manifolds-0.6.0.0: Data/Manifold/Shade.hs
-- |
-- Module : Data.Manifold.Shade
-- Copyright : (c) Justus Sagemüller 2016
-- License : GPL v3
--
-- Maintainer : (@) jsagemue $ uni-koeln.de
-- Stability : experimental
-- Portability : portable
--
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE DeriveFunctor #-}
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE ParallelListComp #-}
{-# LANGUAGE UnicodeSyntax #-}
{-# LANGUAGE PatternSynonyms #-}
{-# LANGUAGE ViewPatterns #-}
{-# LANGUAGE LambdaCase #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE CPP #-}
{-# LANGUAGE TupleSections #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE LiberalTypeSynonyms #-}
{-# LANGUAGE DefaultSignatures #-}
{-# LANGUAGE TemplateHaskell #-}
module Data.Manifold.Shade (
-- * Shades
Shade(..), pattern(:±), Shade'(..), (|±|), IsShade
-- ** Lenses
, shadeCtr, shadeExpanse, shadeNarrowness
-- ** Construction
, fullShade, fullShade', pointsShades, pointsShade's
, pointsCovers, pointsCover's, coverAllAround
-- ** Evaluation
, occlusion, prettyShowsPrecShade', prettyShowShade', LtdErrorShow
-- ** Misc
, factoriseShade, orthoShades, (✠), intersectShade's, linIsoTransformShade
, embedShade, projectShade
, Refinable, subShade', refineShade', convolveShade', coerceShade
, mixShade's, dualShade, dualShade', wellDefinedShade', linearProjectShade
, shadesMerge, pointsShades', pseudoECM, convolveMetric
, WithAny(..), shadeWithAny, shadeWithoutAnything
-- * Misc
, rangeWithinVertices
) where
import Data.List hiding (filter, all, elem, sum, foldr1)
import Data.Maybe
import Data.List.NonEmpty (NonEmpty(..))
import qualified Data.List.NonEmpty as NE
import Data.Semigroup
import Control.DeepSeq
import Data.MemoTrie
import Data.VectorSpace
import Data.AffineSpace
import Math.LinearMap.Category
import Data.Tagged
import Linear (_x,_y,_z,_w)
import Data.Manifold.Types
import Data.Manifold.Types.Primitive ((^))
import Data.Manifold.PseudoAffine
import Data.Manifold.Riemannian
import Data.Manifold.WithBoundary
import Data.Manifold.Atlas
import Data.Function.Affine
import Data.Manifold.Function.Quadratic
import Data.Embedding
import Control.Lens (Lens', (^.), view, _1, _2, mapping, (&))
import Control.Lens.TH
import qualified Prelude as Hask hiding(foldl, sum, sequence)
import qualified Control.Applicative as Hask
import qualified Data.Foldable as Hask
import Data.Foldable (all, elem, toList, sum, foldr1)
import Control.Category.Constrained.Prelude hiding
((^), all, elem, sum, forM, Foldable(..), foldr1, Traversable, traverse)
import Control.Arrow.Constrained
import Control.Monad.Constrained hiding (forM)
import GHC.Generics (Generic)
import Text.Show.Number
import qualified Text.Show.Pragmatic as SP
-- | A 'Shade' is a very crude description of a region within a manifold. It
-- can be interpreted as either an ellipsoid shape, or as the Gaussian peak
-- of a normal distribution (use <http://hackage.haskell.org/package/manifold-random>
-- for actually sampling from that distribution).
--
-- For a /precise/ description of an arbitrarily-shaped connected subset of a manifold,
-- there is 'Region', whose implementation is vastly more complex.
data Shade x where
Shade :: (Semimanifold x, SimpleSpace (Needle x))
=> { _shadeCtr :: !x
, _shadeExpanse :: !(Metric' x) } -> Shade x
deriving instance (Show x, Show (Metric' x), WithField ℝ PseudoAffine x)
=> Show (Shade x)
-- | A “co-shade” can describe ellipsoid regions as well, but unlike
-- 'Shade' it can be unlimited / infinitely wide in some directions.
-- It does OTOH need to have nonzero thickness, which 'Shade' needs not.
data Shade' x = Shade' { _shade'Ctr :: !x
, _shade'Narrowness :: !(Metric x) }
class IsShade shade where
-- type (*) shade :: *->*
-- | Access the center of a 'Shade' or a 'Shade''.
shadeCtr :: Lens' (shade x) x
-- -- | Convert between 'Shade' and 'Shade' (which must be neither singular nor infinite).
-- unsafeDualShade :: WithField ℝ Manifold x => shade x -> shade* x
-- | Check the statistical likelihood-density of a point being within a shade.
-- This is taken as a normal distribution.
occlusion :: ( PseudoAffine x, SimpleSpace (Needle x)
, s ~ (Scalar (Needle x)), RealFloat' s )
=> shade x -> x -> s
factoriseShade :: ( PseudoAffine x, SimpleSpace (Needle x)
, PseudoAffine y, SimpleSpace (Needle y)
, Scalar (Needle x) ~ Scalar (Needle y) )
=> shade (x,y) -> (shade x, shade y)
coerceShade :: ( Manifold x, Manifold y, LocallyCoercible x y
, SimpleSpace (Needle y) ) => shade x -> shade y
-- | ASCII version of '✠'.
orthoShades :: ( PseudoAffine x, SimpleSpace (Needle x)
, PseudoAffine y, SimpleSpace (Needle y)
, Scalar (Needle x) ~ Scalar (Needle y) )
=> shade x -> shade y -> shade (x,y)
linIsoTransformShade :: ( SimpleSpace x, SimpleSpace y, Scalar x ~ Scalar y
, Num' (Scalar x) )
=> (x+>y) -> shade x -> shade y
-- | Squash a shade down into a lower dimensional space.
projectShade :: ( Semimanifold x, Semimanifold y
, Object (Affine s) x, Object (Affine s) y
, SimpleSpace (Needle x), SemiInner (Needle y) )
=> Embedding (Affine s) x y
-> shade y -> shade x
-- | Include a shade in a higher-dimensional space. Notice that this behaves
-- fundamentally different for 'Shade' and 'Shade''. For 'Shade', it gives
-- a “flat image” of the region, whereas for 'Shade'' it gives an “extrusion
-- pillar” pointing in the projection's orthogonal complement.
embedShade :: ( Semimanifold x, Semimanifold y
, Object (Affine s) x, Object (Affine s) y
, SemiInner (Needle x), SimpleSpace (Needle y) )
=> Embedding (Affine s) x y
-> shade x -> shade y
linearProjectShade :: ∀ x y s
. (Num' s, LinearSpace x, SimpleSpace y, Scalar x ~ s, Scalar y ~ s)
=> (x+>y) -> Shade x -> Shade y
linearProjectShade = case ( linearManifoldWitness :: LinearManifoldWitness x
, linearManifoldWitness :: LinearManifoldWitness y
, dualSpaceWitness :: DualSpaceWitness x
, dualSpaceWitness :: DualSpaceWitness y ) of
( LinearManifoldWitness
,LinearManifoldWitness
,DualSpaceWitness, DualSpaceWitness )
-> \f (Shade x ex) -> Shade (f $ x) (transformVariance f ex)
infixl 5 ✠
-- | Combine two shades on independent subspaces to a shade with the same
-- properties on the subspaces (see 'factoriseShade') and no covariance.
