src/Grid.hs

   1 {-# LANGUAGE BangPatterns #-}
   2 -- | The Grid module contains the Grid type, its tests, and the 'zoom'
   3 --   function used to build the interpolation.
   4 module Grid (
   5   cube_at,
   6   grid_tests,
   7   slow_tests,
   8   zoom
   9   )
  10 where
  11
  12 import qualified Data.Array.Repa as R
  13 import qualified Data.Array.Repa.Operators.Traversal as R (unsafeTraverse)
  14 import Test.HUnit (Assertion, assertEqual)
  15 import Test.Framework (Test, testGroup)
  16 import Test.Framework.Providers.HUnit (testCase)
  17 import Test.Framework.Providers.QuickCheck2 (testProperty)
  18 import Test.QuickCheck ((==>),
  19                         Arbitrary(..),
  20                         Gen,
  21                         Property,
  22                         choose)
  23 import Assertions (assertAlmostEqual, assertTrue)
  24 import Comparisons ((~=))
  25 import Cube (Cube(Cube),
  26              find_containing_tetrahedron,
  27              tetrahedra,
  28              tetrahedron)
  29 import Examples (trilinear, trilinear9x9x9, zeros)
  30 import FunctionValues (make_values, value_at)
  31 import Point (Point(..))
  32 import ScaleFactor (ScaleFactor)
  33 import Tetrahedron (
  34   Tetrahedron(v0,v1,v2,v3),
  35   c,
  36   polynomial,
  37   )
  38 import Values (Values3D, dims, empty3d, zoom_shape)
  39
  40
  41 -- | Our problem is defined on a Grid. The grid size is given by the
  42 --   positive number h, which we have defined to always be 1 for
  43 --   performance reasons (and simplicity). The function values are the
  44 --   values of the function at the grid points, which are distance h=1
  45 --   from one another in each direction (x,y,z).
  46 data Grid = Grid { function_values :: Values3D }
  47           deriving (Show)
  48
  49
  50 instance Arbitrary Grid where
  51     arbitrary = do
  52       fvs <- arbitrary :: Gen Values3D
  53       return $ Grid fvs
  54
  55
  56
  57 -- | Takes a grid and a position as an argument and returns the cube
  58 --   centered on that position. If there is no cube there, well, you
  59 --   shouldn't have done that. The omitted "otherwise" case actually
  60 --   does improve performance.
  61 cube_at :: Grid -> Int -> Int -> Int -> Cube
  62 cube_at !g !i !j !k =
  63    Cube i j k fvs' tet_vol
  64    where
  65      fvs = function_values g
  66      fvs' = make_values fvs i j k
  67      tet_vol = 1/24
  68
  69
  70 --   The first cube along any axis covers (-1/2, 1/2). The second
  71 --   covers (1/2, 3/2).  The third, (3/2, 5/2), and so on.
  72 --
  73 --   We translate the (x,y,z) coordinates forward by 1/2 so that the
  74 --   first covers (0, 1), the second covers (1, 2), etc. This makes
  75 --   it easy to figure out which cube contains the given point.
  76 calculate_containing_cube_coordinate :: Grid -> Double -> Int
  77 calculate_containing_cube_coordinate g coord
  78     -- Don't use a cube on the boundary if we can help it. This
  79     -- returns cube #1 if we would have returned cube #0 and cube #1
  80     -- exists.
  81     | coord < offset = 0
  82     | coord == offset && (xsize > 1 && ysize > 1 && zsize > 1) = 1
  83     | otherwise = (ceiling (coord + offset)) - 1
  84     where
  85       (xsize, ysize, zsize) = dims (function_values g)
  86       offset = 1/2
  87
  88
  89 -- | Takes a 'Grid', and returns a 'Cube' containing the given 'Point'.
  90 --   Since our grid is rectangular, we can figure this out without having
  91 --   to check every cube.
  92 find_containing_cube :: Grid -> Point -> Cube
  93 find_containing_cube g (Point x y z) =
  94     cube_at g i j k
  95     where
  96       i = calculate_containing_cube_coordinate g x
  97       j = calculate_containing_cube_coordinate g y
  98       k = calculate_containing_cube_coordinate g z
  99
 100
 101 zoom_lookup :: Values3D -> ScaleFactor -> a -> (R.DIM3 -> Double)
 102 zoom_lookup v3d scale_factor _ =
 103     zoom_result v3d scale_factor
 104
 105
 106 zoom_result :: Values3D -> ScaleFactor -> R.DIM3 -> Double
 107 zoom_result v3d (sfx, sfy, sfz) (R.Z R.:. m R.:. n R.:. o) =
 108   f p
 109   where
 110     g = Grid v3d
 111     offset = 1/2
 112     m' = (fromIntegral m) / (fromIntegral sfx) - offset
 113     n' = (fromIntegral n) / (fromIntegral sfy) - offset
 114     o' = (fromIntegral o) / (fromIntegral sfz) - offset
 115     p  = Point m' n' o'
 116     cube = find_containing_cube g p
 117     t = find_containing_tetrahedron cube p
 118     f = polynomial t
 119
 120
 121 zoom :: Values3D -> ScaleFactor -> Values3D
 122 zoom v3d scale_factor
 123     | xsize == 0 || ysize == 0 || zsize == 0 = empty3d
 124     | otherwise =
 125         R.computeS $ R.unsafeTraverse v3d transExtent f
 126           where
 127             (xsize, ysize, zsize) = dims v3d
 128             transExtent = zoom_shape scale_factor
 129             f = zoom_lookup v3d scale_factor
 130
 131
 132 -- | Check all coefficients of tetrahedron0 belonging to the cube
 133 --   centered on (1,1,1) with a grid constructed from the trilinear
 134 --   values. See example one in the paper.
 135 --
 136 --   We also verify that the four vertices on face0 of the cube are
 137 --   in the correct location.
 138 --
 139 trilinear_c0_t0_tests :: Test.Framework.Test
 140 trilinear_c0_t0_tests =
 141   testGroup "trilinear c0 t0"
 142     [testGroup "coefficients"
 143       [testCase "c0030 is correct" test_trilinear_c0030,
 144        testCase "c0003 is correct" test_trilinear_c0003,
 145        testCase "c0021 is correct" test_trilinear_c0021,
 146        testCase "c0012 is correct" test_trilinear_c0012,
 147        testCase "c0120 is correct" test_trilinear_c0120,
 148        testCase "c0102 is correct" test_trilinear_c0102,
 149        testCase "c0111 is correct" test_trilinear_c0111,
 150        testCase "c0210 is correct" test_trilinear_c0210,
 151        testCase "c0201 is correct" test_trilinear_c0201,
 152        testCase "c0300 is correct" test_trilinear_c0300,
 153        testCase "c1020 is correct" test_trilinear_c1020,
 154        testCase "c1002 is correct" test_trilinear_c1002,
 155        testCase "c1011 is correct" test_trilinear_c1011,
 156        testCase "c1110 is correct" test_trilinear_c1110,
 157        testCase "c1101 is correct" test_trilinear_c1101,
 158        testCase "c1200 is correct" test_trilinear_c1200,
 159        testCase "c2010 is correct" test_trilinear_c2010,
 160        testCase "c2001 is correct" test_trilinear_c2001,
 161        testCase "c2100 is correct" test_trilinear_c2100,
 162        testCase "c3000 is correct" test_trilinear_c3000],
 163
 164     testGroup "face0 vertices"
 165       [testCase "v0 is correct" test_trilinear_f0_t0_v0,
 166        testCase "v1 is correct" test_trilinear_f0_t0_v1,
 167        testCase "v2 is correct" test_trilinear_f0_t0_v2,
 168        testCase "v3 is correct" test_trilinear_f0_t0_v3]
 169     ]
 170   where
 171     g = Grid trilinear
 172     cube = cube_at g 1 1 1
 173     t = tetrahedron cube 0
 174
 175     test_trilinear_c0030 :: Assertion
 176     test_trilinear_c0030 =
 177       assertAlmostEqual "c0030 is correct" (c t 0 0 3 0) (17/8)
 178
 179     test_trilinear_c0003 :: Assertion
 180     test_trilinear_c0003 =
 181       assertAlmostEqual "c0003 is correct" (c t 0 0 0 3) (27/8)
 182
 183     test_trilinear_c0021 :: Assertion
 184     test_trilinear_c0021 =
 185       assertAlmostEqual "c0021 is correct" (c t 0 0 2 1) (61/24)
 186
 187     test_trilinear_c0012 :: Assertion
 188     test_trilinear_c0012 =
 189       assertAlmostEqual "c0012 is correct" (c t 0 0 1 2) (71/24)
 190
 191     test_trilinear_c0120 :: Assertion
 192     test_trilinear_c0120 =
 193       assertAlmostEqual "c0120 is correct" (c t 0 1 2 0) (55/24)
 194
 195     test_trilinear_c0102 :: Assertion
 196     test_trilinear_c0102 =
 197       assertAlmostEqual "c0102 is correct" (c t 0 1 0 2) (73/24)
 198
 199     test_trilinear_c0111 :: Assertion
 200     test_trilinear_c0111 =
 201       assertAlmostEqual "c0111 is correct" (c t 0 1 1 1) (8/3)
 202
 203     test_trilinear_c0210 :: Assertion
 204     test_trilinear_c0210 =
 205       assertAlmostEqual "c0210 is correct" (c t 0 2 1 0) (29/12)
 206
 207     test_trilinear_c0201 :: Assertion
 208     test_trilinear_c0201 =
 209       assertAlmostEqual "c0201 is correct" (c t 0 2 0 1) (11/4)
 210
 211     test_trilinear_c0300 :: Assertion
 212     test_trilinear_c0300 =
 213       assertAlmostEqual "c0300 is correct" (c t 0 3 0 0) (5/2)
 214
 215     test_trilinear_c1020 :: Assertion
 216     test_trilinear_c1020 =
 217       assertAlmostEqual "c1020 is correct" (c t 1 0 2 0) (8/3)
 218
 219     test_trilinear_c1002 :: Assertion
 220     test_trilinear_c1002 =
 221       assertAlmostEqual "c1002 is correct" (c t 1 0 0 2) (23/6)
 222
 223     test_trilinear_c1011 :: Assertion
 224     test_trilinear_c1011 =
 225       assertAlmostEqual "c1011 is correct" (c t 1 0 1 1) (13/4)
 226
 227     test_trilinear_c1110 :: Assertion
 228     test_trilinear_c1110 =
 229       assertAlmostEqual "c1110 is correct" (c t 1 1 1 0) (23/8)
 230
 231     test_trilinear_c1101 :: Assertion
 232     test_trilinear_c1101 =
 233       assertAlmostEqual "c1101 is correct" (c t 1 1 0 1) (27/8)
 234
 235     test_trilinear_c1200 :: Assertion
 236     test_trilinear_c1200 =
 237       assertAlmostEqual "c1200 is correct" (c t 1 2 0 0) 3
 238
 239     test_trilinear_c2010 :: Assertion
 240     test_trilinear_c2010 =
 241       assertAlmostEqual "c2010 is correct" (c t 2 0 1 0) (10/3)
 242
 243     test_trilinear_c2001 :: Assertion
 244     test_trilinear_c2001 =
 245       assertAlmostEqual "c2001 is correct" (c t 2 0 0 1) 4
 246
 247     test_trilinear_c2100 :: Assertion
 248     test_trilinear_c2100 =
 249       assertAlmostEqual "c2100 is correct" (c t 2 1 0 0) (7/2)
 250
 251     test_trilinear_c3000 :: Assertion
 252     test_trilinear_c3000 =
 253       assertAlmostEqual "c3000 is correct" (c t 3 0 0 0) 4
 254
 255     test_trilinear_f0_t0_v0 :: Assertion
 256     test_trilinear_f0_t0_v0 =
 257       assertEqual "v0 is correct" (v0 t) (Point 1 1 1)
 258
 259     test_trilinear_f0_t0_v1 :: Assertion
 260     test_trilinear_f0_t0_v1 =
 261       assertEqual "v1 is correct" (v1 t) (Point 0.5 1 1)
 262
 263     test_trilinear_f0_t0_v2 :: Assertion
 264     test_trilinear_f0_t0_v2 =
 265       assertEqual "v2 is correct" (v2 t) (Point 0.5 0.5 1.5)
 266
 267     test_trilinear_f0_t0_v3 :: Assertion
 268     test_trilinear_f0_t0_v3 =
 269       assertEqual "v3 is correct" (v3 t) (Point 0.5 1.5 1.5)
 270
 271
 272 test_trilinear_reproduced :: Assertion
 273 test_trilinear_reproduced =
 274     assertTrue "trilinears are reproduced correctly" $
 275              and [p (Point i' j' k') ~= value_at trilinear i j k
 276                     | i <- [0..2],
 277                       j <- [0..2],
 278                       k <- [0..2],
 279                       c0 <- cs,
 280                       t <- tetrahedra c0,
 281                       let p = polynomial t,
 282                       let i' = fromIntegral i,
 283                       let j' = fromIntegral j,
 284                       let k' = fromIntegral k]
 285     where
 286       g = Grid trilinear
 287       cs = [ cube_at g ci cj ck | ci <- [0..2], cj <- [0..2], ck <- [0..2] ]
 288
 289
 290 test_zeros_reproduced :: Assertion
 291 test_zeros_reproduced =
 292     assertTrue "the zero function is reproduced correctly" $
 293              and [p (Point i' j' k') ~= value_at zeros i j k
 294                     | i <- [0..2],
 295                       j <- [0..2],
 296                       k <- [0..2],
 297                       let i' = fromIntegral i,
 298                       let j' = fromIntegral j,
 299                       let k' = fromIntegral k,
 300                       c0 <- cs,
 301                       t0 <- tetrahedra c0,
 302                       let p = polynomial t0 ]
 303     where
 304       g = Grid zeros
 305       cs = [ cube_at g ci cj ck | ci <- [0..2], cj <- [0..2], ck <- [0..2] ]
 306
 307
 308 -- | Make sure we can reproduce a 9x9x9 trilinear from the 3x3x3 one.
 309 test_trilinear9x9x9_reproduced :: Assertion
 310 test_trilinear9x9x9_reproduced =
 311     assertTrue "trilinear 9x9x9 is reproduced correctly" $
 312       and [p (Point i' j' k') ~= value_at trilinear9x9x9 i j k
 313             | i <- [0..8],
 314               j <- [0..8],
 315               k <- [0..8],
 316               t <- tetrahedra c0,
 317               let p = polynomial t,
 318               let i' = (fromIntegral i) * 0.5,
 319               let j' = (fromIntegral j) * 0.5,
 320               let k' = (fromIntegral k) * 0.5]
 321     where
 322       g = Grid trilinear
 323       c0 = cube_at g 1 1 1
 324
 325
 326
 327 prop_cube_indices_never_go_out_of_bounds :: Grid -> Gen Bool
 328 prop_cube_indices_never_go_out_of_bounds g =
 329   do
 330     let coordmin = negate (1/2)
 331
 332     let (xsize, ysize, zsize) = dims $ function_values g
 333     let xmax = (fromIntegral xsize) - (1/2)
 334     let ymax = (fromIntegral ysize) - (1/2)
 335     let zmax = (fromIntegral zsize) - (1/2)
 336
 337     x <- choose (coordmin, xmax)
 338     y <- choose (coordmin, ymax)
 339     z <- choose (coordmin, zmax)
 340
 341     let idx_x = calculate_containing_cube_coordinate g x
 342     let idx_y = calculate_containing_cube_coordinate g y
 343     let idx_z = calculate_containing_cube_coordinate g z
 344
 345     return $
 346       idx_x >= 0 &&
 347       idx_x <= xsize - 1 &&
 348       idx_y >= 0 &&
 349       idx_y <= ysize - 1 &&
 350       idx_z >= 0 &&
 351       idx_z <= zsize - 1
 352
 353
 354 -- | Given in Sorokina and Zeilfelder, p. 80, (2.9). Note that the
 355 --   third and fourth indices of c-t10 have been switched. This is
 356 --   because we store the triangles oriented such that their volume is
 357 --   positive. If T and T-tilde share \<v1,v2,v3\> and v0,v0-tilde point
 358 --   in opposite directions, one of them has to have negative volume!
 359 prop_c0120_identity :: Grid -> Property
 360 prop_c0120_identity g =
 361   xsize >= 3 && ysize >= 3 && zsize >= 3 ==>
 362     c t0 0 1 2 0 ~= (c t0 1 0 2 0 + c t10 1 0 0 2) / 2
 363   where
 364     fvs = function_values g
 365     (xsize, ysize, zsize) = dims fvs
 366     cube0 = cube_at g 1 1 1
 367     cube1 = cube_at g 0 1 1
 368     t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
 369     t10 = tetrahedron cube1 10
 370
 371
 372 -- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
 373 --   'prop_c0120_identity'.
 374 prop_c0111_identity :: Grid -> Property
 375 prop_c0111_identity g =
 376   xsize >= 3 && ysize >= 3 && zsize >= 3 ==>
 377     c t0 0 1 1 1 ~= (c t0 1 0 1 1 + c t10 1 0 1 1) / 2
 378   where
 379     fvs = function_values g
 380     (xsize, ysize, zsize) = dims fvs
 381     cube0 = cube_at g 1 1 1
 382     cube1 = cube_at g 0 1 1
 383     t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
 384     t10 = tetrahedron cube1 10
 385
 386
 387 -- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
 388 --   'prop_c0120_identity'.
 389 prop_c0201_identity :: Grid -> Property
 390 prop_c0201_identity g =
 391   xsize >= 3 && ysize >= 3 && zsize >= 3 ==>
 392     c t0 0 2 0 1 ~= (c t0 1 1 0 1 + c t10 1 1 1 0) / 2
 393   where
 394     fvs = function_values g
 395     (xsize, ysize, zsize) = dims fvs
 396     cube0 = cube_at g 1 1 1
 397     cube1 = cube_at g 0 1 1
 398     t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
 399     t10 = tetrahedron cube1 10
 400
 401
 402 -- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
 403 --   'prop_c0120_identity'.
 404 prop_c0102_identity :: Grid -> Property
 405 prop_c0102_identity g =
 406   xsize >= 3 && ysize >= 3 && zsize >= 3 ==>
 407     c t0 0 1 0 2 ~= (c t0 1 0 0 2 + c t10 1 0 2 0) / 2
 408   where
 409     fvs = function_values g
 410     (xsize, ysize, zsize) = dims fvs
 411     cube0 = cube_at g 1 1 1
 412     cube1 = cube_at g 0 1 1
 413     t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
 414     t10 = tetrahedron cube1 10
 415
 416
 417 -- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
 418 --   'prop_c0120_identity'.
 419 prop_c0210_identity :: Grid -> Property
 420 prop_c0210_identity g =
 421   xsize >= 3 && ysize >= 3 && zsize >= 3 ==>
 422     c t0 0 2 1 0 ~= (c t0 1 1 1 0 + c t10 1 1 0 1) / 2
 423   where
 424     fvs = function_values g
 425     (xsize, ysize, zsize) = dims fvs
 426     cube0 = cube_at g 1 1 1
 427     cube1 = cube_at g 0 1 1
 428     t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
 429     t10 = tetrahedron cube1 10
 430
 431
 432 -- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
 433 --   'prop_c0120_identity'.
 434 prop_c0300_identity :: Grid -> Property
 435 prop_c0300_identity g =
 436   xsize >= 3 && ysize >= 3 && zsize >= 3 ==>
 437     c t0 0 3 0 0 ~= (c t0 1 2 0 0 + c t10 1 2 0 0) / 2
 438   where
 439     fvs = function_values g
 440     (xsize, ysize, zsize) = dims fvs
 441     cube0 = cube_at g 1 1 1
 442     cube1 = cube_at g 0 1 1
 443     t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
 444     t10 = tetrahedron cube1 10
 445
 446
 447 -- | All of the properties from Section (2.9), p. 80. These require a
 448 --   grid since they refer to two adjacent cubes.
 449 p80_29_properties :: Test.Framework.Test
 450 p80_29_properties =
 451   testGroup "p. 80, Section (2.9) Properties" [
 452     testProperty "c0120 identity" prop_c0120_identity,
 453     testProperty "c0111 identity" prop_c0111_identity,
 454     testProperty "c0201 identity" prop_c0201_identity,
 455     testProperty "c0102 identity" prop_c0102_identity,
 456     testProperty "c0210 identity" prop_c0210_identity,
 457     testProperty "c0300 identity" prop_c0300_identity ]
 458
 459
 460 grid_tests :: Test.Framework.Test
 461 grid_tests =
 462     testGroup "Grid Tests" [
 463       trilinear_c0_t0_tests,
 464       p80_29_properties,
 465       testProperty "cube indices within bounds"
 466         prop_cube_indices_never_go_out_of_bounds ]
 467
 468
 469 -- Do the slow tests last so we can stop paying attention.
 470 slow_tests :: Test.Framework.Test
 471 slow_tests =
 472     testGroup "Slow Tests" [
 473       testCase "trilinear reproduced" test_trilinear_reproduced,
 474       testCase "trilinear9x9x9 reproduced" test_trilinear9x9x9_reproduced,
 475       testCase "zeros reproduced" test_zeros_reproduced ]