use hmatrix for least squares solving

* this fixes the convergence problems of cubic Bézier curve fitting

MetaBrush.cabal

@@ -68,7 +68,7 @@ library
       , Math.Bezier.Quadratic
       , Math.Bezier.Stroke
       , Math.Epsilon
-      , Math.Linear.SVD
+      , Math.Linear.Solve
       , Math.Module
       , Math.Roots
       , Math.Vector2D
@@ -76,8 +76,12 @@ library
     build-depends:
         groups-generic
           ^>= 0.1.0.0
+      , hmatrix
+          ^>= 0.20.0.0
       , vector
           ^>= 0.12.1.2
+      , QuickCheck
+          ^>= 2.14.1
 
 executable MetaBrush
 

cabal.project

@@ -9,3 +9,10 @@ source-repository-package
   location: https://github.com/thestr4ng3r/gi-cairo-render 
   tag: 8727c43cdf91aeedffc9cb4c5575f56660a86399
   subdir: gi-cairo-render
+
+-- latest version of hmatrix
+source-repository-package
+  type: git
+  location: https://github.com/haskell-numerics/hmatrix
+  tag: 08138810946c7eae2254feeb33269cd962d5e0c8
+  subdir: packages/base

src/lib/Math/Bezier/Cubic/Fit.hs

@@ -16,6 +16,10 @@ import Data.Complex
   ( Complex(..) )
 import Data.Foldable
   ( for_ )
+import Data.Functor
+  ( ($>) )
+import Data.Semigroup
+  ( Arg(..), Max(..), ArgMax )
 
 -- acts
 import Data.Act
@@ -42,8 +46,8 @@ import Math.Bezier.Cubic
   ( Bezier(..), bezier, ddist )
 import Math.Epsilon
   ( epsilon )
-import Math.Linear.SVD
+import Math.Linear.Solve
-  ( lsolve )
+  ( linearSolve )
 import Math.Module
   ( Module((*^), (^-^))
   , Inner((^.^)), quadrance
@@ -63,34 +67,42 @@ import Math.Vector2D
 --   * ends at \( r \) with tangent \( \textrm{t}_r \),
 --   * best fits the intermediate sequence of points \( \left ( q_i \right )_{i=1}^n \).
 --
+-- This function also returns \( \textrm{ArgMax}\ t_\textrm{max}\ d^2_\textrm{max}: \)
+-- the parameter and squared distance of the worst-fitting point.
+-- It is guaranteed that all points to fit lie within the tubular neighbourhood
+-- of radius \( d_\textrm{max} \) of the fitted curve.
+--
 -- /Note/: the order of the intermediate points is important.
 --
 -- Proceeds by fitting a cubic Bézier curve \( B(t) \), \( 0 \leqslant t \leqslant 1 \),
 -- with given endpoints and tangents, which minimises the sum of squares functional
 -- 
--- \[ \sum_{i=1}^n \left \| C(t_i) - q_i \right \|^2. \]
+-- \[ \sum_{i=1}^n \Big \| B(t_i) - q_i \Big \|^2. \]
 --
 -- The values of the parameters \( \left ( t_i \right )_{i=1}^n \) are recursively estimated,
--- starting from uniform parametrisation.
+-- starting from uniform parametrisation (this will be the fit if `maxIters` is 0).
 --
 -- The iteration ends when any of the following conditions are satisfied:
 --
---   * each new estimated parameter values \( t_i' \) differs from
+--   * each new estimated parameter value \( t_i' \) differs from
 --     its previous value \( t_i \) by less than \( \texttt{t_tol} \),
---   * each point \( C(t_i) \) is within squared distance \( \texttt{sq_dist_tol} \)
+--   * each on-curve point \( B(t_i) \) is within distance \( \texttt{dist_tol} \)
---     of the point \( q_i \) it is associated with,
+--     of its corresponding point to fit \( q_i \),
---   * the maximum iteration limit \( \texttt{maxCount} \) has been reached.
+--   * the maximum iteration limit \( \texttt{maxIters} \) has been reached.
 fitPiece
   :: Double             -- ^ \( \texttt{t_tol} \), the tolerance for the Bézier parameter
-  -> Double             -- ^ \( \texttt{sq_dist_tol} \), tolerance for the squared distance
+  -> Double             -- ^ \( \texttt{dist_tol} \), tolerance for the distance
-  -> Int                -- ^ \( \texttt{maxCount} \), maximum number of iterations
+  -> Int                -- ^ \( \texttt{maxIters} \), maximum number of iterations
   -> Point2D Double     -- ^ \( p \), start point
   -> Vector2D Double    -- ^ \( \textrm{t}_p \), start tangent vector (length is ignored)
   -> [ Point2D Double ] -- ^ \( \left ( q_i \right )_{i=1}^n \), points to fit
   -> Point2D Double     -- ^ \( r \), end point
   -> Vector2D Double    -- ^ \( \textrm{t}_r \), end tangent vector (length is ignored)
-  -> Bezier ( Point2D Double )
+  -> ( Bezier ( Point2D Double ), ArgMax Double Double )
-fitPiece t_tol sq_dist_tol maxCount p tp qs r tr = piece
+fitPiece t_tol dist_tol maxIters p tp qs r tr =
+  runST do
+    ts <- Unboxed.Vector.unsafeThaw ( Unboxed.Vector.generate n uniform )
+    loop ts 0
   where
     n :: Int
     n = length qs
@@ -103,12 +115,7 @@ fitPiece t_tol sq_dist_tol maxCount p tp qs r tr = piece
     f2 t = h0 t *^ ( MkVector2D p )
     f3 t = h3 t *^ ( MkVector2D r )
     
