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8 - 透明の物体
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 10-3」まで完了。
進捗状況としては、ひとまず「section 10-3」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
-- https://raytracing.github.io/books/RayTracingInOneWeekend.html
-- section 10-3 Total Internal Reflection with Haskell!!
main :: IO ()
main = do
let
-- Image
aspect_ratio = 16.0 / 9
image_width = 400
image_height = round $ fromIntegral image_width / aspect_ratio
samples_per_pixel = 100
max_depth = 50
-- World
material_ground = make_shared $ MAT_Lambertian {albedo_Lamb = V3 0.8 0.8 0.0}
material_center = make_shared $ MAT_Lambertian {albedo_Lamb = V3 0.7 0.3 0.3}
material_center2 = make_shared $ MAT_Dielectric {ref_idx = 1.7}
material_left = make_shared $ MAT_Metal {albedo_Metal = V3 0.8 0.8 0.8, fuzz = 0.2}
material_right = make_shared $ MAT_Metal {albedo_Metal = V3 0.8 0.6 0.2, fuzz = 0.0}
world = ((((([]
--`add` RT_Sphere{center = V3 0.0 0.0 (-1.0), radius = 0.5, mat_Sphere = material_center})
--`add` RT_Sphere{center = V3 (-0.95) 0.0 (-1.0), radius = 0.5, mat_Sphere = material_left})
--`add` RT_Sphere{center = V3 1.05 0.0 (-1.0), radius = 0.5, mat_Sphere = material_right})
`add` RT_Torus{
centerOfTorus = V3 (-0.55) 0.0 (-2.1),
majorRadius = 0.4,
minorRadius = 0.1,
orientationOfTorus = normalize $ V3 (0.5) 1.0 1.5,
mat_Torus = material_left
})
`add` RT_Torus{
centerOfTorus = V3 1.05 0.0 (-1.0),
majorRadius = 0.3,
minorRadius = 0.2,
orientationOfTorus = normalize $ V3 (-0.5) (-0.2) (-3),
mat_Torus = material_right
})
`add` RT_Sphere{center = V3 (-3.51) 2.4 (-5.9), radius = 2.9, mat_Sphere = material_center})
`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,
mat_Torus = material_center2
})
`add` RT_Sphere{center = V3 0 (-10000.5) (-1), radius = 10000, mat_Sphere = material_ground})
-- 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,
mat :: MaterialData,
t :: Double,
front_face :: Bool
} deriving (Show)
set_face_normal :: HitRecord -> Ray -> V3 Double -> HitRecord
set_face_normal this r outward_normal =
let
isOutside = (dir r `dot` outward_normal) < 0
in
HitRecord {
p = p this,
front_face = isOutside,
normal = if isOutside then outward_normal else -outward_normal,
t = t this,
mat = mat this
}
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,
mat_Sphere :: MaterialData
} 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,
mat = mat_Sphere obj
} 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
mat_Torus :: MaterialData
} 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 $ set_face_normal HitRecord {
p = c1 + x,
normal = zero,
t = k,
front_face = False,
mat = mat_Torus obj
} r ((x - (r1 *^ (normalize $ x - (n `dot` x) *^ n))) ^/ r2)
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 (ret, gen1) = scatter (mat record') r record' gen
in
case ret of
Just (scattered, attenuation) ->
attenuation * (ray_color scattered objects (pred depth) gen1)
Nothing ->
zero
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
reflect :: V3 Double -> V3 Double -> V3 Double
reflect v n = v - (2 * (n `dot` v)) *^ n
refract :: V3 Double -> V3 Double -> Double -> V3 Double
refract uv n eta_over_eta' =
r_out_perp + r_out_parallel
where
cos_theta = (-uv) `dot` n
r_out_perp = eta_over_eta' *^ (uv + cos_theta *^ n)
r_out_parallel = (sqrt $ abs (1 - quadrance r_out_perp)) *^ (-n)
--------------------
-- Material Class --
--------------------
type MaterialData = (MAT_Lambertian + MAT_Metal) + MAT_Dielectric
class Material a where
make_shared :: a -> MaterialData
scatter :: a -> Ray -> HitRecord -> StdGen -> (Maybe (Ray, V3 Double), StdGen)
instance (Material a, Material b) => Material (a + b) where
make_shared = coPair(make_shared, make_shared)
scatter = coPair(scatter, scatter)
-- Lambertian
data MAT_Lambertian = MAT_Lambertian {
albedo_Lamb :: V3 Double
} deriving (Show)
instance Material MAT_Lambertian where
make_shared = Inj1 -: Inj1
scatter this r_in record gen =
let
(rand1, gen1) = random_unit_vector gen
scattered_direction = normal record + rand1
scattered = Ray{orig = p record, dir = scattered_direction}
attenuation = albedo_Lamb this
in
(Just (scattered, attenuation), gen1)
-- Metal
data MAT_Metal = MAT_Metal {
albedo_Metal :: V3 Double,
fuzz :: Double
} deriving (Show)
instance Material MAT_Metal where
make_shared = Inj2 -: Inj1
scatter this r_in record gen =
let
(rand1, gen1) = random_in_unit_sphere gen
reflected = reflect (normalize $ dir r_in) (normal record)
scattered = Ray{orig = p record, dir = reflected + (fuzz this) *^ rand1}
attenuation = albedo_Metal this
in
if (dir scattered `dot` normal record) > 0 then
(Just (scattered, attenuation), gen1)
else
(Nothing, gen1)
-- Dielectric
data MAT_Dielectric = MAT_Dielectric {
ref_idx :: Double
} deriving (Show)
instance Material MAT_Dielectric where
make_shared = Inj2
scatter this r_in record gen =
let
attenuation = V3 1 1 1
eta_over_eta' = if front_face record then 1 / ref_idx this else ref_idx this
unit_direction = normalize $ dir r_in
cos_theta = min (-unit_direction `dot` normal record) 1
sin_theta = sqrt $ 1 - cos_theta ^ 2
in
if eta_over_eta' * sin_theta > 1 then
let
reflected = reflect unit_direction (normal record)
scattered = Ray{orig = p record, dir = reflected}
in
(Just (scattered, attenuation), gen)
else
let
refracted = refract unit_direction (normal record) eta_over_eta'
scattered = Ray{orig = p record, dir = refracted}
in
(Just (scattered, attenuation), gen)
-----------------------------------
-- Category Theoretic Operations --
-----------------------------------
-- Diagrammatic-order composition
(-:) = flip (.)
-- Sum objects and injections
data (+) a b = Inj1 a | Inj2 b
instance (Show a, Show b) => Show (a + b) where
show = coPair(show -: (++ ";inj1"), show -: (++ ";inj2"))
-- 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: