Computer graphics III Rendering equation and its solution
![Computer graphics III – Rendering equation and its solution Jaroslav Křivánek, MFF UK Jaroslav. Computer graphics III – Rendering equation and its solution Jaroslav Křivánek, MFF UK Jaroslav.](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-1.jpg)
![Global illumination – GI Direct illumination Global = direct + indirect 2 Global illumination – GI Direct illumination Global = direct + indirect 2](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-2.jpg)
![Review: Reflection equation n “Sum” (integral) of contributions over the hemisphere: n Lo(x, wo) Review: Reflection equation n “Sum” (integral) of contributions over the hemisphere: n Lo(x, wo)](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-3.jpg)
![From local reflection to global light transport n Reflection equation (local reflection) n Where From local reflection to global light transport n Reflection equation (local reflection) n Where](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-4.jpg)
![From local reflection to global light transport n Plug for Li into the reflection From local reflection to global light transport n Plug for Li into the reflection](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-5.jpg)
![Rendering equation n Remove the subscript “o” from the outgoing radiance: n Description of Rendering equation n Remove the subscript “o” from the outgoing radiance: n Description of](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-6.jpg)
![Reflection equation vs. Rendering equation Similar form – different meaning n Reflection equation q Reflection equation vs. Rendering equation Similar form – different meaning n Reflection equation q](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-7.jpg)
![Rendering Equation – Kajiya 1986 CG III (NPGR 010) - J. Křivánek 2015 Rendering Equation – Kajiya 1986 CG III (NPGR 010) - J. Křivánek 2015](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-8.jpg)
![Path tracing sketch Path tracing sketch](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-9.jpg)
![Recursive unwinding of the RE n n Angular form of the RE To calculate Recursive unwinding of the RE n n Angular form of the RE To calculate](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-10.jpg)
![Path tracing, v. zero (recursive form) get. Li (x, ω): y = trace. Ray(x, Path tracing, v. zero (recursive form) get. Li (x, ω): y = trace. Ray(x,](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-11.jpg)
![Back to theory: Angular and area form of the rendering equation Back to theory: Angular and area form of the rendering equation](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-12.jpg)
![Angular vs. area form of the RE n Angular form q n integral over Angular vs. area form of the RE n Angular form q n integral over](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-13.jpg)
![Angular vs. area form of the RE n Area form q Integral over the Angular vs. area form of the RE n Area form q Integral over the](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-14.jpg)
![Angular form n Add radiance contributions to a point from all directions n For Angular form n Add radiance contributions to a point from all directions n For](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-15.jpg)
![Area form n Sum up contributions to a point from all other points on Area form n Sum up contributions to a point from all other points on](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-16.jpg)
![Most rendering algorithms = (approximate) solution of the RE n Local illumination (Open. GL) Most rendering algorithms = (approximate) solution of the RE n Local illumination (Open. GL)](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-17.jpg)
![Most rendering algorithms = (approximate) solution of the RE n Ray tracing [Whitted, ’ Most rendering algorithms = (approximate) solution of the RE n Ray tracing [Whitted, ’](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-18.jpg)
![Most rendering algorithms = (approximate) solution of the RE n Path tracing [Kajiya, ’ Most rendering algorithms = (approximate) solution of the RE n Path tracing [Kajiya, ’](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-19.jpg)
![From the rendering equation to finite element radiosity From the rendering equation to finite element radiosity](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-20.jpg)
![From the rendering equation to radiosity n Start from the area form of the From the rendering equation to radiosity n Start from the area form of the](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-21.jpg)
![From the rendering equation to radiosity n Diffuse surfaces only q q The BRDF From the rendering equation to radiosity n Diffuse surfaces only q q The BRDF](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-22.jpg)
![From the rendering equation to radiosity n Spatially constant (flat) radiosity B of the From the rendering equation to radiosity n Spatially constant (flat) radiosity B of the](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-23.jpg)
![From the rendering equation to radiosity n Spatially constant (flat) radiosity of the receiving From the rendering equation to radiosity n Spatially constant (flat) radiosity of the receiving](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-24.jpg)
![Classic radiosity equation n System of linear equations n Form factors n Conclusion: the Classic radiosity equation n System of linear equations n Form factors n Conclusion: the](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-25.jpg)
![Radiosity method n Classical radiosity 1. 2. n Stochastic radiosity q q n Form Radiosity method n Classical radiosity 1. 2. n Stochastic radiosity q q n Form](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-26.jpg)
![The operator form of the RE The operator form of the RE](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-27.jpg)
![RE is a Fredhom integral equation of the 2 nd kind General form the RE is a Fredhom integral equation of the 2 nd kind General form the](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-28.jpg)
![Linear operators FIX NOTATION n (L shoul Linear operators act on functions q (as Linear operators FIX NOTATION n (L shoul Linear operators act on functions q (as](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-29.jpg)
![Transport operator n Rendering equation CG III (NPGR 010) - J. Křivánek 2015 Transport operator n Rendering equation CG III (NPGR 010) - J. Křivánek 2015](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-30.jpg)
![Solution of the RE in the operator form n Rendering equation n Formal solution Solution of the RE in the operator form n Rendering equation n Formal solution](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-31.jpg)
![Expansion of the rendering equation n Recursive substitution L n n-fold repetition yields the Expansion of the rendering equation n Recursive substitution L n n-fold repetition yields the](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-32.jpg)
![Expansion of the rendering equation n If T is a contraction (tj. ||T|| < Expansion of the rendering equation n If T is a contraction (tj. ||T|| <](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-33.jpg)
![A different derivation of the Neumann series n Formal solution of the rendering equation A different derivation of the Neumann series n Formal solution of the rendering equation](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-34.jpg)
![Rendering equation n Solution: Neumann series CG III (NPGR 010) - J. Křivánek 2015 Rendering equation n Solution: Neumann series CG III (NPGR 010) - J. Křivánek 2015](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-35.jpg)
![Progressive approximation CG III (NPGR 010) - J. Křivánek 2015 Progressive approximation CG III (NPGR 010) - J. Křivánek 2015](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-36.jpg)
![Progressive approximation n Each application of T corresponds to one step of reflection & Progressive approximation n Each application of T corresponds to one step of reflection &](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-37.jpg)
![Contractivity of T n Holds for all physically correct models q Follows from the Contractivity of T n Holds for all physically correct models q Follows from the](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-38.jpg)
![Alright, so what have we achieved? Rendering equation n Solution through the Neumann series Alright, so what have we achieved? Rendering equation n Solution through the Neumann series](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-39.jpg)
![What exact integral are we evaluating, then? CG III (NPGR 010) - J. Křivánek What exact integral are we evaluating, then? CG III (NPGR 010) - J. Křivánek](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-40.jpg)
![Paths vs. recursion: Same thing, depends on how we look at it n Paths Paths vs. recursion: Same thing, depends on how we look at it n Paths](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-41.jpg)
![Recursive interpretation n n We’ve seen this already, right? But unlike at the beginning Recursive interpretation n n We’ve seen this already, right? But unlike at the beginning](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-42.jpg)
![Path tracing, v. 0. 1 (recursive form) get. Li (x, ω): y = nearest. Path tracing, v. 0. 1 (recursive form) get. Li (x, ω): y = nearest.](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-43.jpg)
![Path tracing, v. 2012, Arnold Renderer © 2012 Columbia Pictures Industries, Inc. All Rights Path tracing, v. 2012, Arnold Renderer © 2012 Columbia Pictures Industries, Inc. All Rights](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-44.jpg)
![](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-45.jpg)
![Path tracing [Kajiya 86] n Only one secondary ray at each intersection q n Path tracing [Kajiya 86] n Only one secondary ray at each intersection q n](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-46.jpg)
- Slides: 46
![Computer graphics III Rendering equation and its solution Jaroslav Křivánek MFF UK Jaroslav Computer graphics III – Rendering equation and its solution Jaroslav Křivánek, MFF UK Jaroslav.](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-1.jpg)
Computer graphics III – Rendering equation and its solution Jaroslav Křivánek, MFF UK Jaroslav. Krivanek@mff. cuni. cz
![Global illumination GI Direct illumination Global direct indirect 2 Global illumination – GI Direct illumination Global = direct + indirect 2](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-2.jpg)
Global illumination – GI Direct illumination Global = direct + indirect 2
![Review Reflection equation n Sum integral of contributions over the hemisphere n Lox wo Review: Reflection equation n “Sum” (integral) of contributions over the hemisphere: n Lo(x, wo)](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-3.jpg)
Review: Reflection equation n “Sum” (integral) of contributions over the hemisphere: n Lo(x, wo) qo qi Li(x, wi) Emitted radiance dwi Lr(x, wo) Total outgoing rad. Reflected rad.
![From local reflection to global light transport n Reflection equation local reflection n Where From local reflection to global light transport n Reflection equation (local reflection) n Where](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-4.jpg)
From local reflection to global light transport n Reflection equation (local reflection) n Where does the incoming radiance Li(x, wi) come from? q From other places in the scene ! Lo( r(x, wi), -wi) = Li(x, wi) Ray casting function CG III (NPGR 010) - J. Křivánek 2015 x r(x, wi)
![From local reflection to global light transport n Plug for Li into the reflection From local reflection to global light transport n Plug for Li into the reflection](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-5.jpg)
From local reflection to global light transport n Plug for Li into the reflection equation n Incoming radiance Li drops out Outgoing radiance Lo at x described in terms of Lo at other points in the scene n CG III (NPGR 010) - J. Křivánek 2015
![Rendering equation n Remove the subscript o from the outgoing radiance n Description of Rendering equation n Remove the subscript “o” from the outgoing radiance: n Description of](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-6.jpg)
Rendering equation n Remove the subscript “o” from the outgoing radiance: n Description of the steady state = energy balance in the scene Rendering = calculate L(x, wo) for all points visible through pixels, such that it fulfils the rendering equation n CG III (NPGR 010) - J. Křivánek 2015
![Reflection equation vs Rendering equation Similar form different meaning n Reflection equation q Reflection equation vs. Rendering equation Similar form – different meaning n Reflection equation q](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-7.jpg)
Reflection equation vs. Rendering equation Similar form – different meaning n Reflection equation q q Describes local light reflection at a single point Integral that can be used to calculate the outgoing radiance if we know the incoming radiance n Rendering equation q q Condition on the global distribution of light in scene Integral equation – unknown quantity L on both sides
![Rendering Equation Kajiya 1986 CG III NPGR 010 J Křivánek 2015 Rendering Equation – Kajiya 1986 CG III (NPGR 010) - J. Křivánek 2015](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-8.jpg)
Rendering Equation – Kajiya 1986 CG III (NPGR 010) - J. Křivánek 2015
![Path tracing sketch Path tracing sketch](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-9.jpg)
Path tracing sketch
![Recursive unwinding of the RE n n Angular form of the RE To calculate Recursive unwinding of the RE n n Angular form of the RE To calculate](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-10.jpg)
Recursive unwinding of the RE n n Angular form of the RE To calculate L(x, wo), we need to calculate L(r(x, w’), -w’) for all directions w’ around the point x For the calculation of each L(r(x, w’), -w’), we need to do the same thing recursively, etc. r( r(x, w’) w’’ w’ wo x CG III (NPGR 010) - J. Křivánek 2015 r(x, w’)
![Path tracing v zero recursive form get Li x ω y trace Rayx Path tracing, v. zero (recursive form) get. Li (x, ω): y = trace. Ray(x,](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-11.jpg)
Path tracing, v. zero (recursive form) get. Li (x, ω): y = trace. Ray(x, ω) return Le(y, –ω) + Lr (y, –ω) // emitted radiance // reflected radiance Lr(x, ω): ω′ = gen. Uniform. Hemisphere. Random. Dir( n(x) ) return 2 p * brdf(x, ω, ω′) * dot(n(x), ω′) * get. Li(x, ω′) CG III (NPGR 010) - J. Křivánek 2015
![Back to theory Angular and area form of the rendering equation Back to theory: Angular and area form of the rendering equation](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-12.jpg)
Back to theory: Angular and area form of the rendering equation
![Angular vs area form of the RE n Angular form q n integral over Angular vs. area form of the RE n Angular form q n integral over](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-13.jpg)
Angular vs. area form of the RE n Angular form q n integral over the hemisphere in incoming directions Substitution CG III (NPGR 010) - J. Křivánek 2015
![Angular vs area form of the RE n Area form q Integral over the Angular vs. area form of the RE n Area form q Integral over the](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-14.jpg)
Angular vs. area form of the RE n Area form q Integral over the scene surface geometry term CG III (NPGR 010) - J. Křivánek 2015 visibility 1 … y visible from x 0 … otherwise
![Angular form n Add radiance contributions to a point from all directions n For Angular form n Add radiance contributions to a point from all directions n For](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-15.jpg)
Angular form n Add radiance contributions to a point from all directions n For each direction, find the nearest surface n Implementation in stochastic path tracing: q For a given x, generate random direction(s), for each find the nearest intersection, return the outgoing radiance at that intersection and multiply it with the cosine-weighted BRDF. Average the result of this calculation over all the generated directions over the hemisphere. CG III (NPGR 010) - J. Křivánek 2015
![Area form n Sum up contributions to a point from all other points on Area form n Sum up contributions to a point from all other points on](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-16.jpg)
Area form n Sum up contributions to a point from all other points on the scene surface n Contribution added only if the two points are mutually visible n Implementation in stochastic path tracing: q n Generate randomly point y on scene geometry. Test visibility between x and y. If mutually visible, add the outgoing radiance at y modulated by the geometry factor. Typical use: direct illumination calculation for area light sources CG III (NPGR 010) - J. Křivánek 2015
![Most rendering algorithms approximate solution of the RE n Local illumination Open GL Most rendering algorithms = (approximate) solution of the RE n Local illumination (Open. GL)](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-17.jpg)
Most rendering algorithms = (approximate) solution of the RE n Local illumination (Open. GL) q q n Only point sources, integral becomes a sum Does not calculate equilibrium radiance, is not really a solution of the RE Finite element methods (radiosity) [Goral, ’ 84] q q q Discretize scene surface (finite elements) Disregard directionality of reflections: everything is assumed to be diffuse Cannot reproduce glossy reflections CG III (NPGR 010) - J. Křivánek 2015
![Most rendering algorithms approximate solution of the RE n Ray tracing Whitted Most rendering algorithms = (approximate) solution of the RE n Ray tracing [Whitted, ’](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-18.jpg)
Most rendering algorithms = (approximate) solution of the RE n Ray tracing [Whitted, ’ 80] q q q n Direct illumination on diffuse and glossy surfaces due to point sources Indirect illumination only on ideal mirror reflection / refractions Cannot calculate indirect illumination on diffuse and glossy scenes, soft shadows etc. … Distributed ray tracing [Cook, ’ 84] q q Estimate the local reflection using the MC method Can calculate soft shadows, glossy reflections, camera defocus blur, etc. CG III (NPGR 010) - J. Křivánek 2015
![Most rendering algorithms approximate solution of the RE n Path tracing Kajiya Most rendering algorithms = (approximate) solution of the RE n Path tracing [Kajiya, ’](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-19.jpg)
Most rendering algorithms = (approximate) solution of the RE n Path tracing [Kajiya, ’ 86] q q q True solution of the RE via the Monte Carlo method Tracing of random paths (random walks) from the camera Can calculate indirect illumination of higher order CG III (NPGR 010) - J. Křivánek 2015
![From the rendering equation to finite element radiosity From the rendering equation to finite element radiosity](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-20.jpg)
From the rendering equation to finite element radiosity
![From the rendering equation to radiosity n Start from the area form of the From the rendering equation to radiosity n Start from the area form of the](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-21.jpg)
From the rendering equation to radiosity n Start from the area form of the RE: n The Radiosity method– assumptions q q Only diffuse surfaces (BRDF constant in wi and wo) Radiosity (i. e. radiant exitance) is spatially constant (flat) over the individual elements CG III (NPGR 010) - J. Křivánek 2015
![From the rendering equation to radiosity n Diffuse surfaces only q q The BRDF From the rendering equation to radiosity n Diffuse surfaces only q q The BRDF](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-22.jpg)
From the rendering equation to radiosity n Diffuse surfaces only q q The BRDF is constant in wi and wo Outgoing radiance is independent of wo and it is equal to radiosity B divided by p CG III (NPGR 010) - J. Křivánek 2015
![From the rendering equation to radiosity n Spatially constant flat radiosity B of the From the rendering equation to radiosity n Spatially constant (flat) radiosity B of the](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-23.jpg)
From the rendering equation to radiosity n Spatially constant (flat) radiosity B of the contributing surface elements Radiosity of the j-th element Geometry factor between surface element j and point x CG III (NPGR 010) - J. Křivánek 2015
![From the rendering equation to radiosity n Spatially constant flat radiosity of the receiving From the rendering equation to radiosity n Spatially constant (flat) radiosity of the receiving](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-24.jpg)
From the rendering equation to radiosity n Spatially constant (flat) radiosity of the receiving surface element i: q Average radiosity over the element … form factor CG III (NPGR 010) - J. Křivánek 2015
![Classic radiosity equation n System of linear equations n Form factors n Conclusion the Classic radiosity equation n System of linear equations n Form factors n Conclusion: the](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-25.jpg)
Classic radiosity equation n System of linear equations n Form factors n Conclusion: the radiosity method is nothing but a way to solve the RE under a specific set of assumptions CG III (NPGR 010) - J. Křivánek 2015
![Radiosity method n Classical radiosity 1 2 n Stochastic radiosity q q n Form Radiosity method n Classical radiosity 1. 2. n Stochastic radiosity q q n Form](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-26.jpg)
Radiosity method n Classical radiosity 1. 2. n Stochastic radiosity q q n Form facto calculation (Monte Carlo, hemicube, …) Solve the linear system (Gathering, Shooting, …) Avoids explicit calculation of form factors Metoda Monte Carlo Radiosity is not practical, not used q q Scene subdivision -> sensitive to the quality of the geometry model (but in reality, models are always broken) High memory consumption, complex implementation CG III (NPGR 010) - J. Křivánek 2015
![The operator form of the RE The operator form of the RE](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-27.jpg)
The operator form of the RE
![RE is a Fredhom integral equation of the 2 nd kind General form the RE is a Fredhom integral equation of the 2 nd kind General form the](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-28.jpg)
RE is a Fredhom integral equation of the 2 nd kind General form the Fredholm integral equation of the 2 nd kind unknown functions Rendering equation: CG III (NPGR 010) - J. Křivánek 2015 equation “kernel”
![Linear operators FIX NOTATION n L shoul Linear operators act on functions q as Linear operators FIX NOTATION n (L shoul Linear operators act on functions q (as](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-29.jpg)
Linear operators FIX NOTATION n (L shoul Linear operators act on functions q (as matrices act on vectors) n The operator is linear if the “acting” is a linear operation n Examples of linear operators CG III (NPGR 010) - J. Křivánek 2015
![Transport operator n Rendering equation CG III NPGR 010 J Křivánek 2015 Transport operator n Rendering equation CG III (NPGR 010) - J. Křivánek 2015](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-30.jpg)
Transport operator n Rendering equation CG III (NPGR 010) - J. Křivánek 2015
![Solution of the RE in the operator form n Rendering equation n Formal solution Solution of the RE in the operator form n Rendering equation n Formal solution](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-31.jpg)
Solution of the RE in the operator form n Rendering equation n Formal solution n unusable in practice – the inverse cannot be explicitly calculated CG III (NPGR 010) - J. Křivánek 2015
![Expansion of the rendering equation n Recursive substitution L n nfold repetition yields the Expansion of the rendering equation n Recursive substitution L n n-fold repetition yields the](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-32.jpg)
Expansion of the rendering equation n Recursive substitution L n n-fold repetition yields the Neumann series CG III (NPGR 010) - J. Křivánek 2015
![Expansion of the rendering equation n If T is a contraction tj T Expansion of the rendering equation n If T is a contraction (tj. ||T|| <](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-33.jpg)
Expansion of the rendering equation n If T is a contraction (tj. ||T|| < 1, which holds for the RE), then n Solution of the rendering equation is then given by CG III (NPGR 010) - J. Křivánek 2015
![A different derivation of the Neumann series n Formal solution of the rendering equation A different derivation of the Neumann series n Formal solution of the rendering equation](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-34.jpg)
A different derivation of the Neumann series n Formal solution of the rendering equation n Proposition n Proof CG III (NPGR 010) - J. Křivánek 2015
![Rendering equation n Solution Neumann series CG III NPGR 010 J Křivánek 2015 Rendering equation n Solution: Neumann series CG III (NPGR 010) - J. Křivánek 2015](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-35.jpg)
Rendering equation n Solution: Neumann series CG III (NPGR 010) - J. Křivánek 2015
![Progressive approximation CG III NPGR 010 J Křivánek 2015 Progressive approximation CG III (NPGR 010) - J. Křivánek 2015](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-36.jpg)
Progressive approximation CG III (NPGR 010) - J. Křivánek 2015
![Progressive approximation n Each application of T corresponds to one step of reflection Progressive approximation n Each application of T corresponds to one step of reflection &](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-37.jpg)
Progressive approximation n Each application of T corresponds to one step of reflection & light propagation one-bounce indirect illumination emission direct illumination Open. GL shading CG III (NPGR 010) - J. Křivánek 2015 two-bounce indirect illumination
![Contractivity of T n Holds for all physically correct models q Follows from the Contractivity of T n Holds for all physically correct models q Follows from the](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-38.jpg)
Contractivity of T n Holds for all physically correct models q Follows from the conservation of energy n It means that repetitive application of the operator lower the remaining light energy (makes sense, since reflection/refraction cannot create energy) n Scenes with white or highly specular surfaces q q reflectivity close to 1 to achieve convergence, we need to simulate more bounces of light CG III (NPGR 010) - J. Křivánek 2015
![Alright so what have we achieved Rendering equation n Solution through the Neumann series Alright, so what have we achieved? Rendering equation n Solution through the Neumann series](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-39.jpg)
Alright, so what have we achieved? Rendering equation n Solution through the Neumann series We have replaced an integral equation by a sum of simple integrals Great we know how to calculate integrals numerically (the Monte Carlo method), which means that we know how to solve the RE, and that means that we can render images, yay! Recursive application to T corresponds to the recursive ray tracing from the camera CG III (NPGR 010) - J. Křivánek 2015
![What exact integral are we evaluating then CG III NPGR 010 J Křivánek What exact integral are we evaluating, then? CG III (NPGR 010) - J. Křivánek](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-40.jpg)
What exact integral are we evaluating, then? CG III (NPGR 010) - J. Křivánek 2015
![Paths vs recursion Same thing depends on how we look at it n Paths Paths vs. recursion: Same thing, depends on how we look at it n Paths](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-41.jpg)
Paths vs. recursion: Same thing, depends on how we look at it n Paths in a high-dimensional path space n Recursive solution of a series of nested (hemi)spherical integrals: CG III (NPGR 010) - J. Křivánek 2015
![Recursive interpretation n n Weve seen this already right But unlike at the beginning Recursive interpretation n n We’ve seen this already, right? But unlike at the beginning](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-42.jpg)
Recursive interpretation n n We’ve seen this already, right? But unlike at the beginning of the lecture, by now we know this actually solves the RE. Angular form of the RE To calculate L(x, wo) I need to calculate L(r(x, w’), -w’) for all directions w’ around the point x. For the calculation of each L(r(x, w’), -w’) I need to do the same thing recursively etc. r( r(x, w’) w’’ w’ wo x CG III (NPGR 010) - J. Křivánek 2015 r(x, w’)
![Path tracing v 0 1 recursive form get Li x ω y nearest Path tracing, v. 0. 1 (recursive form) get. Li (x, ω): y = nearest.](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-43.jpg)
Path tracing, v. 0. 1 (recursive form) get. Li (x, ω): y = nearest. Intersect (x, ω) return get. Le(y, –ω) + get. Lr (y, –ω) // emitted radiance // reflected radiance get. Lr(x, ωinc): [ωgen, pdfgen] = gen. Rnd. Dir. Brdf. Is(ωinc, normal(x) ) return 1/pdfgen * get. Li(x, ωgen) * brdf(x, ωinc, ωgen) * dot(normal(x), ωgen) CG III (NPGR 010) - J. Křivánek
![Path tracing v 2012 Arnold Renderer 2012 Columbia Pictures Industries Inc All Rights Path tracing, v. 2012, Arnold Renderer © 2012 Columbia Pictures Industries, Inc. All Rights](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-44.jpg)
Path tracing, v. 2012, Arnold Renderer © 2012 Columbia Pictures Industries, Inc. All Rights Reserved. CG III (NPGR 010) - J. Křivánek 2015
![](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-45.jpg)
![Path tracing Kajiya 86 n Only one secondary ray at each intersection q n Path tracing [Kajiya 86] n Only one secondary ray at each intersection q n](https://slidetodoc.com/presentation_image_h/7309dd970a8a35badcfd818debd0b9b8/image-46.jpg)
Path tracing [Kajiya 86] n Only one secondary ray at each intersection q n Direct illumination: two strategies q q n n Random selection of interaction (diffuse reflection, refraction, etc. , …) Hope that the generated secondary ray hits the light source, or Explicitly pick a point on the light source Trace hundreds of paths through each pixel and average the result Advantage over distributed ray tracing: now branching of the ray tree means no explosion of the number of rays with recursion depth CG III (NPGR 010) - J. Křivánek 2015
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