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1

Bi-polynomial Modeling of Low-frequency
Reflectances
Boxin Shi, Student Member, IEEE, Ping Tan, Member, IEEE, Yasuyuki Matsushita, Senior Member, IEEE,
and Katsushi Ikeuchi, Fellow, IEEE
Abstract—We present a bi-polynomial reflectance model that can precisely represent the low-frequency component of reflectance.
Most existing reflectance models aim at accurately representing the complete reflectance domain for photo-realistic rendering purposes.
In contrast, our bi-polynomial model is developed for the purpose of accurately solving inverse problems by effectively discarding thehigh-frequency component while retaining nonlinear variations in the low-frequency part. The bi-polynomial reflectance model is usefulfor estimating reflectance and shape of an object. Experimental evaluation in comparison with other parametric reflectance models
demonstrates that the proposed model achieves better performance in reflectometry and photometric stereo applications.
Index Terms—Low-frequency reflectance, Parametric BRDF model, Radiometric image analysis, Reflectometry, Photometric stereo

F

1

I NTRODUCTION

P

ARAMETRIC modeling of reflectances plays an importantrole in both rendering and inverse problems in radiometric image analysis. The vast majority of existing parametricreflectance models are developed for the purpose of photorealistic rendering. They are designed to have an accurate
representation of specular components and successfully applied to forward problems in computer graphics. However,these reflectance models are not necessarily suitable for inverse
problems in computer vision, such as reflectance and shapeestimation. In fact, many of these reflectance models severelycomplicate the inverse problems by introducing high nonlinearity when they are directly used in the computation, and as
a result, the solution methods are forced to involve unstableand expensive nonlinear (or even non-convex) optimization
procedures. While one could use a simplistic model to avoid
such a problem, e.g., the Lambert’s reflectance model, theaccuracy of estimates suffers from its discrepancy from thereal-world reflectance. Therefore, it is desired to develop areflectance model that well represents real-world reflectances
while retaining simplicity for inverse problems.
In forward rendering problems, one of the key challenges
is to accurately model specular reflection that exhibits highfrequency reflectance variations in the incident or exitantangular domain. Since the specular component significantly
varies across materials, in order to faithfully represent it, thespecular term of a reflectance model tends to become complex
and highly nonlinear. While it is essential for photo-realistic
rendering in computer graphics, explicit modeling of high• B. Shi and K. Ikeuchi are with Institute of Industrial Science, the Universityof Tokyo, Japan.
Email: {shi, ki}@cvl.iis.u-tokyo.ac.jp• P. Tan is with Department of Electrical and Computer Engineering,
National University of Singapore.
Email: eletp@nus.edu.sg• Y. Matsushita is with Microsoft Research Asia, Beijing, China.
Email: yasumat@microsoft.com

frequency specular reflections is seldom necessary for inverse
problems in computer vision, particularly when sparse lighting(such as a directional light) is used. In fact, in an image, mostof the pixels exhibit low-frequency reflections (close to diffuse
reflectances) for most materials under sparse lighting. For
example, high-frequency (strong specular) reflections are only
observed in a sparse manner at points where the surface normal
is close to the bisector of viewing and lighting directions.
Recent robust photometric stereo methods [1], [2] are built
upon similar observation, where high-frequency reflectances
are treated as outliers.
Motivated by this observation, we develop a compact parametric Bidirectional Reflectance Distribution Function (BRDF) model for radiometric image analysis using a bi-polynomial
representation. We design this model with two goals by
restricting it to isotropic BRDFs. First, it should be able tofaithfully represent low-frequency reflectances of a broad classof materials. Second, it should make the solution of inverse
problems tractable. Our model is built upon a factorized
form of bivariate BRDF models for isotropic materials [3],[4], where the BRDF is represented as a product of two
univariate functions of half and difference angles. We approximate these univariate functions by low-order polynomials.
In the preliminary version of this work [5], we employed
quadratic functions for these univariate functions. In this work,
we extend it to a general bi-polynomial model and perform
detailed analysis and validations across varying polynomial
orders. We further perform comprehensive comparisons withvarious BRDF models by applying this model to reflectometry
and photometric stereo. We show that accurate results can beobtained by analyzing the low-frequency reflectances with our
proposed model.
The rest of this paper is organized as follows: In thenext section, we discuss previous works in related areas.
In Section 3, we define low-frequency reflectance and show
its approximation by using low-intensity observations. In Section 4, we introduce the proposed bi-polynomial model. We
then assess our model by fitting measured BRDFs and compareIEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE

2

with existing parametric models. In Section 5 and Section 6,
we show applications of our model to reflectometry andphotometric stereo problems. Section 7 concludes the paper.

and assume the Ward reflectance model with spatially varying
diffuse reflection and homogeneous specular reflection. Boivin
and Gagalowicz [24] use the same BRDF model, but their
method deals with only a single image in a hierarchical anditerative framework. Hara et al.’s method [25] uses multiple
point light sources to estimate both illumination distribution
and reflectance represented by the Torrance-Sparrow model. There are recent works of reflectometry that use nonparametric bivariate BRDFs with a discrete table representation. Romeiro et al. propose reflectometry with/without
measured illumination [4], [19] by assuming that the isotropic
BRDFs can be represented using a 2-D table. Their evaluationshows that the 2-D representation is accurate for a majority of
the isotropic BRDFs.
In a shape estimation context, early photometric stereo
works [26], [27] are based on the Lambert’s reflectancemodel. Although the computation is simple, their performancedegrades on many real surfaces, which often exhibit nonLambertian reflectance. Some methods use four light sources
to avoid shadow and specularity [28], [29]. By using more
images and recent robust estimation techniques, the outliers
that deviate from the Lambertian assumptions can be efficiently detected and discarded through rank minimization [1]and sparse regression [2]. To make use of all the observed
data, more sophisticated parametric BRDF models have also
been used, such as the methods based on the Torrance-Sparrowmodel [30], [31], the Ward model [32], [33], and other multilobe models [34]. However, all these methods assume thediffuse reflectance to be Lambertian, which is not true forreal surfaces.
To deal with more general materials, especially those with
non-Lambertian diffuse reflection, some recent methods solve
the photometric stereo problem with reflectance symmetries,
such as isotropy and/or reciprocity. Alldrin et al. [35] exploit isotropy to estimate the azimuth angle of normals. By
assuming bivariate BRDFs, they estimate the elevation angleof normals and surface reflectances iteratively [18]. By further
assuming that the BRDFs can be projected as a 1-D monotonic
function, the elevation angle can also be estimated without
using iterative optimization [36]. A theory of reflectance symmetries and its application to photometric stereo is summarizedin [37]. Based on those properties, surface reconstruction
methods using a special lighting rig and multi-view images
are introduced in [38] and [39]. Photometric stereo can also
be applied to general diffuse surfaces by considering someconsensus properties [40]. Given hundreds of images, surface
normal can be estimated by exploring the similarity of radianceprofiles [41], [42] and attached shadow codes [43], under unknown illumination and unknown reflectance. Given thousandsof images, it is even possible to apply photometric stereo
to general anisotropic surfaces [44]. Although these methods
can deal with a great range of general materials, they usuallyrequire special imaging setup or complicated optimization. In
comparison, our bi-polynomial representation is compact yet
accurate for many real materials. As a result, data capture andoptimization becomes simple with our method.
Given photometric stereo images, biquadratic polynomials
are useful for representing the images that may include self-

