Radiosity+Importance Lecture 4: Radiometry and • Radiosity+Importance: Bidirectional Rendering Equation Chapter 2 in Advanced GI Fall 2004 Kavita Bala Computer Science Cornell University © Kavita Bala, Computer Science, Cornell University Importance Radiosity (IR) Radiosity, Importance • Motivation –O(k2 + n) is too slow – HR oversolves globally, undersolves locally • Insight: Exploit view dependence • Importance: Direct or indirect contribution of patch to image from this view point Radiosity: Forward Importance: Backward © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University IR Intuition IR Algorithm • Importance: adjoint formulation of radiosity • Solves for dual system simultaneously • Importance is shot by treating eye as light source eye importance • Importance R proportional A on image q i i radiosity p © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University 1 Radiosity Equation IR Intuition • Radiosity for each polygon i •I1 = R1 + ρ2F21I2 + ρ3F31I3 +… N ∀i : Bi = Be,i + ρi ∑ B j F(i → j) j=1 Contribution of 1 through 2 • Linear system Direct contribution of 1 – Βi : radiosity of patch i (unknown) – Βe,i : emission of patch i (known) Contribution of 1 to eye – ρ : reflectivity of patch i (known) I ρ2F21I2 2 –F(i→j): form-factor (coefficients of matrix) eye R1 3 1 ρ3F31I3 © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University Importance Radiosity Motivation • Elegant formulation of bidirectional propagation – Replaces ad-hoc solutions scene eye • IR restricted to one viewpoint – Need to unmesh as viewpoint moves ??? © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University What is the behavior of light? Models of Light • Physics of light • Geometric Optics – Emission •Radiometry – Reflection / Refraction – Absorption • Material properties • Simplest model • Rendering Equation • Size of objects > wavelength of light © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University 2 Radiometry Digression: Hemispheres • Hemisphere = two-dimensional surface • Radiometry: measurement of light energy • Direction = point on (unit) sphere • Defines relation between π –Power θ dθ θ ∈[0, ] – Energy 2 – Radiance ϕ ∈[0,2π ] – Radiosity ϕ dϕ © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University Digression: Solid angles Digression: Solid angles 2D 3D 2D 3D L A L R L R A A θ= Ω= R R2 L A R θ= R Ω= R R2 Full circle = 2π radians Full sphere = 4π steradians Full circle = 2π radians Full sphere = 4π steradians © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University Digression: Solid angle Hemispherical coordinates • Full sphere = 4π steradian = 12.566 sr • Direction = point on (unit) sphere • Dodecahedron = 12-sided regular polyhedron; 1 face = 1 sr dA θ dθ dA = (r sinθdϕ)(rdθ ) ϕ dϕ © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University 3 Hemispherical coordinates Hemispherical integration • Area of hemisphere: • Defined a measure over hemisphere • dω = direction vector • Differential solid angle dA dω = = sinθdθdϕ r 2 © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University Hemispherical integration Power • Area of hemisphere: • Energy: Symbol: Q; unit: Joules • Power: Energy per unit time (dQ/dt) 2ππ/ 2 ∫ dω = ∫∫dϕ sinθdθ – Aka. “radiant flux” in this context Ω x 0 0 • Symbol: P or Φ; unit: Watts (Joules / sec) 2π π / 2 – Photons per second = dϕ − cosθ ∫ []0 0 – All further quantities are 2π derivatives of P = ∫ dϕ (flux densities) 0 = 2π © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University Irradiance Irradiance example • Power per unit area (dP/dA) • Uniform point source illuminates small – That is, area density of power surface dA from distance r – It is defined with respect to a surface – Think of it as a piece of a sphere 2 • Symbol: E; unit: W / m –Power P is uniformly spread – Measurable as power on a over the area of the sphere small-area detector – Area power density exiting a dA dP P dP = P ; E = = surface is called 4πr 2 dA 4πr 2 radiant exitance (M) or radiosity (B) but has dA dA’ the same units dP dP E'= = = E cosθ dA' dA/ cosθ © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University 4 Radiance Radiance • Radiance is radiant energy at x in • Power per unit (solid angle times area) direction θ: 5D function – Counts photons that (a) go through a little area around x perpendicular to Θ and (b) are –:L(x → Θ) Power traveling in directions that fall in a little solid per unit projected surface area angle around Θ per unit solid angle Θ – Irradiance per unit solid angle d 2P L(x → Θ) = nd dA⊥dω –A 2 derivative of P Θ dA⊥ – units: Watt / m2.sr x © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University Radiance: Projected area Why is radiance important? d 2P • L(x → Θ) = ⊥ • Invariant along a straight line (in vacuum) dA dωΘ • Why per unit projected surface area? x dA cosθ 2 θ Θ x1 x dA θ dA © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University Invariance of Radiance Why is radiance important? • Response of a sensor (camera, human y eye) is proportional to radiance ny eye nx L(x → y) = L(y ← x) x • Pixel values in image proportional to radiance received from that direction © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University 5 Wavelength Dependence Relationships • Each particle has a wavelength h • Radiance is the fundamental quantity E = d 2P λ L(x → Θ) = dA⊥dω • All radiometric quantities depend on Θ wavelength •Power: P = L(x → Θ) ⋅ cosθ ⋅ dω ⋅ dA ∫∫ Θ Area Solid • Spectral radiance: L(x → Θ,λ) Angle • Radiosity: • Radiance: L(x → Θ) = L(x → Θ,λ)dλ ∫ B = L(x → Θ)⋅cosθ ⋅dω ∫ Θ Solid Angle © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University Example: Diffuse emitter Sun Example: radiance • Diffuse emitter: light source with equal • Power: 3.91 x 1026 W radiance everywhere • Surface Area: 6.07 x 1018 m2 d 2P L(x → Θ) = ⊥ dA dωΘ • Power = Radiance.Surface Area.π P = L(x → Θ) ⋅ cosθ ⋅ dω ⋅ dA ∫∫ Θ • Radiance = Power/(Surface Area.π) Area Solid Angle = L dA cosθ ⋅ dω ∫∫ Θ 7 2 Area Solid • Radiance = 2.05 x 10 W/ m .sr Angle = L ⋅ Area ⋅π © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University Sun Example Sun Example: Power on Earth • Power reaching earth on a 1m2 square: P = L dA cosθ ⋅ dω ∫∫ Θ Area Solid Angle • Assume cosθ = 1 (sun in zenith) P = L dA dω ∫∫Θ Area Solid Same radiance on Earth and Mars? Angle © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University 6 Sun Example: Power on Earth Sun Example: Power on Mars Power = Radiance.Area.Solid Angle Power = Radiance.Area.Solid Angle 2 2 Solid Angle = Projected AreaSun/(distanceearth_sun) Solid Angle = Projected AreaSun/(distancemars_sun) = 6.7 10-5 sr = 2.92 10-5 sr P = (2.05 x 107 W/ m2.sr) x (1 m2 ) x (6.7 10-5 sr) P = (2.05 x 107 W/ m2.sr) x (1 m2 ) x (2.92 10-5 sr) = 1373.5 Watt = 598.6 Watt © Kavita Bala, Computer Science, Cornell University © Kavita Bala, Computer Science, Cornell University 7.