雑記帳
Haskell でレイトレーシングのチュートリアルを追いかける その5 - 様々な拡散のモデル式
引き続きこのサイトのチュートリアルに則って、レイトレーシングによる画像の生成に挑戦。
進捗状況としては、ひとまず「section 8-5, 8-6」まで完了。
進捗状況としては、ひとまず「section 8-5, 8-6」まで完了。
コードの実行結果
(画像はファイルサイズを小さくしたい関係で JPEG 形式にコンバートしているため、各ピクセルの値は実際に得られた値とは微妙に異なる。)
ソースコード
{-# LANGUAGE TypeOperators #-}
module Main where
import Data.Char
import Data.Functor
import Control.Monad
import Control.Lens
import System.Random
import Linear.Vector
import Linear.Metric
import Linear.V3
import Linear.Quaternion
-- section 8-5, 8-6 with Haskell!!
-- https://raytracing.github.io/books/RayTracingInOneWeekend.html with Haskell!! --
-----------------
-- Entry point --
-----------------
main :: IO ()
main = do
let
-- Image
aspect_ratio = 16.0 / 9
image_width = 256
image_height = round $ fromIntegral image_width / aspect_ratio
samples_per_pixel = 100
max_depth = 50
-- World
world = ((([]
`add` RT_Sphere{center = V3 (-0.1) 2.4 (-5.9), radius = 2.9})
`add` RT_Torus{
centerOfTorus = V3 0.05 0.2 (-1.2),
majorRadius = 0.35,
minorRadius = 0.15,
orientationOfTorus = normalize $ V3 (-0.5) 1.2 1.9
})
`add` RT_Sphere{center = V3 0 (-10000.5) (-1), radius = 10000})
-- Camera
camera = Camera {
viewport_height = 2.0,
viewport_width = aspect_ratio * viewport_height camera,
focal_length = 1.0,
origin = zero,
horizontal = viewport_width camera *^ unit _x,
vertical = viewport_height camera *^ unit _y,
lower_left_corner =
origin camera - horizontal camera ^/2 - vertical camera ^/2
- focal_length camera *^ unit _z
}
-- Render
img_data <- return $ "P3\n" ++ show image_width ++ " " ++ show image_height ++ "\n255\n"
putStr $ img_data
foldr (>>) (return ()) $ (fmap $ ($) $ \((j, i), seed) ->
let
f h currentData gen = if h < samples_per_pixel then
let
(randNum1, newGen1) = random gen :: (Double, StdGen)
(randNum2, newGen2) = random newGen1 :: (Double, StdGen)
u = (fromIntegral i + randNum1) / (fromIntegral image_width - 1.0)
v = (fromIntegral j + randNum2) / (fromIntegral image_height - 1.0)
r = get_ray camera (u, v)
pixcel_color = ray_color r world max_depth newGen2
in
f (succ h) (currentData + pixcel_color) newGen2
else
currentData
in
write_color (f 0 zero (mkStdGen seed)) samples_per_pixel) $
zip ((,) <$> [image_height - 1, image_height - 2 .. 0] <*> [0, 1 .. image_width - 1])
((randomRs (0, 536870912) (mkStdGen 21)) :: [Int])
---------------------
-- A Ray Data Type --
---------------------
data Ray = Ray {
orig :: V3 Double, -- Origin of this ray (As a position in 3D Euclidean space)
dir :: V3 Double -- Direction of this ray (As a direction vector in 3D Euclidean space)
} deriving (Show)
at' :: Ray -> Double -> V3 Double
at' r t = (orig r) + t *^ (dir r)
------------------------
-- A Camera Data Type --
------------------------
data Camera = Camera {
viewport_height :: Double,
viewport_width :: Double,
focal_length :: Double,
origin :: V3 Double,
horizontal :: V3 Double,
vertical :: V3 Double,
lower_left_corner :: V3 Double
} deriving (Show)
get_ray :: Camera -> (Double, Double) -> Ray
get_ray this (u, v) =
Ray {
orig = origin this,
dir = lower_left_corner this + u *^ horizontal this + v *^ vertical this - origin this
}
----------------------
-- A Hittable Class --
----------------------
type HittableData = (RT_Sphere + RT_Torus) + RT_Sphere -- Third RT_Sphere is just dummies
class Hittable a where
toSum :: a -> HittableData
hit :: a -> Ray -> Double -> Double -> Maybe HitRecord