2

R ELATED W ORKS

Reflectance Modeling. To precisely represent the appearanceof real-world materials, various parametric BRDF models havebeen developed over the decades. These BRDF models canbe categorized into physically-based and empirical models.
Physically-based models, such as the Torrance-Sparrow [6]and Cook-Torrance [7] models, are mostly designed basedon the microfacet theory. They assume that surfaces consistof shiny V -grooves with consideration of geometric attenuation (masking, shadowing, and inter-reflections) and Fresnel
effects. Based on a similar microfacet theory, the Oren-Nayarmodel [8] captures the reflectance of rough surfaces. The
empirical models such as the Phong [9] and Blinn-Phong [10]models are widely used because of their computational efficiency. The Lafortune model [11] uses generalized cosine
lobes. This model can fit measured reflectance data withhigh precision, but with unintuitive parameters. There arealso models that bridge these two categories by partly using
physically motivated terms, e.g., the Ward [12] and Ashikhmin
models [13]. The Ward model is also based on the microfacet
theory, but omits the Fresnel and geometric attenuation terms.
The Ashikhmin model is an anisotropic BRDF model witha non-Lambertian diffuse term. An experimental evaluationof various models with measured data can be found in [14].
Generally, parametric models are compact and easy to use forforward problems, but they are only accurate for a limitedclass of materials.
A BRDF can also be represented by a 4-D discrete tableindexed by lighting and viewing directions. For isotropic
materials, this representation can be reduced to a 3-D table [15], [16]. The 3-D table can be re-arranged using halfand difference vectors [17] through a re-parameterization as
suggested in [3]. It can be further reduced to a 2-D table fora wide range of isotropic materials by omitting the rotation ofdifference vector [4], [18], [19]. Although high-quality rendering can be achieved using the discrete table representations,such non-parametric forms are generally unsuitable for inverse
problems because the number of parameters to be estimatedbecomes prohibitively large. To maintain the accuracy while
reducing the complexity, recent approaches use various basis
functions to represent general BRDFs [20], [21], [22]. Our
bi-polynomial model shares a similar goal of simplifying the
BRDF representation for general materials, but we focus onmodeling low-frequency reflectances with a simpler parametricform. Furthermore, we aim to solve inverse problems forradiometric image analysis rather than forward rendering.
Radiometric Image Analysis. Radiometric image analysis
seeks to recover scene properties, such as reflectance andshape, from the recorded scene radiance. Here we briefly
review related work in reflectometry and photometric stereo,i.e., surface reflectance and shape estimation, respectively.
Most of the works in reflectometry are based on parametricreflectance models. Yu et al. [23] use a sparse set of photos

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE•

3

BRDF 1 (alum‐bronze) and 89 (violet‐acrylic)

Tlow

100%

3

50%

5%

0

1
2
3
4
Spherical harmonics order

5

0

RMSE

Fig. 2. Fitting errors of spherical harmonics with differentl = (0, 0, 1)T(1, 0, 1) T
(1, 0, 0)Tn = (0,0, 1)T to observations(1, 0, 1) T under(1, 0,
0)Tordersvarious
Tlow . The result isT
Fig. 1. An
low-frequency
Forillustration
normals onofthe
hemisphere, v =reflectance.
(0, 0, 1)T From
For lightsonthehemisphere,
v
=
(0,
0,
1)
square-rooted for a visualization purpose. The legend
left to right, spheres √illuminated√ by distant lights from shows the mapping between the error magnitude anddirections [0, 0, 1]> , [1/ 2, 0, 1/ 2]> , [1, 0, 0]> and viewed color. The black areas are undefined due to insufficientfrom [0, 0, 1]> are shown.
equations for fitting.
shadowing and interreflections as shown in polynomial texture
maps [45], which are able to interpolate images and synthesize
appearances under new lighting directions. A more recent
approach [46] extends the polynomial texture maps to handle
specularity and shadow. Our model uses a similar mathematical form for representing the low-frequency component ofgeneral isotropic reflectance.

3

L OW- FREQUENCY R EFLECTANCE

Let us consider the reflectance as a function of lighting,viewing, and surface normal directions. We use the term lowfrequency reflectance to denote the reflectance component
that does not abruptly change with the variation of lighting
directions. Note that this definition does not limit the lowfrequency component to diffuse reflectance, because it can
include a wide and blunt specular lobe.
On many real surfaces, strong specularity is only observed
when the surface normal is close to the bisector of lighting
and viewing directions. Thus, the majority of pixels in animage should present low-frequency reflectances under a directional (or sparse) lighting. We show synthetic spheres in
Figure 1 rendered using the measured BRDFs A LUM - BRONZE(top) and V IOLET- ACRYLIC (bottom) from the MERL BRDF
database [17]. The spheres are rendered under different lighting directions but a fixed viewpoint. The majority of pixelsof these renderings have smoothly varying values1 , which werefer to as low-frequency reflectance observations. For a static
scene observed from a fixed camera under a continuously moving light source, the low-frequency reflectances are observedat the pixels where intensities do not show sudden changes
under varying lighting directions. We seek to model such
low-frequency reflectances in a concise form by effectively
discarding the high-frequency reflectances for applications to
inverse problems.
1. These high-dynamic range (HDR) images are tone mapped using themethod of Reinhard et al. [47].