(✠) :: ( IsShade shade, PseudoAffine x, SimpleSpace (Needle x)
, PseudoAffine y, SimpleSpace (Needle y)
, Scalar (Needle x) ~ Scalar (Needle y) )
=> shade x -> shade y -> shade (x,y)
(✠) = orthoShades
instance IsShade Shade where
shadeCtr f (Shade c e) = fmap (`Shade`e) $ f c
occlusion = occ pseudoAffineWitness dualSpaceWitness
where occ :: ∀ x s . ( PseudoAffine x, SimpleSpace (Needle x)
, Scalar (Needle x) ~ s, RealFloat' s )
=> PseudoAffineWitness x -> DualNeedleWitness x -> Shade x -> x -> s
occ (PseudoAffineWitness SemimanifoldWitness) DualSpaceWitness (Shade p₀ δ)
= \p -> case p.-~.p₀ of
(Just vd) | mSq <- normSq δinv vd
, mSq == mSq -- avoid NaN
-> exp (negate mSq)
_ -> zeroV
where δinv = dualNorm δ
factoriseShade = fs dualSpaceWitness dualSpaceWitness
where fs :: ∀ x y . ( PseudoAffine x, SimpleSpace (Needle x)
, PseudoAffine y, SimpleSpace (Needle y)
, Scalar (Needle x) ~ Scalar (Needle y) )
=> DualNeedleWitness x -> DualNeedleWitness y
-> Shade (x,y) -> (Shade x, Shade y)
fs DualSpaceWitness DualSpaceWitness (Shade (x₀,y₀) δxy)
= (Shade x₀ δx, Shade y₀ δy)
where (δx,δy) = summandSpaceNorms δxy
orthoShades = fs dualSpaceWitness dualSpaceWitness
where fs :: ∀ x y . ( PseudoAffine x, SimpleSpace (Needle x)
, PseudoAffine y, SimpleSpace (Needle y)
, Scalar (Needle x) ~ Scalar (Needle y) )
=> DualNeedleWitness x -> DualNeedleWitness y
-> Shade x -> Shade y -> Shade (x,y)
fs DualSpaceWitness DualSpaceWitness (Shade x δx) (Shade y δy)
= Shade (x,y) $ sumSubspaceNorms δx δy
coerceShade = cS dualSpaceWitness dualSpaceWitness
where cS :: ∀ x y . (LocallyCoercible x y, SimpleSpace (Needle y))
=> DualNeedleWitness x -> DualNeedleWitness y -> Shade x -> Shade y
cS DualSpaceWitness DualSpaceWitness
= \(Shade x δxym)
-> Shade (locallyTrivialDiffeomorphism x) (tN δxym)
where tN = case oppositeLocalCoercion :: CanonicalDiffeomorphism y x of
CanonicalDiffeomorphism ->
transformNorm . arr $ coerceNeedle' ([]::[(y,x)])
linIsoTransformShade = lits linearManifoldWitness linearManifoldWitness
dualSpaceWitness dualSpaceWitness
where lits :: ∀ x y . ( LinearSpace x, SimpleSpace y
, Scalar x ~ Scalar y, Num' (Scalar x) )
=> LinearManifoldWitness x -> LinearManifoldWitness y
-> DualSpaceWitness x -> DualSpaceWitness y
-> (x+>y) -> Shade x -> Shade y
lits (LinearManifoldWitness)
(LinearManifoldWitness)
DualSpaceWitness DualSpaceWitness
f (Shade x δx)
= Shade (f $ x) (transformNorm (adjoint $ f) δx)
embedShade = ps' (semimanifoldWitness, semimanifoldWitness)
where ps' :: ∀ s x y . ( Semimanifold y
, Object (Affine s) x, Object (Affine s) y
, SemiInner (Needle x), SimpleSpace (Needle y) )
=> (SemimanifoldWitness x, SemimanifoldWitness y)
-> Embedding (Affine s) x y
-> Shade x -> Shade y
ps' (SemimanifoldWitness, SemimanifoldWitness)
(Embedding q _) (Shade x e) = Shade y (transformVariance j e)
where y = q $ x
(_,j) = evalAffine q x
projectShade = ps' (semimanifoldWitness, semimanifoldWitness)
where ps' :: ∀ s x y . ( Semimanifold x
, Object (Affine s) x, Object (Affine s) y
, SimpleSpace (Needle x), SemiInner (Needle y) )
=> (SemimanifoldWitness x, SemimanifoldWitness y)
-> Embedding (Affine s) x y
-> Shade y -> Shade x
ps' (SemimanifoldWitness, SemimanifoldWitness)
(Embedding _ q) (Shade x e) = Shade y (transformVariance j e)
where y = q $ x
(_,j) = evalAffine q x
dualShade :: ∀ x . (PseudoAffine x, SimpleSpace (Needle x))
=> Shade x -> Shade' x
dualShade = case dualSpaceWitness :: DualSpaceWitness (Needle x) of
DualSpaceWitness -> \(Shade c e) -> Shade' c $ dualNorm e
dualShade' :: ∀ x . (PseudoAffine x, SimpleSpace (Needle x))
=> Shade' x -> Shade x
dualShade' = case dualSpaceWitness :: DualSpaceWitness (Needle x) of
DualSpaceWitness -> \(Shade' c e) -> Shade c $ dualNorm' e
instance ImpliesMetric Shade where
type MetricRequirement Shade x = (Manifold x, SimpleSpace (Needle x))
inferMetric' (Shade _ e) = e
inferMetric = im dualSpaceWitness
where im :: (Manifold x, SimpleSpace (Needle x))
=> DualNeedleWitness x -> Shade x -> Metric x
im DualSpaceWitness (Shade _ e) = dualNorm e
instance ImpliesMetric Shade' where
type MetricRequirement Shade' x = (Manifold x, SimpleSpace (Needle x))
inferMetric (Shade' _ e) = e
inferMetric' (Shade' _ e) = dualNorm e
shadeExpanse :: Lens' (Shade x) (Metric' x)
shadeExpanse f (Shade c e) = fmap (Shade c) $ f e
instance IsShade Shade' where
shadeCtr f (Shade' c e) = fmap (`Shade'`e) $ f c
occlusion = occ pseudoAffineWitness
where occ :: ∀ x s . ( PseudoAffine x, SimpleSpace (Needle x)
, Scalar (Needle x) ~ s, RealFloat' s )
=> PseudoAffineWitness x -> Shade' x -> x -> s
occ (PseudoAffineWitness (SemimanifoldWitness)) (Shade' p₀ δinv) p
= case p.-~.p₀ of
(Just vd) | mSq <- normSq δinv vd
, mSq == mSq -- avoid NaN
-> exp (negate mSq)
_ -> zeroV
factoriseShade (Shade' (x₀,y₀) δxy) = (Shade' x₀ δx, Shade' y₀ δy)
where (δx,δy) = summandSpaceNorms δxy
orthoShades (Shade' x δx) (Shade' y δy) = Shade' (x,y) $ sumSubspaceNorms δx δy
coerceShade = cS
where cS :: ∀ x y . (LocallyCoercible x y) => Shade' x -> Shade' y
cS = \(Shade' x δxym) -> Shade' (locallyTrivialDiffeomorphism x) (tN δxym)
where tN = case oppositeLocalCoercion :: CanonicalDiffeomorphism y x of
CanonicalDiffeomorphism ->
transformNorm . arr $ coerceNeedle ([]::[(y,x)])
linIsoTransformShade = lits linearManifoldWitness linearManifoldWitness
dualSpaceWitness dualSpaceWitness
where lits :: ∀ x y . ( SimpleSpace x, SimpleSpace y
, Scalar x ~ Scalar y, RealFloat' (Scalar x) )
=> LinearManifoldWitness x -> LinearManifoldWitness y
-> DualSpaceWitness x -> DualSpaceWitness y
-> (x+>y) -> Shade' x -> Shade' y
lits (LinearManifoldWitness)
(LinearManifoldWitness)
DualSpaceWitness DualSpaceWitness
f (Shade' x δx)
= Shade' (f $ x) (transformNorm (pseudoInverse f) δx)
embedShade = ps (semimanifoldWitness, semimanifoldWitness)
