Importance Sampling Microfacet-Based BSDFs using the Distribution of Visible Normals Eric Heitz 1 & Eugene d’Eon 2 1 INRIA ; CNRS ; Univ. Grenoble Alpes 2 The Jig Lab
Introduction Dielectric material (e.g. water, plastic) 1
Introduction Rough dielectric material 2
Introduction Conductor material (e.g. metal) 3
Introduction Rough conductor material 4
Introduction Microfacet theory: statistical model of the rough interface and its interaction with light. → microfacet normals 5
Introduction State of the art in microfacet theory Walter et al., Microfacet Models for Refraction through Rough Surfaces , EGSR 2007 � Physical model → the equations. f r ( ω i , ω o ) = F ( ω i , ω hr ) G 2 ( ω i , ω o , ω hr ) D ( ω hr ) BRDF 4 | ω i · ω g || ω o · ω g | | ω i · ω ht || ω o · ω ht | n 2 0 (1 − F ( ω i , ω ht )) G 2 ( ω i , ω o , ω ht ) D ( ω ht ) BTDF f t ( ω i , ω o ) = | ω i · ω g || ω o · ω g | 2 ( n i ( ω i · ω ht ) + n o ( ω o · ω ht ) ) � Importance sampling → how to solve the equations. Where does Choose the light path a direction continue? at random. 6
Introduction Available in Mitsuba (Physically Based Renderer) Mitsuba’s documentation 7
Introduction Rendering with Mitsuba (path tracing+MIS) 32 spp 128 spp A dielectric glass plate ( n = 1.5 ) with anisotropic Beckmann roughness ( α x = 0.05 , α y = 0.4 ). 8
Introduction Rendering with Mitsuba (path tracing+MIS) 512 spp 4096 spp A dielectric glass plate ( n = 1.5 ) with anisotropic Beckmann roughness ( α x = 0.05 , α y = 0.4 ). 9
Introduction What if we had perfect BSDF importance sampling? Precomputed data (inverse CDF) 10
Introduction What if we had perfect BSDF importance sampling? Perfect importance sampling Previous with precomputed data (512 spp) (512 spp) � This is NOT what we propose to do. But this shows that there is clearly room for improvement. 11
Introduction OK, so why not just use perfect BSDF importance sampling? Precomputed data Precomputed data Precomputed data (inverse CDF) (inverse CDF) (inverse CDF) 12
Introduction Per-BSDF precomputed data? � OK for “toy-scenes” with a low number of BSDFs. only 1 BSDF in this scene � Not affordable for complex scenes with a huge number of different BSDFs. 13
Introduction Example: Textured Assets � Parametric BRDF models (roughness parameters α x , α y ) � Per-texel parameters � GigaBytes of textures Varying isotropic GGX Roughness map ( α ) ∞ number of BSDFs in this scene Courtesy of Christian Bense 14
Introduction Example: Textured Assets � Parametric BRDF models (roughness parameters α x , α y ) � Per-texel parameters � GigaBytes of textures Varying anisotropic Beckmann Roughness map ( α x , α y ) ∞ 2 number of BSDFs in this scene Courtesy of WETA Digital 14
Introduction We propose A better BSDF importance sampling scheme with lower variance, not perfect, but that can be used in production rendering. Fulfilled requirements � Parametric BSDF models used in production (typically Beckmann and GGX) � Varying anisotropic roughness parameters α x , α y � No assumption on the kind of integrator (bidir, MLT, etc.) � Plug & Play (trivial update of an existing implementation of the previous method) 15
Introduction Our approach We investigate the physical “meaning” of the BSDF model and use it to our advantage. f r ( ω i , ω o ) = F ( ω i , ω h r ) G 2 ( ω i , ω o , ω h r ) D ( ω h r ) 4 | ω i · ω g || ω o · ω g | n 2 f t ( ω i , ω o ) = | ω i · ω h t || ω o · ω h t | 0 (1 − F ( ω i , ω h t )) G 2 ( ω i , ω o , ω h t ) D ( ω h t ) � 2 | ω i · ω g || ω o · ω g | � n i ( ω i · ω h t ) + n o ( ω o · ω h t ) Looks complicated. But the model behind the equations is actually very simple and intuitive. 16
The Problem with the Previous Method 17
The Problem with the Previous Method How to sample outgoing directions? 18
The Problem with the Previous Method Available information: the distribution of normals. 18
The Problem with the Previous Method Use it to generate a random normal sample. 18
The Problem with the Previous Method Apply a transport operator on the normal (“reflect” or ”transmit”). 18
The Problem with the Previous Method Importance sampling is based on the distribution of normals. 19
The Problem with the Previous Method But in the physical model, the incident ray can only intersect normals that are visible . 19
The Problem with the Previous Method In the physical model, the transport operators are applied only on visible normals. 19
The Problem with the Previous Method This difference is what makes importance sampling inefficient. 20
The Problem with the Previous Method Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals. 20
The Problem with the Previous Method Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals. 20
The Problem with the Previous Method Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals. 20
The Problem with the Previous Method Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals. 20
The Problem with the Previous Method Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals. 20
The Problem with the Previous Method Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals. 20
The Problem with the Previous Method Problem 1: sampling space wasting. 21
The Problem with the Previous Method Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals. 22
The Problem with the Previous Method Samples are generated with the distribution of normals, but are weighted by the distribution of visible normals. 22
The Problem with the Previous Method Problem 2: high sampling weights. firefly artifacts → 23
Our Method: Main Idea 24
Our Method: Main Idea Importance sampling with the distribution of normals distribution of visible normals. 25
Our Method: Main Idea Our importance sampling method does exactly what is represented in this picture. 26
Our Method: Main Idea Only one component of the model is missing: most of the rays leave the surface... 27
Our Method: Main Idea ...but other are occluded (microsurface shadowing). 27
Our Method: Main Idea Previous weights Our weights masking*shadowing*projected area shadowing [0, ∞ [ [0,1] 28
Our Method: Main Idea Previous 512 spp (88.9s) Our 408 spp (87.1s) A dielectric glass plate ( n = 1.5 ) with anisotropic Beckmann roughness ( α x = 0.05 , α y = 0.4 ). same rendering time 29
Our Method: Main Idea Previous 256 spp (194.5s) Our 192 spp (198.9s) A rough dielectric with isotropic GGX roughness ( α = 0.05 ). same rendering time 30
Our Method: Main Idea Previous 32 spp (2.9s) Our 28 spp (2.8s) A rough conductor with isotropic GGX roughness ( α = 0.10 ). same rendering time 31
Our Method: Main Idea Previous 8192 spp (31.1min) Our 1024 spp (5.2m) A rough conductor with isotropic GGX roughness ( α = 0.15 ). 8 × less samples per pixel 6 × less rendering time 32
Our Method: in Practice 33
Our Method: in Practice The distribution of visible normals D ω i ( ω m ) = G 1 ( ω i , ω m ) 〈 ω i , ω m 〉 D ( ω m , α x , α y ) cos θ i � D ( ω m , α x , α y ) distribution of normals � ( α x , α y ) roughness parameters � G 1 ( ω i , ω m ) masking function � 〈 ω i , ω m 〉 cosine projection factor � Sampling this PDF is not simple! 34
Our Method: in Practice microfacet normals ⇔ microsurface slope 35
Our Method: in Practice scaling the roughness ⇔ stretching the microsurface 35
Our Method: in Practice The distribution of visible normals is stretch-invariant. 36
Our Method: in Practice If a sampling scheme is available for let’s say α x = 1 and α y = 1 ... ...it can be used for any other α x and α y . 37
Our Method: in Practice Example: we want to sample D ω i for α = 1 2 . 38
Our Method: in Practice First, we stretch into a configuration where α = 1 . 38
Our Method: in Practice Now, since α = 1 we know how to sample D ω i . 38
Our Method: in Practice Finally, we go back to the initial configuration. 38
Our Method: in Practice Implementing the sampling scheme for α = 1 ↓ 39
Our Method: in Practice Spherical coordinates ω m ⇔ ( θ m , φ m ) The PDF of θ m depends on θ i and φ m The PDF of φ m depends on θ i and θ m 2D PDFs difficult � 40
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