Page 1

A Microfacet-based Reflectance Model for Photometric Stereo

with Highly Specular Surfaces

Lixiong Chen ∗1, Yinqiang Zheng1, Boxin Shi2, Art Subpa-Asa3, and Imari Sato1

1National Institute of Informatics, Tokyo, Japan

2Artificial Intelligence Research Center, National Institute of AIST, Tokyo, Japan

3Tokyo Institute of Technology, Tokyo, Japan

Abstract

A precise, stable and invertible model for surface re-

flectance is the key to the success of photometric stereo

with real world materials. Recent developments in the field

have enabled shape recovery techniques for surfaces of var-

ious types, but an effective solution to directly estimating

the surface normal in the presence of highly specular re-

flectance remains elusive. In this paper, we derive an ana-

lytical isotropic microfacet-based reflectance model, based

on which a physically interpretable approximate is tailored

for highly specular surfaces. With this approximate, we

identify the equivalence between the surface recovery prob-

lem and the ellipsoid of revolution fitting problem, where the

latter can be described as a system of polynomials. Addi-

tionally, we devise a fast, non-iterative and globally optimal

solver for this problem. Experimental results on both syn-

thetic and real images validate our model and demonstrate

that our solution can stably deliver superior performance

in its targeted application domain.

1. Introduction

The object appearance is a compound of illumination,

surface reflectance and surface shape. Under directional

light l, the appearance I of a surface point with normal n ob-

served from direction v is described as a product of shading

signal n⊺l and the reflectance signal specified by the Bidi-

rectional Reflectance Distribution Function (BRDF) ρ(·):

I = ρ( v, l, n) n⊺l.

(1)

Photometric stereo aims at inversely recovering n given

multiple observations with different lighting directions.

*Part of this work was finished when the author was a Master student of

McGill University, Montreal, Canada, and was visiting National Institute

of Informatics (NII), Japan, as an intern funded in part by the NII MOU

internship program.

(a) chrome steel

(b) metallic paint

(c) two layer gold

Figure 1: Examples of highly specular reflectance targeted

in this paper are pervasive in the real world, but effectively

estimating the shape of the specular surfaces is a challeng-

ing task.

Therefore, finding a proper tradeoff between the expres-

siveness and complexity of Equation 1 is critical. For ex-

ample, the Lambertian model [43] allows straightforward

normal recovery, but a pure Lambertian surface rarely ex-

ists in real world objects. In recent years, the photomet-

ric stereo has seen dramatic development in shape recovery

techniques for surfaces made of a great variety of materi-

als [9, 15, 16, 22, 37, 44]. After all, sufficient know-how

is already at hand to help extract shape cues from low-

frequency reflectance. However, how to properly handle

highly specular surfaces, like the ones illustrated in Fig-

ure 1, remains to be a hard nut. This is because when sig-

nals representing specular reflectance become predominant,

knowledge about inferring shape solely from specularities

for a typical photometric stereo setup is still limited.

On the contrary, how to render visually realistic

specular/general surfaces has been extensively studied in

the computer graphics community. For example, the micro-

facet reflectance model and its variants [4, 6, 10, 12, 25, 41]

offer significant insight into the formation of specularity

given the detailed knowledge of the surface geometry. Un-

fortunately, it is not straightforward to apply these models

for the inverse problem that seeks to recover the surface ge-

ometry, because analysis is absent to warrant model invert-

3162

ibility and estimation accuracy.

In this paper, we carefully investigate the state-of-the-art

microfacet theory involving the ellipsoid microfacet normal

distribution function (a.k.a. ellipsoid NDF), and have suc-

ceed in deriving an analytical model serviceable for photo-

metric stereo. We further introduce a physical interpretable

approximate that brings appealing algebraic properties for

specular surface normal estimation, without sacrificing its

expressiveness. Essentially, with this approximate, we iden-

tify that the calibrated photometric stereo problem boils

down to an ellipsoid of revolution fitting problem, for which

we devise a fast, non-iterative and globally optimal solver

targeting a system of polynomials. We use empirical re-

sults on both synthetic and real images to justify our theory

and model, and demonstrate that our solution outperforms

its peers when applied to highly specular surfaces. More-

over, based on our model, we also discuss the possibility to

devise a generalized solution to get the best of both worlds.

To sum up, our contributions mainly lie in

1. Deriving an analytical form based on the microfacet

theory and the ellipsoid NDF for photometric stereo;

2. Developing a physically interpretable approximation

for highly specular reflectance, which equates the

problem of normal estimation with an ellipsoid of rev-

olution fitting problem;

3. Designing a fast, non-iterative and globally optimal

solver to stably obtain the normal of specular surfaces.

The remaining of this paper is organized as follows: Sec-

tion 2 discusses the related works, Section 3 presents our

analytical microfacet reflectance model, and its reduction

for highly specular reflectance is derived in Section 4. Sec-

tion 5 explains our normal recovery algorithm, then in Sec-

tion 6 we discuss our experiment results obtained from both

synthetic images and real images. Section 7 concludes this

paper.

2. Related Works

Reflectance models play important roles in realistic ren-

dering (forward problem) and shape inference using photo-

metric stereo (inverse problem).

Reflectance models for rendering always seek to cap-

ture the finest real-world subtleties. Early models are either

empirically based, where appearance is directly compared

with observations [30, 23], or microfacet-based but in a con-

strained parametric form [36, 35, 7, 42, 28]. Recent works,

such as the Ashikhmin-Shirley model for anisotropic ma-

terials [4] and the GGX distribution [8] for general rough

surfaces, acknowledge the dominant effect of the NDF and

leave it as a design choice.

A relevant line of research proposes to represent NDF

directly with tabulated data [3]. This motivates the attempts

to directly capture the real-world reflectance [25, 27], which

in general requires dense sampling [14] and the ability

to process large amount of data. One workaround is to

leverage the prior knowledge of BRDF that leads to semi-

parameterized fitting [11, 6]. Among the proposed models,

we pay more attention to the ellipsoid NDF [40]. In case

of isotropic reflectance, it is analogous to the microfacet

BRDF with GGX/Trowbridge-Reitz distribution [41, 38].

It is worth noting that for rendering energy conservation is

not warranted, but for finer appearance capturing dense light

distribution is always desired [26], and the sampled surface

is assumed to have regular geometry [12] (e.g. flat surface).

Reflectance models for photometric stereo seek to de-

compose surface geometry from the scene radiance, which

usually contains highly complex surface reflectance. The

Lambertian model is widely adopted for its simplicity [43].

To deal with more general materials, specular highlight

could be discarded through outlier rejection [44, 22]. The

general reflectance could also be modeled by parametric

BRDF models [37, 15, 9], or a combined effect of several

easy-to-model components [16].

Recent approaches adopt non-parametric formulation to

handle a broader range of materials. Without explicit mod-

eling of the BRDF, some general reflectance properties,

such as isotropy [2], monotonicity [32], and their combi-

nation with visibility [19], are exploited to infer surface ori-

entation. The BRDFs can also be explicitly represented as

a bivariate function [1, 45], a constrained bivariate regres-

sion [21] or a sparse dictionary-based representation [20].

Overall, BRDF for photometric stereo is a delicate trade off

between generality and complexity [33].

Benchmark evaluation [34] demonstrates that the state-

of-the-art performance can be achieved with data containing

less-specular observations [21, 33]. These approaches work

well for a great diversity of real-world materials, but are

challenged by specularity-dominant observations. In con-

trast, our method aims to attack this challenging problem

with theoretical support.

3. A Microfacet BRDF with Ellipsoid Normal

Distribution Function

The microfacet reflectance model postulates that the sur-

face is made up by a large collection of tiny facets, and the

surface radiance is essentially a composition of microfacet

reflections, where the radiance intensity can be evaluated

[10, 36, 5] as

I( v) = ∫Ω+max( m · l, 0)D( m)G( l, v)ρm( m, l, v) d m.

(2)

As illustrated in Figure 2, Ω+ denotes the the visible up-

per half sphere, ρm( m, l, v) describes the reflectance of a

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Figure 2: The coordinates in which BRDF is defined. By

convention n = (0, 0, 1), and v and lare unit vectors that al-

low to orient arbitrarily above the positive half-sphere. Note

that this is in contrast to the typical setup for photometric

stereo, where v = (0, 0, 1)

specific microfacet with normal m under directional light l

while being perceived along v, D( m) is the microfacet Nor-

mal Distribution Function (NDF) counting the population

of the microfacets of the same orientation, and G( l, v) is

the masking-shadowing term ensuring power conservation.

To model general reflectance, each microfacet can be ef-

fectively assumed to exhibit mirror reflection [41], namely,

ρm( m, l, v) = F(θd)δm( h) dictating that a microfacet con-

tributes to the actual reflection only if its normal m and bi-

sector h = l+ v

| l+ v|

are perfectly aligned, and according to the

Fresnel equations the amount of power it reflects is deter-

mined by the angle θd made by the normal and the incident

light. Hence, Equation 2 can be rewritten as:

I( v, l) = G( l, v)D( h)F(θd).

(3)

Essentially, the microfacet model is to built upon the

construction of a Gauss map that parameterizes the micro-

facet in Euclidean Space R3 with its normal h, where the

NDF evaluates its rate of the change over a unit sphere S2.

In this regard, NDF is inherently the gaussian curvature of

the surface that the Gauss map applies to. For example, a

planar surface with zero Gaussian curvature leads to a Dirac

delta NDF that only spikes along the normal of the plane.

So, with identical setting given above, the NDF can also be

implicitly defined as the inverse of the Gaussian curvature

of the illusory surface covered by the microfacets. More-

over, recent study [40] demonstrates that the ellipsoidal mi-

crofacet arrangement and the general CGX NDF are equiv-

alent. Whereas the success of the latter has been widely

acknowledged in rendering, in the following we present

the appealing algebraic properties the former manifests for

shape analysis.

(a) specular surface

(b) diffusive surface

Figure 3: The ellipsoid NDF describes that the microfacets

can be re-arranged through translation to cover the upper

surface of an ellipsoid. A “flatter” ellipsoid indicates that

more microfacets are aligned with the surface, representing

a smoother material.

3.1. Ellipsoid NDF for Isotropic Reflectance

Implicitly defining the NDF over an ellipsoid offers sev-

eral algebraically appealing properties. As illustrated in

Figure 3, if Ω+ denotes an arbitrarily defined unit area of

the physical surface under examination, and the microfacets

can be geometrically translated to cover the upper half of a

ellipsoid, then there exists a unique parametrization of sur-

face point p by the surface normal S p

|S p|

= h, and the fol-

lowing is always satisfied:

p⊺S p = 1,

(4)

where S is a 3-by-3 matrix and can always be re-scaled to

normalize the RHS of the equation to 1. It has the following

properties to characterize the shape of the ellipsoid:

1. S is symmetric and positively definite;

2. In the case of isotropic reflectance, S denotes an el-

lipsoid of revolution, so its eigenvalues satisfy that

λ3 ≥ λ2 = λ1 > 0;

3. Correspondingly, the lengths of the major and the mi-

nor axes are 1

√λ1

and 1

√λ3

, respectively;

4. The arrangement of the microfacetets has to be physi-

cally consistent with the surface geometry, so the mi-

nor axis is aligned with the surface normal. Namely,

S n = λ3n.

Spectral theorem states that S = λ1u u⊺ + λ2v v⊺ +

λ3n n⊺, where u, v, and n are its eigenvectors. Correspond-

ingly, S−1 = 1

λ1

u u⊺+ 1

λ2

v v⊺+ 1

λ3

n n⊺. Also, let |S| denote

the determinant of S, and Kg = |S|( h⊺S−1h)2 denote the

Gaussian curvature of the microfacet-parameterized ellip-

soidal surface [17], the ellipsoid NDF D( h) for isotropic

reflectance (λ1 = λ2) thus can be expressed as:

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Figure 4: The shadowing function guarantees that the total

area receiving illumination over a surface of unit area does

not exceed l⊺n. In our model, we remove the restriction that

the region has to be in the upper sphere Ω+, so the entire

intersected area is considered.

D( h) =

1

Kg

=

1

λ2

1λ3(( h⊺u)2+( h⊺v)2

λ1

+ ( h⊺n)2

λ3

)2

=

1

λ3(1 − ( h⊺n)2 + λ1

λ3

( h⊺n)2)2

.

(5)

3.2. The Masking-Shadowing Function

The masking-shadowing function G( l, v) is introduced

to impose a physical constraint that the visible and the il-

luminated area must not exceed the projected area along

the perceived direction v or the illumination direction l

(Figure 4), respectively. In the case where v is fixed,

G( l, v) = G(θi) has to satisfy the following for isotropic

reflectance [18]:

n⊺l = ∫Ω+max( h · l, 0)D( h)G(θi) d h,

(6)

where max( h · l, 0) is to ensure the the microfacet lies

in the shadow vanishes. Because in a typical photometric

stereo setup a large population of the microfacets are illu-

minated by a moving directional light over the upper hemi-

sphere, this highly nonlinear term is only significant when a

light significantly deviates away from the normal. So, with

the premise that lights are distributed sufficiently, we re-

lax this expression by removing this operator. Therefore,

by plugging the widely adopted Smith Microsurface Pro-

file [35] and the ellipsoid NDF with ∫ ( h · l)D( h) d h =

π√l⊺S l|S|−1 [39] into Equation 6, we arrive at the follow-

ing derivation for the shadowing function:

G( l) =

l⊺n

π

λ1√λ3

√l⊺S l

=

l⊺n

π

λ1√λ3

√λ1(1 − ( l⊺n)2) + λ3( l⊺n)2

.

(7)

3.3. The Fresnel Term

In theory, the Fresnel term F(θd) only starts to vary dra-

matically as θd → π

2

, so in general it does not encode suf-

ficient information for shape analysis unless both view and

light are at the grazing angles, which only occasionally oc-

curs when lights are located at numerous locations for pho-

tometric stereo. Therefore, this term can be safely taken as

an unknown constant.

3.4. A General Reflectance Model

Combining Equation 3, 5 and 7 and letting λ = λ1

λ3

leads

to:

I( l) = C

λ

(1 − (1 − λ)( h⊺n)2)2

l⊺n

√(λ + (λ − 1)( l⊺n)2)

,

(8)

where C is an unknown product subsuming the camera gain,

the Fresnel term and 1

π; more importantly, we notice that λ

is a factor independent of the geometry and illumination, as

well as a sole term characterizing the material’s reflectance

property. Being the square of the ratio of the minor axis

length 1

√λ3

to the major axis length 1

√λ1

, λ successfully

decouples the pixel-wise material evaluation from the ac-

tual imaging process: since 1

√λ1

is the radius of a circular

patch orthographically imaged to a specific pixel, λ is es-

sentially the “normalized” shape descriptor of the ellipsoid,

regardless how large the “volume” that the ellipsoid occu-

pies, which is also independent of the camera pose and light

intensity.

Algebraically, λ plays a central role in identifying the

type of surface reflectance. Since microfacet arrangement

has to be consistent with the surface geometry, we have

1

λ3

≤ 1

λ1, hence λ ∈ (0, 1]. When λ → 1, it results

in a sphere, I( l) → C

√λ l⊺n, which corresponds to the

ideal diffusive case, because microfacets are arranged along

an arbitrary direction with equal probability (Figure 3b).

On contrary, when λ → 0, G( l) → 1, the material re-

flectance becomes more conspicuous till only specularities

are present. Since the former has been extensively studied

in the existing literature, here we look into the details of the

latter.

4. Physics Driven Approximate Model for

Specular Reflectance

λ → 0 leads to a description for perfect mirror reflec-

tion, and when G( l) → 1, the surface radiance has to be

evaluated under two scenarios:

1. h⊺n = 1. By Equation 2, I( l) → ∫Ω+D( h) d h →

C ∫Ω+

δn( h) d h = C, where δ( h) is the dirac delta

function describing the infinite impulse due to 1

λ

;

2. h⊺n = 1. I( l) =

Cλ

(1−( h⊺n)2)2

→ 0, which is a direct

simplification from Equation 8.

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In the first case the light is directly observed through the

ideal mirror reflection whereas no diffusive radiance can be

captured when h falls off from n in the second case.

Therefore, by letting λ take a sufficiently small value, we

obtain a reflectance function for highly specular materials:

I( l) ≈

Cλ

(1 − (1 − λ)( h⊺n)2)2

,

(9)

which can be rearranged into

√I( l)

Cλ(1 − ( h⊺n)2 + λ( h⊺n)2) ≈ 1,

(10)

and by defining ˆS = u u⊺ + v v⊺ + λ n n⊺, we can further

simplify Equation 10 as

((

I( l)

Cλ

)

1

4 h)T ˆS((

I( l)

Cλ

)

1

4 h) ≈ 1.

(11)

Equation 11 is essentially the standard equation for an

ellipsoid of revolution ˆS centered at the origin, where the

original ellipsoid S and ˆS are co-axial. Fitting an ellipsoid

requires at least 4 points on its surface, which can be easily

satisfied for photometric stereo. After all, as the directional

light relocates, distinct appearances can be obtained except

for nearly perfect mirror reflection, for which only the ap-

pearance of the illuminant is directly seen from one specific

location. In our analysis we rule out this extreme case.

Moreover, we expect our solver in the following section

to become less accurate when applied to diffusive materi-

als. As discussed in Section 3.4, in diffusive cases λ → 1,

the ellipsoid degenerates to a sphere without elongation (see

Figure 3b). Algebraically, this means that Equation 11 can

be satisfied by a set of non-unique ˆS. Fortunately, the value

of λ itself serves as a good measure for estimation confi-

dence, so we can safely “roll back” to the existing solvers

implemented for low-frequency reflectance if large values

for λ are detected.

5. Optimal Ellipsoid of Revolution Fitting for

Normal Estimation

Given the illumination direction l (equivalently the half

vector h) and the image radiance I( l), photometric stereo

seeks to recover the surface normal n. In our scenario, this

boils down to fitting an unknown surface of an ellipsoid of

revolution in R3 using points h on a spherical surface S2

whose lengths are re-scaled by (

I( l)

Cλ )1

4

. After the ellipsoid

is determined, by detecting its elongation the surface normal

can also be obtained.

Fitting an ellipsoid can be formulated into an energy

minimization problem, and we are able to retrieve the global

minimum from the solutions to a system of polynomials.

For simplification we denote P = √I( l) and ω = √ 1

Cλ

.

To get around the unit norm constraint on n, we also let

n = √(1 − λ)ω n, hence for each of the k observations,

Equation 10 can be rewritten as

Pi (ω − nT hihi

⊺

n) = 1,i = 1, 2, ··· , k,

(12)

where i is the observation index.

By averaging all k equations, we have ¯P =

∑k

i=1 P

k

,

¯H =

∑k

i=1 Pihihi

⊺

k

, and

ω =

1 + nT ¯n

¯P

.

(13)

Moreover, combining Equation 13 into Equation 12 leads to

nT (Pihihi

⊺

− Pi

¯H

¯P )

n =

Pi

¯P −

1,

(14)

a quadratic polynomial with respect to n = [ˆn1, ˆn2, ˆn3]T .

Therefore, all k equations in Equation 14 can be organized

into the matrix form

Mx = M[n2

1, n1n2, n1n3, n2

2, n2n3, n2

3]T = b,

(15)

so M ∈ Rk×6 and b ∈ Rk are established.

However, due to measurement noise and model approx-

imation, we do not enforce the equality in Equation 15

to hold strictly, instead we try to find the optimal n =

[ ˆn1, ˆn2, ˆn3]T that minimizes the following energy function

f(n) = Mx − b2

2 = xT MT Mx − 2bT Mx + bT b.

(16)

Since the cost function in Equation 16 is nonconvex, we

try to find its global minimizer by retrieving all its station-

ary points. Specifically, we solve the three-variable cubic

equations defined by the partial derivatives as

∂f

∂n1

= 0,

∂f

∂n2

= 0,

∂f

∂n3

= 0,

(17)

which is a three-variable cubic polynomial system that has

27 solutions. Since the system is homogeneous, the solu-

tions are positive-negative symmetric. Therefore, we only

need to examine 13 independent solutions. These facts

motivate us to develop a solver based on the symmetric

Gröbner basis [24]. To our best knowledge, this is the first

example for such technique to apply to photometric stereo.

Finally, λ and C can be determined consecutively using the

length of n.

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(a) 60 lights (b) 150 lights (c) 250 lights (d) 500 lights

Figure 5: Distribution of lights with various densities. As

illustrated, the lights from the behind make no effect on im-

age formation.

6. Experiments

We validate our model and evaluate our method on both

synthetic and real images. Since shadowing and inter-

reflections are not considered, we use PBRT [29] and

MERL BRDF [25] to render spheres as our synthetic input.

In the experiments, we compare our results with the results

produced by the Least Square (LS), Constrained Bivariate

Regression (CBR) [21] and Biquadratic (Bi-Quad) [33],

where the latter two offer the state-of-the-art performance

according to the benchmark test in [34]. We locate the di-

rection of the lights using a set of spiral points [31]. We also

apply our method on the helmet model from the USC “Light

Stage Data Gallery” [13] to evaluate its stability on specu-

lar materials, as well as on the “DiLiGenT” benchmark data

set [34] to examine its performance in diffusive cases.

6.1. Evaluation with Synthetic Images

We perform two types of experiments on synthetic im-

ages. First, we compare the angular estimation accuracy

obtained by the four methods under sparse (i.e. 60 lights)

and dense (i.e. 500 lights) light distributions, respectively,

then we investigate how the light density interacts with our

solver by applying it to 4 different lighting configurations

(i.e. 60, 150, 250, 500 lights, Figure 5). For CBR, we set

N1 = 2 and N2 = 4, and “retroreflective” on. Unless other-

wise stated, in our setup all methods take the original input

as is, and only positive pixel values are considered. This

means that only a subset of the lights contribute the actual

computation for a specific pixel.

6.1.1 Complementary Accuracy Traces

With respect to materials, our solution produces an accuracy

trace complementary to those produced by the other three

solutions targeting on low-frequency reflectance. Figure 6a

and 6b present the angular estimation errors in degrees over

the 100 materials in MERL, under both dense light distri-

bution (500 lights) and sparse light distribution (60 lights).

From the plots we draw two observations: (1) All methods

perform inferiorly over a specific set of materials, and this

Figure 7: Some materials exhibit highly localized specular-

ity, so its appearance is sensitive to light density. In terms

of model fitting, the pixels that carry specular signals (e.g.

point A) are more likely to be correctly estimated by our

solver than those do not (e.g. point B). Left: Estimation

error for specular green phenolic. Middle: A closer view

over the region showing both accurate and inaccurate es-

timations. Right: Appearances produced by four distinct

lights. Point B is “by-passed” by all lights so it does not

carry specular signals.

division performance over the materials is almost indepen-

dent of light density; (2) The peer methods outperform on

surface with diffusive reflectance, and our proposed solver

delivers better performance for specular surfaces with a few

exceptions (Section 6.1.2), which is consistent with our pre-

diction made in Section 3.4.

6.1.2 Impact of Light Density

We also study how the light density affects the accuracy of

our solver. Figure 6c compares the accuracy obtained un-

der various illumination densities: 500, 250, 150, 60 lights,

respectively. The main observation we make is that our so-

lution produces stable output for various lighting densities,

but the denser the distribution, the higher the accuracy is

achieved. More importantly, the gain brought by applying

denser light distribution is more significant for some partic-

ular specular surfaces, which are highlighted in red. For ex-

ample, “specular green phenolic”, as indicated in Figure 7,

exhibits extremely localized specularity with large estima-

tion error. This is because when light distribution is suf-

ficiently dense, many pixels (e.g. point B) do not exhibit

specular property at all, as compared with the pixels (e.g.

point A) covered by specularities, and they are less accu-

rately estimated as a consequence of mis-fitting the model.

Though uncommon in a photometric stereo setup, we ex-

pect to see that with a denser light distribution, estimation

on the specular materials highlighted in red in Figure 6 shall

continue to improve.

6.2. Evaluation with Real Images

We apply our solution to two image sets, “helmet side

right” from “Light Stage Data Gallery” and “DiLiGenT”.

The “helmet” image set contains specular appearances cap-

tured under 253 directional lights, and the “DiLiGenT” data

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(a) Mean estimation error in degrees produced by the four methods on sampled MERL reflectance using 60 lights.

(b) Mean estimation error in degrees produced by the four methods on sampled MERL reflectance using 500 lights.

(c) Mean estimation error in degrees produced by the our method on sampled MERL reflectance using 60, 150, 250 and 500 lights.

Figure 6: Performance evaluation on synthetic images rendered by PBRT using MERL. Materials that exhibit extremely

localized specularities are highlighted in red, the estimate accuracy of which rely on the light density for our method.

set mainly represents diffusive materials.

6.2.1 Performance on Specular Surfaces

Figure 8 visualizes the estimated normals with their respec-

tive +x, +y and +z components, with (1) upper sphere

lights only and (2) all 253 lights. We do not make quali-

tative comparison because ground truth is unavailable, but

qualitatively our method shows reasonable and consistent

results, indicating its stability. Besides, the convex shape of

the model is clearly illustrated by +x, +y and +z compo-

nents together. We also note that CBR delivers much more

reasonable results when only upper sphere lights are se-

lected (Figure 9). In general, light distribution makes a ma-

jor impact on estimation accuracy. It is reported that exist-

ing approaches shall perform better with properly adjusted

“position threshold” [34], so we will look into the details of

this factor in our future work.

6.2.2 Detection of Diffusive Reflectance

Among the ten models in “DiLiGenT”, “ball”, “reading”,

“cow” and “harvest” represent relatively more specular ma-

terials. Figure 10 compares the estimation error produced

by CBR, Bi-Quad and ours, together with the median value

of λ we obtained for each model. Here we assume that each

object is made of homogenous material, so a rough cross-

pixel analysis is allowed. It is interesting to observe that

except for “ball” and “reading”, the lower the λ value we

detect, the better the performance our method delivers. The

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+x

+y

+z

Normal

Figure 8: Normal maps obtained by our method for “helmet

front left”. Row 1: with upper sphere lights only. Row 2:

with all 253 lights.

Figure 9: Normal maps obtained by CBR and Bi-Quad.

From left to right: CBR with upper sphere lights only and

with all 253 lights; Bi-Quad with upper sphere lights only

and with all 253 lights.

Figure 10: Average estimation error in degrees produced by

the three methods. The black dotted line indicates the cor-

responding λ obtained for each model. Our model predicts

that the trace of λ should be consistent with the trace of

estimations error for homogeneous materials.

inferior performance on “ball” is due to limited light distri-

bution as discussed in Section 6c, and “reading” is inaccu-

rately estimated because surface non-convexity has caused

a significant amount of specular inter-reflections.

However, we would also like to point out that though λ

can correctly indicate the smoothness of the surface, a fail-

safe “switch” that allows us to roll back to the existing so-

lutions for low-frequency reflectance remains to be absent.

So, how to properly incorporate λ into a solution that han-

dles general reflectance remains as part of our future work.

7. Conclusion

In this paper, we derive a novel analytical microfacet-

based isotropic reflectance model from the ellipsoid nor-

mal distribution function. We also introduce a physically

interpretable approximate of our model that is particularly

serviceable for specular reflectance analysis. With this ap-

proximate, we identify that the problem for specular surface

normal recovery is essentially an ellipsoid of revolution fit-

ting problem, where the latter can be described by a system

of polynomials. And in order to solve this problem, we also

devise a fast, non-iterative and globally optimal solver. Ex-

periments on both synthetic and real images demonstrates

the superiority of our model and algorithm on the targeted

specular surfaces.

Currently, our approximate model and its polynomial

system solver are tailored for highly specular surfaces. To

extend the success into shape recovery of general isotropic

surfaces is left as our future work.

Acknowledgement

This work was supported in part by JSPS KAKENHI

Grant Number JP15H05918 and a project commissioned

by the New Energy and Industrial Technology Development

Organization (NEDO).

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