Importance Sampling Microfacet-Based BSDFs using the Distribution of Visible Normals

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Importance Sampling Microfacet-Based BSDFs using

the Distribution of Visible Normals

Eric Heitz, Eugene d’Eon

To cite this version:

Eric Heitz, Eugene d’Eon. Importance Sampling Microfacet-Based BSDFs using the Distribution of

Visible Normals. Computer Graphics Forum, 2014. �hal-00996995v1�

Eurographics Symposium on Rendering 2014

Wojciech Jarosz and Pieter Peers

(Guest Editors)

Volume 33 (2014), Number 4

Importance Sampling Microfacet-Based BSDFs using the

Distribution of Visible Normals

E. Heitz

and E. d’Eon

INRIA ; CNRS ; Univ. Grenoble Alpes

Jig Lab, Wellington, New Zealand

Previous: BSDF Importance sampling using the

distribution of normals. 64 spp (10.0s)

Our: BSDF Importance sampling using the

distribution of visible normals. 58 spp (10.4s)

Figure 1: A dielectric glass plate (n = 1.5) with anisotropic GGX roughness (α

= 0.05, α

= 0.4) on all faces (with the Smith

masking function). For a similar sample budget and the same render time, our method (right) signiﬁcantly reduces the variance

and converges faster than the common technique used in previous work (left).

Abstract

We present a new approach to microfacet-based BSDF importance sampling. Previously proposed sampling

schemes for popular analytic BSDFs typically begin by choosing a microfacet normal at random in a way that is

independent of direction of incident light. To sample the full BSDF using these normals requires arbitrarily large

sample weights leading to possible ﬁreﬂies. Additionally, at grazing angles nearly half of the sampled normals

face away from the incident ray and must be rejected, making the sampling scheme inefﬁcient. Instead, we show

how to use the distribution of visible normals directly to generate samples, where normals are weighted by their

projection factor toward the incident direction. In this way, no backfacing normals are sampled and the sample

weights contain only the shadowing factor of outgoing rays (and additionally a Fresnel term for conductors). Ar-

bitrarily large sample weights are avoided and variance is reduced. Since the BSDF depends on the microsurface

model, we describe our sampling algorithm for two models: the V-cavity and the Smith models. We demonstrate

results for both isotropic and anisotropic rough conductors and dielectrics with Beckmann and GGX distributions.

Categories and Subject Descriptors (according to ACM CCS): I.3.7 [Computer Graphics]: Three-Dimensional

Graphics and Realism—Keywords: Microfacet BSDF, Importance Sampling

 2014 The Author(s)

Computer Graphics Forum

 2014 The Eurographics Association and John

Wiley & Sons Ltd. Published by John Wiley & Sons Ltd.

E. Heitz & E. d’Eon / Importance Sampling Microfacet-Based BSDFs using the Distribution of Visible Normals

1. Introduction

Rendering realistic surfaces requires the use of complex ma-

terial models. The most common way to deﬁne surface re-

ﬂectance is the Bidirectional Scattering Distribution Func-

tions (BSDF), which incorporates light transports at a sur-

face’s interface such as reﬂection, described by a Bidi-

rectionnal Reﬂectance Distribution Functions (BRDF), or

transmission, described by a Bidirectional Transmittance

Distribution Functions (BTDF). Microfacet theory is the

most common theoretical framework for deriving analytic

physically-based BSDFs for rough surfaces. For Monte

Carlo rendering algorithms to make images efﬁciently using

BSDFs, an efﬁcient importance sampling must be provided.

In this paper we present a new strategy for deriving impor-

tance sampling for analytic microfacet BSDFs. In constrast

to previous approaches, we exploit the structure of the mi-

crofacet theory to directly sample from the distribution of

visible normals, which produces higher quality and lower

variance samples in comparison to previous approaches.

A key element of our approach is to move away from nor-

mal distributions directly and use, instead, the distribution

of slopes for the microsurface. In slope space, we note two

key invariance relationships that produce an optimal sam-

pling strategy for any surface roughness or anisotropy, pro-

vided the optimal strategy for an isotropic surface with unit

roughness is known.

In Section

2, we review the microfacet-based BSDF

model and we explain why the common associated impor-

tance sampling scheme is not optimal. In Section 3, we show

that importance sampling the BSDF with the distribution of

visible normals makes a better use of the sampling space

and produces sample weights in [0,1]. This method does

not create ﬁreﬂy artifacts and leads to a signiﬁcative vari-

ance reduction (Figure

2). To make this practical, in Sec-

tion 4, we show how to importance sample the distribution of

visible normals analytically with the V-cavity microsurface

model. In Section 5, we show how to importance sample the

distribution of visible normals with the Smith microsurface

model. We propose analytic sampling procedures for Beck-

mann and GGX distributions in the associated supplemen-

tal material. For other distributions, if an analytic sampling

procedure is not available we propose to use precomputed

data. Thanks to the shape invariance property of the dis-

tribution of visible slopes, we can do the precomputations

for only one BSDF with roughness α = 1 and we use the

same data with arbitrary roughness (α 6= 1) and anisotropy

(α

6= α

). In this way, we importance sample an entire class

of anisotropic parametric BSDFs with a small amount of pre-

computed data.

