Radiosity Radiosity Radiosity Radiosity Motivation: what is - PDF document

Radiosity Radiosity Radiosity Radiosity � Motivation: what is missing in ray Motivation: what is missing in ray- -traced images? traced images? � � Indirect illumination effects Indirect illumination effects � � Color bleeding Color bleeding � � Soft shadows Soft shadows � � Radiosity Radiosity is a physically is a physically- -based illumination algorithm capable based illumination algorithm capable � of simulating the above phenomena in a scene made of ideal of simulating the above phenomena in a scene made of ideal diffuse surfaces. diffuse surfaces. � Books: Books: � � Cohen and Wallace, Cohen and Wallace, Radiosity Radiosity and Realistic Image Synthesis, and Realistic Image Synthesis, � Academic Press Professional 1993. Academic Press Professional 1993. � Sillion Sillion and and Puech Puech, , Radiosity Radiosity and Global Illumination, Morgan and Global Illumination, Morgan- - � Kaufmann Kaufmann, 1994. , 1994. Radiosity Radiosity in a Nutshell in a Nutshell Indirect illumination effects Indirect illumination effects � Break surfaces into many small elements Break surfaces into many small elements � Light source � Formulate and solve a linear system of equations Formulate and solve a linear system of equations � that models the equilibrium of inter that models the equilibrium of inter- -reflected reflected Eye light in a scene. light in a scene. � The solution gives us the amount of light leaving The solution gives us the amount of light leaving Diffuse Reflection � each point on each surface in the scene. each point on each surface in the scene. � Once solution is computed, the shaded elements Once solution is computed, the shaded elements � can be quickly rendered from any viewpoint. can be quickly rendered from any viewpoint. Meshing (partition into elements) Meshing (partition into elements) Radiosity Radiosity Change geometry Input geometry Form-Factors Change light Solution or colors Render Change view 1

Radiometric quantities The Radiosity Radiosity Equation Equation Radiometric quantities The � Radiant energy [J] Radiant energy [J] � Assume that surfaces in the scene have been Assume that surfaces in the scene have been � � discretized discretized into into n n small elements. small elements. � Radiant power (flux): radiant energy per second [W] Radiant power (flux): radiant energy per second [W] � � Assume that each element emits/reflects light Assume that each element emits/reflects light � � Irradiance (flux density): incident radiant power per Irradiance (flux density): incident radiant power per uniformly across its surface. uniformly across its surface. � unit area [W/m unit area [W/m 2 2 ] ] � Define the Define the radiosity radiosity B B as the total hemispherical as the total hemispherical � flux density (W/m 2 flux density (W/m 2 ) leaving a surface. ) leaving a surface. � Radiosity Radiosity (flux density): outgoing radiant power per (flux density): outgoing radiant power per � unit area [W/m 2 unit area [W/m 2 ] ] � Let Let’ ’s write down an expression describing the s write down an expression describing the � total flux (light power) leaving element i total flux (light power) leaving element i in the in the � Radiance (angular flux density): radiant power per Radiance (angular flux density): radiant power per � scene: scene: 2 sr unit projected area per unit solid angle [W/(m unit projected area per unit solid angle [W/(m 2 sr)] )] total flux = emitted flux + reflected flux total flux = emitted flux + reflected flux The Radiosity The Radiosity Equation Equation The Form Factor The Form Factor � Total flux leaving element i: Total flux leaving element i: � The form factor The form factor F F ji ji tells us how much of the flux tells us how much of the flux � B i A � i leaving element j actually reaches element i. leaving element j actually reaches element i. � Total flux emitted by element i: Total flux emitted by element i: E i A � i � Total reflected flux: Total reflected flux: θ θ � cos cos 1 ∫ ∫ = x y F V ( x , y ) dA dA � (reflectance of element i)*(the total incoming flux) (reflectance of element i)*(the total incoming flux) � ij π − 2 x y A x y ∈ ∈ � total incoming flux = sum of contributions from all other total incoming flux = sum of contributions from all other i x A y A i j � elements in the scene elements in the scene ∑ ρ B A F i j j ji j � The full The full radiosity radiosity equation is then: equation is then: � n ∑ = + ρ B A E A B A F i i i i i j j ji = j 1 Form- Form -Factor Computation Factor Computation Properties of Form Factors Properties of Form Factors � Reciprocity: Reciprocity: = � A F A F i ij j ji � Additivity Additivity: : = + F F F � ∪ ) i ( j k ij ik � Conservation of energy in a closed environment: Conservation of energy in a closed environment: � n ∑ = F 1 ij = j 1 2

The Radiosity Radiosity Equation Equation Finally... The Finally... n ∑ � The The radiosity radiosity equation equation � A linear system of n equations in n unknowns: A linear system of n equations in n unknowns: = + ρ � B A E A B A F � i i i i i j j ji = j 1 − ρ − ρ − ρ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 F F L F B E n ∑ � Divide equation by A Divide equation by A i : = + ρ A 1 11 1 12 1 1 n 1 1 i : j ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ B E B F � − ρ − ρ M i i i j A ji F 1 F B E i ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = = 2 21 2 22 2 2 j 1 ⎢ M O M ⎥ ⎢ M ⎥ ⎢ M ⎥ n ∑ � Apply form Apply form- -factor reciprocity: factor reciprocity: = + ρ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ B E B F � − ρ − ρ ⎣ L L ⎦ ⎣ ⎦ ⎣ ⎦ i i i j ij F 1 F B E = n n 1 n nn n n j 1 � We can write this using matrix notation: We can write this using matrix notation: � ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ B E B 1 1 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = + M M M ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ B ⎦ ⎣ E ⎦ ⎣ B ⎦ n n n The Radiosity The Radiosity Method Method Examples: Examples: � Take as input a geometric model of the scene, with Take as input a geometric model of the scene, with � emission and reflection properties of each surface emission and reflection properties of each surface � Step 1 Step 1 - - Meshing Meshing: : Discretize Discretize input surfaces into a input surfaces into a � mesh of small elements mesh of small elements � Step 2 Step 2 - - Setup Setup: Compute the form factors : Compute the form factors F F ij � ij � Step 3 Step 3 - - Solution Solution: Solve the resulting linear : Solve the resulting linear � system of equations system of equations � Step 4 Step 4 - - Display Display: Render shaded elements from : Render shaded elements from � any desired view point. any desired view point. � These steps are often interleaved in practice. These steps are often interleaved in practice. � Examples: Examples: Examples: Examples: 3