friday-0.2.0.1: src/Vision/Image/Filter/Internal.hs
{-# LANGUAGE BangPatterns
, FlexibleContexts
, FlexibleInstances
, GADTs
, MultiParamTypeClasses
, TypeFamilies
, TupleSections
, ScopedTypeVariables #-}
-- | Provides high level functions to define and apply filters on images.
--
-- Filters are operations on images on which the surrounding of each processed
-- pixel is considered according to a kernel.
--
-- Please use 'Vision.Image.Filter' if you only want to apply filter to images.
--
-- To apply a filter to an image, use the 'apply' method:
--
-- @
-- let -- Creates a filter which will blur the image. Uses a 'Double' as
-- -- accumulator of the Gaussian kernel.
-- filter :: 'Blur' GreyPixel Double GreyPixel
-- filter = 'gaussianBlur' 2 Nothing
-- in 'apply' filter img :: Grey
-- @
--
-- The @radius@ argument of some filters is used to determine the kernel size.
-- A radius as of 1 means a kernel of size 3, 2 a kernel of size 5 and so on.
--
-- The @acc@ type argument of some filters defines the type which will be used
-- to store the accumulated value of the kernel (e.g. by setting @acc@ to
-- 'Double' in the computation of a Gaussian blur, the kernel average will be
-- computed using a 'Double').
module Vision.Image.Filter.Internal (
-- * Types
Filterable (..), Filter (..)
, BoxFilter, BoxFilter1, SeparableFilter, SeparableFilter1
, KernelAnchor (..)
, Kernel (..)
, SeparableKernel (..), SeparatelyFiltrable (..)
, FilterFold (..), FilterFold1 (..)
, BorderInterpolate (..)
-- * Functions
, kernelAnchor, borderInterpolate
-- * Filters
-- ** Morphological operators
, Morphological, dilate, erode
-- ** Blur
, Blur, blur, gaussianBlur
-- ** Derivation
, Derivative, DerivativeType (..), scharr, sobel
-- ** Others
, Mean, mean
) where
import Data.List
import Data.Ratio
import Data.Word
import Foreign.Storable (Storable)
import qualified Data.Vector.Storable as V
import Vision.Image.Class (MaskedImage (..), Image (..), FromFunction (..), (!))
import Vision.Image.Type (Manifest, Delayed)
import Vision.Primitive (Z (..), (:.) (..), DIM1, Point, Size, ix1, ix2)
-- Types -----------------------------------------------------------------------
-- | Provides an implementation to execute a type of filter.
--
-- 'src' is the original image, 'res' the resulting image and 'f' the filter.
class Filterable src res f where
-- | Applies the given filter on the given image.
apply :: f -> src -> res
data Filter src kernel init fold acc res = Filter {
fKernelSize :: !Size
, fKernelCenter :: !KernelAnchor
-- | See 'Kernel' and 'SeparableKernel'.
, fKernel :: !kernel
-- | Computes a constant value for each pixel before applying the kernel.
--
-- This value will be passed to 'fKernel' functions and to the 'fPost'
-- function.
-- For most filters, @fInit@ will be @\_ _ -> ()@.
, fInit :: !(Point -> src -> init)
-- | Defines how the accumulated value is initialized.
--
-- See 'FilterFold' and 'FilterFold1'.
, fFold :: !fold
-- | This function will be executed after the iteration of the kernel on a
-- given point.
--
-- Can be used to normalize the accumulated value, for example.
, fPost :: !(Point -> src -> init -> acc -> res)
, fInterpol :: !(BorderInterpolate src)
}
-- | 2D filters which are initialized with a value.
type BoxFilter src init acc res = Filter src (Kernel src init acc) init
(FilterFold acc) acc res
-- | 2D filters which are not initialized with a value.
type BoxFilter1 src init res = Filter src (Kernel src init src) init
FilterFold1 src res
-- | Separable 2D filters which are initialized with a value.
type SeparableFilter src init acc res = Filter src
(SeparableKernel src init acc)
init (FilterFold acc) acc res
-- | Separable 2D filters which are not initialized with a value.
type SeparableFilter1 src init res = Filter src
(SeparableKernel src init src)
init FilterFold1 src res
-- | Defines how the center of the kernel will be determined.
data KernelAnchor = KernelAnchor !Point | KernelAnchorCenter
-- | A simple 2D kernel.
--
-- The kernel function accepts the coordinates in the kernel, the value of the
-- pixel at these coordinates ('src'), the current accumulated value and returns
-- a new accumulated value.
--
-- Non-separable filters computational complexity grows quadratically according
-- to the size of the sides of the kernel.
newtype Kernel src init acc = Kernel (init -> Point -> src -> acc -> acc)
-- | Some kernels can be factorized in two uni-dimensional kernels (horizontal
-- and vertical).
--
-- Separable filters computational complexity grows linearly according to the
-- size of the sides of the kernel.
--
-- See <http://http://en.wikipedia.org/wiki/Separable_filter>.
data SeparableKernel src init acc = SeparableKernel {
-- | Vertical (column) kernel.
skVertical :: !(init -> DIM1 -> src -> acc -> acc)
-- | Horizontal (row) kernel.
, skHorizontal :: !(init -> DIM1 -> acc -> acc -> acc)
}
-- | Used to determine the type of the accumulator image used when computing
-- separable filters.
--
-- 'src' and 'res' are respectively the source and the result image types while
-- 'acc' is the pixel type of the accumulator.
class ( Image (SeparableFilterAccumulator src res acc)
, ImagePixel (SeparableFilterAccumulator src res acc) ~ acc
, FromFunction (SeparableFilterAccumulator src res acc)
, FromFunctionPixel (SeparableFilterAccumulator src res acc) ~ acc)
=> SeparatelyFiltrable src res acc where
type SeparableFilterAccumulator src res acc
instance Storable acc => SeparatelyFiltrable src (Manifest p) acc where
type SeparableFilterAccumulator src (Manifest p) acc = Manifest acc
instance Storable acc => SeparatelyFiltrable src (Delayed p) acc where
type SeparableFilterAccumulator src (Delayed p) acc = Delayed acc
-- | Uses the result of the provided function as the initial value of the
-- kernel's accumulator, depending on the center coordinates in the image.
--
-- For most filters, the function will always return the same value (i.e.
-- defined as @const 0@), but this kind of initialization could be required for
-- some filters.
data FilterFold acc = FilterFold (Point -> acc)
-- | Uses the first pixel in the kernel as initial value. The kernel must not be
-- empty and the accumulator type must be the same as the source pixel type.
--
-- This kind of initialization is needed by morphological filters.
data FilterFold1 = FilterFold1
-- | Defines how image boundaries are extrapolated by the algorithms.
--
-- '|' characters in examples are image borders.
data BorderInterpolate a =
-- | Replicates the first and last pixels of the image.
--
-- > aaaaaa|abcdefgh|hhhhhhh
BorderReplicate
-- | Reflects the border of the image.
--
-- > fedcba|abcdefgh|hgfedcb
| BorderReflect
-- | Considers that the last pixel of the image is before the first one.
--
-- > cdefgh|abcdefgh|abcdefg
| BorderWrap
-- | Assigns a constant value to out of image pixels.
--
-- > iiiiii|abcdefgh|iiiiiii with some specified 'i'
| BorderConstant !a
-- Instances -------------------------------------------------------------------
-- Following implementations share a lot of similar processing. However, GHC
-- fails to specialise and optimise correctly when goXXX functions are top-level
-- functions, even with static argument transformations.
-- | Box filters initialized with a given value.
instance (Image src, FromFunction res, src_pix ~ ImagePixel src
, res_pix ~ FromFunctionPixel res)
=> Filterable src res (BoxFilter src_pix init acc res_pix) where
apply !(Filter ksize anchor (Kernel kernel) initF fold post interpol) !img =
let !(FilterFold fAcc) = fold
in fromFunction size $ \(!pt@(Z :. iy :. ix)) ->
let pix = img ! pt
!ini = initF pt pix
!acc = fAcc pt
!iy0 = iy - kcy
!ix0 = ix - kcx
!safe = iy0 >= 0 && iy0 + kh <= ih
&& ix0 >= 0 && ix0 + kw <= iw
in post pt pix ini $!
if safe then goColumnSafe ini (iy0 * iw) ix0 0 acc
else goColumn ini iy0 ix0 0 acc
where
!size@(Z :. ih :. iw) = shape img
!(Z :. kh :. kw) = ksize
!(Z :. kcy :. kcx) = kernelAnchor anchor ksize
goColumn !ini !iy !ix !ky !acc
| ky < kh = case borderInterpolate interpol ih iy of
Left iy' -> goLine ini iy (iy' * iw) ix ix ky 0 acc
Right val -> goLineConst ini iy ix ky 0 val acc
| otherwise = acc
goColumnSafe !ini !linearIY !ix !ky !acc
| ky < kh = goLineSafe ini linearIY ix ix ky 0 acc
| otherwise = acc
goLine !ini !iy !linearIY !ix0 !ix !ky !kx !acc
| kx < kw =
let !val = case borderInterpolate interpol iw ix of
Left ix' -> img `linearIndex` (linearIY + ix')
Right val' -> val'
!acc' = kernel ini (ix2 ky kx) val acc
in goLine ini iy linearIY ix0 (ix + 1) ky (kx + 1) acc'
| otherwise = goColumn ini (iy + 1) ix0 (ky + 1) acc
goLineSafe !ini !linearIY !ix0 !ix !ky !kx !acc
| kx < kw =
let !val = img `linearIndex` (linearIY + ix)
!acc' = kernel ini (ix2 ky kx) val acc
in goLineSafe ini linearIY ix0 (ix + 1) ky (kx + 1) acc'
| otherwise = goColumnSafe ini (linearIY + iw) ix0 (ky + 1) acc
goLineConst !ini !iy !ix !ky !kx !val !acc
| kx < kw = let !acc' = kernel ini (ix2 ky kx) val acc
in goLineConst ini iy ix ky (kx + 1) val acc'
| otherwise = goColumn ini (iy + 1) ix (ky + 1) acc
{-# INLINE apply #-}
-- | Box filters initialized using the first pixel of the kernel.
instance (Image src, FromFunction res, src_pix ~ ImagePixel src
, res_pix ~ FromFunctionPixel res)
=> Filterable src res (BoxFilter1 src_pix init res_pix) where
apply !(Filter ksize anchor (Kernel kernel) initF _ post interpol) !img
| kh == 0 || kw == 0 =
error "Using FilterFold1 with an empty kernel."
| otherwise =
fromFunction size $ \(!pt@(Z :. iy :. ix)) ->
let pix = img ! pt
!ini = initF pt pix
!iy0 = iy - kcy
!ix0 = ix - kcx
!safe = iy0 >= 0 && iy0 + kh <= ih
&& ix0 >= 0 && ix0 + kw <= iw
in post pt pix ini $! if safe then goColumn1Safe ini iy0 ix0
else goColumn1 ini iy0 ix0
where
!size@(Z :. ih :. iw) = shape img
!(Z :. kh :. kw) = ksize
!(Z :. kcy :. kcx) = kernelAnchor anchor ksize
goColumn1 !ini !iy !ix =
case borderInterpolate interpol ih iy of
Left iy' ->
let !linearIY = iy' * iw
!acc = safeIndex linearIY ix
in goLine ini iy linearIY ix (ix + 1) 0 1 acc
Right val -> goLineConst ini iy ix 0 1 val val
goColumn1Safe !ini !iy !ix =
let !linearIY = iy * iw
!acc = img `linearIndex` (linearIY + ix)
in goLineSafe ini linearIY ix (ix + 1) 0 1 acc
goColumn !ini !iy !ix !ky !acc
| ky < kh = case borderInterpolate interpol ih iy of
Left iy' -> goLine ini iy (iy' * iw) ix ix ky 0 acc
Right val -> goLineConst ini iy ix ky 0 val acc
| otherwise = acc
goColumnSafe !ini !linearIY !ix !ky !acc
| ky < kh = goLineSafe ini linearIY ix ix ky 0 acc
| otherwise = acc
goLine !ini !iy !linearIY !ix0 !ix !ky !kx !acc
| kx < kw =
let !val = safeIndex linearIY ix
!acc' = kernel ini (ix2 ky kx) val acc
in goLine ini iy linearIY ix0 (ix + 1) ky (kx + 1) acc'
| otherwise = goColumn ini (iy + 1) ix0 (ky + 1) acc
goLineSafe !ini !linearIY !ix0 !ix !ky !kx !acc
| kx < kw =
let !val = img `linearIndex` (linearIY + ix)
!acc' = kernel ini (ix2 ky kx) val acc
in goLineSafe ini linearIY ix0 (ix + 1) ky (kx + 1) acc'
| otherwise = goColumnSafe ini (linearIY + iw) ix0 (ky + 1) acc
goLineConst !ini !iy !ix !ky !kx !val !acc
| kx < kw = let !acc' = kernel ini (ix2 ky kx) val acc
in goLineConst ini iy ix ky (kx + 1) val acc'
| otherwise = goColumn ini (iy + 1) ix (ky + 1) acc
safeIndex !linearIY !ix =
case borderInterpolate interpol iw ix of
Left ix' -> img `linearIndex` (linearIY + ix')
Right val -> val
{-# INLINE apply #-}
-- | Separable filters initialized with a given value.
instance ( Image src, FromFunction res
, src_pix ~ ImagePixel src
, res_pix ~ FromFunctionPixel res
, SeparatelyFiltrable src res acc)
=> Filterable src res (SeparableFilter src_pix init acc res_pix)
where
apply !f !img =
fst $! wrapper img f
where
wrapper :: (Image src, FromFunction res)
=> src
-> SeparableFilter (ImagePixel src) init acc (FromFunctionPixel res)
-> (res, SeparableFilterAccumulator src res acc)
wrapper !src !(Filter ksize anchor kernel initF fold post interpol) =
(res, tmp)
where
!size@(Z :. ih :. iw) = shape src
!(Z :. kh :. kw) = ksize
!(Z :. kcy :. kcx) = kernelAnchor anchor ksize
!(SeparableKernel vert horiz) = kernel
!(FilterFold fAcc) = fold
!tmp = fromFunction size $ \(!pt@(Z :. iy :. ix)) ->
let pix = src ! pt
!ini = initF pt pix
!acc0 = fAcc pt
!iy0 = iy - kcy
in if iy0 >= 0 && iy0 + kh <= ih
then goColumnSafe ini iy0 ix 0 acc0
else goColumn ini iy0 ix 0 acc0
!res = fromFunction size $ \(!pt@(Z :. iy :. ix)) ->
let pix = src ! pt
!ini = initF pt pix
!acc0 = fAcc pt
!ix0 = ix - kcx
in post pt pix ini $!
if ix0 >= 0 && ix0 + kw <= iw
then goLineSafe ini (iy * iw) ix0 0 acc0
else goLine ini acc0 (iy * iw) ix0 0
acc0
goColumn !ini !iy !ix !ky !acc
| ky < kh =
let !val = case borderInterpolate interpol ih iy of
Left iy' -> src ! ix2 iy' ix
Right val' -> val'
!acc' = vert ini (ix1 ky) val acc
in goColumn ini (iy + 1) ix (ky + 1) acc'
| otherwise = acc
goColumnSafe !ini !iy !ix !ky !acc
| ky < kh =
let !val = src ! ix2 iy ix
!acc' = vert ini (ix1 ky) val acc
in goColumnSafe ini (iy + 1) ix (ky + 1) acc'
| otherwise = acc
goLine !ini !acc0 !linearIY !ix !kx !acc
| kx < kw =
let !val =
case borderInterpolate interpol iw ix of
Left ix' -> tmp `linearIndex` (linearIY + ix')
Right val' -> constCol ini acc0 val'
!acc' = horiz ini (ix1 kx) val acc
in goLine ini acc0 linearIY (ix + 1) (kx + 1) acc'
| otherwise = acc
goLineSafe !ini !linearIY !ix !kx !acc
| kx < kw =
let !val = tmp `linearIndex` (linearIY + ix)
!acc' = horiz ini (ix1 kx) val acc
in goLineSafe ini linearIY (ix + 1) (kx + 1) acc'
| otherwise = acc
-- Computes the value of an out of image column when the
-- interpolation method is BorderConstant.
constCol !ini !acc0 !constVal =
foldl' (\acc ky -> vert ini (ix1 ky) constVal acc) acc0
[0..kh-1]
{-# INLINE wrapper #-}
{-# INLINE apply #-}
-- | Separable filters initialized using the first pixel of the kernel.
instance ( Image src, FromFunction res
, src_pix ~ ImagePixel src
, res_pix ~ FromFunctionPixel res
, SeparatelyFiltrable src res src_pix)
=> Filterable src res (SeparableFilter1 src_pix init res_pix)
where
apply !f !img =
fst $! wrapper img f
where
wrapper :: (Image src, FromFunction res, acc ~ ImagePixel src
, FromFunction (SeparableFilterAccumulator src res acc)
, FromFunctionPixel (SeparableFilterAccumulator src res acc) ~ acc
, Image (SeparableFilterAccumulator src res acc)
, ImagePixel (SeparableFilterAccumulator src res acc) ~ acc)
=> src
-> SeparableFilter1 (ImagePixel src) init (FromFunctionPixel res)
-> (res, SeparableFilterAccumulator src res acc)
wrapper !src !(Filter ksize anchor kernel initF _ post interpol)
| kh == 0 || kw == 0 =
error "Using FilterFold1 with an empty kernel."
| otherwise =
(res, tmp)
where
!size@(Z :. ih :. iw) = shape src
!(Z :. kh :. kw) = ksize
!(Z :. kcy :. kcx) = kernelAnchor anchor ksize
!(SeparableKernel vert horiz) = kernel
!tmp = fromFunction size $ \(!pt@(Z :. iy :. ix)) ->
let pix = src ! pt
!ini = initF pt pix
!iy0 = iy - kcy
in if iy0 >= 0 && iy0 + kh <= ih
then goColumn1Safe ini iy0 ix
else goColumn1 ini iy0 ix
!res = fromFunction size $ \(!pt@(Z :. iy :. ix)) ->
let pix = src ! pt
!ini = initF pt pix
!ix0 = ix - kcx
in post pt pix ini $!
if ix0 >= 0 && ix0 + kw <= iw
then goLine1Safe ini (iy * iw) ix0
else goLine1 ini (iy * iw) ix0
goColumn1 !ini !iy !ix =
case borderInterpolate interpol ih iy of
Left iy' ->
let !acc = src ! ix2 iy' ix
in goColumn ini (iy + 1) ix 1 acc
Right val ->
goColumn ini (iy + 1) ix 1 val
goColumn1Safe !ini !iy !ix =
let !linearIY = iy * iw
!acc = src `linearIndex` (linearIY + ix)
in goColumnSafe ini (linearIY + iw) ix 1 acc
goColumn !ini !iy !ix !ky !acc
| ky < kh =
let !val = case borderInterpolate interpol ih iy of
Left iy' -> src ! ix2 iy' ix
Right val' -> val'
!acc' = vert ini (ix1 ky) val acc
in goColumn ini (iy + 1) ix (ky + 1) acc'
| otherwise = acc
goColumnSafe !ini !linearIY !ix !ky !acc
| ky < kh =
let !val = src `linearIndex` (linearIY + ix)
!acc' = vert ini (ix1 ky) val acc
in goColumnSafe ini (linearIY + iw) ix (ky + 1) acc'
| otherwise = acc
goLine1 !ini !linearIY !ix =
let !acc =
case borderInterpolate interpol iw ix of
Left ix' -> tmp `linearIndex` (linearIY + ix')
Right val -> columnConst ini val
in goLine ini linearIY (ix + 1) 1 acc
goLine1Safe !ini !linearIY !ix =
let !linearIX = linearIY + ix
!acc = tmp `linearIndex` linearIX
in goLineSafe ini (linearIX + 1) 1 acc
goLine !ini !linearIY !ix !kx !acc
| kx < kw =
let !val =
case borderInterpolate interpol iw ix of
Left ix' -> tmp `linearIndex` (linearIY + ix')
Right val' -> columnConst ini val'
!acc' = horiz ini (ix1 kx) val acc
in goLine ini linearIY (ix + 1) (kx + 1) acc'
| otherwise = acc
goLineSafe !ini !linearIX !kx !acc
| kx < kw =
let !val = tmp `linearIndex` linearIX
!acc' = horiz ini (ix1 kx) val acc
in goLineSafe ini (linearIX + 1) (kx + 1) acc'
| otherwise = acc
columnConst !ini !constVal = goColumnConst ini 1 constVal constVal
goColumnConst !ini !ky !constVal !acc
| ky < kh = goColumnConst ini (ky + 1) constVal
(vert ini (ix1 ky) acc constVal)
| otherwise = acc
{-# INLINE wrapper #-}
{-# INLINE apply #-}
-- Functions -------------------------------------------------------------------
-- | Given a method to compute the kernel anchor and the size of the kernel,
-- returns the anchor of the kernel as coordinates.
kernelAnchor :: KernelAnchor -> Size -> Point
kernelAnchor (KernelAnchor ix) _ = ix
kernelAnchor (KernelAnchorCenter) (Z :. kh :. kw) = ix2 (round $ (kh - 1) % 2)
(round $ (kw - 1) % 2)
-- | Given a method of interpolation, the number of pixel in the dimension and
-- an index in this dimension, returns either the index of the interpolated
-- pixel or a constant value.
borderInterpolate :: BorderInterpolate a
-> Int -- ^ The size of the dimension.
-> Int -- ^ The index in the dimension.
-> Either Int a
borderInterpolate !interpol !len !ix
| word ix < word len = Left ix
| otherwise =
case interpol of
BorderReplicate | ix < 0 -> Left 0
| otherwise -> Left $! len - 1
BorderReflect -> Left $! goReflect ix
BorderWrap -> Left $! ix `mod` len
BorderConstant i -> Right i
where
goReflect !ix' | ix' < 0 = goReflect (-ix' - 1)
| ix' >= len = goReflect ((len - 1) - (ix' - len))
| otherwise = ix'
{-# INLINE borderInterpolate #-}
-- Morphological operators -----------------------------------------------------
type Morphological pix = SeparableFilter1 pix () pix
dilate :: Ord pix => Int -> Morphological pix
dilate radius =
Filter (ix2 size size) KernelAnchorCenter (SeparableKernel kernel kernel)
(\_ _ -> ()) FilterFold1 (\_ _ _ !acc -> acc) BorderReplicate
where
!size = radius * 2 + 1
kernel _ _ = max
{-# INLINE dilate #-}
erode :: Ord pix => Int -> Morphological pix
erode radius =
Filter (ix2 size size) KernelAnchorCenter (SeparableKernel kernel kernel)
(\_ _ -> ()) FilterFold1 (\_ _ _ !acc -> acc) BorderReplicate
where
!size = radius * 2 + 1
kernel _ _ = min
{-# INLINE erode #-}
-- Blur ------------------------------------------------------------------------
type Blur src acc res = SeparableFilter src () acc res
-- | Blurs the image by averaging the pixel inside the kernel.
--
-- Considers using a type for 'acc' with
-- @maxBound acc >= maxBound src * (kernel size)²@.
blur :: (Integral src, Integral acc, Num res)
=> Int -- ^ Blur radius.
-> Blur src acc res
blur radius =
Filter (ix2 size size) KernelAnchorCenter (SeparableKernel vert horiz)
(\_ _ -> ()) (FilterFold (const 0)) post BorderReplicate
where
!size = radius * 2 + 1
!nPixs = fromIntegral $ square size
vert _ _ !val !acc = acc + fromIntegral val
horiz _ _ !acc' !acc = acc + acc'
post _ _ _ !acc = fromIntegral $ acc `div` nPixs
{-# INLINE blur #-}
-- | Blurs the image by averaging the pixel inside the kernel using a Gaussian
-- function.
--
-- See <http://en.wikipedia.org/wiki/Gaussian_blur>
gaussianBlur :: (Integral src, Floating acc, RealFrac acc, Storable acc
, Integral res)
=> Int -- ^ Blur radius.
-> Maybe acc
-- ^ Sigma value of the Gaussian function. If not given, will be
-- automatically computed from the radius so that the kernel
-- fits 3σ of the distribution.
-> Blur src acc res
gaussianBlur !radius !mSig =
Filter (ix2 size size) KernelAnchorCenter (SeparableKernel vert horiz)
(\_ _ -> ()) (FilterFold (const 0)) (\_ _ _ !acc -> round acc)
BorderReplicate
where
!size = radius * 2 + 1
-- If σ is not provided, tries to fit 3σ in the kernel.
!sig = case mSig of Just s -> s
Nothing -> (0.5 + fromIntegral radius) / 3
vert _ !(Z :. y) !val !acc = let !coeff = kernelVec V.! y
in acc + fromIntegral val * coeff
horiz _ !(Z :. x) !val !acc = let !coeff = kernelVec V.! x
in acc + val * coeff
!kernelVec =
-- Creates a vector of Gaussian values and normalizes it so its sum
-- equals 1.
let !unormalized = V.generate size $ \x ->
gaussian $! fromIntegral $! abs $! x - radius
!kernelSum = V.sum unormalized
in V.map (/ kernelSum) unormalized
gaussian !x = invSigSqrt2Pi * exp (inv2xSig2 * square x)
-- Pre-computed terms of the Gaussian function.
!invSigSqrt2Pi = 1 / (sig * sqrt (2 * pi))
!inv2xSig2 = -1 / (2 * square sig)
{-# INLINE gaussianBlur #-}
-- Derivation ------------------------------------------------------------------
type Derivative src res = SeparableFilter src () res res
data DerivativeType = DerivativeX | DerivativeY
-- | Estimates the first derivative using the Scharr's 3x3 kernel.
--
-- Convolves the following kernel for the X derivative:
--
-- @
-- -3 0 3
-- -10 0 10
-- -3 0 3
-- @
--
-- And this kernel for the Y derivative:
--
-- @
-- -3 -10 -3
-- 0 0 0
-- 3 10 3
-- @
--
-- Considers using a signed integer type for 'res' with
-- @maxBound res >= 16 * maxBound src@.
scharr :: (Integral src, Integral res)
=> DerivativeType -> Derivative src res
scharr der =
let !kernel =
case der of
DerivativeX -> SeparableKernel kernel1 kernel2
DerivativeY -> SeparableKernel kernel2 kernel1
in Filter (ix2 3 3) KernelAnchorCenter kernel (\_ _ -> ())
(FilterFold (const 0)) (\_ _ _ !acc -> acc) BorderReplicate
where
kernel1 _ !(Z :. 1) !val !acc = acc + 10 * fromIntegral val
kernel1 _ !(Z :. _) !val !acc = acc + 3 * fromIntegral val
kernel2 _ !(Z :. 0) !val !acc = acc - fromIntegral val
kernel2 _ !(Z :. 1) !_ !acc = acc
kernel2 _ !(Z :. ~2) !val !acc = acc + fromIntegral val
{-# INLINE scharr #-}
-- | Estimates the first derivative using a Sobel's kernel.
--
-- Prefer 'scharr' when radius equals @1@ as Scharr's kernel is more accurate
-- and is implemented faster.
--
-- Considers using a signed integer type for 'res' which is significantly larger
-- than 'src', especially for large kernels.
sobel :: (Integral src, Integral res, Storable res)
=> Int -- ^ Kernel radius.
-> DerivativeType
-> Derivative src res
sobel radius der =
Filter (ix2 size size) KernelAnchorCenter (SeparableKernel vert horiz)
(\_ _ -> ()) (FilterFold (const 0)) (\_ _ _ !acc -> acc)
BorderReplicate
where
!size = radius * 2 + 1
vert _ !(Z :. x) !val !acc = let !coeff = vec1 V.! x
in acc + fromIntegral val * coeff
horiz _ !(Z :. x) !val !acc = let !coeff = vec2 V.! x
in acc + fromIntegral val * coeff
!radius' = fromIntegral radius
(!vec1, !vec2) = case der of DerivativeX -> (vec1', vec2')
DerivativeY -> (vec2', vec1')
!vec1' = let pows = [ 2^i | i <- [0..radius'] ]
in V.fromList $ pows ++ (tail (reverse pows))
!vec2' = V.fromList $ map negate [1..radius'] ++ [0] ++ [1..radius']
{-# INLINE sobel #-}
-- Others ----------------------------------------------------------------------
type Mean src acc res = SeparableFilter src () acc res
-- | Computes the average of a kernel of the given size.
--
-- This is similar to 'blur' but with a rectangular kernel and a 'Fractional'
-- result.
mean :: (Integral src, Integral acc, Fractional res)
=> Size -> SeparableFilter src () acc res
mean size@(Z :. h :. w) =
Filter size KernelAnchorCenter (SeparableKernel vert horiz) (\_ _ -> ())
(FilterFold (const 0)) post BorderReplicate
where
vert _ _ !val !acc = acc + fromIntegral val
horiz _ _ !acc' !acc = acc + acc'
!nPixsFactor = 1 / (fromIntegral $! h * w)
post _ _ _ !acc = fromIntegral acc * nPixsFactor
{-# INLINE mean #-}
-- -----------------------------------------------------------------------------
square :: Num a => a -> a
square a = a * a
word :: Integral a => a -> Word
word = fromIntegral