-    piece :: Bezier ( Point2D Double )
+    loop :: forall s. Unboxed.MVector s Double -> Int -> ST s ( Bezier ( Point2D Double ), ArgMax Double Double )
-    piece = runST do
-      ts <- Unboxed.Vector.unsafeThaw ( Unboxed.Vector.generate n uniform )
-      loop ts 0
-
-    loop :: forall s. Unboxed.MVector s Double -> Int -> ST s ( Bezier ( Point2D Double ) )
     loop ts count = do
       let
         hermiteParameters :: Mat22 Double -> Vector2D Double -> Int -> [ Point2D Double ] -> ST s ( Vector2D Double )
@@ -129,7 +136,7 @@ fitPiece t_tol sq_dist_tol maxCount p tp qs r tr = piece
             b1'  = b1  + ( q'  ^.^ f0i )
             b2'  = b2  + ( q'  ^.^ f1i )
           hermiteParameters ( Mat22 a11' a12' a21' a22' ) ( Vector2D b1' b2' ) ( i + 1 ) rest
-        hermiteParameters a b _ [] = pure ( lsolve a b )
+        hermiteParameters a b _ [] = pure ( linearSolve a b )
       
       Vector2D s1 s2 <- hermiteParameters ( Mat22 0 0 0 0 ) ( Vector2D 0 0 ) 0 qs
 
@@ -141,37 +148,37 @@ fitPiece t_tol sq_dist_tol maxCount p tp qs r tr = piece
         bez :: Bezier ( Point2D Double )
         bez = Bezier p cp1 cp2 r
 
-      if count >= maxCount
-      then pure bez
-      else do
-
       -- Run one iteration of Laguerre's method to improve the parameter values t_i,
       -- so that t_i' is a better approximation of the parameter
       -- at which the curve is closest to q_i.
-        ( ts_ok, pts_ok ) <- ( `execStateT` ( True, True ) ) $ for_ ( zip qs [ 0 .. ] ) \( q, i ) -> do
+      ( dts_changed, argmax_sq_dist ) <- ( `execStateT` ( False, Max ( Arg 0 0 ) ) ) $ for_ ( zip qs [ 0 .. ] ) \( q, i ) -> do
         ti <- lift ( Unboxed.MVector.unsafeRead ts i )
         let
           poly :: [ Complex Double ]
           poly = map (:+ 0) $ ddist @( Vector2D Double ) bez q
-            ti' :: Double
+        ti' <- case laguerre epsilon 1 poly ( ti :+ 0 ) of
-            ti' = case laguerre epsilon 1 poly ( ti :+ 0 ) of
           x :+ y
             |  abs y > epsilon
             || isNaN x
             || isNaN y
-                -> ti
+            -> modify' ( first ( const True ) ) $> ti
             | otherwise
-                -> x
+            -> pure x
-
+        let
-          when ( abs ( ti' - ti ) > t_tol )
+          dt, sq_dist :: Double
-            ( modify' ( first  ( const False ) ) )
+          dt = abs ( ti' - ti )
-          when ( quadrance @( Vector2D Double ) q ( bezier @( Vector2D Double ) bez ti' ) > sq_dist_tol )
+          sq_dist = quadrance @( Vector2D Double ) q ( bezier @( Vector2D Double ) bez ti' )
-            ( modify' ( second ( const False ) ) )
+        when ( dt > t_tol )
+          ( modify' ( first ( const True ) ) )
+        modify' ( second ( <> Max ( Arg ti' sq_dist ) ) )
         lift ( Unboxed.MVector.unsafeWrite ts i ti' )
 
-        if ts_ok || pts_ok 
+      case argmax_sq_dist of
-        then pure bez
+        Max ( Arg _ max_sq_dist )
-        else loop ts ( count + 1 )
+          |  count < maxIters
+          && ( dts_changed || max_sq_dist > dist_tol ^ ( 2 :: Int ) )
+          -> loop ts ( count + 1 )
+        _ -> pure ( bez, argmax_sq_dist )
 
 -- | Cubic Hermite polynomial.
 h0, h1, h2, h3 :: Num t => t -> t

src/lib/Math/Linear/SVD.hs

@@ -1,107 +0,0 @@
-{-# LANGUAGE DerivingStrategies  #-}
-{-# LANGUAGE NamedFieldPuns      #-}
-{-# LANGUAGE RecordWildCards     #-}
-{-# LANGUAGE ScopedTypeVariables #-}
-
-module Math.Linear.SVD
-  ( SVD(..), svd
-  , pinv
-  , lsolve
-  )
-  where
-
--- base
-import Data.Complex
-  ( Complex(..), magnitude )
-
--- MetaBrush
-import Math.Epsilon
-  ( epsilon )
-import Math.Vector2D
-  ( Vector2D(..), Mat22(..) )
-
---------------------------------------------------------------------------------
-
-data SVD a
-  = SVD
-  { u   :: !( Complex a )
-  , sv1 :: !a
-  , sv2 :: !a
-  , v   :: !( Complex a )
-  }
-  deriving stock Show
-
--- | Singular value decomposition of a real 2x2 matrix.
-svd :: forall a. ( RealFloat a, Show a ) => Mat22 a -> SVD a
-svd ( Mat22 a b c d )
-  | abs f < tol
-  || any isNaN [ c1, c2, s1, s2 ]
-  || magnitude u < 1 - tol
-  || magnitude v < 1 - tol
-  = SVD { u = 1, v = 1, .. }
-  | otherwise
-  = SVD { .. }
-  where
-    tol = sqrt epsilon
-
-    det = a * d - b * c
-    q = a * a + b * b + c * c + d * d
-
-    n1 = q + 2 * det
-    n2 = q - 2 * det
-
-    r1 = sqrt n1
-    r2 = sqrt n2
-
-    sv1 = 0.5 * ( r1 + r2 )
-    sv2 = 0.5 * ( r1 - r2 )
-
-    k = a * a - d * d
-    l = b * b - c * c
-    f = n1 * n2
-
-
-    i = 0.5 / sqrt f
-
-    ip = ( k + l ) * i
-    im = ( k - l ) * i
-
-    c1 =                            sqrt ( 0.5 + ip )
-    c2 =                            sqrt ( 0.5 + im )
-    s1 = signum ( a * c + b * d ) * sqrt ( 0.5 - ip )
-    s2 = signum ( a * b + c * d ) * sqrt ( 0.5 - im )
-
-    u = c1 :+ s1
-    v = c2 :+ s2
-
--- | Pseudo-inverse of a real 2x2 matrix.
-pinv :: forall a. ( RealFloat a, Show a ) => Mat22 a -> Mat22 a
-pinv mat = case svd mat of
-  SVD { u = c1 :+ s1, sv1, sv2, v = c2 :+ s2 } ->
-    Mat22
-      ( c1 * c2 * rsv1 + s1 * s2 * rsv2 ) ( s1 * c2 * rsv1 - c1 * s2 * rsv2 )
-      ( c1 * s2 * rsv1 - s1 * c2 * rsv2 ) ( s1 * s2 * rsv1 + c1 * c2 * rsv2 )
-    where
-      rsv1, rsv2 :: a
-      rsv1
-        | sv1 < epsilon
-        = sv1
-        | otherwise
-        = recip sv1
-      rsv2
-        | abs sv2 < epsilon
-        = sv2
-        | otherwise
-        = recip sv2
-
--- | Solve a 2x2 system of linear equations.
-lsolve :: forall a. ( RealFloat a, Show a ) => Mat22 a -> Vector2D a -> Vector2D a
-lsolve mat ( Vector2D x y ) = Vector2D x' y'
-  where
-    x', y', a11, a12, a21, a22 :: a
-    Mat22
-      a11 a12
-      a21 a22
-        = pinv mat
-    x' = a11 * x + a12 * y
-    y' = a21 * x + a22 * y

src/lib/Math/Linear/Solve.hs

@@ -0,0 +1,21 @@
+module Math.Linear.Solve
+  ( linearSolve )
+  where
+
+-- hmatrix
+import qualified Numeric.LinearAlgebra      as LAPACK
+  ( linearSolveLS )
+import qualified Numeric.LinearAlgebra.Data as HMatrix
+  ( Matrix, col, matrix, atIndex )
+
+-- MetaBrush
+import Math.Vector2D
+  ( Vector2D(..), Mat22(..) )
+
+--------------------------------------------------------------------------------
+
+linearSolve :: Mat22 Double -> Vector2D Double -> Vector2D Double
+linearSolve ( Mat22 a b c d ) ( Vector2D p q ) = Vector2D ( sol `HMatrix.atIndex` (0,0) ) ( sol `HMatrix.atIndex` (1,0) )
+  where
+    sol :: HMatrix.Matrix Double
+    sol = LAPACK.linearSolveLS ( HMatrix.matrix 2 [a,b,c,d] ) ( HMatrix.col [p,q] )

src/lib/Math/Vector2D.hs

@@ -2,6 +2,7 @@
 {-# LANGUAGE DeriveTraversable          #-}
 {-# LANGUAGE DerivingVia                #-}
 {-# LANGUAGE FlexibleInstances          #-}
+{-# LANGUAGE GeneralizedNewtypeDeriving #-}
 {-# LANGUAGE MultiParamTypeClasses      #-}
 {-# LANGUAGE PatternSynonyms            #-}
 
@@ -15,7 +16,7 @@ module Math.Vector2D
 import Data.Monoid
   ( Sum(..) )
 import GHC.Generics
-  ( Generic )
+  ( Generic, Generic1 )
 
 -- acts
 import Data.Act
@@ -23,7 +24,7 @@ import Data.Act
 
 -- generic-data
 import Generic.Data
-  ( GenericProduct(..) )
+  ( Generically1(..), GenericProduct(..) )
 
 -- groups
 import Data.Group
@@ -40,12 +41,15 @@ import Math.Module
 --------------------------------------------------------------------------------
 
 data Point2D a = Point2D !a !a
-  deriving stock ( Show, Eq, Generic, Functor, Foldable, Traversable )
+  deriving stock ( Show, Eq, Generic, Generic1, Functor, Foldable, Traversable )
   deriving ( Act ( Vector2D a ), Torsor ( Vector2D a ) )
     via Vector2D a
+  deriving Applicative
+    via Generically1 Point2D
 
 newtype Vector2D a = MkVector2D { tip :: Point2D a }
-  deriving stock ( Show, Eq, Functor, Foldable, Traversable )
+  deriving stock   ( Show, Generic, Generic1, Foldable, Traversable )
+  deriving newtype ( Eq, Functor, Applicative )
   deriving ( Semigroup, Monoid, Group )
     via GenericProduct ( Point2D ( Sum a ) )
 
@@ -70,4 +74,6 @@ cross ( Vector2D x1 y1 ) ( Vector2D x2 y2 )
 
 data Mat22 a
   = Mat22 !a !a !a !a
-  deriving stock ( Show, Eq, Generic, Functor, Foldable, Traversable )
+  deriving stock ( Show, Eq, Generic, Generic1, Functor, Foldable, Traversable )
+  deriving Applicative
+    via Generically1 Mat22