To capture low-frequency reflectances, we need a method
to identify these observations. As we will show below, thelow-frequency reflectances have strong correlation with lowintensity observations. Hence, we can simply use an intensity threshold Tlow to extract observations of low-frequency
reflectance in practice. For example, given a set of photometric stereo images, we may draw an intensity profile foreach pixel under varying lighting directions. After discarding
observations in shadow, we sort all the remaining observationsin an ascending order and keep only those ranked below apercentage Tlow .
To evaluate the effectiveness of this simple method, we fit
spherical harmonics of different orders to our low-intensity
observations and assess the fitting error. We perform fitting
for the intensity profile at each pixel, where the observation
is a function of lighting directions. The intensity profile y canbe fit by spherical harmonics ỹ represented asỹ(θ, φ) =

b
l
X
X

Clm Ylm (θ, φ),(1)

l=0 m=−lwhere (θ, φ) are elevation angle and azimuth angle of alighting direction, b is the order of spherical harmonics, Y isspherical harmonics basis functions, and C is the coefficient.
The coefficient C can be solved for by linear least squares. For
experimental validation, we fit Equation (1) to the observations
thresholded by Tlow using all of the 100 measured materials
in the MERL database [17]. Specifically, we fix the viewing
direction as [0, 0, 1]> and sample 1620 normals by uniformly
choosing 36 longitudes and 45 altitudes on the hemisphere. For
each of these normals, we use 100 lighting directions randomly
sampled from the hemisphere. In other words, we have 1620
intensity profiles, and the length of each is 100. We then vary
Tlow from 10% to 100% with a step of 10%, and vary theorder of spherical harmonics b from 0 to 5. For each intensity
profile of length f , given fixed b and Tlow , we compute therelative Root Mean Square Error (RMSE) [4], which is definedIEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCEnl

as polynomial functions of cos θh and cos θd , respectively. As
a result, our BRDF model becomes a bi-polynomial function
represented as

v

h
θh
θd

4

ρ(θh , θd ) 'ρ1 (θh )ρ2 (θd ) = ρ˜1 (n> h)ρ˜2 (l> h)

θd

=

k
X

Ai (n> h)i

i=0

o

k
X

Bj (l> h)j ,

(3)

j=0

where k is the order of polynomials. The above equation canbe further expanded by the following relaxation

Fig. 3. The definitions of θh and θd .

ρ̃(x, y) =

k X
k
X

Cij xi y j ,(4)

i=0 j=0asv
u f
X (yi − y˜i )21u
RMSE = t
,
f i=1
yi2

(2)

where y and ỹ are original and spherical harmonics fit values.
For each fixed b and Tlow , we perform this evaluation for
all 1620 normals and 100 BRDFs, and visualize the mean
RMSE in Figure 2. From the error distribution, we can seethat low-order spherical harmonics closely fit the observations
selected by a small Tlow below 50%, as indicated by theblue rectangle in the figure. Note that our representationin Equation (1) is different from the spherical harmonics
based BRDF representation in [48], which is not intended
for inverse problems. We use Equation (1) only to verify theeffectiveness of the intensity thresholding for selecting lowfrequency reflectances.

4

T HE B I - POLYNOMIAL BRDF M ODELThis section describes the proposed bi-polynomial BRDF
model. Given surface normal, lighting, and viewing directions n, l, and v, we can calculate the half vector ash = (l + v)/kl + vk, which is the bisector of l and v. Following the notations of [3], we use θh to denote the half angle
between n and h, and θd for the difference angle between l(or v) and h as illustrated in Figure 3. Hence, the following
relationships hold: n> h = cos θh and l> h = cos θd .
Our reflectance model is built upon the bivariate BRDF
model of [3], where it is shown that most of the isotropic
BRDFs can be represented as a bivariate function ρ(θh , θd ).
This representation is evaluated by [4] with a large numberof measured BRDFs [17] for the development of passivereflectometry. It is further discussed in [3] that any isotropic
BRDF based on the microfacet theory should consist of a
univariate function of θh , and its Fresnel term should be a
univariate function of θd . As shown in [13] and [49], themasking and shadowing terms in a microfacet-based BRDF
model vary smoothly and are actually close to constant.
These analyses motivate us to further simplify the bivariate
function ρ(θh , θd ) as a factorized form ρ1 (θh )ρ2 (θd ). Similar
simplification has been used in [50] to assist material capturing
and editing.
To obtain a compact parametric model suitable for inverse
problems, we represent the factorized terms ρ1 (θh ) and ρ2 (θd )where Cij = Ai Bj and x = n> h, y = l> h for notation
simplicity.
In this paper, we focus on discussing the bilinear, biquadratic and bicubic models, i.e., k = 1, 2, and 3. Let us take thebiquadratic model (k = 2) as an example. It can be expressedasρ˜1 (n> h)ρ˜2 (l> h) = A2 (n> h)2 + A1 (n> h) + A0
(5)
B2 (l> h)2 + B1 (l> h) + B0 ,with its linear relaxation asρ̃(x, y) =C22 x2 y 2 + C21 x2 y + C20 x2 + C12 xy 2 +C11 xy + C10 x + C02 y 2 + C01 y + C00 .

(6)

In the biquadratic case, there are 9 reflectance parameters in
the relaxed linear model in total, and we denote them in avector form as
x = [C22 , C21 , . . . , C00 ]> ∈ R9×1 .

(7)Note that the conversion from Equation (5) to Equation (6) is
unique, but the other direction is not, and Equation (6) may notalways have the product form of Equation (5). Equation (6)is a linear function of its parameters [C22 , C21 , · · · , C00 ]> ,while Equation (5) is a bilinear function of [A2 , A1 , A0 ]> and[B2 , B1 , B0 ]> . Hence, the relaxed model is easier to fit. The
bilinear and bicubic model can be defined similarly, with 4 and
16 reflectance parameters in their relaxed forms, respectively.
It is also straightforward to express the factorized terms ρ˜1and ρ˜2 in Equation (3) by higher-order polynomials, or evenuse different orders of ρ˜1 and ρ˜2 . But we experimentally foundthat models with higher orders had little advantage in modeling accuracy and were suffered from instability. Additional
discussions about the choice of orders of polynomials are left
for the experiment section.
There are also other possible parameterizations of thebivariate function. We choose polynomials mainly for two
reasons: 1) As discussed in [45], polynomials are goodat representing smooth intensity variations (low-frequency)
caused by different lightings; 2) Polynomials make the inverse problem tractable. For example, one might use theDiscrete Cosine Transform (DCT) as an alternative in thedomain of (θh , θd ) to represent the bivariate BRDF, which also
yields high modeling precision. However, when recovering
the unknown surface normal n, the problem becomes highly
nonlinear about n, because a DCT basis has the form of

BRDF fitting error (Isotropic)
Normal: thetaN 2 : 2: 90, phiN 10 : 10 : 360
Light: Random 100
MERL 100 BRDFs

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RMSE (instead of MAE)

cos(θh /k) = cos(arccos(n> h)/k), where k is a non-zero
integer. In contrast, our polynomial model is much simpler
and only involves terms like (n> h)k .
4.1 Relationship with other reflectance models
The bi-polynomial model can accurately represent the lowfrequency component of conventional dichromatic reflectancemodels [51], which represent reflectance as a summation ofLambertian diffuse and specular terms. Since the specular term
is mostly concentrated in the high-frequency component, thelow-frequency component of these models is largely Lambertian and can be represented well by our bi-polynomial
model. For example, the biquadratic model can be degradedto the Lambert’s model if we set A2 = A1 = B2 = B1 = 0
in Equation (5). The bi-polynomial model can also represent
the low-frequency component of other BRDF models that
rely only on n> h. For instance, the Blinn-Phong model [10]can be represented with the bi-polynomial model by setting
coefficients that are related to l> h (except B0 ) as zero.
The Cook-Torrance model [7] is widely used to represent
various surface reflectances. It consists of a Lambertian diffuse, and a specular term. Its specular component Sc can bedenoted as
DF G
ks
.
(8)
SC =
π (n> l)(n> v)
The terms D, F , and G are the microfacet distribution,
Fresnel, and geometrical attenuation terms, respectively. The
microfacet distribution D is represented as


