Volume 41 (2022), Number 4

Eurographics Symposium on Rendering 2022
A. Ghosh and L.-Y. Wei(Guest Editors)

Microsurface Transformations
A. Atanasov1,2

V. Koylazov1

R. Dimov1

A. Wilkie2

1 Chaos Software, Bulgaria
2 Charles University, Czech Republic

1 0
0 1

!

1 2
0 1

!

5 0
0 1

!




1
0


10u4 1Figure 1: A demonstration how different 2D transformations (top right insets) of the microsurface affect the reflection off a rough silver
plate. Top left: The original unmodified surface with shape-variant GTR distribution with tail γ = 1.7 and roughness α = 0.0121. A 10 × 10
grid covers the unit texture space to serve as a reference. Top right: The microsurface is stretched 5 times horizontally. This makes the surface
smoother in that direction, so the highlight is compressed accordingly. Bottom left: A skew transform M with (a, b, c, d) = (1, 0, 2, 1), see
Equation (9). Bottom right: An example of a non-linear transformation: T (u, v) = (u, v + 2u5 ). The Jacobian matrix JT (top right inset) isa local linear approximation of T and we apply it as a local tangential transform in Equation (9). Notice that the prolonged shape of thehighlight to the right is due to more surface compression in that region.
Abstract
We derive a general result in microfacet theory: given an arbitrary microsurface defined via standard microfacet statistics, weshow how to construct the statistics of its linearly transformed counterparts. A common use case of such transformations is
to generate anisotropic versions of a given surface. Traditional anisotropic derivations based on varying the roughness of anisotropic distribution in an ellipse have a general closed-form formula only for the subclass of shape-invariant distributions.
While our formulation is equivalent to these specific constructs, it is more general in two aspects: it leads to simple closedform solutions for all distributions, including shape-variant ones, and works for all invertible 2D transform matrices. The
latter is of particular importance in case of deformation of the macrosurface, since it can be approximated locally by a linear
transformation in the tangent plane. We demonstrate our results using the Generalized Trowbridge-Reitz (GTR) distribution
which is shape-invariant only in the special case of the popular Trowbridge-Reitz (GGX) distribution.
CCS Concepts• Computing methodologies → Rendering; Reflectance modeling;© 2022 The Author(s)Computer Graphics Forum © 2022 The Eurographics Association and John
Wiley & Sons Ltd. Published by John Wiley & Sons Ltd.

A. Atanasov, V. Koylazov, R. Dimov & A. Wilkie / Microsurface Transformations

1. Introduction
Microfacet theory is based on geometric optics [BS63; TS67]
and it has been adopted in computer graphics as a standard technique [Bli77; CT82; WMLT07]. It captures the statistical behavior of rough surfaces: microfacet surfaces (microsurfaces) arerepresented by a microfacet distribution function and a propershadowing term is necessary for physically-correct microsurface
configurations [Smi67; APS00; Hei14]. A practically viable andphysically-based microfacet model requires both analytic formulas for evaluating and sampling the microfacet distribution, anda simple and accurate formula for the shadowing term. Sampling
equations for the distribution of visible normals [HD14] are also
highly desirable. Unfortunately, they are available only for the twomost popular microfacet distributions [RBSM19] - the Beckmann
distribution [BS63] and the Trowbridge-Reitz distribution [TR75].
The latter distribution was popularized by Walter et al. [WMLT07]
with the alias "GGX". The important effect of multiple scatteringin microfacet surfaces has been addressed [HHdD16], but this isout of the scope of our work. We restrict our results to the first scattering event. Also, our work is based entirely in geometric optics,
and diffraction effects are ignored [HP17]. Lastly, a model that canbe extended to an anisotropic microfacet distribution is valuable inpractical applications, as this functionality adds substantial additional control over object appearance, besides the basic parametersof the microfacet distribution function: many real-world objects exhibit anisotropic reflections [Kaj85], so models that are capable ofdescribing their appearance are needed.
The current state of the art for creating analytic anisotropic distributions from existing analytic isotropic ones is based on varying the roughness of the isotropic distribution in an ellipse. Thisidea can be traced back at least to the elliptical Gaussian model ofWard [War92]. It was later adapted to microfacet models [APS00;
KSK10] and has been used since then in modern rendering systems [NVZJ19; PJH16; Bur12]. Heitz defined the class of shapeinvariant isotropic distributions for which this construction is always possible [Hei14]. Specifically, this class of distributions has
considerable practical advantages:• Derivation of normalized anisotropic distributions by applying different roughness values in x and y direction. For shapeinvariant distributions this is equivalent to a non-uniform scalingof the distribution.
• Extension of the 1D isotropic shadowing function to anisotropic
without increasing its dimension.
• Extension of the isotropic sampling equations (including distribution of visible normals if available) to anisotropy.
Our work generalizes all of these advantages to all microfacet distributions and to all 2D invertible transformations, beyond simple scaling. We investigate how the microfacet distribution and the
shadowing function of an existing microsurface change when the
microsurface is linearly transformed in the tangent plane.
The related previous work is presented in the next section. Necessary preliminary concepts from microfacet theory are provided
in Section 3. Our method is derived in Section 4: we first investigate which linear transformations of the microsurface result invalid microsurfaces, then we show how to construct the statistics ofthe transformed microsurfaces and finally we make the connectionof our technique with the classical anisotropy for shape-invariant
distributions. In Section 5 we demonstrate our technique by constructing anisotropic modifications of shape-variant distributions.
To enable the visualization of deforming objects we show how to
track the local deformation of an object using a tension texture.

2. Previous work
Kajiya derived anisotropic reflection models from the equationsof electromagnetism [Kaj85]. He pointed out that statistical models are desirable, but the anisotropic integrals are "extremely complex". Poulin proposed an anisotropic model based on sampling
microscopic cylinders layered onto the surface [PF90]. Ashikhmin
et al. [APS00] introduced various anisotropic models to microfacet
theory, and developed matching shadowing equations.
Nagano et al. [NFA*15] studied the complex appearance of human skin under deformation. In addition to the meso-structure detail represented by scanned normal maps they showed that microstructure dynamics play a vital role in the overall appearance.
Measurement of skin samples indicated that stretched skin appear shinier while compressed skin have rougher look. They incorporated this measurement data into their shader by blurring thestretched portions of the surface and sharpening the compressed
ones. From appearance point of view this aspect is similar to
our work, although we present a mathematical framework for microfacet surfaces that is not based on measurements. Dupuy etal. [DHI*13] presented a filtering technique for displaced surfaces
based on non-centred anisotropic Beckmann distributions. It is amulti-scale representation for mapped surfaces that, similarly to ourwork, can predict scaled surface appearance. However, our work is
focused on microsurface-related behavior for all distributions while
their work is targeted at filtering Beckmann surfaces. Non-centred
distributions can be used to modulate an arbitrary microsurface by anormal or displacement map, and they can be applied to our model
as well. The shifted distribution remains normalized and only the
shadowing function requires alteration [Hei14].
There are a number of approaches for filtering reflectance frommicrostructures [HSRG07; WDR11; JHY*14; WZYR19], including a large amount of work for rendering glints following the framework of Yan et al. [YHJ*14]. A model capable of large spectrum of
appearances based on tabulated piecewise linear distributions waspresented by Ribardière et al. [RBSM19]. Our work is targeted at
modifying existing distributions and as such is orthogonal to thesemethods.
GGX is probably the microfacet distribution that is currently
most widely used in practice, as it has been consistently verified
to be a better all-around match for acquired data [TR75; Bli77;
WMLT07; DHI*15] than the Beckmann distribution: the main reason for this seems to be the long tails of the distribution. However, many real materials have even longer tails, and therefore Disney developed the Generalized Trowbridge-Reitz (GTR) distribution as a natural extension to the GGX distribution [Bur12]: thisexposes an exponent γ that provides additional control of the tail.
This is especially attractive for exponents less than two, whichcorrespond to heavier tails that those of GGX. For example, a
GTR exponent γ = 23 results in the Henyey-Greenstein distribu© 2022 The Author(s)Computer Graphics Forum © 2022 The Eurographics Association and John Wiley & Sons Ltd.

A. Atanasov, V. Koylazov, R. Dimov & A. Wilkie / Microsurface Transformations

tion [HG41] evaluated at half-angles, while exponent of γ = 1 corresponds to the Berry distribution [Ber23] which is employed in
the coat layer of Disney’s material [Bur12]. We are not aware ofother tail-controllable distributions that are used in the industry. Interestingly, the GTR distribution is shape-variant for all tail exponents γ ̸= 2, and the elliptical anisotropic formulation cannot be
integrated analytically in these cases [Bur12]. This makes it a practical use case to demonstrate our technique on.
Löw et al. [LKYU12] introduce isotropic models suited for
glossy reflections that allow control of the tail of the distribution.
Work has been done to develop shape-invariant distributions that
admit tail control. The STD [RBMS17] was proposed to span Beckmann and GGX distributions although the heavier than GGX tailcontrol was limited. The Hyper-Cauchy distribution was utilized
in the optics literature [WOB*06] and was later used for BRDF
fitting [BM14]. Ribardière et al. presented a hierarchical classification of analytic distributions that generalize Beckmann and GGX
distributions until they reach the most general Skewed Generalized
T-Distribution (SGTD) [RBSM19]. They note that only the distributions in the lower levels of this hierarchy are shape-invariant,and therefore are guaranteed to have an anisotropic form. With ourframework an anisotropic version of any SGTD can easily be constructed. Notably, the GTR distribution is not part of this classification. Recently, Barla et al. [BPV18] presented a technique thatcombines two BRDFs to provide more control on the tails of the
distribution. Similarly, our technique modifies existing BRDFs andcan be employed on top of their technique when their input BRDFs
are microfacet-based.
It has been shown that genetic programming could be used
to find new analytic BRDFs, including microfacet ones that provide better fits to acquired data compared to existing microfacet BRDFs [BLPW14]. All of their microfacet distributions are
isotropic, and it is unlikely for their technique to generate shapeinvariant distributions. Our framework can directly extend them toanisotropy.
Heitz et al. [HDHN16] introduced Linearly Transformed Spherical Distributions (LTSD), and derived a closed-form formula forthem. However, their formula does not lead to normalized microfacet distributions, while our framework for linearly transformed
microsurfaces ensures this. In Section 4, we derive the Jacobianthat ensures the normalization of the transformed microfacet distributions, and discuss the differences to the LTSD formulation.
Finally, there are examples of shape-variant distributions used in
the industry, like Phong [WMLT07], Sheen [EK17], discrete GGX
and Beckmann [AK16], and GTR [Bur12].
3. Preliminaries
In this section we review the relevant background. The main notation is collected in Table 1.
3.1. Microsurface statistics
In microfacet theory the shaded surface is referred to as the macrosurface, while the microscopic details are represented by an infinite collection of microfacets which form a continuous microsurface [WMLT07]. Features of the macrosurface are assumed to be© 2022 The Author(s)Computer Graphics Forum © 2022 The Eurographics Association and John Wiley & Sons Ltd.

Table 1: Table of notation.
2

H
v
n
m
D
Dv
G1
P22
α
αx , αy
γ
M
DM
DvM
G1M
P22M

Unit hemisphere(vx , vy , vz ) = (sin θv cos φv , sin θv sin φv , cos θv ) ∈ H2
Macrosurface normal (0, 0, 1)
Microsurface normal (micro-normal)
Microfacet distribution function
Distribution of visible normals from direction v
Monodirectional shadowing function
Slope distribution
Roughness of the microfacet distribution
Roughness values along x and y for anisotropy
Tail falloff of the microfacet distribution
Microsurface transformation matrix
Microfacet distribution of the transformed surface
Visible normals distribution for the transformed surface
Shadowing function of the transformed surface
Slope distribution of the transformed surface

much larger than the microfacets, so from the standpoint of the microsurface the macrosurface can be assumed to be locally flat. In
our exposition all vectors are in the local frame of the macrosurface, and we assume its local normal as n = (0, 0, 1). A differential
area dA on the macrosurface is projected along a direction v ∈ H2
to differential area (v · n)dA, Figure 2.
The microsurface is fully determined by its profile (commonly
Smith) and microfacet distribution D(m) [Hei14]. The latter is defined in a coordinate system that is aligned with the macrosurface.
For a given micro-normal m ∈ H2 it evaluates to the differential
area of the microsurface oriented with m. In a valid configuration
the area of a non-flat microsurface is always greater than the macrosurface area, and the signed projected area of the microsurface must
be equal to the projected area of the macrosurface along an arbitrary
direction v ∈ H2 [WMLT07], Figure 2 a). This property enforces anormalization constraint on the microfacet distribution:
Z
H2

D(m)(v · m)dm = (v · n).

(1)

Pointwise evaluation of microfacet-based BRDFs requires anauxiliary function derived from D(m). The monodirectional shadowing G1 (o, m) yields the fraction of the microfacets with normals
m that are visible from direction o, Figure 2 b). It is subject to anormalization constraint [Smi67; APS00; Hei14]:Z
H2

D(m)G1 (v, m) max(0, v · m)dm = (v · n).

(2)This equation is equivalent to the Weak White Furnace Test for
specular microfacets [Hei14] when the integration changes fromv+i
normals m to vectors on the sphere i with m = ||v+i||[TS67].
Notice the similarity between the two constraints. The negative
projected area from the back-facing microfacets in Equation (1)(v · m) < 0 is clamped in Equation (2), and instead the shadowing
is compensated by the G1 term. This resemblance is illustrated withthe two diagrams in Figure 2.
Microfacet BRDFs require a tool to evaluate what portion of microfacets with normal m are seen from both the incoming i andA. Atanasov, V. Koylazov, R. Dimov & A. Wilkie / Microsurface Transformations

A
v)d
(n.
+

-

A
v)d
(n.

m

+

v

n

m

+

+

×G 1

v

dA

dA

a) Constraint on D, Equation (1)b) Constraint on G1 , Equation (2)nFigure 2: A differential area dA on the macrosurface with normal n. The projection of this area along direction v ∈ H2 is (n · v)dA. The
microsurface (green) is continuous and it is composed of microfacets with micro-normals m. Microsurface statistics are subject to constraints.
a) Microfacet distribution D: The projected areas along v of all forward-facing microfacets (blue) minus the projected areas of all backwardfacing microfacets must be equal to the projected area of the macrosurface (n · v)dA. b) Shadowing function G1 : When the total projectedarea along v of forward-facing microfacets (blue) is greater than the projected area of the macrosurface (n · v)dA the shadowing factor G1
must diminish it to compensate for the shadowed regions (red).

outgoing o light directions. This is achieved by the shadowingmasking function G(o, i, m) which is based on the shadowing
G1 [WMLT07; Hei14]. In this paper, we exclusively use the heightcorrelated shadowing-masking function [RDP05; Hei14].

ment α12 with the ellipse

We use these two constraints to validate our transformed microsurfaces. First we verify that D has a proper normalization. It issufficient to ensure that the projected microfacets along the macrosurface normal do not overlap and perfectly cover the macrosurface. Hence, Equation (1) is substituted with v := n:This leads to a distribution which is not properly normalized. A
normalization factor can be computed if Equation (3) can be integrated in closed-form. This is not always possible - for example theGTR distribution does not have closed-form normalization except
for the special case of GGX distribution [Bur12].

Z
H2

D(m)(n · m)dm = 1.

(3)

Then to validate G1 we must ensure that Equation (2) is fulfilled
for each v ∈ H2 .
Finally, a microfacet distribution can be converted to a distribution of slopes


my
mx
P22 − , −= D(m) cos4 θm ,
(4)
mz
mz
m

y
x
where − mmz and − mz are the slopes of a microfacet with normalm = (mx , my , mz ). The factor cos3 θm is for the change of measurefrom solid angle to slopes and the factor cos θm is for the projectionof the microfacets onto the macrosurface [WMLT07; Hei14].

3.2. Anisotropy
The classical anisotropy approach presented here is based on
Heitz’s extended report [Hei14]. Commonly, microfacet distributions D(m) have roughness parameter to control the spread of the
distribution which is usually denoted by α. Traditionally, isotropic
distributions are extended to anisotropy by varying the roughness
in the xy tangential plane. This is done by specifying roughness αxalong x axis and αy along y axis and replacing the isotropic argu-

cos2 φm sin2 φm
+
.
α2x
α2y

(5)

3.3. Shape-invariant distributions
Shape-invariant isotropic distributions have the property that theirshape does not change when the roughness changes. They are an
important sub-class of distributions: for all roughness values the
distribution curves are scaled copies of each other. Such distributions take the form:
1tan θm
1
f,(6)
D(m) =
α(m · n)4 α2
where the function f is a 1D distribution. The anisotropic version
based on Equation (5) iss!
1
cos2 φm sin2 φm
1
f
tan
θ
+
. (7)
D(m) =
m(m · n)4 αx αy
α2xα2y
To complete the anisotropic microsurface configuration the shadowing function G1 (v, m) must be evaluated for the projected roughness onto the direction vqαv = cos2 φv α2x + sin2 φv α2y .
(8)
4. Our framework
We take a different approach to altering microfacet distributions.
By investigating how linear transformations of the microsurface af© 2022 The Author(s)Computer Graphics Forum © 2022 The Eurographics Association and John Wiley & Sons Ltd.

A. Atanasov, V. Koylazov, R. Dimov & A. Wilkie / Microsurface Transformations

fect its statistics we arrive at a simple and general framework for
applying modifications to all valid microsurfaces.
4.1. Transforming the microsurface
Although the microfacets are infinitely small, we can still think of
the microsurface as a 3D object which is aligned with the macrosurface. For instance, Heitz et al. [HHdD16] validated their multiple
scattering model by ray-tracing a pre-generated Beckmann surface.
So it stands to reason that the microsurface can be transformed by a3 × 3 matrix, just like regular geometry. We consider all such transformations that lead to a valid microsurface. These are all invertible
matrices of the form


a c 0
(9)
M = b d 0 .
0 0 1
• M performs an arbitrary linear transformation on the microsurface in the tangential plane, parallel to the macrosurface, Figure 3 b).
• The first two column vectors of M are restricted to the tangential
plane, because the microsurface must remain aligned with themacrosurface, Figure 3 c).
• The first two row vectors of M are also contained in the tangential plane, because vertical shear could break the microsurface
configuration. Microfacet normals m could turn to the negative
hemisphere, Figure 3 d).
• Scaling the height of the microsurface h times results in the same
microsurface statistics as if we scale the xy plane by a factorof h1 . This is observed in the slopes of the transformed surface
amx +bmy
cm +dmand − xmz y where the scaling of the height hmz
mz
is identical to modifying M using ( ah , hb , hc , dh ). Therefore, we al-−ways set the last entry of M to 1.

Compression

⨉½
a)

b)

Misalignment

c)

Shear

d)

Figure 3: Valid (top row) and broken (bottom row) microsurface
configurations. Compression: a)→ b): Linear transformations parallel to the macrosurface lead to new valid microsurfaces. Misalignment: a)→ c): Linear transformations outside the tangential plane break the micro-macrosurface alignment, as some microfacets sink below the macrosurface (red). Shear: b)→ d): Vertical
shear parallel to xy plane moves some microfacets into the negative
hemisphere (red).

transformed by the inverse transpose of that matrix (M −1 )T . Thistransformation is derived by enforcing the transformed tangents tobe orthogonal to the transformed normals as they should be by definition [PJH16]. Indeed this is directly verified((M −1 )T m)T (Mt) = mT (M −1 M)t = mT t = 0,
where t is a tangent to a microfacet with normal m.
Figure 1 demonstrates our method with non-uniform scaling andskew transforms for shape-variant GTR. Note that the effect ofstretching in Figure 1 (top right) is observed using the ellipticalanisotropic formula for shape-invariant distributions, however inthis shape-variant case the linear stretching of the microsurface
does not result in linear stretching in the roughness parameter. The
effect of a general 2D linear transform in Figure 1 (bottom left) beyond stretching has not been shown so far even for the simpler caseof shape-invariant distributions. Furthermore, in Figure 1 (bottom
right) we demonstrate our method for a non-linear 2D transformation. The Jacobian matrix of this transformation is a local linear
approximation that we use within our framework.

4.2. Statistics of the transformed microsurface
In order to compute the statistics of the transformed microsurface
we transform the arguments of the initial valid microsurface statistics. All arguments are unit vectors so a normalization is required
after the arguments are transformed.
Shadowing function G1M (v, m): When the vector v is transformed with the microsurface the shadowing configuration does notchange. Note that the micro-normal m must be transformed with
the inverse transpose of M!
Mv
(M −1 )T m
G1M,
= G1 (v, m),(11)||Mv|| ||(M −1 )T m||
and therefore we can express the shadowing of the transformed microsurface G1M in terms of G1 by inverting the matrices before thearguments and normalizing!
M −1 v
MT m,
G1M (v, m) = G1.
(12)||M −1 v|| ||M T m||
This is a generalization of the masking probability invariance [Hei14]. Indeed, if an object is lit by a directional light andsome parts of it are self-shadowed, then a linear transformation applied both to the object and the light direction will preserve theshadowed portions. This is because the linear transformations preserve the projection along the transformed light direction.
Microfacet distribution DM (m): The input normal m is transformed similarly, but the resulting distribution is not normalized ingeneral. To find proper normalization we investigate the non-linear
transformation N : H2 → H2 that applies linear transformation toa normal and normalizes the resultu = N(m) =A basic fact in computer graphics is that when an object is transformed using a matrix M, the normal vectors to the surface must be© 2022 The Author(s)Computer Graphics Forum © 2022 The Eurographics Association and John Wiley & Sons Ltd.

(10)

MT m
.
||M T m||(13)

N is a bijection because M T is invertible. The product of M T andA. Atanasov, V. Koylazov, R. Dimov & A. Wilkie / Microsurface Transformations

the micro-normal m = (mx , my , mz ) is M T m = (amx + bmy , cmx +dmy , mz ) and has lengthq||M T m|| = (amx + bmy )2 + (cmx + dmy )2 + m2z .
(14)Note that the microfacet distribution is two-dimensional: the normal m is on the hemisphere, and can be represented by the first twocomponents m
qx and my . The z component is projected on the hemisphere mz = 1 − m2x − m2y . In order to find a new distribution DM
by transforming the argument of D with N we need to normalize by
the absolute value of the Jacobian determinant of N [PJH16]
DM (m) = | det JN |D(u),

(15)

∂ux
∂mx
∂uy
∂mx

(16)

where
det JN =

∂ux
∂my
∂uy
∂my

=

∂ux ∂uy∂ux ∂uy−.
∂mx ∂my ∂my ∂mx

We proceed to compute the partial derivatives
2

2

|detM

М

|.dA

dA

dA

y
x

Figure 4: The microsurface projection onto xy plane. Left: A
square patch (black) on the macrosurface has differential dA. Portion of the microsurface that is inside this patch is coloured withgreen while the rest is coloured red. Right: When this microsurface
is transformed using the matrix M, the portion of the microsurface which was previously in dA is transformed accordingly (graydashed region). The transform M changes the area by a factor of| det M| = | det M T |.

2

my (ad − bcd − a) + mx my (acd + b − bc ) + a
∂ux
=
∂mx
||M T m||3(17)

m2 (bc2 − acd − b) + mx my (bcd + a − ad 2 ) + b
∂ux
= x
∂my
||M T m||3

(18)

n

θm

m2y (cb2 − abd − c) + mx my (abc + d − da2 ) + c
∂uy
=
∂mx
||M T m||3(19)

m2 (da2 − abc − d) + mx my (abd + c − cb2 ) + d
∂uy
= x
.
∂my
||M T m||3(20)

After simplification of Equation (16) using Mathematica [Wol16]
we arrive at a concise resultdet M
ad − bc=.
(21)
det JN =||M T m||4||M T m||4
We use this result in Equation (15) to obtain the microfacet distribution of the transformed microsurface
DM (m) =| det M|D(u).
||M T m||4

Furthermore, we observe in Figure 5 thatcos θm.
cos θu

θu

u
cosθm
cosθu

Figure 5: Geometry of micro-normal transformation. Let m(green) be a micro-normal from the original microsurface. The
transformed vector M T m (blue, dashed) has the same z-component
as the original vector m because the transformation M T acts only
MT m
on the xy plane. This vector is normalized and labelled u = ||M
T m|| .
From the two similar right triangles with hypotenuses M T m and uθm
we find that ||M T m|| = cos
cos θu .

(22)

We provide geometrical intuition behind Equation (22). The matrix M transforms the xy plane and changes the projected microsurface area by a factor of | det M| = | det M T |. The microfacet distribution measures the microsurface differential area relative to themacrosurface differential area, see Figure 4. Therefore, this factor
compensates for the change of projected area.

||M T m|| =

МТm

m

(23)

Therefore, in Equation (22) the factor ||M T1m||4 serves the purposeto change the Jacobian cos13 θ and the inverse projection cos1θu inuEquation (4), both with respect to the transformed micro-normalu, to the Jacobian cos13 θ and the inverse projection cos1θm with rem
spect to the original micro-normal m. Equation (22) is similar to the
Linearly Transformed Spherical Distributions (LTSD) [HDHN16],
however their formula does not include the inverse projections.
Hence, our formulation given with Equations (22) and (9) defines

a different family of distributions which we name by analogy Linearly Transformed Microfacet Distributions (LTMD).
Finally, we express the slope distribution of the transformed microsurface P22M from the original slope distribution P22 by using,in that order, Equations (4), (22) and (23)


my
mx
= DM (m) cos4 θm
P22M − , −mzmz| det M|=
D(u) cos4 θm||M T m||4
= | det M|

cos4 θuD(u) cos4 θm
cos4 θm(24)= | det M|D(u) cos4 θu


uy
ux
= | det M|P22 − , −.
uzuz
Consequently, Equation (22) can be also derived from the slope
distribution.
© 2022 The Author(s)Computer Graphics Forum © 2022 The Eurographics Association and John Wiley & Sons Ltd.

A. Atanasov, V. Koylazov, R. Dimov & A. Wilkie / Microsurface Transformations

Sampling micro-normals: in order to implement practical microfacet materials we need sampling equations for the microfacet
distribution. In our work we assume that the original microsurface before transformation has sampling equations at least forD(m)(n · m). Based on these, a micro-normal u′ is sampledu′ ∝ D(m)(n · m).

(25)

Then it is transformed to the modified microsurface using the inverse transpose of Mm′ =(M −1 )T u′∝ DM (m)(n · m).
||(M −1 )T u′ ||(26)

If the original microfacet distribution has sampling equations forthe distribution of visible normals [HD14]
Dv (m) =G1 (v, m)(v · m)D(m)
v·n

(27)

then we can use them to sample the distribution of visible normalsof the transformed surface
DvM (m) =G1M (v, m)(v · m)DM (m)
.
v·n

(28)

To do that for a given direction v we first transform it to the space′
M −1 vof the original surface w = ||M−1 v|| , generate a visible normal uw
using the available equations and then transform it to the modified(M −1 )T u′
surface using the inverse transpose m′v = ||(M −1 )T uw′ || .
w

4.3. Numerical validation
In addition to the proof presented in this section we also validated
our technique numerically. Specifically, we applied random transformations M on a large variety of microsurfaces with random parameters and verified that:• The normalization constraint in Equation (3) is fulfilled for alltransformed microfacet distributions DM .
• The shadowing constraint in Equation (2) with random directions v ∈ H2 is fulfilled for all transformed shadowing functions
G1M .
The list of tested distributions include GTR [Bur12], anisotropic
Beckmann and GGX [Hei14], STD [RBMS17], Phong [WMLT07],
Sheen [EK17], Discrete GGX distribution [JHY*14; AK16]. We
provide the source code for this numerical validation.

4.4. Equivalence with classical anisotropy for shape-invariant
distributions
Suppose that we have a shape-invariant isotropic distribution D(m)with roughness α, defined by Equation (6). To construct the traditional anisotropic form we replace α with two roughness values αx
and αy and use Equation (7). In our framework this is equivalent toapplying the diagonal transformation matrix
α

0 0
αx
α
0 .
M=0
(29)
αy
0
0 1
© 2022 The Author(s)Computer Graphics Forum © 2022 The Eurographics Association and John Wiley & Sons Ltd.

Then substituting Equation (6) in Equation (22)
| det M|1
1
tan θuDM (m) =fα||M T m||4 cos4 θu α2
 s
α2
1  1 u2x + u2y 1αx αy= cos4 θfm cos4 θu α2αu2z
cos4 θus(30)!
1 α2 m2x α2 m2y1f=+
α α2x m2z α2y m2z
cos4 θm αx αys!
sin θm cos2 φm sin2 φm
1f+=cos θmcos4 θm αx αy
α2xα2ywe get to Equation (7) when we first convert tan θu to cartesian coordinates, express u in terms of m from Equation (13) and convert
back to spherical coordinates, see Table 1.
An important point is that for shape-variant distributions thescaling of the microsurface is not equivalent to the scaling of theroughness coefficient. Scaling operations of the microsurface arealways possible within our framework, while formulas for scalingof the roughness coefficient cannot always be derived.
4.5. Discussion
In summary, for a given microsurface (D, G1 ) and a linear transformation M given in Equation (9) our technique defines the transformed microsurface (DM , G1M ) using Equations (22) and (12). It
is noteworthy that in our derivation we do not make any assumptions on the input microsurface. Therefore the input distribution
can be both shape-invariant and shape-variant, as well as isotropic
or anisotropic. This means that it can be used as a black box - it canbe implemented in a base class and can be applied regardless of theunderlying microfacet distribution, including stochastic [JHY*14]
or data-driven [APS00; YHJ*14; RBSM19] approaches. Furthermore, the solution is in closed-form and its performance depends
mainly on the expressions for D and G1 .
5. Results
We demonstrate our technique using the standard microfacet BRDF
for specular microfacets [WMLT07] with an appropriate Fresnel
term for dielectrics and conductors. All materials use the GTR distribution with different tail exponents. Dimov derived an accurate
Smith shadowing term for GTR tails γ ∈ [0, 4] that is exact at the
integers [Dim15] which we provide in Appendix A. Due to the lackof sampling formula for the distribution of visible GTR normals the
sampling efficiency deteriorates with roughness because of the increased shadowing. We propose a sampling strategy to improve thesampling of rough materials in Appendix B.
5.1. Deforming objects
In Figure 6 we apply our framework in a scene with deforming
objects. We track the deformations using a tension texture, basedon which we derive the local linear transformations and render thecorresponding anisotropic microsurface. The purpose of the tension
texture is to compute a 2D tension matrix that contains the local

A. Atanasov, V. Koylazov, R. Dimov & A. Wilkie / Microsurface Transformations

Isotropic

Ours

Figure 6: Decorative objects made from elastic dielectric material with microfacet BRDFs. All birds and all dogs have materials withshape-variant GTR distributions with roughness values α = 0.25 and α = 0.16, and tail exponents γ = 2.2 and γ = 1.4, respectively. We
demonstrate how the objects reflections change due to deformation. When the object is stretched the microsurface becomes smoother andtherefore the reflection highlight becomes sharper. When the microsurface is compressed the reflection spreads out. Top row: A standard
isotropic material. Bottom row: Our method that tracks the deformation via a tension matrix and computes the corresponding anisotropic
reflection. The first and last objects in the composition are at "rest" state while all objects in between are deformed. Middle row: Alternating
zoomed-in regions of isotropic (top row) and deformed anisotropic (bottom row). A helpful analogy to understand the highlight changes that
occur with our model is a rubber balloon: as it gets inflated (stretched out), it gets to be smoother and more and more shiny. Our model canreplicate this behaviour even for anisotropic stretching.

surface transformation relative to a predefined “rest” state. In order
to compute this, we need a stable local space for each shading pointthat is consistent between the rest state and the current deformed
state of the mesh. Then, at each shading point, we transform theedges of the currently shaded triangle to this local space both forthe deformed mesh and for the “rest” mesh. Finally we compute atransformation in local space that converts the “rest” edges to thecurrent deformed edges.

dimensional, because we are only interested in the projection to thetangent UV space. Then the 2 × 2 tension matrix is T = DR−1 ,where R = (r1 r2 ) and D = (d1 d2 ), see Figure 7. For shading, we
embed the tension matrix in the matrix M, see Equation (9). The
tension texture is applied to the deformed object and keeps a reference to the "rest" object. During texture evaluation the matrices R
and D are computed for the shaded triangle and the tension matrix
T is returned.

We use the UV mapping coordinates assigned to the surface as
the stable local space. For each shading point, we can computean orthonormal matrix that converts from world space to the local UVW space for the deformed mesh, and another orthonormal
matrix that converts from world space to the local UVW space forthe “rest” mesh. Using these two matrices, we can rotate the edgesof the currently shaded triangle in the deformed mesh and the edgesof the same triangle in the "rest" mesh to local space. Note that ingeneral the local UVW spaces are not orthonormal [PJH16] but weforce them to be by preserving the U direction and then orthogonalizing and normalizing the matrices. In this way we retain the
shape of the triangles and only align them with the U direction in
the tangent plane. Let r1 and r2 be the transformed edges in local space of the "rest" triangle, and d1 and d2 be the transformed
edges in local space of the deformed triangle. These edges are 2-

This method computes per-face tension matrices and requires
relatively fine geometry tessellation to achieve smooth results. Another option is to pre-process the "rest" mesh and the deformed
mesh and to store a weighted average of the face tension matrices
in the vertices. Upon rendering, vertex tension matrices are interpolated along the triangles to facilitate continuous shading.

5.2. Stretch anisotropy
We have established that traditional anisotropy can be reproduced
as a special case of our framework by using a non-uniform scaling
matrix, see Equation (29). For our stretch anisotropy we use a convenient scaling transform that is defined by a single scaling factor© 2022 The Author(s)Computer Graphics Forum © 2022 The Eurographics Association and John Wiley & Sons Ltd.

A. Atanasov, V. Koylazov, R. Dimov & A. Wilkie / Microsurface Transformations

Deformation

T=DR-1
r1

d1

v

r2

d2

u
R=(r1 r2)

D=(d1 d2)

Figure 7: A triangle of the "rest" mesh (blue) and its corresponding triangle in the deformed mesh (red). The edges of these triangles are rotated to UV space (yellow): the edges of the "rest"
triangle r1 and r2 form the matrix R that converts to the space ofthe "rest" triangle, and the edges of the deformed triangle d1 andd2 form the matrix D that converts to the space of the deformed triangle. Therefore, the matrix T = DR−1 encodes the local tensionof the deformation.

s ∈ (−1, 1):



1

, 1 − s, 1 , s ∈ [0, 1)
diag 1−s
Ms =
.



diag 1 + s, 1 , 1 , s ∈ (−1, 0)
1+s

(31)

For positive values of s it stretches the x-coordinate and shrinks
the y-coordinate while for negative values of s it does the opposite.
Moreover, det Ms = 1, MsT = Ms , M0 = I and Ms−1 is constructedby swapping the first two diagonal entries of Ms . Note that a 2D
rotation can be concatenated with the matrix Ms , but we choose
to keep the matrix simple, and perform anisotropy rotation via theshading frame rotation.
We showcase our technique in Figure 8. The vases have brushed
aluminum material, and all of them have anisotropic shape-variant
distributions that do not have analytic elliptical anisotropic equations, given in Equation (5).

6. Conclusion
We propose a change in how one goes about the construction ofanisotropic microfacet distributions. Performing these derivations
based on the transformations that need to be done to the microgeometry, instead of based on the directional variation of surface
roughness, offers desirable advantages: our approach works withall planar linear transformations (including skew), and with all
slope distribution functions, regardless of whether they are shapeinvariant, or not. This offers artists a wider range of creative control,
both in terms of the highlight shapes they can create, and the typesof surfaces that can be made anisotropic.
© 2022 The Author(s)Computer Graphics Forum © 2022 The Eurographics Association and John Wiley & Sons Ltd.

Figure 8: Brushed aluminum vases rendered with GTR distribution. Columns from left to right have roughness values α =0.01, 0.04, 0.16. The top row has the Berry distribution γ = 1 andthe bottom row has a tail exponent γ = 3.5. All materials haveshape-variant anisotropic distributions with scaling factor s = 52 ,see Equation (31).

7. Acknowledgements
We thank members from Chaos Bulgaria R&D team for proofreading the manuscript and the anonymous reviewers for constructive feedback. We also acknowledge funding from Charles University grant SVV-260588 and the Czech Science Foundation (GAČR)
grant 22-22875S.
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Appendix A: A Smith shadowing term for GTR γ ∈ [0, 4], whichis exact for the integer tails
Burley [Bur12] introduced the isotropic GTR distribution
Dγ (m) =1(γ − 1)(α2 − 1),π(1 − (α2 )1−γ ) (1 + (α2 − 1) cos2 θm )γ(32)

where α ∈ [0, 1] controls the roughness and γ ≥ 0 is the tail exponent. The distribution has a 00 singularity at γ = 1, and it converges
to the Berry distribution [Ber23; Bur12] at the limit
D1 (m) = lim Dγ (m) =γ→1(α2 − 1)1
. (33)
π log α2 (1 + (α2 − 1) cos2 θm )© 2022 The Author(s)Computer Graphics Forum © 2022 The Eurographics Association and John Wiley & Sons Ltd.

S0 (µ) =
1+

2
q

1
+1
µ2

,

µ log α2, 2
α (µ+B1 )A1 − B1 + µ logµ+A1pp
where A1 = µ2 + α2 and B1 = µ2 + 1,
S1 (µ) =S2 (µ) =1+S3 (µ) =where A3 =S4 (µ) =

2
q 2

,

(35)

(36)

(37)

α
+1
µ2

4B3 µA3,
α2 (3α2 + 1) + 2µB3 (µ + A3 )(38)p
µ2 + α2 and B3 = α2 + 1 and

2A4 µB34,332
2
A4 µ(B4 + µ ) + 3α (α (5α4 + 2α2 + 1) + 4µ2 (2α4 + α2 + 1))(39)

where A4 = 8α4 + 8α2 + 8 and B4 =p
µ2 + α2 .

In Figure 9 b) we show these shadowing curves for a fixed roughness α = 0.6. In order to compute an accurate approximation for all
non-integer tails γ ∈ (0, 4) all five integer formulas are evaluatedand a cubic spline interpolation [dBoo01] is performed. Natural
boundary conditions are suggested in order to minimize the oscillations. We have verified that with this approach the constraint inEquation (2) is closely met.
Appendix B: An improved sampling for the GTR distribution
Rendering microfacet BRDF with GTR distribution with highroughness has inherent high variance due to the lack of sampling
equations from the distribution of visible normals [HD14]. The reduced effectiveness of the sampling from D [WMLT07] is due toincreased shadowing. When the roughness is approaching 1 the single scattering term gets darker and more diffuse. We found thatstochastic mixing of the sampling strategy with uniform sampling
improves the sampling of rough surfaces. A random number andthe roughness α determine whether uniform sampling or samplingfrom D will be used for the direction. The balance heuristic isfinally applied for probability computation[VG95]. Render timecomparison can be seen in Figure 10.

A. Atanasov, V. Koylazov, R. Dimov & A. Wilkie / Microsurface Transformations
2

1

S0
S1
S2
S3
S4

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D2
D3
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a) GTR Microfacet distribution Dγ

μ 4

3

2

b) GTR Average visibility Sγ

render time reduction with our sampling

Figure 9: Plots of the microfacet distribution Dγ and the average visibility Sγ for γ = {0, 1, 2, 3, 4} and for a fixed roughness α =0.6. Notethat limα→0,γ>0 (Dγ , Sγ ) = (δ, 1) and limα→1 (Dγ , Sγ ) = (D0 , S0 ).

γ=0.1

γ=1

γ=2

0%

0%

-10%

-10%

-20%

-20%

-30%

γ=4

-30%
0.01

0.3

0.6

0.9

0%

0%

-10%

-5%

-20%

-10%

-30%

0.01

0.3

0.6

0.9

0.01

0.3

0.6

0.9

-15%
0.01

0.3

0.6

0.9

roughness
Figure 10: Left: Torus knot with GTR roughness α = 0.3, tail exponent γ = 1 and anisotropy s = 0.75, lit by a rectangular light. Right:
Charts for the torus knot scene showing how much render times improve for the same image quality when switching from classic sampling toour sampling described in Appendix B. We use variance-based image sampler which samples each pixel until a given noise level is reached.
Top left chart is for isotropic material (s = 0) with 95 average samples per pixel (aspp) for the classical sampling vs. 84 aspp for oursampling. The top right, bottom left and bottom right charts represent anisotropic materials with s = {0.25, 0.5, 0.75} and 91 vs. 79 aspp,
95 vs. 89 aspp and 131 vs. 119 aspp, respectively. Therefore, our sampling technique reduces the number of samples per pixel by 10% on
average for this scene. Moreover, the charts demonstrate that this technique helps with very different distribution tails: GTR with γ = 0.1 has
much heavier tail than GGX while the tail of GTR with γ = 4 is much "lighter", see Figure 9 a).

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