2. Background and Previous Work

2.1. Microfacet Theory

Microfacet theory was originally developed in the early work

of Torrance et al. [

TS67, CT82]. Several alternative models

our

Figure 2: An isotropic rough conductor with GGX distribu-

tion (α = 0.05) and the Smith masking function (64 samples

per pixel).

or approximations are built on top of this theory [

War92,

vGSK98, AS00, KSK01]. This paper is built on the top of

two major previous works:

• The extension of microfacet theory to transmittance by

Stam [

Sta01]. Closely related, Walter et al. [WMLT07]

describe BSDF importance sample strategies for different

distribution of normals.

• Heitz’ investigation of the properties of the masking-

shadowing function in microfacet-based BRDFs [Hei14].

He shows how the masking function is related to the mi-

crosurface model and to the correct normalization of the

BRDF.

In the following, we review microfacet theory as presented

in these two previous works.

Figure 3: The physically-based BSDF model is constructed

by applying the operators reﬂect and transmit on the distri-

bution of visible normals. The ratio of reﬂection and trans-

mission is given by the Fresnel term F.

 2014 The Author(s)

Computer Graphics Forum

 2014 The Eurographics Association and John Wiley & Sons Ltd.

E. Heitz & E. d’Eon / Importance Sampling Microfacet-Based BSDFs using the Distribution of Visible Normals

= (0,0,1) geometric normal

= (x

) microfacet normal

= (x

) incident direction

†

(ω

·ω

> 0)

= (x

) outgoing direction

D(ω

) distribution of normals

(ω

)

distribution of visible normals

(ω

,ω

) masking function

(ω

,ω

) masking-shadowing function

·ω

dot product

|ω

·ω

| absolute value of the dot product

hω

,ω

i clamped dot product

(a)

Heaviside function:

1 if a > 0 and 0 if a ≤ 0

Table 1: Notation.

The Distribution of Normals D describes the statistical

distribution of the normals ω

of which the microsurface

is made. The area of the microsurface projected onto the ge-

ometric normal ω

is the unit area of the geometric surface:

Ω

(ω

·ω

)D(ω

)dω

= 1. (1)

Parametric models typically specify roughness parameters

and α

in two orthogonal tangent directions (α

= α

for isotropic BSDFs). When these parameters need to be in-

cluded explicitly we write D(ω

,α

) instead of D(ω

The Distribution of Visible Normals D

describes the

statistical distribution of normals ω

visible from the inci-

dent direction

†

and is deﬁned by

(ω

) =

(ω

,ω

)|ω

·ω

|D(ω

)

|ω

·ω

, (2)

where the term |ω

·ω

| is the projected area of the normal

in direction ω

, |ω

·ω

| is the projected area of the ge-

ometric surface in direction ω

, and G

is the masking func-

tion. The relation between D and D

is illustrated in Fig-

ure

3. Heitz [Hei14] shows that in a mathematically well-

deﬁned BSDF model, the masking function G

should be

such that the distribution is normalized:

Ω

(ω

)dω

= 1.

Our approach is to use the knowledge of the BSDF model to

improve the sampling strategy. We can do it only for math-

ematically well-deﬁned models. Two common microsurface

models have masking functions G

that satisfy this criteria:

the V-cavity and the Smith models.

†

To ease the notation, we assume that the incident direction ω

located in the upper part of the hemisphere, i.e. ω

· ω

> 0.

The V-cavity Microsurface Model was introduced by Tor-

rance et al. [TS67, CT82]. It assumes a continuous distribu-

tion of separate microsurfaces rather than just one micro-

surface [Hei14]. Each microsurface is composed of two nor-

mals ω

= (x

) and ω

′

= (−x

,−y

) (see Fig-

ure

11 in Appendix B) and the contribution of each surface is

weighted by PDF (ω

·ω

)D(ω

) in the BSDF. The mask-

ing function of a V-cavity microsurface is given by

(ω,ω

) = χ



ω ·ω



min



1,2

|ω

·ω

||ω ·ω

|ω,ω



(3)

where the heaviside function ensures that backfacing mi-

crofacets are discarded for the correct side of the microsur-

face. The traditional masking-shadowing function accounts

for height correlation in the BRDF: ω

is in the upper hemi-

sphere (ω

·ω

> 0) and masking and shadowing occur on

the same side of the microfacet and are correlated. We gen-

eralize this to the BTDF: ω

is in the lower side of the mi-

crosurface and masking and shadowing occur on opposite

sides of the microfacet and are anticorrelated. The masking-

shadowing function is

(ω

,ω

) =



min(G

(ω

,ω

),G

(ω

,ω

)), if ω

·ω

> 0,

max(G

(ω

,ω

) + G

(ω

,ω

) −1,0), otherwise.

Kelemen et al. proposed a simpliﬁed version of this

masking-shadowing function to aid improve efﬁciency and

permit importance sampling [

KSK01]. However, their func-

tion does not satisfy Equation (2), which makes their model

physically ill-deﬁned. In Section 4, we will show that using

the V-cavitity function can actually lead to a simple impor-

tance sampling scheme, which we ﬁnd to be more efﬁcient

than the Kelemen model.

The Smith Microsurface Model. The Smith masking func-

tion was originally derived from a raytracing formula-

tion [

Smi67] for Beckmann distributions D and general-

ized to other distributions by Brown [Bro80]. Recently,

Heitz showed that the Smith masking function can be de-

rived from the equation of the conservation of the projected

area [Hei14]. The Smith microsurface model is the most ac-

curate model for compactly describing geometrical-optics

reﬂectance and transmission from explicit random height

ﬁelds (such as an ocean surface), and is better than the

V-cavity model for approximating reﬂectance of measured

material [Hei14] . However, speciﬁc masking functions G

must be derived for each new distribution of normals. An-

alytic solutions are available for Beckmann [

Smi67] and

GGX [WMLT07]. Heitz showed that the same formulas

can be used with the anisotropic extensions of D [Hei14].

Different forms of the Smith masking-shadowing function

are available, the most simple is the non-correlated form:

(ω

,ω

) = G

(ω

,ω

(ω

,ω

 2014 The Author(s)

Computer Graphics Forum

 2014 The Eurographics Association and John Wiley & Sons Ltd.

E. Heitz & E. d’Eon / Importance Sampling Microfacet-Based BSDFs using the Distribution of Visible Normals

The Scattering Model is given by the local illumination

equation:

L(ω

) =

Ω

L(ω

)|ω

·ω

| f (ω

,ω

)dω

, (4)

where f is the BSDF and L(ω

) is the radiance in

the outgoing direction. The construction of the BSDF

model is detailed by Walter et al. [

WMLT07] and is il-

lustrated in Figure 3(b): the distribution of visible nor-

mals is transformed into a distribution of outgoing di-

rections ω

by reﬂection and transmission. The BSDF

f (ω

,ω

) = f

(ω

,ω

) + f

(ω

,ω

) is the sum of the BRDF

(ω

,ω

) =

F(ω

,ω

(ω

,ω

)D(ω

)

4|ω

·ω

||ω

·ω

and the BTDF

(ω

,ω

) =

|ω

·ω

||ω

·ω

|ω

·ω

||ω

·ω

(1 −F(ω

,ω

))G

(ω

,ω

)D(ω

)

(ω

·ω

) + n

(ω

·ω

))

where ω

and ω

are the half vectors of reﬂection and

transmission respectively, and n

and n

are the indices of

refraction of the incident and transmitted sides respectively

(we refer the reader to [

WMLT07] for more details).

2.2. Importance Sampling the BSDF

Typically, importance sampling is used to solve Equa-

tion (4). The most common importance sampling strategy

for parametric BSDFs uses the distribution of normals to

generate the samples [War92,AS00,APS00,KSK01,Ash07,

WMLT07]. In this paper, we call it the “previous method”

and it is illustrated in Algorithm 1. In order to importance

sample the cosine weighted BSDF |ω

·ω

| f (ω

,ω

), pre-

vious algorithms start by generating normals ω

with the

PDF (ω

·ω

)D(ω

) rather than sampling outgoing direc-

tions ω

directly. Then, a light transport operator is applied

on the sampled normal to generate the outgoing direction.

The transport operators, reﬂect

= 2|ω

·ω

|ω

−ω

and transmit



nc −sign(ω

·ω

)

1 + n(c

−1)



−nω

with c = (ω

·ω

), and n =

are chosen randomly and with a probability depending on the

Fresnel term: F(ω

,ω

) is the reﬂection probability and 1−

F(ω

,ω

) is the transmission probability. The PDF resulting

from this procedure is the PDF used to generate the normals

multiplied by the Jacobian of the chosen operator:



∂ω



4|ω

·ω



∂ω



|ω

·ω

(ω

·ω

) + n

(ω

·ω

))

Since the same Jacobians are present in the sampling PDF

and in the BRDF/BTDF expressions, they simplify out in the

sample weight expression. Also, since reﬂection and trans-

mission are sampled randomly according to the Fresnel term,

the value of Fresnel is present in the sampling PDF as well as

in the BRDF/BTDF, and simpliﬁes out in the sample weight

expression. In both cases the ﬁnal expression for the sample

weight reduces to

weight(ω

) =

|ω

·ω

(ω

,ω

)

|ω

·ω

||ω

·ω

Discussion This sampling algorithm is almost perfect at

normal incidence but we can see in Figure

4 that it has

two main problems at grazing angles. The ﬁrst problem is

that up to half of the generated normals ω

are backfacing

(ω

·ω

< 0). The associated weights are 0 and so half of

the sample space is wasted. The second problem at graz-

ing angles is that the sample weights can be arbitrarily high

with the Smith model leading to unwanted variance, espe-

cially for paths with multiple surface interactions (though

the sample weights are at most 2 with the V-cavity model).

This is responsible for the ﬁreﬂy artifacts in Figure

2. Wal-

ter et al. [WMLT07] notice this problem and propose using a

BSDF with a slightly different roughness α

′

for sampling the

NDF at grazing angles: α

′

= (1.2−0.2

|ω

·ω

|α). This is

denoted as “Walter’s trick” in Figure 4. Walter’s trick signi-

ﬁcatively reduces the maximal weight with Beckmann dis-

tributions but does not help much with GGX (in our experi-

ence, despite trying offset values greater than 1.2) and also

does not alleviate the problem of wasted samples.

Algorithm 1 Sample ω

(previous)

← Sample (ω

·ω

)D(ω

)

if U < F(ω

,ω

) then

← reﬂect(ω

,ω

)

else

← transmit(ω

,ω

)

end if

weight(ω

) ←

|ω

·ω

(ω

,ω

)

|ω

·ω

| |ω

·ω

Algorithm 2 Sample ω

(our)

← Sample D

(ω

) ⊲ Algorithm

3 or 4

if U < F(ω

,ω

) then

← reﬂect(ω

,ω

)

else

← transmit(ω

,ω

)

end if

weight(ω

) ←

(ω

,ω

)

(ω

,ω

)

 2014 The Author(s)

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 2014 The Eurographics Association and John Wiley & Sons Ltd.

E. Heitz & E. d’Eon / Importance Sampling Microfacet-Based BSDFs using the Distribution of Visible Normals

previous our

PDF(ω

) (ω

·ω

)D(ω

) D

(ω

)

weight(ω

)

|ω

·ω

| G

(ω

,ω

)

|ω

·ω

||ω

·ω

(ω

,ω

)

(ω

,ω

)

V-cavities G weight(ω

) ∈ [0,2] weight(ω

) ∈ [0,1]

Smith G weight(ω

) ∈ [0,∞[ weight(ω

) ∈ [0,1]

Table 2: Properties of the importance sampling techniques.

(ω

·ω

)D(ω

)

(Walter’s trick)

our

(ω

)

V-cavity Beckmann (α = 0.3)

Smith Beckmann (α = 0.3)

Smith GGX (α = 0.1)

Figure 4: The sampling weights with different PDFs

at grazing angle (θ

= 1.5). The sampling space is

parametrized by the two random numbers (U

) used to

generate ω

. Our sampling scheme completely ﬁlls the sam-

ple space with weights very near 1.0. (The plot of our method

with the V-cavity model is noisy because we use a supple-

mental random number U

, see Section

4).

3. Our Importance Sampling Scheme

The core idea behind our new sampling scheme is to use

the distribution of visible normals D

as the sampling PDF

rather than the view-independent PDF (ω

·ω

)D(ω

), as

shown in Algorithms

2. The differences with the previous

method are highlighted in red. By doing this we are directly

simulating what is illustrated in Figure 3(b): an incident ray

intersects a random normal chosen within the distribution

of visible normals for this ray and is reﬂected or transmitted.

Only shadowing needs to be added, since it does not directly

emerge from this model but is there to enforce single

scattering on the microsurface [

Hei14]. The value of the

weight

(ω

,ω

)

(ω

,ω

)

is the amount of visible shadowing.

Note that when masking and shadowing are not cor-

related, i.e. G

(ω

,ω

) = G

(ω

,ω

(ω

,ω

then the weight comes down to shadowing alone

(ω

,ω

)/G

(ω

,ω

) = G

(ω

,ω

). This makes

sense, because if masking and shadowing are not correlated,

then the visible shadowing is simply the shadowing.

Table

2 compares the previous importance sampling

method with our method for the V-cavity and the Smith mi-

crosurface models. Since G

≤ G

the weight in our method

is always less than 1: ﬁreﬂy artifacts do not appear (see Fig-

ure 2). This can be observed in Figure 4. We also see that

our method does not waste the sampling space like previ-

ous methods. This is because we never generate backfacing

normals.

Sections

4 and 5 are dedicated to implement method

Sample D

(ω

) for the V-cavity and the Smith microsur-

face models, respectively.

4. Sampling D

(ω

) with the V-cavity Model

Importance sampling D

(ω

) with the V-cavity model is

summarized in Algorithm

3. We start by sampling from the

PDF (ω

·ω

)D(ω

) as in the previous method. This gives

us a random microsurface with dual normals ω

and ω

′

(see Figure

11). Then, we choose randomly between ω

′

using selection probabilities proportional to hω

,ω

and hω

,ω

′

i respectively, which correspond to their pro-

jected areas toward ω

. This random choice, weighted by the

projected area, is what transforms the PDF (ω

·ω

)D(ω

)

into D

(ω

). In practice, we implement this by a random

swap of ω

by ω

′

with probability

hω

,ω

′

hω

,ω

i+hω

,ω

′

Algorithm 3 Sampling D

(ω

) (V-cavity model)

← Sample (ω

·ω

)D(ω

) ⊲ 1. choose microsurface

if U <

hω

,ω

′

hω

,ω

i+hω

,ω

′

then ⊲ 2. choose normal

← ω

′

end if

Figure 5: Illustration of Algorithm 3.

Figure

5 illustrates how the algorithm works. When the

incident ray approaches the geometric surface, it starts

by choosing a V-cavitity microsurface. Since the V-cavity

model assumes separate microsurfaces, they have all the

same visibility and can be chosen with the view-independent

PDF (ω

·ω

)D(ω

). Then, on the chosen V-cavity micro-

surface, the ray must choose between ω

and ω

′

with a

probability weighted by their respective visibilities. We in-

corporate this second choice in the ﬁnal PDF with a random

swap between the two normals. Thus, Our algorithm requires

an additional random number U. The formal proof for this

algorithm is given in Appendix

 2014 The Author(s)

Computer Graphics Forum

 2014 The Eurographics Association and John Wiley & Sons Ltd.

E. Heitz & E. d’Eon / Importance Sampling Microfacet-Based BSDFs using the Distribution of Visible Normals

5. Sampling D

(ω

) with the Smith Model

5.1 we show that we can sample from the distribution of

visible normals D

by sampling from the distribution of vis-

ible slopes P

. Then, in 5.2, we will show that the shape of

is stretch invariant. In practice, this means that a single

analytic or tabulated inversion can be used with any rough-

ness and anisotropy. These computations are detailed in

5.3.

The complete algorithm we use to sample D

is summa-

rized and discussed in

5.4.

5.1. Formulation in Slope Space

In Equation (2) we have already deﬁned D

, but in this sec-

tion we use an alternative equivalent formulation:

(ω

) =

hω

,ω

iD(ω

)

Ω

hω

,ω

iD(ω

)dω

hω

,ω

iD(ω

), (5)

where c =

Ω

hω

,ω

iD(ω

)dω

is the normalization fac-

tor of the PDF. This normalization factor does not appear in

Equation (

2) but is implicitly contained in the masking func-

tion G

[Hei14]. In the following, we use Heitz’s [Hei14]

formulation of Equation (5) in slope space. We recall that

˜m = (x

˜m

) = (−

,−

) are the slopes associated to the

normal ω

= (x

) and reciprocally ω

(−x

˜m

,−y

˜m

,1)

√

˜m

If the microsurface is a heightﬁeld, the distribution of nor-

mals is linked to the distribution of slopes P

˜m

) by

(ω

·ω

)D(ω

) =



∂ ˜m



( ˜m = (x

˜m

)),

where



∂ ˜m



(ω

·ω

)

is the Jacobian of the slope to nor-

mal transformation. The distributions of visible normals and

of visible slopes are linked in the same way by

(ω

) =



∂ ˜m

∂ω



( ˜m = (x

˜m

)). (6)

In slope space, Equation (

5) is written:

˜m

) =

˜m

+ y

˜m

+ z

)(x

˜m

+ y

˜m

+ z

˜m

), (7)

where the normalization factor in slope space is:

c =

R R

˜m

+ y

˜m

+ z

)(x

˜m

+ y

˜m

+ z

˜m

)dx

˜m

→ If we sample P

and transform the output slope into a

normal ω

, the resulting PDF is D

. This is the ﬁrst idea of

Algorithm

5.2. Stretch Invariance

Stretching a microsurface is illustrated in Figure

6. The

slopes of the microsurface are divided by the stretching co-

efﬁcients: (x

˜m

) becomes (

˜m

). The stretching opera-

tor S

,λ

(−) that transforms the normalized incident vector

= (x

) into another normalized incident vector

‡

deﬁned as

,λ

(ω

) =

(λ

,λ

)

(λ

)

+ (λ

)

+ z

Heitz [

Hei14] calls P

shape-invariant if it has the same

shape after stretching, i.e. if for any λ

,λ

> 0

˜m

,α

) =



˜m



, (8)

where P

˜m

) is written P

˜m

,α

) to specify

the roughness parameters explicitly. This is usually the case

when the distribution comes from a distribution of slopes.

Heitz showed that if P

is shape-invariant, then the mask-

ing function G

is stretch-invariant. We introduce a stronger

property: if the distribution of slopes P

is shape-invariant,

then the distribution of visible slopes P

is also shape-

invariant. This is written

˜m

,α

) =

,λ

(ω

)



˜m



(9)

and is illustrated in Figure

7. The formal proof of this re-

sult is provided in Appendix A. A practical consequence of

Equation (9) is that the distribution of visible slopes can al-

ways be expressed with an isotropic roughness α (we choose

arbitrarily α = 1):

˜m

,α

) =

,α

(ω

)



˜m

,1,1



The factor

in the right-hand side is the Jacobian of the

transformation from



˜m



to (x

˜m

→ If a sampling algorithm is available for (

˜m

)

with PDF P

,α

(ω

)

(

˜m

,1,1), we can use it to sample

˜m

) with PDF P

˜m

,α

). This is the second

idea of Algorithm

We note that a similar observation was made by Becker

and Max [

BM93]: they show that the distribution of visi-

ble normals is invariant to height-stretching, which is equiv-

alent to scaling the slope amplitude k ˜mk, where the slope

coordinates x

˜m

and y

˜m

are scaled by the same factor. This al-

lows for isotropic roughness changes only. Instead of height-

stretching we use slope-stretching, where x

˜m

and y

˜m

can be

scaled separately, which permits fully anisotropic roughness

changes.

‡

Note that the normals are not modiﬁed in the same way because

they are not vectors but covectors. The normals after stretching are

given by the new slopes: ω

(−

˜m

,−

˜m

,1)



˜m





˜m



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E. Heitz & E. d’Eon / Importance Sampling Microfacet-Based BSDFs using the Distribution of Visible Normals

Figure 6: Stretching the conﬁguration.

˜m

) × stretching ∝ P

˜m

,1,1)

˜m

) × stretching ∝P

(ω

)

˜m

,1,1)

Figure 7: Stretch invariance in slope space of P

and P

(Beckmann distribution).

5.3. Sampling P

˜m

,1,1)

After stretching the conﬁguration, the distribution of visible

slopes is P

˜m

,1,1). In order to sample (x

˜m

) with

PDF P

˜m

,1,1), we ﬁrst assume that the frame is such

as ω

= (x

,0,z

). This is possible since we are now using an

isotropic distribution of normals (α

,α

= 1). The expres-

sion of Equation (

7) simpliﬁes to

˜m

,1,1) =

˜m

+ z

)(x

˜m

+ z

˜m

This expression emphasizes that the projection factor

˜m

+ z

)(x

˜m

+ z

) is not related to the slope compo-

nent y

˜m

. We use this property to sample P

in two steps a.

and b.

5.3.1. a. Sampling P

2−

˜m

,1,1)

We start by sampling the marginalized random variable x

˜m

given by the PDF

2−

˜m

,1,1)

+∞

−∞

˜m

+ z

)(x

˜m

+ z

˜m

,1,1) dy

˜m

+ z

)(x

˜m

+ z

)

+∞

−∞

˜m

,1,1) dy

˜m

+ z

)(x

˜m

+ z

2−

˜m

,1,1), (10)

where P

2−

˜m

,1,1) is the marginalized PDF of x

˜m

. We in-

vert

˜m

−∞

2−

′

˜m

,1,1) dx

′

˜m

⇒ x

˜m

= C

2−

−1

,1,1), (11)

where C

2−

is the cumulative distribution fonction (CDF) as-

sociated to the marginalized PDF P

2−

This equation can be solved analytically for Beckmann

and GGX distributions (see our supplemental material). For

other distributions, if an analytic solution is not available,

we propose to precompute the solutions and store them in a

2D table x

˜m

= T

˜m

[θ

]. Indeed, x

˜m

depends only on two

variables: θ

(because x

= sinθ

and z

= cosθ

) and U

5.3.2. b. Sampling P

2|2

˜m

,1,1)

Then, we sample the conditional random variable y

˜m

given by the PDF

2|2

˜m

,1,1)

˜m

+ z

)(x

˜m

+ z

˜m

,1,1)

+∞

−∞

˜m

+ z

)(x

˜m

+ z

˜m

′

˜m

,1,1) dy

′

˜m

,1,1)

+∞

−∞

˜m

′

˜m

,1,1) dy

′

˜m

= P

2|2

˜m

,1,1). (12)

This shows that the sampling of y

˜m

does not depend on the

incident direction ω

. Hence, we invert

˜m

−∞

2|2

′

˜m

,1,1) dy

′

˜m

⇒ y

˜m

= C

2|2

−1

˜m

,1,1), (13)

where C

2|2

˜m

,1,1) the CDF associated to the view-

independent conditional PDF P

2|2

This equation can be solved analytically for Beckmann

and GGX distributions (see our supplemental material). For

other distributions, if an analytic solution is not available, we

propose to precompute the solutions and store them in a 2D

table y

˜m

= T

˜m

]. Indeed, y

˜m

depends only on two

variables: x

˜m

and U

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E. Heitz & E. d’Eon / Importance Sampling Microfacet-Based BSDFs using the Distribution of Visible Normals

5.4. Sampling D

(ω

)

In Algorithm

4, we sample (x

˜m

) with PDF

,α

(ω

)

˜m

,1,1) by sampling x

˜m

with marginal-

ized PDF P

2−

˜m

,1,1) and y

˜m

with conditional

PDF P

2|2

˜m

,1,1). Then, we transform this

slope vector back to the canonical frame where

= (cosφ

sinθ

,sinφ

sinθ

,cosθ

) by applying a ro-

tation of angle φ

and we unstretch the slopes by multiplying

by respectively α

and α

. The sampled PDF at this point is

˜m

,α

). Finally, we transform the slopes into a

normal. The sampled PDF is D

(ω

,α

Algorithm 4 Sample D

(ω

) (Smith proﬁle)

′

← S

,α

(ω

) ⊲ 1. stretch

˜m

←C

2−

′

−1

,1,1) ⊲ 2.a. sample P

2−

′

˜m

,1, 1)

˜m

←C

2|2

−1

˜m

,1,1) ⊲ 2.b. sample P

2|2

˜m

,1, 1)



˜m



←



cosφ

′

−sinφ

′

sinφ

′

cosφ

′



˜m



⊲ 3. rotate



˜m



←



˜m



⊲ 4. unstretch

←

(−x

˜m

,−y

˜m

,1)

√

˜m

⊲ 5. compute normal

Figure 8: Illustration of Algorithm 4.

Analytic Computation of C

2−

−1

and C

2|2

−1

The corner-

stone of Algorithm

4 is the inversion of the CDFs C

2−

and

2|2

. We found this to be possible analytically for Beckmann

and GGX distributions, for which we provide the detailed

derivations of the inversion procedures and their C++ imple-

mentations in the associated supplemental material.

Precomputation of C

2−

−1

and C

2|2

−1

If an analytic in-

version procedure of the CDFs C

2−

and C

2|2

is not avail-

able, we propose to use precomputed data T

˜m

[θ

] and

˜m

] parametrized in slope space. One advantage of

the slope space is that the equation of P

becomes sepa-

rable: χ

˜m

+ z

)(x

˜m

+ z

) does not depend on y

and

thus can be pulled out one of the two integrals. This prop-

erty is used in Equations (

10) and (12) and makes the pre-

computations easier than they would have been with spheri-

cal coordinates. The problem of using spherical coordinates

was discussed by Becker and Max [BM93]: they store ta-

bles parametrized in (θ,φ) and they either assume that θ and

φ can be sampled independently, which is approximate and

leads to biased sampling, or they need to store a 3D table

to sample θ for a given φ (or the reciprocal). This would

require an enormous 3D table and would be impractical

(>1GB). However, in slope space, Equation (12) shows that

2|2

˜m

,1,1) = P

2|2

˜m

,1,1) and the inversion does

not depend on θ

, which allows us to use a 2D precomputed

table indexed by x

˜m

and U

. Thus, we need to precompute

only two 2D tables T

˜m

[θ

] and T

˜m

]. More im-

portantly, thanks to the invariance property in slope space,

we can use the same data with different roughnesses and

anisotropy. In our implementation, T

˜m

and T

˜m

are both of

size 1024

. Popular heavy-tailed distributions of slopes like

GGX have very long tails but also a very sharp peak at the

same time, and so require a 1024

table to capture them well.

The total memory footprint for every parametric anisotropic

BSDF with the same shape is then 32MB.

Limitation Note that this technique works only when the

distribution of normals D is built from a shape-invariant dis-

tribution of slopes P

. For instance, this is the case for

the Beckmann [

Smi67] and GGX [WMLT07], but not for

the Phong distribution (which lacks a Smith masking func-

tion [

WMLT07]).

6. Results and Validation

In Figure 1 we show a dielectric glass plate with anisotropic

GGX roughness (Smith masking) on all faces. The illumi-

nant is constant uniform distant white radiance from all di-

rections (a white environment map). In all renders we use

forward path tracing with unbounded path lengths. Multi-

ple importance sampling (MIS) mixes BSDF and next-event-

estimation (light sampling) at each path vertex. Figure 1 uses

64 independent samples per pixel. Our new BSDF sampling

scheme runs in roughly the same time as prior work and pro-

vides a signiﬁcant reduction in variance. Figure

9 shows a

conﬁguration where ﬁreﬂies are still present with the previ-

ous method even with 4000 samples per pixel. Due to the de-

fault Gaussian ﬁlter in Mitsuba, a single bright sample may

contribute signiﬁcantly to several pixels (appearing as a 2x2

bright pixel block).

Figure 10 compares the convergence of importance sam-

pling with the previous method to our method in a conﬁg-

uration shown in Figure 2, i.e. a rough conductor lit by an

environment map. The plot shows that both methods con-

verge to the same value, which was expected because the

previous method and our method are both mathematically

well-deﬁned. However, the convergence rate is not the same.

In the two ﬁrst rows we used the Smith model. We see

that the convergence rate is very sensitive to the fact that

the sampling weight can be arbitrary high with the previous

method. This produces ﬁreﬂy artifacts that take a long time

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E. Heitz & E. d’Eon / Importance Sampling Microfacet-Based BSDFs using the Distribution of Visible Normals

to disappear. In contrary, our method does not produce these

artifacts since all BSDF sample weights are bounded to 1.

In the two last rows we compare the methods used with the

V-cavity model. While the previous method does not suffer

from very high sampling weights (they are bounded to 2),

the convergence of our method is faster. This is because we

do not waste the sampling space by generating backfacing

normals with weight 0.

More results and more convergence comparisons are

available in the associated supplemental material.

Previous (Walter’s trick) Our

4000spp, 12min 3844spp, 12min

Figure 9: A dielectric glass plate (n = 1.5) with isotropic

Beckmann roughness (α = 0.1) on all faces (with the Smith

masking function). Fireﬂies are persistent with the previ-

ous method even with a large number of samples (4000 per

pixel).

7. Conclusion

We have presented a new importance sampling scheme for

microfacet-based BSDFs. The scope of our scheme is broad,

encompassing many popular isotropic and anisotropic ana-

lytic BRDF and BSDF models and allows a variety of nor-

mal distribution functions. The performance of our scheme

is comparable to previous methods and is suitable for both

ofﬂine and GPU rendering. We have shown that importance

sampling the BSDF using the distribution of visible normals

rather than the distribution of normals is practical and makes

convergence faster, signiﬁcantly reducing artifacts at grazing

incidence angles, especially for signiﬁcant roughness levels.

With the V-cavity microsurface model, this is implemented

with a simple one conditional swap of the normal. With the

much more accurate Smith microsurface model, we provide

analytic sampling algorithms for Beckmann and GGX dis-

tributions in our supplemental material. Otherwise, we have

shown that a small set of precomputed data can be used to

sample BSDFs with any roughness and anisotropy, due to

key shape invariance properties of the distribution of visible

slopes.

Importance Sampling - Smith GGX (α = 0.05)

our

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

10 100 1000

number of samples

our

Multiple Importance Sampling - Smith GGX (α = 0.05)

our

0.05

0.1

0.15

0.2

0.25

10 100 1000

number of samples

our

Importance Sampling - V-cavity GGX (α = 0.05)

our

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

10 100 1000

number of samples

our

Multiple Importance Sampling - V-cavity GGX (α = 0.05)

our

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

10 100 1000

number of samples

our

Figure 10: Comparisons of our importance sampling tech-

nique to previous methods (with and without MIS). The im-

ages are computed with 8 samples per pixel. The plots shows

the convergence of several running estimates of the same

pixel of the pedestal for both techniques. The pixel variance

is seen directly in the spread of the various estimates.

References

[APS00] ASHIKMIN M., PREMOŽE S., SHIRLEY P.: A

microfacet-based brdf generator. In Proceedings of the 27th

annual conference on Computer graphics and interactive tech-

niques (2000), SIGGRAPH ’00, pp. 65–74.

[AS00] ASHIKHMIN M., SHIRLEY P.: An anisotropic phong brdf

model. Journal of Graphics Tools 5 (2000), 25–32.

2, 4

[Ash07] ASHIKHMIN M.: Distribution-based BRDFs. Tech. rep.,

2007.

[BM93] BECKER B. G., MAX N. L.: Smooth transitions between

bump rendering algorithms. In Proceedings of the 20th Annual

Conference on Computer Graphics and Interactive Techniques

(1993), SIGGRAPH ’93, pp. 183–190.

6, 8

[Bro80] BROWN G.: Shadowing by non-gaussian random sur-

faces. IEEE Trans. on Antennas and Propagation 28, 6 (Nov

1980), 788–790.

[CT82] COOK R. L., TORRANCE K. E.: A reﬂectance model for

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Computer Graphics Forum

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computer graphics. ACM Trans. Graph. 1, 1 (Jan. 1982), 7–24.

2, 3

[Hei14] HEITZ E.: Understanding the masking-shadowing func-

tion in microfacet-based brdfs. Journal of Computer Graphics

Techniques (JCGT) 3, 2 (2014). To appear.

2, 3, 5, 6

[KSK01] KELE MEN C., SZI RMAY-KALOS L.: A microfacet

based coupled specular-matte brdf model with importance sam-

pling. In Eurographics Short Presentations (2001).

2, 3, 4

[Smi67] SMITH B.: Geometrical shadowing of a random rough

surface. IEEE Trans. on Antennas and Propagation 15 (1967),

668–671.

3, 8

[Sta01] STA M J.: An illumination model for a skin layer bounded

by rough surfaces. In Rendering Techniques (2001), pp. 39–52.

[TS67] TORRANCE K. E., SPARROW E. M. : Theory for off-

specular reﬂection from roughened surfaces. J. Opt. Soc. Am. 57,

9 (Sep 1967), 1105–1112.

2, 3

[vGSK98] VAN GINNE KEN B., STAVRI DI M., KOENDERINK

J. J.: Diffuse and specular reﬂectance from rough surfaces. Jour-

nal of Appl. Opt. 37, 1 (Jan 1998), 130–139.

[War92] WARD G . J.: Measuring and modeling anisotropic re-

ﬂection. SIGGRAPH Comput. Graph. 26, 2 (July 1992), 265–

272.

2, 4

[WMLT07] WALTER B., MARSCHNER S. R., LI H., TORRANCE

K. E.: Microfacet models for refraction through rough sur-

faces. In Proc. Eurographics Symposium on Rendering (2007),

EGSR’07, pp. 195–206.

2, 3, 4, 8

Appendix A: Shape Invariance of P

We demonstrate the result of Equation (

9):

˜m

,α

)

(−x

˜m

−y

˜m

+ z

)(−x

˜m

−y

˜m

+ z

˜m

,α

)

Z Z

(−x

˜m

−y

˜m

+ z

)(−x

˜m

−y

˜m

+ z

˜m

,α

)dx

˜m

(−λ

˜m

−λ

˜m

) (−λ

˜m

−λ

˜m

)

√

(λ

)

+(α

)

(

˜m

)

Z Z

(−λ

˜m

−λ

˜m

+ z

)(−λ

˜m

−λ

˜m

+ z

)

(λ

)

+ (λ

)

+ z

(

˜m

)dx

˜m

(−λ

˜m

−λ

˜m

) (−λ

˜m

−λ

˜m

)

√

(λ

)

+(α

)

(

˜m

)

Z Z

(−λ

˜m

−λ

˜m

+ z

)(−λ

˜m

−λ

˜m

+ z

)

(λ

)

+ (λ

)

+ z

(

˜m

,λ

(ω

)



˜m



where we start to write Equation (

7). In the second step we

develop by applying Equation (8) and by multiplying the

numerator and the denominator by

√

(λ

)

+(α

)

. In

the thrid step, at the denominator we make the change of

variable:

˜m

= d

˜m

Appendix B: Proof for Algorithm

To verify that Algorithm 3 samples PDF D

(ω

), we will

expand G

from Equation (

3) in the deﬁnition of D

(ω

)

from Equation (

2). The min in Equation (3) is used to differ-

entiate between 2 cases illustrated in Figure

11.

Case (a) If ω

′

is backfacing, then the shadowing occuring

on ω

is G

(ω

,ω

) = 2

(ω

·ω

) (ω

·ω

)

hω

,ω

, and by using the fact

that hω

,ω

′

i = 0 and |ω

·ω

| > 0 we get

(ω

) = 2

|ω

·ω

||ω

·ω

|ω

,ω

|ω

,ω

|D(ω

)

|ω

·ω

= 2

|ω

·ω

|ω

·ω

|ω

·ω

|D(ω

)

= 2

hω

,ω

hω

,ω

i+ 0

|ω

·ω

|D(ω

)

= 2

hω

,ω

hω

,ω

i+ hω

,ω

′

(ω

·ω

)D(ω

Case (b) If ω

and ω

′

are both not backfacing for ω

, then

there is no masking (G

= 1) and Equation (

2) becomes

(ω

) =

hω

,ω

iD(ω

)

|ω

·ω

= 2

hω

,ω

hω

,ω

i+ hω

,ω

′

(ω

·ω

)D(ω

because |ω

·ω

| =

hω

,ω

i+hω

,ω

′

2hω

,ω

Conclusion In either case the distribution of visible

normals is given by the same expression. The factor

hω

,ω

hω

,ω

i+hω

,ω

′

represents the repartition of visibility be-

tween ω

and ω

′

. In Algorithm

3, the conditional re-

placement of ω

by ω

′

multiplies the original PDF (ω

)D(ω

) by the factor

hω

,ω

hω

,ω

i+hω

,ω

′

, and the factor 2 is

present in the sampled PDF because ω

can be chosen for 2

microsurfaces (the one associated to himself and the one to

′

) and reciprocally for ω

′

Figure 11: Masking conﬁgurations with the V-cavity model.

Either one microfacet is backfacing and completely masked

(a) or the two microfacets are completely visible (b).

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