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15 - 最終レンダリング
遂に本編の最終ポイントとなる画像生成に到達したー！！
メインプログラムは前回のものと基本的に同じで、空間内にボールをランダムに配置する処理を追加したくらいの変更だけだったけど、それだけで生成画像がスゴいことになった！！！
コードの実行結果
余談
最初は小さなボールを全てトーラスに置き換えて描画してみようかなと考えていたけど
- パフォーマンスを考慮しないコーディングを Haskell で行っている (i.e. C/C++ で書かれたものと比べたら圧倒的に遅いはず)
- 球面ですらなかなか時間がかかっている中、球面と比較してかなり計算コストの高いトーラスで全て置き換えたら処理に要する時間がとんでもないことになりそう
- そもそもトーラスに置き換えるとなると、レイに対する反射や屈折を追跡する最大回数を多くする必要もありそうだし、そうなると更に時間がかかることになる
など色々あって、大きなボールだけをトーラスにするくらいが現実的かなという結論に行きついた。
因みにこの設定でも出力画像を得るのにかなりの時間がかかった。(厳密に測定したわけではないけど、実行開始時の大体の時刻と画像ファイルの最終更新時刻から考えて、3時間半程かかっている。)
通常のリアルタイム3DCGで用いられる描画テクだと透明のドーナツをリアルに描画するのって難しそうだけど、こんなにも綺麗に描画できるレイトレ素晴らしい。
参照
既に何度もリンクを貼っているけど、今回は「本編の最後」という切りの良いタイミングということで、再度リンクを貼り付けておく。
めっちゃ楽しかったので、興味ある人は是非！
ソースコード
{-# LANGUAGE TypeOperators #-}
module Main where
import Data.Complex
import Control.Monad.Fix
import Control.Lens
import System.Random
import Linear.Vector
import Linear.Metric
import Linear.V3
-- https://raytracing.github.io/books/RayTracingInOneWeekend.html
main :: IO ()
main = do
let
-- Image
aspect_ratio = 3 / 2
image_width = 1200
image_height = round $ fromIntegral image_width / aspect_ratio
samples_per_pixel = 500
max_depth = 100
-- World
world = random_scene
-- Camera
lookfrom = V3 13 (-3) 2
lookat = V3 0 0 0
dist_to_focus = norm $ lookat - lookfrom
aperture = 0.1
camera = cam(lookfrom, lookat, V3 0 0 1, 20, aspect_ratio, aperture, dist_to_focus)
img_data = "P3\n" ++ show image_width ++ " " ++ show image_height ++ "\n255\n"
putStr $ img_data
foldr (>>) (return ()) $ do
let
indices = [image_height - 1, image_height - 2 .. 0] `prod` [0 .. image_width - 1]
seeds = (randomRs (0, 536870912) (mkStdGen 21) :: [Int])
((j,i), seed) <- zip indices seeds
return $ do
let
pixcel_color = getColor(0, 0, mkStdGen seed)
getColor = fix $ \rec (s, pixcel_color', gen_current) -> do
let
(randNum1, gen1) = random gen_current
(randNum2, gen2) = random gen1
u = (fromIntegral i + randNum1) / (fromIntegral image_width - 1.0)
v = (fromIntegral j + randNum2) / (fromIntegral image_height - 1.0)
(r,gen3) = get_ray camera (u, v) gen2
(d,gen4) = ray_color r world max_depth gen3
if s < samples_per_pixel then
rec(succ s, pixcel_color' + d, gen4)
else
pixcel_color'
write_color pixcel_color samples_per_pixel
random_scene :: [HittableData]
random_scene =
let
k = 2*11
numOfUsedRndsPerLoop = 13
rnds = randoms (mkStdGen 210) :: [Double]
tinyBalls = do
i <- [0..k-1]
j <- [0..k-1]
let
a = fromIntegral $ i - k `div` 2
b = fromIntegral $ j - k `div` 2
rnd idx = rnds !! (numOfUsedRndsPerLoop*(k*i + j) + idx)
choose_mat = rnd(0)
c = vec3(a + 0.9*rnd(1), b + 0.9*rnd(2), 0.2)
if ((norm $ c - vec3(4, 0.2, 0)) > 0.9) then
if choose_mat < 0.8 then
let
albedo = vec3(rnd(3)*rnd(6),rnd(4)*rnd(7),rnd(5)*rnd(8))
material = make_shared $ MAT_Lambertian {albedo_Lamb = albedo}
in
return $ toSum $ RT_Sphere{
center = c,
radius = 0.2,
mat_Sphere = material
}
else if choose_mat < 0.95 then
let
albedo = vec3((fmap $ unitTo(0.5,1)) $ (rnd(9), rnd(10), rnd(11)))
fuzz' = unitTo(0,0.5) $ rnd(12)
material = make_shared $ MAT_Metal {albedo_Metal = albedo, fuzz = fuzz'}
in
return $ toSum $ RT_Sphere{
center = c,
radius = 0.2,
mat_Sphere = material
}
else
let
material = make_shared $ MAT_Dielectric {ref_idx = 1.5}
in
return $ toSum $ RT_Sphere{
center = c,
radius = 0.2,
mat_Sphere = material
}
else
[]
material_ground = make_shared $ MAT_Lambertian {albedo_Lamb = V3 0.5 0.5 0.5}
material1 = make_shared $ MAT_Dielectric {ref_idx = 1.5}
material2 = make_shared $ MAT_Lambertian {albedo_Lamb = V3 0.4 0.2 0.1}
material3 = make_shared $ MAT_Metal {albedo_Metal = V3 0.7 0.6 0.5, fuzz = 0.0}
ground = toSum $ RT_Sphere{center = V3 0 1 (-1000), radius = 1000, mat_Sphere = material_ground}
bigDonut1 = toSum $ RT_Torus{
centerOfTorus = vec3(0, 0, 1),
majorRadius = 0.75,
minorRadius = 0.25,
orientationOfTorus = normalize $ V3 0.5 0.3 1.0,
mat_Torus = material1
}
bigDonut2 = toSum $ RT_Torus{
centerOfTorus = vec3(-4, 0, 1),
majorRadius = 0.75,
minorRadius = 0.25,
orientationOfTorus = normalize $ V3 0.4 (-1.5) 1.0,
mat_Torus = material2
}
bigDonut3 = toSum $ RT_Torus{
centerOfTorus = vec3(4, 0, 1),
majorRadius = 0.75,
minorRadius = 0.25,
orientationOfTorus = normalize $ V3 (-0.7) (-0.7) 0.9,
mat_Torus = material3
}
in
[ground, bigDonut1, bigDonut2, bigDonut3] ++ tinyBalls
where
vec3(x1,x2,x3) = V3 x1 x2 x3
unitTo (a,b) x = a + (b-a)*x
data Ray = Ray {
orig :: V3 Double,
dir :: V3 Double
} deriving (Show)
at' :: Ray -> Double -> V3 Double
at' r t = (orig r) + t *^ (dir r)
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,
axis_u :: V3 Double,
axis_v :: V3 Double,
axis_w :: V3 Double,
lens_radius :: Double
} deriving (Show)
cam (lookfrom,lookat,vup,vfov,aspect_ratio,aperture,focus_dist) =
Camera {
viewport_height = viewport_height',
viewport_width = viewport_width',
origin = origin',
horizontal = horizontal',
vertical = vertical',
lower_left_corner = lower_left_corner',
axis_u = u,
axis_v = v,
axis_w = w,
lens_radius = lens_radius'
}
where
theta = deg2rad(vfov)
h = tan(theta/2)
viewport_height' = 2.0 * h
viewport_width' = aspect_ratio * viewport_height'
w = normalize $ lookat - lookfrom
u = normalize $ w `cross` vup
v = u `cross` w
origin' = lookfrom
horizontal' = (focus_dist * viewport_width') *^ u
vertical' = (focus_dist * viewport_height') *^ v
lower_left_corner' =
origin' - horizontal' ^/2 - vertical' ^/2
+ focus_dist *^ w
lens_radius' = aperture / 2
get_ray :: Camera -> (Double, Double) -> StdGen -> (Ray, StdGen)
get_ray this (s, t) gen0 = (r,gen1)
where
(in_unit_disk, gen1) = random_in_unit_sphere gen0
rd = lens_radius this *^ in_unit_disk
offset = (rd ^._x) *^ axis_u this + (rd ^._y) *^ axis_v this
r =
Ray {
orig = origin this + offset,
dir = lower_left_corner this + s *^ horizontal this + t *^ vertical this - (origin this + offset)
}
type HittableData = (RT_Sphere + RT_Torus) + RT_Sphere
class Hittable a where
toSum :: a -> HittableData
hit :: a -> Ray -> Double -> Double -> Maybe HitRecord
instance (Hittable a, Hittable b) => Hittable (Either 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 = 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),
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,
mat_Torus :: MaterialData
} deriving (Show)
instance Hittable RT_Torus where
toSum = inj2 -: inj1
hit obj r t_min t_max =
let
p0 = orig r
a = dir r
a_norm = norm a
c1 = centerOfTorus obj
r1 = majorRadius obj
r2 = minorRadius obj
n = orientationOfTorus obj
s = getIntersection_forTorus (p0,a,c1,r1,r2,n)
oc = p0 - c1
a_sq = quadrance (dir r)
half_b = oc `dot` dir r
c = quadrance oc - (r1 + r2 + 0.01) ^ 2
discriminant = half_b ^ 2 - a_sq*c
in
if discriminant > 0 then
if null s then
Nothing
else
let
k = minimum s
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
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
putStr $ f(v' ^._x) ++ " " ++ f(v' ^._y) ++ " " ++ f(v' ^._z) ++ "\n"
ray_color :: Ray -> [HittableData] -> Int -> StdGen -> (V3 Double, StdGen)
ray_color r objects depth gen =
if depth <= 0 then
(zero, gen)
else
let
record = hitSomething objects r 0.001 infinity
in
case record of
Just record' ->
let
(ret, gen1) = scatter (mat record') r record' gen
in
case ret of
Just (scattered, attenuation) ->
let
(col_t , gen_t) = ray_color scattered objects (pred depth) gen1
in
(attenuation * col_t , gen_t)
Nothing ->
(zero, gen1)
Nothing ->
let
unit_direction = normalize $ (dir r)
s = 0.5 * (unit_direction ^._z + 1.0)
in
(lerp s (V3 0.5 0.7 1.0) (V3 1.0 1.0 1.0), gen)
--------------------
-- 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)
random_in_unit_disk :: StdGen -> (V3 Double, StdGen)
random_in_unit_disk gen0 =
let
(rand1,gen1) = randomR (-1, 1) gen0 :: (Double, StdGen)
(rand2,gen2) = randomR (-1, 1) gen1 :: (Double, StdGen)
v = V3 rand1 rand2 0
in
if quadrance v >= 1 then
random_in_unit_sphere gen2
else
(v, gen2)
---------------
-- 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
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 = min 1 ((-uv) `dot` n)
r_out_perp = eta_over_eta' *^ (uv + cos_theta *^ n)
r_out_parallel = (sqrt $ abs (1 - quadrance r_out_perp)) *^ (-n)
reflectance :: Double -> Double -> Double
reflectance cosine ref_idx =
let
r0 = ((1 - ref_idx) / (1 + ref_idx)) ^ 2
in
r0 + (1-r0) * (1 - cosine) ^ 5
near_zero (V3 r1 r2 r3) =
(abs(r1) < s) && (abs(r2) < s) && (abs(r3) < s)
where
s = 1.0E-7
--------------------
-- 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 (Either 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 =
if near_zero(normal record + rand1) then
normal record
else
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
(rand1, gen1) = random gen
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 1 (-unit_direction `dot` normal record)
sin_theta = sqrt $ 1 - cos_theta ^ 2
cannot_refract = eta_over_eta' * sin_theta > 1
in
if cannot_refract || (reflectance cos_theta eta_over_eta' > rand1) then
let
reflected = reflect unit_direction (normal record)
scattered = Ray{orig = p record, dir = reflected}
in
(Just (scattered, attenuation), gen1)
else
let
refracted = refract unit_direction (normal record) eta_over_eta'
scattered = Ray{orig = p record, dir = refracted}
in
(Just (scattered, attenuation), gen)
getIntersection_forTorus :: (V3 Double, V3 Double, V3 Double, Double, Double, V3 Double) -> [Double]
getIntersection_forTorus = solveQuarticEq . genCoefficients
genCoefficients (x0,a,c,r1,r2,n) = (b4,b3,b2,b1,b0)
where
d0 = x0 - c
k = (r1^2) - (r2^2)
a_sq = quadrance a
d0_sq = quadrance d0
b4 = a_sq^2
b3 = 4*(d0 `dot` a)*a_sq
b2 = 2*d0_sq*a_sq+4*((d0 `dot` a)^2) + 2*k*a_sq - 4*(r1^2)*a_sq + 4*(r1^2)*(n `dot` a)^2
b1 = 4*d0_sq*(d0 `dot` a)+4*k*(d0 `dot` a) - 8*(r1^2)*(d0 `dot` a) + 8*(r1^2)*(n `dot` d0)*(n `dot` a)
b0 = d0_sq*d0_sq+2*k*d0_sq+k^2 - 4*(r1^2)*d0_sq + 4*(r1^2)*(n `dot` d0)^2
solveQuarticEq (a4,a3,a2,a1,a0) =
let
sol = do
(x_Re :+ x_Im) <- [x1,x2,x3,x4]
if (abs(x_Im) < 1.0E-8) && (1.0E-8 <= x_Re) then
return x_Re
else
[]
in
sol
where
l1 = (toCmp $ k3/4)/sqrt(k4)
l2 = (toCmp $ (cbrt(2)*k5)/(3*a4))/k8 + k8/(toCmp $ 3*cbrt(2)*a4)
l3 = (toCmp $ (a3^2)/(2*a4^2) - (4*a2)/(3*a4)) - l2
k1 = l1 + l3
k2 = -l1 + l3
k3 = -((a3/a4)^3) + (4*a2*a3)/(a4^2) - (8*a1)/a4
k4 = (toCmp $ ((a3/(2*a4))^2) - (2*a2)/(3*a4)) + l2
k5 = a2^2 - 3*a1*a3 + 12*a0*a4
k6 = 2*a2^3 - 9*a1*a2*a3 + 27*a0*a3^2 + 27*a1^2*a4 - 72*a0*a2*a4
k7 = -4*k5^3 + k6^2
k8 = cbrt((toCmp $ k6) + sqrt(toCmp $ k7))
l4 = toCmp $ -a3/(4*a4)
l5 = sqrt(k2)/2
l6 = sqrt(k1)/2
l7 = sqrt(k4)/2
x1 = l4 - l5 - l7
x2 = l4 + l5 - l7
x3 = l4 - l6 + l7
x4 = l4 + l6 + l7
cbrt x = x ** (1/3)
toCmp x = x :+ 0
prod x y = x >>= (\u -> zip (repeat u) y)
(-:) = flip (.)
type (+) a b = Either a b
inj1 :: a -> a + b
inj1 = Left
inj2 :: b -> a + b
inj2 = Right
coPair :: (a1 -> b, a2 -> b) -> (a1 + a2 -> b)
coPair = uncurry eitherタグ一覧: