 51, 52, 53, 54 ] , [ 55, 56, 57, 58, 59, 60, 61, 62, 63 ] , [ 64, 65, 66, 67, 68, 69, 70, 71, 72 ] , [ 73, 74, 75, 76, 77, 78, 79, 80, 81 ] , [ 82, 83, 84, 85, 86, 87, 88, 89, 90 ] ]>>>
computeWithStrideAs P (Stride 3) $ mapStencil Edge (maxStencil (Sz 3)) a
Array P Seq (Sz (3 :. 3)) [ [ 30, 33, 36 ] , [ 57, 60, 63 ] , [ 84, 87, 90 ] ]
Since: 0.4.3
minStencil :: (Bounded e, Ord e, Index ix) => Sz ix -> Stencil ix e e Source #
Create a stencil centered at 0 that will extract the maximum value in the region of supplied size.
Example
Here is a sample implementation of min pooling.
>>>
import Data.Massiv.Array as A
>>>
a <- computeAs P <$> resizeM (Sz2 9 9) (Ix1 10 ..: 91)
>>>
a
Array P Seq (Sz (9 :. 9)) [ [ 10, 11, 12, 13, 14, 15, 16, 17, 18 ] , [ 19, 20, 21, 22, 23, 24, 25, 26, 27 ] , [ 28, 29, 30, 31, 32, 33, 34, 35, 36 ] , [ 37, 38, 39, 40, 41, 42, 43, 44, 45 ] , [ 46, 47, 48, 49, 50, 51, 52, 53, 54 ] , [ 55, 56, 57, 58, 59, 60, 61, 62, 63 ] , [ 64, 65, 66, 67, 68, 69, 70, 71, 72 ] , [ 73, 74, 75, 76, 77, 78, 79, 80, 81 ] , [ 82, 83, 84, 85, 86, 87, 88, 89, 90 ] ]>>>
computeWithStrideAs P (Stride 3) $ mapStencil Edge (minStencil (Sz 3)) a
Array P Seq (Sz (3 :. 3)) [ [ 10, 13, 16 ] , [ 37, 40, 43 ] , [ 64, 67, 70 ] ]
Since: 0.4.3
foldlStencil :: Index ix => (a -> e -> a) -> a -> Sz ix -> Stencil ix e a Source #
Stencil that does a left fold in a row-major order. Regardless of the supplied size
resulting stencil will be centered at zero, although by using Padding
it is possible
to overcome this limitation.
Examples
>>>
import Data.Massiv.Array as A
>>>
a = computeAs P $ iterateN (Sz2 3 4) (+1) (10 :: Int)
>>>
a
Array P Seq (Sz (3 :. 4)) [ [ 11, 12, 13, 14 ] , [ 15, 16, 17, 18 ] , [ 19, 20, 21, 22 ] ]>>>
applyStencil noPadding (foldlStencil (flip (:)) [] (Sz2 3 2)) a
Array DW Seq (Sz (1 :. 3)) [ [ [20,19,16,15,12,11], [21,20,17,16,13,12], [22,21,18,17,14,13] ] ]>>>
applyStencil (Padding (Sz2 1 0) 0 (Fill 10)) (foldlStencil (flip (:)) [] (Sz2 3 2)) a
Array DW Seq (Sz (2 :. 3)) [ [ [16,15,12,11,10,10], [17,16,13,12,10,10], [18,17,14,13,10,10] ] , [ [20,19,16,15,12,11], [21,20,17,16,13,12], [22,21,18,17,14,13] ] ]
Since: 0.4.3
foldrStencil :: Index ix => (e -> a -> a) -> a -> Sz ix -> Stencil ix e a Source #
Stencil that does a right fold in a row-major order. Regardless of the supplied size
resulting stencil will be centered at zero, although by using Padding
it is possible
to overcome this limitation.
Examples
>>>
import Data.Massiv.Array as A
>>>
a = computeAs P $ iterateN (Sz2 3 4) (+1) (10 :: Int)
>>>
a
Array P Seq (Sz (3 :. 4)) [ [ 11, 12, 13, 14 ] , [ 15, 16, 17, 18 ] , [ 19, 20, 21, 22 ] ]>>>
applyStencil noPadding (foldrStencil (:) [] (Sz2 2 3)) a
Array DW Seq (Sz (2 :. 2)) [ [ [11,12,13,15,16,17], [12,13,14,16,17,18] ] , [ [15,16,17,19,20,21], [16,17,18,20,21,22] ] ]
Since: 0.4.3
foldStencil :: (Monoid e, Index ix) => Sz ix -> Stencil ix e e Source #
Create a stencil that will fold all elements in the region monoidally.
Since: 0.4.3
Profunctor
dimapStencil :: (c -> d) -> (a -> b) -> Stencil ix d a -> Stencil ix c b Source #
A Profunctor dimap. Same caviat applies as in lmapStencil
Since: 0.2.3
lmapStencil :: (c -> d) -> Stencil ix d a -> Stencil ix c a Source #
A contravariant map of a second type parameter. In other words map a function over each element of the array, that the stencil will be applied to.
Note: This map can be very inefficient, since for stencils larger than 1 element in size, the supllied function will be repeatedly applied to the same element. It is better to simply map that function over the source array instead.
Since: 0.2.3
rmapStencil :: (a -> b) -> Stencil ix e a -> Stencil ix e b Source #
Convolution
makeConvolutionStencil :: (Index ix, Num e) => Sz ix -> ix -> ((ix -> e -> e -> e) -> e -> e) -> Stencil ix e e Source #
Create a convolution stencil by specifying border resolution technique and an accumulator function.
Note - Using makeUnsafeConvolutionStencil
will be
slightly faster, therefore it is recommended to switch from this function, after manual
verification that the created stencil behaves as expected.
Examples
Here is how to create a 2D horizontal Sobel Stencil:
sobelX :: Num e => Stencil Ix2 e e sobelX = makeConvolutionStencil (Sz2 3 3) (1 :. 1) $ f -> f (-1 :. -1) (-1) . f (-1 :. 1) 1 . f ( 0 :. -1) (-2) . f ( 0 :. 1) 2 . f ( 1 :. -1) (-1) . f ( 1 :. 1) 1 {-# INLINE sobelX #-}
Since: 0.1.0
makeConvolutionStencilFromKernel :: (Manifest r e, Index ix, Num e) => Array r ix e -> Stencil ix e e Source #
Make a stencil out of a Kernel Array. This Stencil
will be slower than if
makeConvolutionStencil
is used, but sometimes we just really don't know the
kernel at compile time.
Since: 0.1.0
makeCorrelationStencil :: (Index ix, Num e) => Sz ix -> ix -> ((ix -> e -> e -> e) -> e -> e) -> Stencil ix e e Source #
Make a cross-correlation stencil
Note - Using makeUnsafeCorrelationStencil
will be
much faster, therefore it is recommended to switch from this function, after manual
verification that the created stencil behaves as expected.
Since: 0.1.5
makeCorrelationStencilFromKernel :: (Manifest r e, Index ix, Num e) => Array r ix e -> Stencil ix e e Source #
Make a cross-correlation stencil out of a
Kernel Array. This Stencil
will be slower than if makeCorrelationStencil
is used, but
sometimes we just really don't know the kernel at compile time.
Since: 0.1.5