instance (Hittable a, Hittable b) => Hittable (a + b) where
toSum = coPair(toSum, toSum)
hit = coPair(hit, hit)
add :: Hittable a => [HittableData] -> a -> [HittableData]
add list obj = (toSum obj) : list
data HitRecord = HitRecord {
p :: V3 Double,
normal :: V3 Double,
t :: Double,
front_face :: Bool
} deriving (Show)
set_face_normal :: HitRecord -> Ray -> V3 Double -> HitRecord
set_face_normal this r outward_normal = HitRecord {
p = p this,
normal = if dir r `dot` outward_normal < 0 then outward_normal else -outward_normal,
t = t this,
front_face = (dir r `dot` outward_normal < 0)
}
hitSomething :: [HittableData] -> Ray -> Double -> Double -> Maybe HitRecord
hitSomething list r t_min t_max =
let
f (list', r', closest_so_far, currentRecord) =
case list' of
x:xs ->
let
temp = hit x r' t_min closest_so_far
in
case temp of
Just a ->
f $ (xs, r', t a, temp)
Nothing ->
f $ (xs, r', closest_so_far, currentRecord)
[] ->
currentRecord
in
f $ (list, r, t_max, Nothing)
----------------------
-- Hittable Objects --
----------------------
-- Sphere
data RT_Sphere = RT_Sphere {
center :: V3 Double,
radius :: Double
} deriving (Show)
instance Hittable RT_Sphere where
toSum = Inj1 -: Inj1
hit obj r t_min t_max =
let
p0 = orig r
c1 = center obj
r1 = radius obj
oc = p0 - c1
a = quadrance (dir r)
half_b = oc `dot` dir r
c = quadrance oc - (radius obj) ^ 2
discriminant = half_b ^ 2 - a*c in
if discriminant > 0 then
let
root = sqrt discriminant
f k = case k of
x:xs ->
if t_min < x && x < t_max then
return $ set_face_normal HitRecord {
p = at' r x,
normal = zero,
t = x,
front_face = False
} r ((at' r x - c1) ^/ r1)
else
f $ xs
[] ->
Nothing
in
f $ [(-half_b - root) / a, (-half_b + root) / a]
else
Nothing
-- Torus
data RT_Torus = RT_Torus {
centerOfTorus :: V3 Double,
majorRadius :: Double,
minorRadius :: Double,
orientationOfTorus :: V3 Double -- [BEWARB] this pseudo-vector must be normalized
} deriving (Show)
instance Hittable RT_Torus where
toSum = Inj2 -: Inj1
hit obj r t_min t_max =
let
p0 = orig r
c1 = centerOfTorus obj
r1 = majorRadius obj
r2 = minorRadius obj
n = orientationOfTorus obj
oc = p0 - c1
a = quadrance (dir r)
half_b = oc `dot` dir r
c = quadrance oc - (r1 + r2 + 0.001) ^ 2
discriminant = half_b ^ 2 - a*c in
if discriminant > 0 then
let
root = sqrt discriminant
s = newton's_method
50
(max 0.0001 ((-half_b - root) / a + 0.0001), max 0.0001 ((-half_b + root) / a))
(max 0.0001 ((-half_b - root) / a + 0.0001), 0.0001)
(findIntersection_forTorus obj r)
(findIntersection_forTorus' obj r)
in
s >>= (\k ->
let
x = at' r k - c1
in
if t_min < k && k < t_max then
return $ HitRecord {
p = c1 + x,
normal = (x - (r1 *^ (normalize $ x - (n `dot` x) *^ n))) ^/ r2,
t = k,
front_face = False
}
else
Nothing
)
else
Nothing
findIntersection_forTorus :: RT_Torus -> Ray -> Double -> Double
findIntersection_forTorus obj r t =
(quadrance u + r1 ^2 - r2 ^ 2) ^ 2 - 4 * (r1 ^ 2) * (u `dot` (u - (n `dot` u) *^ n))
-- quadrance u - 2 * r1 * (sqrt $ u `dot` (u - (n `dot` u) *^ n)) + r1 ^2 - r2 ^ 2
where
a = dir r
u = at' r t - centerOfTorus obj
n = orientationOfTorus obj
r1 = majorRadius obj
r2 = minorRadius obj
findIntersection_forTorus' :: RT_Torus -> Ray -> Double -> Double
findIntersection_forTorus' obj r t =
4 * (a `dot` u) * (quadrance u + r1 ^ 2 - r2 ^ 2)
- 8 * (r1^2) * (u `dot` (a - (n `dot` a) *^ n))
-- 2 * (a `dot` u)