where ps :: ∀ s x y . ( Object (Affine s) x, Object (Affine s) y
, SemiInner (Needle x), SemiInner (Needle y) )
=> (SemimanifoldWitness x, SemimanifoldWitness y)
-> Embedding (Affine s) x y
-> Shade' x -> Shade' y
ps (SemimanifoldWitness, SemimanifoldWitness)
(Embedding q p) (Shade' x e) = Shade' y (transformNorm j e)
where y = q $ x
(_,j) = evalAffine p y
projectShade = ps (semimanifoldWitness, semimanifoldWitness)
where ps :: ∀ s x y . ( Object (Affine s) x, Object (Affine s) y
, SemiInner (Needle x), SemiInner (Needle y) )
=> (SemimanifoldWitness x, SemimanifoldWitness y)
-> Embedding (Affine s) x y
-> Shade' y -> Shade' x
ps (SemimanifoldWitness, SemimanifoldWitness)
(Embedding p q) (Shade' x e) = Shade' y (transformNorm j e)
where y = q $ x
(_,j) = evalAffine p y
shadeNarrowness :: Lens' (Shade' x) (Metric x)
shadeNarrowness f (Shade' c e) = fmap (Shade' c) $ f e
newtype ShadeNeedle x = ShadeNeedle { shadeCtrDiff :: Needle x
-- TODO add shade-spread information
}
deriving instance (AdditiveGroup (Needle x)) => AdditiveGroup (ShadeNeedle x)
deriving instance (VectorSpace (Needle x)) => VectorSpace (ShadeNeedle x)
instance (VectorSpace (Needle x)) => Semimanifold (ShadeNeedle x) where
type Needle (ShadeNeedle x) = ShadeNeedle x
(.+~^) = (^+^)
instance ∀ x . (PseudoAffine x, VectorSpace (Needle x)) => Semimanifold (Shade x) where
type Needle (Shade x) = ShadeNeedle x
(.+~^) = case semimanifoldWitness :: SemimanifoldWitness x of
SemimanifoldWitness
-> \(Shade c e) (ShadeNeedle v) -> Shade (c.+~^v) e
(.-~^) = case semimanifoldWitness :: SemimanifoldWitness x of
SemimanifoldWitness
-> \(Shade c e) (ShadeNeedle v) -> Shade (c.-~^v) e
semimanifoldWitness = case semimanifoldWitness :: SemimanifoldWitness x of
(SemimanifoldWitness)
-> SemimanifoldWitness
data ShadeHalfNeedle x = ShadeHalfNeedle -- TODO add shade-spread information
instance AdditiveMonoid (ShadeHalfNeedle x) where
zeroHV = undefined
addHVs = undefined
instance ( VectorSpace (Needle x)
) => HalfSpace (ShadeHalfNeedle x) where
type FullSubspace (ShadeHalfNeedle x) = Needle x
type Ray (ShadeHalfNeedle x) = Ray x
type MirrorJoin (ShadeHalfNeedle x) = Needle x
scaleNonNeg = undefined
fromFullSubspace = undefined
projectToFullSubspace = undefined
fullSubspaceIsVectorSpace _ = undefined
rayIsHalfSpace _ = undefined
mirrorJoinIsVectorSpace _ = undefined
fromPositiveHalf = undefined
fromNegativeHalf = undefined
instance ( AffineSpace x, Manifold x, Diff x ~ Needle x
, Atlas x, HasTrie (ChartIndex x) -- ??
, LinearSpace (Needle x), LinearSpace (Needle' x)
, Num' (Scalar (Needle x))
) => SemimanifoldWithBoundary (Shade x) where
type Interior (Shade x) = Shade' x
type Boundary (Shade x) = x
type HalfNeedle (Shade x) = ShadeHalfNeedle x
extendToBoundary = undefined
smfdWBoundWitness = undefined
needleIsOpenMfd _ = undefined
scalarIsOpenMfd _ = undefined
instance ( AffineSpace x, Manifold x, Diff x ~ Needle x
, Atlas x, HasTrie (ChartIndex x) -- ??
, Geodesic x
, LinearSpace (Needle x), LinearSpace (Needle' x)
, Scalar (Needle x) ~ ℝ
) => Geodesic (Shade x) where
geodesicBetween = gb dualSpaceWitness
where gb :: DualNeedleWitness x -> Shade x -> Shade x -> Maybe (D¹ -> Shade x)
gb DualSpaceWitness (Shade c (Norm e)) (Shade ζ (Norm η)) = pure interp
where interp t@(D¹ q) = Shade (pinterp t)
(Norm . arr . lerp ed ηd $ (q+1)/2)
ed@(LinearMap _) = arr e
ηd@(LinearMap _) = arr η
Just pinterp = geodesicBetween c ζ
newtype Shade'Needle x = Shade'Needle { shade'CtrDiff :: Needle x
-- TODO add shade-spread information
}
deriving instance (AdditiveGroup (Needle x)) => AdditiveGroup (Shade'Needle x)
deriving instance (VectorSpace (Needle x)) => VectorSpace (Shade'Needle x)
instance (VectorSpace (Needle x)) => Semimanifold (Shade'Needle x) where
type Needle (Shade'Needle x) = Shade'Needle x
(.+~^) = (^+^)
instance (AffineManifold x) => Semimanifold (Shade' x) where
type Needle (Shade' x) = Shade'Needle x
Shade' c e .+~^ Shade'Needle v = Shade' (c.+~^v) e
Shade' c e .-~^ Shade'Needle v = Shade' (c.-~^v) e
semimanifoldWitness = case semimanifoldWitness :: SemimanifoldWitness x of
SemimanifoldWitness -> SemimanifoldWitness
data Shade'HalfNeedle x = Shade'HalfNeedle -- TODO add shade-spread information
instance AdditiveMonoid (Shade'HalfNeedle x) where
zeroHV = undefined
addHVs = undefined
instance ( VectorSpace (Needle x)
) => HalfSpace (Shade'HalfNeedle x) where
type FullSubspace (Shade'HalfNeedle x) = Needle x
type Ray (Shade'HalfNeedle x) = Ray x
type MirrorJoin (Shade'HalfNeedle x) = Needle x
scaleNonNeg = undefined
fromFullSubspace = undefined
projectToFullSubspace = undefined
fullSubspaceIsVectorSpace _ = undefined
rayIsHalfSpace _ = undefined
mirrorJoinIsVectorSpace _ = undefined
fromPositiveHalf = undefined
fromNegativeHalf = undefined
instance ( AffineSpace x, Manifold x, Diff x ~ Needle x
, Atlas' x
, LinearSpace (Needle x), LinearSpace (Needle' x)
) => SemimanifoldWithBoundary (Shade' x) where
type Interior (Shade' x) = Shade x
type Boundary (Shade' x) = x
type HalfNeedle (Shade' x) = Shade'HalfNeedle x
extendToBoundary = undefined
smfdWBoundWitness = undefined
needleIsOpenMfd _ = undefined
scalarIsOpenMfd _ = undefined
instance ∀ x . (WithField ℝ AffineManifold x, Geodesic x, SimpleSpace (Needle x))
=> Geodesic (Shade' x) where
geodesicBetween (Shade' c e) (Shade' ζ η) = pure interp
where sharedSpan = sharedNormSpanningSystem e η
interp t = Shade' (pinterp t)
(spanNorm [ v ^/ (alerpB 1 (recip qη) t)
| (v,qη) <- sharedSpan ])
Just pinterp = geodesicBetween c ζ
fullShade :: (Semimanifold x, SimpleSpace (Needle x))
=> x -> Metric' x -> Shade x
fullShade ctr expa = Shade ctr expa
fullShade' :: WithField ℝ SimpleSpace x => x -> Metric x -> Shade' x
fullShade' ctr expa = Shade' ctr expa
infixl 6 :±, |±|
-- | Span a 'Shade' from a center point and multiple deviation-vectors.
#if GLASGOW_HASKELL < 800
pattern (:±) :: ()
#else
pattern (:±) :: (Semimanifold x, SimpleSpace (Needle x))
#endif
=> (Semimanifold x, SimpleSpace (Needle x))
=> x -> [Needle x] -> Shade x
pattern x :± shs <- (Shade x (varianceSpanningSystem -> shs))
where x :± shs = fullShade x $ spanVariance shs
-- | Similar to ':±', but instead of expanding the shade, each vector /restricts/ it.
-- Iff these form a orthogonal basis (in whatever sense applicable), then both
-- methods will be equivalent.
--
-- Note that '|±|' is only possible, as such, in an inner-product space; in
-- general you need reciprocal vectors ('Needle'') to define a 'Shade''.
(|±|) :: ∀ x . WithField ℝ EuclidSpace x => x -> [Needle x] -> Shade' x
x|±|shs = Shade' x $ spanNorm [v^/(v<.>v) | v<-shs]
-- | Attempt to find a 'Shade' that describes the distribution of given points.
-- At least in an affine space (and thus locally in any manifold), this can be used to
-- estimate the parameters of a normal distribution from which some points were
-- sampled. Note that some points will be “outside” of the shade,
-- as happens for a normal distribution with some statistical likelyhood.
-- (Use 'pointsCovers' if you need to prevent that.)
--
-- For /nonconnected/ manifolds it will be necessary to yield separate shades
-- for each connected component. And for an empty input list, there is no shade!
-- Hence the result type is a list.
pointsShades :: (WithField ℝ PseudoAffine x, SimpleSpace (Needle x))
=> [x] -> [Shade x]
pointsShades = map snd . pointsShades' mempty . map ((,()))
coverAllAround :: ∀ x s . ( Fractional' s, WithField s PseudoAffine x
, SimpleSpace (Needle x) )
=> x -> [Needle x] -> Shade x
coverAllAround x₀ offs = Shade x₀
$ guaranteeIn dualSpaceWitness offs
(scaleNorm (1/fromIntegral (length offs)) $ spanVariance offs)
where guaranteeIn :: DualNeedleWitness x -> [Needle x] -> Metric' x -> Metric' x
guaranteeIn w@DualSpaceWitness offs ex
= case offs >>= \v -> guard ((ex'|$|v) > 1) >> [(v, spanVariance [v])] of
[] -> ex
outs -> guaranteeIn w (fst<$>outs)
( densifyNorm $
ex <> scaleNorm
(sqrt . recip . fromIntegral
$ 2 * length outs)
(mconcat $ snd<$>outs)
)
where ex' = dualNorm ex
-- | Like 'pointsShades', but ensure that all points are actually in
-- the shade, i.e. if @['Shade' x₀ ex]@ is the result then
-- @'metric' (recipMetric ex) (p-x₀) ≤ 1@ for all @p@ in the list.
pointsCovers :: ∀ x . (WithField ℝ PseudoAffine x, SimpleSpace (Needle x))
=> [x] -> [Shade x]
pointsCovers = case pseudoAffineWitness :: PseudoAffineWitness x of
(PseudoAffineWitness (SemimanifoldWitness)) ->
\ps -> map (\(ps', Shade x₀ _)
-> coverAllAround x₀ [v | (p,())<-ps'
, let Just v
= p.-~.x₀])
(pointsShades' mempty ((,())<$>ps)
:: [([(x,())], Shade x)])
pointsShade's :: ∀ x . (WithField ℝ PseudoAffine x, SimpleSpace (Needle x))
=> [x] -> [Shade' x]
pointsShade's = case dualSpaceWitness :: DualNeedleWitness x of
DualSpaceWitness -> map (\(Shade c e :: Shade x) -> Shade' c $ dualNorm e) . pointsShades
pointsCover's :: ∀ x . (WithField ℝ PseudoAffine x, SimpleSpace (Needle x))
=> [x] -> [Shade' x]
pointsCover's = case dualSpaceWitness :: DualNeedleWitness x of
DualSpaceWitness -> map (\(Shade c e :: Shade x) -> Shade' c $ dualNorm e) . pointsCovers
pseudoECM :: ∀ x y p . (WithField ℝ PseudoAffine x, SimpleSpace (Needle x), Hask.Functor p)
=> p x -> NonEmpty (x,y) -> (x, ([(x,y)],[(x,y)]))
pseudoECM = case semimanifoldWitness :: SemimanifoldWitness x of
SemimanifoldWitness ->
\_ ((p₀,y₀) NE.:| psr) -> foldl' ( \(acc, (rb,nr)) (i,(p,y))
-> case (p.-~.acc, acc) of
(Just δ, acci)
-> (acci .+~^ δ^/i, ((p,y):rb, nr))
_ -> (acc, (rb, (p,y):nr)) )
(p₀, mempty)
( zip [1..] $ (p₀,y₀):psr )
pointsShades' :: ∀ x y . (WithField ℝ PseudoAffine x, SimpleSpace (Needle x))
=> Metric' x -> [(x,y)] -> [([(x,y)], Shade x)]
pointsShades' _ [] = []
pointsShades' minExt ps = case (expa, ctr) of
(Just e, c)
-> (ps, fullShade c e) : pointsShades' minExt unreachable
_ -> pointsShades' minExt inc'd
++ pointsShades' minExt unreachable
where (ctr,(inc'd,unreachable)) = pseudoECM ([]::[x]) $ NE.fromList ps
expa = ( (<>minExt) . spanVariance . map (^/ fromIntegral (length ps)) )
<$> mapM ((.-~.ctr) . fst) ps
-- | Attempt to reduce the number of shades to fewer (ideally, a single one).
-- In the simplest cases these should guaranteed cover the same area;
-- for non-flat manifolds it only works in a heuristic sense.
shadesMerge :: ∀ x . (WithField ℝ Manifold x, SimpleSpace (Needle x))
=> ℝ -- ^ How near (inverse normalised distance, relative to shade expanse)
-- two shades must be to be merged. If this is zero, any shades
-- in the same connected region of a manifold are merged.
-> [Shade x] -- ^ A list of /n/ shades.
-> [Shade x] -- ^ /m/ ≤ /n/ shades which cover at least the same area.
shadesMerge fuzz (sh@(Shade c₁ e₁) : shs)
= case extractJust (tryMerge pseudoAffineWitness dualSpaceWitness)
shs of
(Just mg₁, shs') -> shadesMerge fuzz
$ shs'++[mg₁] -- Append to end to prevent undue weighting
-- of first shade and its mergers.
(_, shs') -> sh : shadesMerge fuzz shs'
where tryMerge :: PseudoAffineWitness x -> DualNeedleWitness x
-> Shade x -> Maybe (Shade x)
tryMerge (PseudoAffineWitness (SemimanifoldWitness)) DualSpaceWitness
(Shade c₂ e₂)
| Just v <- c₁.-~.c₂
, [e₁',e₂'] <- dualNorm<$>[e₁, e₂]
, b₁ <- e₂'|$|v
, b₂ <- e₁'|$|v
, fuzz*b₁*b₂ <= b₁ + b₂
= Just $ let cc = c₂ .+~^ v ^/ 2
Just cv₁ = c₁.-~.cc
Just cv₂ = c₂.-~.cc
in Shade cc $ e₁ <> e₂ <> spanVariance [cv₁, cv₂]
| otherwise = Nothing
shadesMerge _ shs = shs
-- | Weakened version of 'intersectShade's'. What this function calculates is
-- rather the /weighted mean/ of ellipsoid regions. If you interpret the
-- shades as uncertain physical measurements with normal distribution,
-- it gives the maximum-likelyhood result for multiple measurements of the
-- same quantity.
mixShade's :: ∀ y . (WithField ℝ Manifold y, SimpleSpace (Needle y))
=> NonEmpty (Shade' y) -> Maybe (Shade' y)
mixShade's = ms pseudoAffineWitness dualSpaceWitness
where ms :: PseudoAffineWitness y -> DualNeedleWitness y
-> NonEmpty (Shade' y) -> Maybe (Shade' y)
ms (PseudoAffineWitness (SemimanifoldWitness)) DualSpaceWitness
(Shade' c₀ (Norm e₁):|shs) = sequenceA ciso >> pure mixed
where ciso = [ci.-~.c₀ | Shade' ci shi <- shs]
cis = [v | Just v <- ciso]
σe = arr . sumV $ e₁ : (applyNorm . _shade'Narrowness<$>shs)
cc = σe \$ sumV [ei $ ci | ci <- cis
| Shade' _ (Norm ei) <- shs]
mixed = Shade' (c₀+^cc) $ densifyNorm ( mconcat
[ Norm $ ei ^/ (1+(normSq ni $ ci^-^cc))
| ni@(Norm ei) <- Norm e₁ : (_shade'Narrowness<$>shs)
| ci <- zeroV : cis
] )
(+^) = (.+~^)
-- cc should minimise the quadratic form
-- β(cc) = ∑ᵢ ⟨cc−cᵢ|eᵢ|cc−cᵢ⟩
-- = ⟨cc|e₁|cc⟩ + ∑ᵢ₌₁… ⟨cc−c₂|e₂|cc−c₂⟩
-- = ⟨cc|e₁|cc⟩ + ∑ᵢ₌₁…( ⟨cc|eᵢ|cc⟩ − 2⋅⟨cᵢ|eᵢ|cc⟩ + ⟨cᵢ|eᵢ|cᵢ⟩ )
-- It is thus
-- β(cc + δ⋅v) − β cc
-- = ⟨cc + δ⋅v|e₁|cc + δ⋅v⟩
-- + ∑ᵢ₌₁…( ⟨cc + δ⋅v|eᵢ|cc + δ⋅v⟩ − 2⋅⟨cᵢ|eᵢ|cc + δ⋅v⟩ + ⟨cᵢ|eᵢ|cᵢ⟩ )
-- − ⟨cc|e₁|cc⟩
-- − ∑ᵢ₌₁…( ⟨cc|eᵢ|cc⟩ + 2⋅⟨cᵢ|eᵢ|cc⟩ − ⟨cᵢ|eᵢ|cᵢ⟩ )
-- = ⟨cc + δ⋅v|e₁|cc + δ⋅v⟩
-- + ∑ᵢ₌₁…( ⟨cc + δ⋅v|eᵢ|cc + δ⋅v⟩ − 2⋅⟨cᵢ|eᵢ|δ⋅v⟩ )
-- − ⟨cc|e₁|cc⟩
-- − ∑ᵢ₌₁…( ⟨cc|eᵢ|cc⟩ )
-- = 2⋅⟨δ⋅v|e₁|cc⟩ + ⟨δ⋅v|e₁|δ⋅v⟩
-- + ∑ᵢ₌₁…( 2⋅⟨δ⋅v|eᵢ|cc⟩ + ⟨δ⋅v|eᵢ|δ⋅v⟩ − 2⋅⟨cᵢ|eᵢ|δ⋅v⟩ )
-- = 2⋅⟨δ⋅v|∑ᵢeᵢ|cc⟩ − 2⋅∑ᵢ₌₁… ⟨cᵢ|eᵢ|δ⋅v⟩ + 𝓞(δ²)
-- This should vanish for all v, which is fulfilled by
-- (∑ᵢeᵢ)|cc⟩ = ∑ᵢ₌₁… eᵢ|cᵢ⟩.
-- | Evaluate the shade as a quadratic form; essentially
-- @
-- minusLogOcclusion sh x = x <.>^ (sh^.shadeExpanse $ x - sh^.shadeCtr)
-- @
-- where 'shadeExpanse' gives a metric (matrix) that characterises the
-- width of the shade.
minusLogOcclusion' :: ∀ x s . ( PseudoAffine x, LinearSpace (Needle x)
, s ~ (Scalar (Needle x)), RealFloat' s )
=> Shade' x -> x -> s
minusLogOcclusion' (Shade' p₀ δinv)
= occ (pseudoAffineWitness :: PseudoAffineWitness x)
(dualSpaceWitness :: DualNeedleWitness x)
where occ (PseudoAffineWitness (SemimanifoldWitness)) DualSpaceWitness
p = case p.-~.p₀ of
(Just vd) | mSq <- normSq δinv vd
, mSq == mSq -- avoid NaN
-> mSq
_ -> 1/0
minusLogOcclusion :: ∀ x s . ( PseudoAffine x, SimpleSpace (Needle x)
, s ~ (Scalar (Needle x)), RealFloat' s )
=> Shade x -> x -> s
minusLogOcclusion (Shade p₀ δ)
= occ (pseudoAffineWitness :: PseudoAffineWitness x)
(dualSpaceWitness :: DualNeedleWitness x)
where occ (PseudoAffineWitness (SemimanifoldWitness)) DualSpaceWitness
= \p -> case p.-~.p₀ of
(Just vd) | mSq <- normSq δinv vd
, mSq == mSq -- avoid NaN
-> mSq
_ -> 1/0
where δinv = dualNorm δ
rangeWithinVertices :: ∀ i m t s
. ( Geodesic i
, Geodesic m
, WithField s AffineManifold (Interior i)
, WithField s AffineManifold (Interior m)
, SimpleSpace (Needle (Interior i))
, SimpleSpace (Needle (Interior m))
, SimpleSpace (Needle' (Interior i))
, SimpleSpace (Needle' (Interior m))
, RealFrac' s
, Hask.Traversable t )
=> (Interior i,Interior m) -> t (i,m)
-> Maybe (Shade (Interior i) -> Shade (Interior m))
rangeWithinVertices (cii,cmi) verts = do
vs <- sequenceA [ fzip ( middleBetween pi ci >>= (toInterior>=>(.-~.cii))
, middleBetween pm cm >>= (toInterior>=>(.-~.cmi)) )
| (pi, pm) <- Hask.toList verts ]
affinSys <- correspondingDirections @(Interior m) @(Interior i)
(cii,cmi) vs
return $ embedShade affinSys
where ci = fromInterior cii
cm = fromInterior cmi
data DebugView x where
DebugView :: ( Show x, Show (Needle x+>Needle' x), LinearShowable (Needle x)
, Needle' x ~ Needle x ) => DebugView x
-- | Class of manifolds which can use 'Shade'' as a basic set type.
-- This is easily possible for vector spaces with the default implementations.
class (WithField ℝ PseudoAffine y, SimpleSpace (Needle y)) => Refinable y where
debugView :: Maybe (DebugView y)
default debugView :: ( Show y, Show (Needle y+>Needle' y)
, Needle' y~Needle y, LinearShowable (Needle y) )
=> Maybe (DebugView y)
debugView = Just DebugView
-- | @a `subShade'` b ≡ True@ means @a@ is fully contained in @b@, i.e. from
-- @'minusLogOcclusion'' a p < 1@ follows also @minusLogOcclusion' b p < 1@.
subShade' :: Shade' y -> Shade' y -> Bool
subShade' (Shade' ac ae) (Shade' tc te)
= case pseudoAffineWitness :: PseudoAffineWitness y of
PseudoAffineWitness (SemimanifoldWitness)
| Just v <- tc.-~.ac
, v² <- normSq te v
, v² <= 1
-> all (\(y',μ) -> case μ of
Nothing -> True -- 'te' has infinite extension in this direction
Just ξ
| ξ<1 -> False -- 'ae' would be vaster than 'te' in this direction
| ω <- abs $ y'<.>^v
-> (ω + 1/ξ)^2 <= 1 - v² + ω^2
-- See @images/constructions/subellipse-check-heuristic.svg@
) $ sharedSeminormSpanningSystem te ae
_ -> False
-- | Intersection between two shades.
refineShade' :: Shade' y -> Shade' y -> Maybe (Shade' y)
refineShade' (Shade' c₀ (Norm e₁)) (Shade' c₀₂ (Norm e₂))
= case ( dualSpaceWitness :: DualNeedleWitness y
, pseudoAffineWitness :: PseudoAffineWitness y ) of
(DualSpaceWitness, PseudoAffineWitness (SemimanifoldWitness))
-> do
c₂ <- c₀₂.-~.c₀
let σe = arr $ e₁^+^e₂
e₁c₂ = e₁ $ c₂
e₂c₂ = e₂ $ c₂
cc = σe \$ e₂c₂
cc₂ = cc ^-^ c₂
e₁cc = e₁ $ cc
e₂cc = e₂ $ cc
α = 2 + e₂c₂<.>^cc₂
guard (α > 0)
let ee = σe ^/ α
c₂e₁c₂ = e₁c₂<.>^c₂
c₂e₂c₂ = e₂c₂<.>^c₂
c₂eec₂ = (c₂e₁c₂ + c₂e₂c₂) / α
return $ case middle . sort
$ quadraticEqnSol c₂e₁c₂
(2 * (e₁cc<.>^c₂))
(e₁cc<.>^cc - 1)
++quadraticEqnSol c₂e₂c₂
(2 * (e₂cc<.>^c₂ - c₂e₂c₂))
(e₂cc<.>^cc - 2 * (e₂c₂<.>^cc) + c₂e₂c₂ - 1) of
[γ₁,γ₂] | abs (γ₁+γ₂) < 2 -> let
cc' = cc ^+^ ((γ₁+γ₂)/2)*^c₂
rγ = abs (γ₁ - γ₂) / 2
η = if rγ * c₂eec₂ /= 0 && 1 - rγ^2 * c₂eec₂ > 0
then sqrt (1 - rγ^2 * c₂eec₂) / (rγ * c₂eec₂)
else 0
in Shade' (c₀.+~^cc')
(Norm (arr ee) <> spanNorm [ee $ c₂^*η])
_ -> Shade' (c₀.+~^cc) (Norm $ arr ee)
where quadraticEqnSol a b c
| a == 0, b /= 0 = [-c/b]
| a /= 0 && disc == 0 = [- b / (2*a)]
| a /= 0 && disc > 0 = [ (σ * sqrt disc - b) / (2*a)
| σ <- [-1, 1] ]
| otherwise = []
where disc = b^2 - 4*a*c
middle (_:x:y:_) = [x,y]
middle l = l
-- ⟨x−c₁|e₁|x−c₁⟩ < 1 ∧ ⟨x−c₂|e₂|x−c₂⟩ < 1
-- We search (cc,ee) such that this implies
-- ⟨x−cc|ee|x−cc⟩ < 1.
-- Let WLOG c₁ = 0, so
-- ⟨x|e₁|x⟩ < 1.
-- cc should minimise the quadratic form
-- β(cc) = ⟨cc−c₁|e₁|cc−c₁⟩ + ⟨cc−c₂|e₂|cc−c₂⟩
-- = ⟨cc|e₁|cc⟩ + ⟨cc−c₂|e₂|cc−c₂⟩
-- = ⟨cc|e₁|cc⟩ + ⟨cc|e₂|cc⟩ − 2⋅⟨c₂|e₂|cc⟩ + ⟨c₂|e₂|c₂⟩
-- It is thus
-- β(cc + δ⋅v) − β cc
-- = ⟨cc + δ⋅v|e₁|cc + δ⋅v⟩ + ⟨cc + δ⋅v|e₂|cc + δ⋅v⟩ − 2⋅⟨c₂|e₂|cc + δ⋅v⟩ + ⟨c₂|e₂|c₂⟩
-- − ⟨cc|e₁|cc⟩ − ⟨cc|e₂|cc⟩ + 2⋅⟨c₂|e₂|cc⟩ − ⟨c₂|e₂|c₂⟩
-- = ⟨cc + δ⋅v|e₁|cc + δ⋅v⟩ + ⟨cc + δ⋅v|e₂|cc + δ⋅v⟩ − 2⋅⟨c₂|e₂|δ⋅v⟩
-- − ⟨cc|e₁|cc⟩ − ⟨cc|e₂|cc⟩
-- = 2⋅⟨δ⋅v|e₁|cc⟩ + ⟨δ⋅v|e₁|δ⋅v⟩ + 2⋅⟨δ⋅v|e₂|cc⟩ + ⟨δ⋅v|e₂|δ⋅v⟩ − 2⋅⟨c₂|e₂|δ⋅v⟩
-- = 2⋅δ⋅⟨v|e₁+e₂|cc⟩ − 2⋅δ⋅⟨v|e₂|c₂⟩ + 𝓞(δ²)
-- This should vanish for all v, which is fulfilled by
-- (e₁+e₂)|cc⟩ = e₂|c₂⟩.
--
-- If we now choose
-- ee = (e₁+e₂) / α
-- then
-- ⟨x−cc|ee|x−cc⟩ ⋅ α
-- = ⟨x−cc|ee|x⟩ ⋅ α − ⟨x−cc|ee|cc⟩ ⋅ α
-- = ⟨x|ee|x−cc⟩ ⋅ α − ⟨x−cc|e₂|c₂⟩
-- = ⟨x|ee|x⟩ ⋅ α − ⟨x|ee|cc⟩ ⋅ α − ⟨x−cc|e₂|c₂⟩
-- = ⟨x|e₁+e₂|x⟩ − ⟨x|e₂|c₂⟩ − ⟨x−cc|e₂|c₂⟩
-- = ⟨x|e₁|x⟩ + ⟨x|e₂|x⟩ − ⟨x|e₂|c₂⟩ − ⟨x−cc|e₂|c₂⟩
-- < 1 + ⟨x|e₂|x−c₂⟩ − ⟨x−cc|e₂|c₂⟩
-- = 1 + ⟨x−c₂|e₂|x−c₂⟩ + ⟨c₂|e₂|x−c₂⟩ − ⟨x−cc|e₂|c₂⟩
-- < 2 + ⟨x−c₂−x+cc|e₂|c₂⟩
-- = 2 + ⟨cc−c₂|e₂|c₂⟩
-- Really we want
-- ⟨x−cc|ee|x−cc⟩ ⋅ α < α
-- So choose α = 2 + ⟨cc−c₂|e₂|c₂⟩.
--
-- The ellipsoid "cc±√ee" captures perfectly the intersection
-- of the boundary of the shades, but it tends to significantly
-- overshoot the interior intersection in perpendicular direction,
-- i.e. in direction of c₂−c₁. E.g.
-- https://github.com/leftaroundabout/manifolds/blob/bc0460b9/manifolds/images/examples/ShadeCombinations/EllipseIntersections.png
-- 1. Really, the relevant points are those where either of the
-- intersector badnesses becomes 1. The intersection shade should
-- be centered between those points. We perform according corrections,
-- but only in c₂ direction, so this can be handled efficiently
-- as a 1D quadratic equation.
-- Consider
-- dⱼ c := ⟨c−cⱼ|eⱼ|c−cⱼ⟩ =! 1
-- dⱼ (cc + γ⋅c₂)
-- = ⟨cc+γ⋅c₂−cⱼ|eⱼ|cc+γ⋅c₂−cⱼ⟩
-- = ⟨cc−cⱼ|eⱼ|cc−cⱼ⟩ + 2⋅γ⋅⟨c₂|eⱼ|cc−cⱼ⟩ + γ²⋅⟨c₂|eⱼ|c₂⟩
-- =! 1
-- So
-- γⱼ = (- b ± √(b²−4⋅a⋅c)) / 2⋅a
-- where a = ⟨c₂|eⱼ|c₂⟩
-- b = 2 ⋅ (⟨c₂|eⱼ|cc⟩ − ⟨c₂|eⱼ|cⱼ⟩)
-- c = ⟨cc|eⱼ|cc⟩ − 2⋅⟨cc|eⱼ|cⱼ⟩ + ⟨cⱼ|eⱼ|cⱼ⟩ − 1
-- The ± sign should be chosen to get the smaller |γ| (otherwise
-- we end up on the wrong side of the shade), i.e.
-- γⱼ = (sgn bⱼ ⋅ √(bⱼ²−4⋅aⱼ⋅cⱼ) − bⱼ) / 2⋅aⱼ
-- 2. Trim the result in that direction to the actual
-- thickness of the lens-shaped intersection: we want
-- ⟨rγ⋅c₂|ee'|rγ⋅c₂⟩ = 1
-- for a squeezed version of ee,
-- ee' = ee + ee|η⋅c₂⟩⟨η⋅c₂|ee
-- ee' = ee + η² ⋅ ee|c₂⟩⟨c₂|ee
-- ⟨rγ⋅c₂|ee'|rγ⋅c₂⟩
-- = rγ² ⋅ (⟨c₂|ee|c₂⟩ + η² ⋅ ⟨c₂|ee|c₂⟩²)
-- = rγ² ⋅ ⟨c₂|ee|c₂⟩ + η² ⋅ rγ² ⋅ ⟨c₂|ee|c₂⟩²
-- η² = (1 − rγ²⋅⟨c₂|ee|c₂⟩) / (rγ² ⋅ ⟨c₂|ee|c₂⟩²)
-- η = √(1 − rγ²⋅⟨c₂|ee|c₂⟩) / (rγ ⋅ ⟨c₂|ee|c₂⟩)
-- With ⟨c₂|ee|c₂⟩ = (⟨c₂|e₁|c₂⟩ + ⟨c₂|e₂|c₂⟩)/α.
-- | If @p@ is in @a@ (red) and @δ@ is in @b@ (green),
-- then @p.+~^δ@ is in @convolveShade' a b@ (blue).
--
-- Example: https://nbviewer.jupyter.org/github/leftaroundabout/manifolds/blob/master/test/ShadeCombinations.ipynb#shadeConvolutions
--
-- <<images/examples/ShadeCombinations/2Dconvolution-skewed.png>>
convolveMetric :: Hask.Functor p => p y -> Metric y -> Metric y -> Metric y
convolveMetric _ ey eδ = case wellDefinedNorm result of
Just r -> r
Nothing -> case debugView :: Maybe (DebugView y) of
Just DebugView -> error $ "Can not convolve norms "
++show (arr (applyNorm ey) :: Needle y+>Needle' y)
++" and "++show (arr (applyNorm eδ) :: Needle y+>Needle' y)
where eδsp = sharedSeminormSpanningSystem ey eδ
result = spanNorm [ f ^* ζ crl | (f,crl) <- eδsp ]
ζ = case filter (>0) . catMaybes $ snd<$>eδsp of
[] -> const 0
nzrelap
-> let cre₁ = 1/minimum nzrelap
cre₂ = maximum nzrelap
edgeFactor = sqrt ( (1 + cre₁)^2 + (1 + cre₂)^2 )
/ (sqrt (1 + cre₁^2) + sqrt (1 + cre₂^2))
in \case
Nothing -> 0
Just 0 -> 0
Just sq -> edgeFactor / (recip sq + 1)
convolveShade' :: Shade' y -> Shade' (Needle y) -> Shade' y
convolveShade' = defaultConvolveShade'
defaultConvolveShade' :: ∀ y . Refinable y => Shade' y -> Shade' (Needle y) -> Shade' y
defaultConvolveShade' = case (pseudoAffineWitness :: PseudoAffineWitness y) of
PseudoAffineWitness (SemimanifoldWitness)
-> \(Shade' y₀ ey) (Shade' δ₀ eδ) -> Shade' (y₀.+~^δ₀)
$ convolveMetric ([]::[y]) ey eδ
instance Refinable ℝ where
refineShade' (Shade' cl el) (Shade' cr er)
= case (normSq el 1, normSq er 1) of
(0, _) -> return $ Shade' cr er
(_, 0) -> return $ Shade' cl el
(ql,qr) | ql>0, qr>0
-> let [rl,rr] = sqrt . recip <$> [ql,qr]
b = maximum $ zipWith (-) [cl,cr] [rl,rr]
t = minimum $ zipWith (+) [cl,cr] [rl,rr]
in guard (b<t) >>
let cm = (b+t)/2
rm = (t-b)/2
in return $ Shade' cm (spanNorm [recip rm])
-- convolveShade' (Shade' y₀ ey) (Shade' δ₀ eδ)
-- = case (metricSq ey 1, metricSq eδ 1) of
-- (wy,wδ) | wy>0, wδ>0
-- -> Shade' (y₀.+~^δ₀)
-- ( projector . recip
-- $ recip (sqrt wy) + recip (sqrt wδ) )
-- (_ , _) -> Shade' y₀ zeroV
instance ∀ a b . ( Refinable a, Refinable b
, Scalar (DualVector (DualVector (Needle b)))
~ Scalar (DualVector (DualVector (Needle a))) )
=> Refinable (a,b) where
debugView = case ( debugView :: Maybe (DebugView a)
, debugView :: Maybe (DebugView b)
, dualSpaceWitness :: DualSpaceWitness (Needle a)
, dualSpaceWitness :: DualSpaceWitness (Needle b) ) of
(Just DebugView, Just DebugView, DualSpaceWitness, DualSpaceWitness)
-> Just DebugView
instance Refinable ℝ⁰
instance Refinable ℝ¹
instance Refinable ℝ²
instance Refinable ℝ³
instance Refinable ℝ⁴
instance ( SimpleSpace a, SimpleSpace b
, Refinable a, Refinable b
, Scalar a ~ ℝ, Scalar b ~ ℝ
, Scalar (DualVector a) ~ ℝ, Scalar (DualVector b) ~ ℝ
, Scalar (DualVector (DualVector a)) ~ ℝ, Scalar (DualVector (DualVector b)) ~ ℝ )
=> Refinable (LinearMap ℝ a b) where
debugView = Nothing
intersectShade's :: ∀ y . Refinable y => NonEmpty (Shade' y) -> Maybe (Shade' y)
intersectShade's (sh:|shs) = Hask.foldrM refineShade' sh shs
-- | Essentially the same as @(x,y)@, but not considered as a product topology.
-- The 'Semimanifold' etc. instances just copy the topology of @x@, ignoring @y@.
data x`WithAny`y
= WithAny { _untopological :: y
, _topological :: !x }
deriving (Hask.Functor, Show, Generic)
instance (NFData x, NFData y) => NFData (WithAny x y)
instance ∀ x y . (Semimanifold x) => Semimanifold (x`WithAny`y) where
type Needle (WithAny x y) = Needle x
WithAny y x .+~^ δx = WithAny y $ x.+~^δx
semimanifoldWitness = case semimanifoldWitness :: SemimanifoldWitness x of
SemimanifoldWitness -> SemimanifoldWitness
instance (PseudoAffine x) => PseudoAffine (x`WithAny`y) where
WithAny _ x .-~! WithAny _ ξ = x.-~!ξ
WithAny _ x .-~. WithAny _ ξ = x.-~.ξ
pseudoAffineWitness = case pseudoAffineWitness :: PseudoAffineWitness x of
PseudoAffineWitness (SemimanifoldWitness)
-> PseudoAffineWitness (SemimanifoldWitness)
instance (AffineSpace x) => AffineSpace (x`WithAny`y) where
type Diff (WithAny x y) = Diff x
WithAny _ x .-. WithAny _ ξ = x.-.ξ
WithAny y x .+^ δx = WithAny y $ x.+^δx
instance (VectorSpace x, Monoid y) => VectorSpace (x`WithAny`y) where
type Scalar (WithAny x y) = Scalar x
μ *^ WithAny y x = WithAny y $ μ*^x
instance (AdditiveGroup x, Monoid y) => AdditiveGroup (x`WithAny`y) where
zeroV = WithAny mempty zeroV
negateV (WithAny y x) = WithAny y $ negateV x
WithAny y x ^+^ WithAny υ ξ = WithAny (mappend y υ) (x^+^ξ)
instance (AdditiveGroup x) => Hask.Applicative (WithAny x) where
pure x = WithAny x zeroV
WithAny f x <*> WithAny t ξ = WithAny (f t) (x^+^ξ)
instance (AdditiveGroup x) => Hask.Monad (WithAny x) where
return x = WithAny x zeroV
WithAny y x >>= f = WithAny r $ x^+^q
where WithAny r q = f y
shadeWithAny :: y -> Shade x -> Shade (x`WithAny`y)
shadeWithAny y (Shade x xe) = Shade (WithAny y x) xe
shadeWithoutAnything :: Semimanifold x => Shade (x`WithAny`y) -> Shade x
shadeWithoutAnything (Shade (WithAny _ b) e) = Shade b e
extractJust :: (a->Maybe b) -> [a] -> (Maybe b, [a])
extractJust f [] = (Nothing,[])
extractJust f (x:xs) | Just r <- f x = (Just r, xs)
| otherwise = second (x:) $ extractJust f xs
prettyShowShade' :: LtdErrorShow x => Shade' x -> String
prettyShowShade' sh = prettyShowsPrecShade' 0 sh []
instance LtdErrorShow x => SP.Show (Shade' x) where
showsPrec = prettyShowsPrecShade'
instance LtdErrorShow x => SP.Show (Shade x) where
showsPrec = prettyShowsPrecShade
wellDefinedShade' :: LinearSpace (Needle x) => Shade' x -> Maybe (Shade' x)
wellDefinedShade' (Shade' c e) = Shade' c <$> wellDefinedNorm e
data LtdErrorShowWitness m where
LtdErrorShowWitness :: (LtdErrorShow m, LtdErrorShow (Needle m))
=> PseudoAffineWitness m -> LtdErrorShowWitness m
class Refinable m => LtdErrorShow m where
ltdErrorShowWitness :: LtdErrorShowWitness m
default ltdErrorShowWitness :: (LtdErrorShow m, LtdErrorShow (Needle m))
=> LtdErrorShowWitness m
ltdErrorShowWitness = LtdErrorShowWitness pseudoAffineWitness
showsPrecShade'_errorLtdC :: Int -> Shade' m -> ShowS
prettyShowsPrecShade :: Int -> Shade m -> ShowS
prettyShowsPrecShade p sh@(Shade c e')
= showParen (p>6) $ v
. (":±["++) . flip (foldr id) (intersperse (',':) u) . (']':)
where v = showsPrecShade'_errorLtdC 6 (Shade' c e :: Shade' m)
u :: [ShowS] = case ltdErrorShowWitness :: LtdErrorShowWitness m of
LtdErrorShowWitness (PseudoAffineWitness SemimanifoldWitness) ->
[ showsPrecShade'_errorLtdC 6 (Shade' δ e :: Shade' (Needle m))
| δ <- varianceSpanningSystem e']
e = dualNorm' e'
prettyShowsPrecShade' :: Int -> Shade' m -> ShowS
prettyShowsPrecShade' p sh@(Shade' c e)
= showParen (p>6) $ v
. ("|±|["++) . flip (foldr id) (intersperse (',':) u) . (']':)
where v = showsPrecShade'_errorLtdC 6 sh
u :: [ShowS] = case ltdErrorShowWitness :: LtdErrorShowWitness m of
LtdErrorShowWitness (PseudoAffineWitness SemimanifoldWitness) ->
[ showsPrecShade'_errorLtdC 6 (Shade' δ e :: Shade' (Needle m))
| δ <- varianceSpanningSystem e']
e' = dualNorm e
instance LtdErrorShow ℝ⁰ where
showsPrecShade'_errorLtdC _ _ = ("zeroV"++)
instance LtdErrorShow ℝ where
showsPrecShade'_errorLtdC _ (Shade' v u) = errorLtdShow (δ/2) v
where δ = case u<$|1 of
σ | σ>0 -> sqrt $ 1/σ
_ -> v*10
instance LtdErrorShow ℝ² where
showsPrecShade'_errorLtdC _ sh = ("V2 "++) . shshx . (' ':) . shshy
where shx = projectShade (lensEmbedding _x) sh :: Shade' ℝ
shy = projectShade (lensEmbedding _y) sh :: Shade' ℝ
shshx = showsPrecShade'_errorLtdC 0 shx
shshy = showsPrecShade'_errorLtdC 0 shy
instance LtdErrorShow ℝ³ where
showsPrecShade'_errorLtdC _ sh = ("V3 "++) . shshx . (' ':) . shshy . (' ':) . shshz
where shx = projectShade (lensEmbedding _x) sh :: Shade' ℝ
shy = projectShade (lensEmbedding _y) sh :: Shade' ℝ
shz = projectShade (lensEmbedding _z) sh :: Shade' ℝ
shshx = showsPrecShade'_errorLtdC 0 shx
shshy = showsPrecShade'_errorLtdC 0 shy
shshz = showsPrecShade'_errorLtdC 0 shz
instance LtdErrorShow ℝ⁴ where
showsPrecShade'_errorLtdC _ sh
= ("V4 "++) . shshx . (' ':) . shshy . (' ':) . shshz . (' ':) . shshw
where shx = projectShade (lensEmbedding _x) sh :: Shade' ℝ
shy = projectShade (lensEmbedding _y) sh :: Shade' ℝ
shz = projectShade (lensEmbedding _z) sh :: Shade' ℝ
shw = projectShade (lensEmbedding _w) sh :: Shade' ℝ
shshx = showsPrecShade'_errorLtdC 0 shx
shshy = showsPrecShade'_errorLtdC 0 shy
shshz = showsPrecShade'_errorLtdC 0 shz
shshw = showsPrecShade'_errorLtdC 0 shw
instance ∀ x y .
( LtdErrorShow x, LtdErrorShow y
, Scalar (DualVector (Needle' x)) ~ Scalar (DualVector (Needle' y)) )
=> LtdErrorShow (x,y) where
ltdErrorShowWitness = case ( ltdErrorShowWitness :: LtdErrorShowWitness x
, ltdErrorShowWitness :: LtdErrorShowWitness y ) of
( LtdErrorShowWitness(PseudoAffineWitness(SemimanifoldWitness))
, LtdErrorShowWitness(PseudoAffineWitness(SemimanifoldWitness)) )
->LtdErrorShowWitness(PseudoAffineWitness(SemimanifoldWitness))
showsPrecShade'_errorLtdC _ sh = ('(':) . shshx . (',':) . shshy . (')':)
where (shx,shy) = factoriseShade sh
shshx = showsPrecShade'_errorLtdC 0 shx
shshy = showsPrecShade'_errorLtdC 0 shy
instance ∀ v .
(HilbertSpace v, SemiInner v, FiniteDimensional v, LtdErrorShow v, Scalar v ~ ℝ)
=> LtdErrorShow (LinearMap ℝ v ℝ) where
showsPrecShade'_errorLtdC p sh = showParen (p>7) $
("().<"++) . showsPrecShade'_errorLtdC 7
(linIsoTransformShade (arr fromLinearForm) sh :: Shade' v)
instance ∀ v .
(HilbertSpace v, SemiInner v, FiniteDimensional v, LtdErrorShow v, Scalar v ~ ℝ)
=> LtdErrorShow (LinearMap ℝ v (ℝ,ℝ)) where
showsPrecShade'_errorLtdC p sh = showParen (p>7) $
( "Left ().<"++) . showsPrecShade'_errorLtdC 7 shx
. ("^+^Right().<"++) . showsPrecShade'_errorLtdC 7 shy
where (shx,shy) = factoriseShade
(linIsoTransformShade (lfun $ \f
-> ( fromLinearForm $ fst . f
, fromLinearForm $ snd . f ) ) sh
:: Shade' (v,v))
instance LtdErrorShow x => Show (Shade' x) where
showsPrec = prettyShowsPrecShade'