The Light Field. Last lecture: Radiometry and photometry

Transcription

1 The Light Field Last lecture: Radiometry and photometry This lecture: Light field = radiance function on rays Conservation of radiance Measurement equation Throughput and counting rays Irradiance calculations Subtitle: Lines and Light From London and Upton CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

2 Field Radiance or Light Field Definition: The field radiance (luminance) at a point in space in a given direction is the power per unit solid angle per unit area perpendicular to the direction da dω L( x, ω) L(x,!) d (x,!) dxd! r( x, ω) CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

3 Spherical Gantry 4D Light Field L( x, y, θ, ϕ) ( θ, ϕ) Captures all the light leaving an object - like a hologram CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

4 Two-Plane Light Field D Array of Cameras D Array of Images L( u, v, s, t) CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

5 Multi-Camera Array 4D Light Field CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

6 Lytro Immerge

7 Lytro Cameras

8 Properties of Radiance

9 Properties of Radiance Fundamental field quantity that characterizes the distribution of light in an environment Radiance is the quantity associated with a ray Rendering is all about computing the radiance Radiance is invariant along a ray Reduces parameters from 6D to 4D Response of a sensor proportional to the radiance Cameras measure radiance CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

10 1st Law: Conservation of Radiance The radiance in the direction of a light ray remains constant as the ray propagates d Φ 1 d Φ d Φ = Φ 1 d L 1 da 1 dω 1 d Φ 1 = L1dω1dA1 dω da L d Φ = LdωdA r da da dω da = = dω da r L = L CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

11 Quiz Does radiance increase under a magnifying glass? No!! CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

12 The Measurement Equation

13 Radiance: nd Law The response of a sensor is proportional to the radiance of the surface visible to the sensor. Ω Aperture A Sensor R = Ldω da = LT A Ω T dω da = A Ω T (throughput) quantifies the gathering power of the device; the higher the throughput the greater the amount of light gathered T is constant, sensor response R proportional to average radiance L CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

14 Quiz Assume a wall has constant radiance Does the response of a sensor when pointed a wall depend on the distance to the wall? No!! CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

15 Throughput Measuring the Number of Rays

16 Beam of Rays Define an infinitesimal beam as the set of rays intersecting two infinitesimal surface elements r( u1, v1, u, v ) da ( u, v ) da ( u, v ) How many rays are in this beam? Defn: The throughput is the number of rays in a beam CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

17 Number of Rays in a Beam Define an infinitesimal beam as the set of rays intersecting two infinitesimal surface elements r( u1, v1, u, v ) da ( u, v ) da ( u, v ) This many: da da = 1 d T x x 1 CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

18 Why is this the Right Measure? da ( u, v ) da ( u, v ) da da = 1 d T x x 1 Suppose we double the area da1 (or da), then the number of rays in the beam should double Suppose we half the distance x1-x, then the number of rays also doubles CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

19 Alternative Definition of Radiance Two physical laws 1. Energy is conserved. The size of a beam does not change as rays propagate Beam Radiance is the ratio of conserved quantities, therefore, the radiance is also conserved CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

20 Changing Ray Coordinates Parameterize rays wrt to receiver r( u, v, θ, φ ) dω ( θ, φ ) da ( u, v ) da d T = 1 da = dω da x x 1 Same value for the throughput, different formula CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

21 Changing Ray Coordinates Parameterize rays wrt to source r( u1, v1, θ1, φ1 ) da ( u, v ) dω ( θ, φ ) d T = da da = da dω x x CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

22 Changing Ray Coordinates Again Tilting the surfaces re-parameterizes the rays! r( u1, v1, u, v ) da ( u, v ) da ( u, v ) d cosθ cosθ T = 1 x x 1 da da 1 CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

23 Number of Rays Hitting a Shape Parameterize rays by Projected area r( x, y, θ, φ) A" ( ω! ) ω! Measuring the number or rays that hit a shape T = dω( θ, ϕ) da( x, y) = dω( θ, ϕ) da( x, y) = = CS348b Lecture 5 S S R A! ( θ, ϕ) dω( θ, ϕ) S 4π A! Sphere:! T = 4π A = 4π R Pat Hanrahan / Matt Pharr, Spring 017

24 Calculated Another Way Parameterize rays by r( u, v, θ, φ) ( u, v) Sphere: ( θ, φ) % & % & T = ' cos θ dω( θ, ϕ) ( ' da( u, v) ( N! ') H ( N! ) (* ') M ( "###$###% " #$##% * π S T = π S = 4π R Crofton s Theorem: 4 π A! = π S A! = S 4 CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

25 Irradiance from a Uniform Area Light Source

26 Irradiance from the Environment d Φ ( x, ω) = L ( x, ω)cosθ da dω i de( x, ω) = L ( x, ω)cosθ dω i i θ dω L ( x, ω) i Light meter E( x) = Li ( x, ω)cosθ dω CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017 H da

27 Irradiance from a Uniform Area Source A E( x) = L cosθ dω H = L Ω = LΩ! cosθ dω Ω! Ω Direct Illumination CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

28 Uniform Disk Source Geometric Derivation Algebraic Derivation r Ω! = cosα π 1 0 cosθ dφ d cosθ h sinα Ω! = π sin α α = = = π π π r cos sin α r + h θ cosα 1 CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

29 Spherical Source Geometric Derivation Algebraic Derivation r Ω! = cosθ dω R = π sin α α = π r R Ω! = π sin α CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

30 The Sun Solar constant (normal incidence at zenith) Irradiance 1353 W/m Illuminance 17,500 lm/m = 17.5 kilolux Solar angle α Ω = π sin α πα =.5 degrees =.004 radians (half angle)! Solar radiance = 6 x 10-5 steradians 3 E W / m 7 W L = = = Ω! 6 10 sr m sr CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

31 Polygonal Source CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

32 Polygonal Source CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

33 Polygonal Source CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

34 Consider 1 Edge γ i Area of sector γ N E θ!! A = γ cosθ = γ N N E CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

35 Lambert s Formula 3 Ai A1 i= 1 n n!! A = γ N N i i i i= 1 i= 1 A A 3 CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017

36 Penumbras and Umbras CS348b Lecture 5 Pat Hanrahan / Matt Pharr, Spring 017