-- - 2 * r1 * (u `dot` (a - (n `dot` a) *^ n)) / (sqrt $ u `dot` (u - (n `dot` u) *^ n))
where
a = dir r
u = at' r t - centerOfTorus obj
n = orientationOfTorus obj
r1 = majorRadius obj
r2 = minorRadius obj
----------------------------------------
-- Computing the color of a given ray --
----------------------------------------
write_color :: V3 Double -> Int -> IO ()
write_color (V3 r g b) spp =
let
v' = V3 (sqrt $ r / fromIntegral spp) (sqrt $ g / fromIntegral spp) (sqrt $ b / fromIntegral spp)
f = show.floor.(256*).(clamp 0 0.999)
in
do
tmp <- return $ f(v' ^._x) ++ " " ++ f(v' ^._y) ++ " " ++ f(v' ^._z) ++ "\n"
putStr $ tmp
ray_color :: Ray -> [HittableData] -> Int -> StdGen -> V3 Double
ray_color r objects depth gen =
if depth <= 0 then
zero
else
let
record = hitSomething objects r 0.0001 infinity
in
case record of
Just record' ->
let
(rand1, gen1) = random_in_hemisphere (normal record') gen
target = (p record' + rand1)
in
0.5 *^ ray_color Ray{orig = p record', dir = target - p record'} objects (pred depth) gen1
Nothing ->
let
unit_direction = normalize $ (dir r)
s = 0.5 * (unit_direction ^._y + 1.0)
in
lerp s (V3 0.5 0.7 1.0) (V3 1.0 1.0 1.0)
--------------------
-- Random numbers --
--------------------
random_in_unit_sphere :: StdGen -> (V3 Double, StdGen)
random_in_unit_sphere gen0 =
let
(rand1,gen1) = randomR (-1, 1) gen0 :: (Double, StdGen)
(rand2,gen2) = randomR (-1, 1) gen1 :: (Double, StdGen)
(rand3,gen3) = randomR (-1, 1) gen2 :: (Double, StdGen)
v = V3 rand1 rand2 rand3
in
if quadrance v >= 1 then
random_in_unit_sphere gen3
else
(v, gen3)
random_unit_vector :: StdGen -> (V3 Double, StdGen)
random_unit_vector gen0 =
let
(a, gen1) = randomR (0, 2*pi) gen0 :: (Double, StdGen)
(z, gen2) = randomR (-1, 1) gen1 :: (Double, StdGen)
r = sqrt $ 1 - z^2
in
(V3 (r*cos(a)) (r*sin(a)) z, gen2)
random_in_hemisphere :: V3 Double -> StdGen -> (V3 Double, StdGen)
random_in_hemisphere normal gen0 =
let
(in_unit_sphere, gen1) = random_in_unit_sphere gen0
in
if in_unit_sphere `dot` normal > 0 then
(in_unit_sphere, gen1)
else
(-in_unit_sphere, gen1)
---------------
-- Utilities --
---------------
infinity :: RealFloat a => a
infinity = encodeFloat (floatRadix 0 - 1) (snd $ floatRange 0)
deg2rad :: Floating a => a -> a
deg2rad degrees = degrees * pi / 180
clamp :: (Ord a, Num a) => a -> a -> a -> a
clamp x y val = (max x).(min y) $ val
isInClosedInterval :: (Ord a, Fractional a) => (a, a) -> a -> Bool
isInClosedInterval (a, b) val = (a <= val && val <= b)
isInOpenInterval :: (Ord a, Fractional a) => (a, a) -> a -> Bool
isInOpenInterval (a, b) val = (a < val && val < b)
newton's_method :: (Ord a, Fractional a) => Int -> (a, a) -> (a, a) -> (a -> a) -> (a -> a) -> Maybe a
newton's_method depth interval (current, prev) f f' =
if (uncurry (/=)) interval && isInClosedInterval interval current && depth > 0 then
if abs(current - prev) < 1.0E-10 then
Just current
else
newton's_method (pred depth) interval (current - (f(current) / f'(current)), current) f f'
else
Nothing
-- Joke
derivativeOf :: (Floating a) => (a -> a) -> Int -> a -> a
derivativeOf f precision =
\x -> (f(x + dx) - f(x)) / dx
where dx = 0.1^precision
-- Diagrammatic-order composition
(-:) = flip (.)
-- Sum objects and injections
data (+) a b = Inj1 a | Inj2 b
-- Dual to pairs
coPair :: (a1 -> b, a2 -> b) -> (a1 + a2 -> b)
coPair (f, g) x = case x of
Inj1 x -> f x
Inj2 x -> g xタグ一覧: