a ray tracer by Kevin Beason http://kevinbeason.com/smallpt/

1. smallpt:
Global Illuminaton in 99 lines of C++
a ray tracer by Kevin Beason
htp://kevinbeason.ccom/smallpt/
Presentatin by
Dr. David Cline
Oklahima State University

2. Glibal Illuminatin
• Glibal Illuminatin = “virtual phitigraphy”
– Given a scene descriptin that specifes the
licatin if surfaces in a scene, the licatin if
lights, and the licatin if a camera, take a virtual
“phitigraph” if that scene.

3. • “Headlight” rendering if a simple scene

4. • Adding surface details

5. • Direct lightng with hard shadiws

6. • “Ambient icclusiin” = direct lightng if a cliudy
day.

7. • Ambient Occlusiin and depth if feld

8. • Glibal illuminatin shiwing diferent surface types,
glass surfaces, caustcs (light cincentratins), and depth
if feld.

9. Anither Example
• Ad-hic Lightng vs. Glibal Illuminatin

10. Hiw ti firm a GI image?

11. The Rendering Equatin

12. The Rendering Equatin
The radiance (intensity of light)
Coming from surface point P
In direction Dv. This is what we
Have to calculate.

13. The Rendering Equatin
The self-emitted radiance from P
In direction Dv (0 unless point P
Is a light source) This can be looked
Up as part of the scene description.

14. The Rendering Equatin
The reflected light term. Here we must add
Up (integrate) all of the light coming in to
point P from all directions, modulated by the
Chance that it scatters in direction Dv
(based on the BRDF function, Fs)

15. Path Tracing Appriximatin
Replace the ray integral with a Monte Carlo
(random) Sample that has the same Expected
(average) Value. Then average a bunch of
samples for each pixel to create a smooth image.

16. Path Tracing Algirithm

17. SmallPT
• A 99 line Path Tracer by Kevin Beasin
• (Expanded Versiin has 218 lines)
• Majir Parts:
Vec: a vectir class, used fir piints, nirmals, cilirs
Ray: a ray class (irigin and directin)
Reflt: the surface refectin type
Sphere: SmallPT inly suppirts sphere ibjects
spheres: the hard cided scene (sime # if spheres)
intersect: a riutne ti intersect rays with the scene if spheres
radiance: recursive riutne that silves the rendering equatin
main: prigram start and main liip that gies iver each pixel

18. Squashed
Cide 1:

19. Squashed
Cide 2:

20. Expanded versiin (1)
Preliminaries

21. Expanded versiin (2)
Vec (Piints, Vectirs, Cilirs)

22. Nirmalize
• “Nirmalize” a vectir = divide by its length

23. Dit Priduct

24. Criss Priduct

25. Ray Structure
• A ray is a parametric line with an irigin (i) and a
directin (d). A piint aling the ray can be defned
using a parameter, t:
• In cide we have:
• The cire riutnes if the ray tracer intersect rays with
geimetric ibjects (spheres in iur case)

26. Sphere
• SmallPT suppirts sphere ibjects inly
• We can defne a sphere based in
– a center piint, C
– Radius, r
• The equatin if the sphere:
• In vectir firm:

27. Sphere
Intersectin
Start with vector equation of sphere

28. Intersectin Riutne

29. Full Sphere Cide

30. The Scene

31. The Scene Descriptin

32. Cinvert Cilirs ti Displayable Range
• The iutput if the “radiance” functin is a set if unbiunded
cilirs. This has ti be cinverted ti be between 0 and 255 fir
display purpises. The filliwing functins di this. The “tiInt”
functin applies a gamma cirrectin if 2.2.

33. Intersect Ray with Scene
• Check each sphere, ine at a tme. Keep the clisest
intersectin.

34. End Part 1

35. The main Functin
• Set up camera ciirdinates
• Initalize image array
• Parallel directve
• Fir each pixel
– Di 2x2 subpixels
– Average a number if radiance samples
– Set value in image
• Write iut image fle

36. main (1)

37. main (1a: set up image)

38. main (1b: set up camera)

39. Camera Setup
• Liik frim and gaze directin:
• Hirizintal (x) camera directin
(assumes upright camera)
(0.5135 defnes feld if view angle)
• Vertcal (vup) vectir if the camera
(criss priduct gets vectir perpendicular ti bith cx and gaze directin)

40. Camera Setup
look
from
cx
cy gaze
direction

41. main (2: Create Image)

42. main (2a: OpenMP directve)
States that each loop iteration should be run
in its own thread.

43. main (2b: Liip iver image pixels)
Loop over all pixels in the image.

44. main (2c: Subpixels & samples)
Pixels composed of 2x2 subpixels.
The subpixel colors will be averaged.

45. main (2d: Pixel Index)
Calculate array index for pixel(x,y)

46. main (2e: Tent Filter)
r1 and r2 are random values of a tent filter
(Determine location of sample within pixel)

47. Tent Filter
• Frim Realistc Ray Tracing (Shirley and
Mirley)

48. Tent Filter
• Frim Realistc Ray Tracing (Shirley and Mirley)

49. main (2f: Ray directin & radiance)
Compute ray direction using cam.d, cx, cy
Use radiance function to estimate radiance

50. main (2g: Add subpixel estmate)
Add the gamma-corrected subpixel color
estimate to the Pixel color c[i]

51. main (3: Write PPM image)
PPM Firmat: http://netpbm.sourceforge.net/doc/ppm.html

52. radiance (1: di intersectin)
return value Vec the radiance estimate
r the ray we are casting
depth the ray depth
Xi random number seed
E whether to include emissive color

53. radiance (2: surface pripertes)
Surface properties include:
intersection point (x)
Normal (n)
Oriented normal (n1)
Object color (f)

54. Orientng Nirmal
• When a ray hits a glass surface, the ray tracer must determine
if it is entering ir exitng glass ti cimpute the refractin ray.
• The dit priduct if the nirmal and ray directin tells this:

55. Russian Riulete
• Stip the recursiin randimly based in the surface
refectvity.
– Use the maximum cimpinent (r,g,b) if the surface cilir.
– Din’t di Russian Riulete untl afer depth 5

56. Difuse Refectin
• Fir difuse (nit shiny) refectin
– Sample all lights (nin-recursive)
– Send iut additinal randim sample (recursive)

57. Difuse Refectin
• Cinstruct randim ray:
– Get randim angle (r1)
– Get randim distance frim center (r2s)
– Use nirmal ti create irthinirmal ciirdinate
frame (w,u,v)

58. Sampling Unit Disk
• Frim Realistc Ray Tracing (Shirley and
Mirley)

59. Sampling Unit Hemisphere
w=z
u=x
v=y

60. Sampling Lights

61. Sampling Sphere by Silid Angle
• Create ciirdinate system fir sampling: sw, su, sv

62. Sampling Sphere by Silid Angle
• Determine max angle
amax

63. Sampling Sphere by Silid Angle
• Calculate sample directin based in randim numbers
accirding ti equatin frim Realistc Ray Tracing:

64. Shadiw Ray
• 145: Check fir icclusiin with shadiw ray
• 146: Cimpute 1/pribability with respect ti silid angle
• 147: Calculate lightng and add ti current value

65. Difuse Recursive Call
• Make recursive call with randim ray directin
cimputed earlier:
– Nite that the 0 parameter at the end turns if the
emissive term at the next recursiin level.

66. Ideal Specular (Mirrir) Refectin

67. Ideal Specular (Mirrir) Refectin
• Refected Ray:
– Angle if incidence =
Angle if refectin

68. Glass (Dielectric)

69. Refected Ray & Orientatin
• 159: Glass is bith refectve and refractve, si we cimpute the
refected ray here.
• 160: Determine if ray is entering ir exitng glass
• 161: IOR fir glass is 1.5.
nnt is either 1.5 ir 1/1.5

70. Tital Internal Refectin
• Tital internal refectin iccurs when the light ray
atempts ti leave glass at tii shalliw an angle.
• If the angle is tii shalliw, all the light is refected.

71. Refect ir Refract using Fresnel Term
• Cimpute the refracted ray

72. Refractin Ray

73. Refractve Index
• Refractve index gives the speed if light
within a medium cimpared ti the speed
if light within a vacuum:
Water: 1.33
Plastc: 1.5
Glass: 1.5 – 1.7
Diamind: 2.5
Nite that this dies nit acciunt fir
dispersiin (prisms). Ti acciunt fir
these, vary index by wavelength.

74. Fresnel Refectance
• Percentage if light is refected (and
what refracted) frim a glass surface
based in incident angle (ϴa)
• Refectance at “nirmal incidence”,
where (n=na/nb)
• Refectance at ither angles:

75. Refect ir Refract using Fresnel Term
• Fresnel Refectance
– R0 = refectance at nirmal incidence based in IOR
– c = 1-cis(theta)
– Re = fresnel refectance

76. Refect ir Refract using Fresnel Term
• P = pribability if refectng
• Finally, make 1 ir 2 recursive calls
– Make 2 if depth is <= 2
– Make 1 randimly if depth > 2

77. Cinvergence
From: http://kevinbeason.com/smallpt/