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Graphics & Visualization Chapter 16 GLOBAL ILLUMINATION ALGORITHMS Graphics & Visualization: Principles & Algorithms Chapter 16

Introduction • Global Illumination Algorithms: Deal with the realistic computation of light transport in a scene  Estimate indirect illumination, i.e. light reaching a point through multiple surface  inter-reflections Compute both d irect illumination and indirect light  • Images are radiometrically accurate and photorealistic • All aspects of the image-generator pipeline are based on physics: The reflection properties of all materials are described by BRDFs  The light sources are radiometrically modeled  The transport of light through the scene is computed accurately  The display of the image uses accurate tone-mapping operators  2 Graphics & Visualization: Principles & Algorithms Chapter 16

The Physics of Light-Object Interaction II • The rendering equation : Is the most fundamental equation for photorealistic synthesis  Expresses the equilibrium of light distribution in a 3D scene  Takes into account:   The radiometric specification of the light sources  The BRDF specifications of all materials • It is an energy balance that expresses how much excitant radiance is present at a given surface point in a certain direction 3 Graphics & Visualization: Principles & Algorithms Chapter 16

Rendering Equation: Hemispherical Integration • BRDF at a surface point x : Expresses excitant radiance L r in direction ( φ r , θ r ) versus incident irradiance  E i from direction ( φ i , θ i ):     dL ( , ) dL ( , )       f ( , , , ) r r r r r r (16.1) r r r i i       dE ( , ) L ( , )cos( ) d i i i i i i i i • Integrate Eq(16.1) over the hemisphere Ω i of all possible differential solid angles d ω i :            dL ( , ) L ( , ) f ( , , , )cos( ) d r r r i i i r r r i i i i             L ( , ) L ( , ) f ( , , , )cos( ) d  r r r i i i r r r i i i i i 4 Graphics & Visualization: Principles & Algorithms Chapter 16

Rendering Equation: Hemispherical Integration (2) • Add a constant term, which corresponds to the self-emitted radiance L e ( φ r , θ r ) of point x • The complete rendering equation is:                L ( , ) L ( , ) L ( , ) f ( , , , )cos( ) d  r r r e r r i i i r r r i i i i i (16.2) 5 Graphics & Visualization: Principles & Algorithms Chapter 16

Rendering Equation: Surface-Area Integration • We can move from solid angle to surface integration domain • The integral is taken over all visible surfaces • Transform d ω i to the corresponding differential surface dA • Let y : the first visible surface point seen from point x in direction ( φ i , θ i )  ( φ i y θ y ) : the direction pointing from y towards x  r xy : the distance between x and y  S visible : the set of all visible surfaces as seen from x  • Then:  cos( ) dA   y d i 2 r xy • Then L r becomes:   cos( )cos( )   i y            L ( , ) L ( , ) L ( , ) f ( , , , ) dA (16.3) r r r e r r S i i i r r r i i 2 r Visible xy 6 Graphics & Visualization: Principles & Algorithms Chapter 16

Rendering Equation: Surface-Area Integration (2) • Radiance remains constant along a straight line (no scattering), so we may replace received radiance with emitted radiance at the source:         L ( , ) L ( , , ) L ( , , ) x y i i i i i i r y y • In equation Eq(16.3): the product of both cosine terms divided by r xy 2   is a geometric coupling term  is dependent on the geometrical relationship between x and y  is independent of the actual radiance distribution or BRDF's defined on the surfaces:   cos( )cos( ) i y  G ( , ) x y 2 r xy 7 Graphics & Visualization: Principles & Algorithms Chapter 16

Rendering Equation: Surface-Area Integration (3) • Substituting all of the above in Eq[16.3]:              L ( , , ) L ( , , ) L ( , , ) f ( , , , ) ( , ) G dA x x y x y r r r e r r S r y y r r r i i Visible 8 Graphics & Visualization: Principles & Algorithms Chapter 16

Rendering Equation: Surface-Area Integration (4) • Integral over all surfaces is preferable to integral over visible surfaces only: Advantage: Single integration domain, identical for all points x  • We introduce a visibility term V( x , y ):  1 , and are mutually invisible x y   V( ) x,y 0 , otherwise  • The rendering equation becomes:              L ( , , ) L ( , , ) L ( , , ) f ( , , , ) ( , ) ( , ) G V dA x x y x y x y r r r e r r r y y r r r i i S S : the integration domain indicating all surface point y 9 Graphics & Visualization: Principles & Algorithms Chapter 16

Rendering Equation: Surface-Area Integration (5) • A special case: Direct illumination from one light source            L ( , , ) L ( , , ) f ( , , , ) ( , ) ( , ) G V dA x y x y x y r r r S e y y r r r i i 1 S 1 : the surface area domain of the light source Direct illumination from more light sources:  - Split the integral in a sum of integrals for each light source L            L ( , , ) L ( , , ) f ( , , , ) ( , ) ( , ) G V dA x y x y x y r r r S e y y r r r i i j  j 1 10 Graphics & Visualization: Principles & Algorithms Chapter 16

Rendering Equation: Surface-Area Integration (6) Direct illumination from one light source  11 Graphics & Visualization: Principles & Algorithms Chapter 16

Environment Map Illumination • The light source is encoded as a (hemi)-spherical environment map • An emitted radiance L e ( φ i , θ i ) is defined for each incoming direction             L ( , ) L ( , ) f ( , , , )cos( ) d  r r r e i i r r r i i i i i • Usually is given as a high dynamic range image • Can contain more than a million pixels 12 Graphics & Visualization: Principles & Algorithms Chapter 16

Discretized Form of the Rendering Equation • For some applications, it is useful to express: Light energy per surface patch (usually individual polygons)  The hemisphere of all outgoing directions  • Achieved by discretizing the rendering equation • We solve a linear system: each equation describes the energy balance of a single patch  Radiosity algorithms • Note that not all of these algorithms use the radiosity B radiometric quantity 13 Graphics & Visualization: Principles & Algorithms Chapter 16

Discretized Form of the Rendering Equation (2) Assumptions for the formulation of radiosity equations: All surfaces in the scene are subdivided in surface patches  The outgoing radiance is similar for all surface points on the patch  The algorithm will compute only the average radiance of all surface  points All surface patches have diffuse reflectance characteristics  Each patch has only 1 radiance value as a final solution  The light sources are considered to be diffuse as well (although  this is not strictly necessary) 14 Graphics & Visualization: Principles & Algorithms Chapter 16

Discretized Form of the Rendering Equation (3) • Radiosity B for a single point x Is the flux per surface area, or  Radiance integrated over the hemisphere of outgoing directions at x  • Average radiosity B i emitted by a surface patch i with area A i : 1        B L ( , , )cos( ) d dA x  i S r r r r i A i x i • On purely diffuse surfaces: self-emitted radiance L e and the BRDF f r do not depend on incoming or  outgoing directions • Rendering equation for a surface point x:        L ( ) L ( ) L ( , , ) f ( )cos( ) d x x x x  r e i i i r i i x 15 Graphics & Visualization: Principles & Algorithms Chapter 16

Discretized Form of the Rendering Equation (4) • The incident radiance L i ( x , φ i , θ i ): depends on incident direction  corresponds to the exitant radiance L r ( y ) emitted towards x by the point y  visible from x along the direction ( φ i , θ i ) • The integral equation without any directions present:    L ( ) L ( ) f ( ) G ( , ) ( , ) V L ( ) dA x x x x y x y y r e r r y S • In a diffuse environment: Radiosity and radiance are related  B ( x ) = π L r ( x ) and B e ( x ) = π L e ( x )  16 Graphics & Visualization: Principles & Algorithms Chapter 16