1
1
−
>
2
(n h)
1
,
D=
exp 
(9)
4m2 (n> h)4m2
where m indicates the surface roughness, and D is clearly afunction of θh [7]. The Fresnel term F is often simplified by
the Schlick’s approximation [13], [52], denoted as
F = ks + (1 − ks )(1 − l> h)5 ,(10)where ks is a constant. Hence, F is a function of θd . Although
the term G is relatively complicated, it varies smoothly andis close to a constant over a large range of exitant angles as
evaluated in [13], [49]. Similar to the formulation of CookTorrance model, our model has both θh and θd terms in a
product form. More importantly, as evaluated in [5], the lowfrequency component of the Cook-Torrance model can beclosely approximated by the biquadratic model. Other models
based on the microfacet theory, such as the Ward model [12],have a similar expression for modeling the specular component, written as!
1 − n>1 h
ks
p
SW =
exp
.
(11)
m2
4πm2 (n> l)(n> v)
The low-frequency component can be purely Lambertian, ora mixture of Lambertian diffuse, and soft specular terms.
When the surface roughness m is large, the specular lobe will
become wide and blunt, thus the diffuse and specular cannot
be separated by simply using a threshold Tlow . Our model isdesigned for dealing with such cases.

0.005
DARK-REDPAINT

WHITEFABRIC2SILVERMETTALICPAINT2CHROME0

Fig. 4. BRDF
fitting
errorsof plot,themodel to*“a(b)” a =sorted BRDF
ID in the right(b) =biquadratic
original ID in the databaseall materials in the MERL database. The colors indicate
error magnitudes. The columns vary with Tlow , and therows correspond to different BRDFs ordered by the mean
fitting errors over columns. Some rendered spheres are
displayed on the left for reference.

4.2

Model validation using measured data

To verify the representation power of the proposed model, we
fit the biquadratic model to the measured BRDFs in the MERL
database. We use a similar experimental setup as Section 3
with the same set of n, l, and v. The threshold Tlow is appliedin the same manner for extracting low-frequency reflectances.
Note that the fitting experiment in Section 3 is performed foreach normal under varying lightings (or for each intensity
profile), while here we fit the BRDF model to observations
collected from different normal directions. For fitting extracted
low-frequency reflectances, we first solve Equation (6) via linear least squares, through which we obtain C22 , C21 , . . . , C00 .
By expanding Equation (5), we then establish a system ofbilinear equations as: C22 = A2 B2 , C21 = A2 B1 , . . . , C00 =A0 B0 , which can be written as C = ab> in a matrix form,
where C ∈ R3×3 stores C22 , . . . , C00 , a = [A2 , A1 , A0 ]> , andb = [B2 , B1 , B0 ]> . This system has multiple solutions. Here,
we use a singular value decomposition (SVD) as C = UΣV>√√
for obtaining the solution as a = u1 σ1 and b = σ1 v1> ,where u1 is the first column vector of U, v1> is the first row
vector of V> , and σ1 is the greatest singular value in Σ. The
solution is optimal in the least squares sense. To better fit themeasured data (not just to C), we further perform an iterative
optimization for refining a and b by first fixing a to updateb, and then b to update a.
The RMSEs of all BRDFs (rows) varying with Tlow
(columns) are shown in Figure 4. From the region within thelight blue rectangle, we can see that our model closely fits
low-frequency reflectances of different materials. As a generaltendency, our model has smaller fitting errors for materials
with broader specular lobes.
4.3

Comparison with other parametric models

There are parametric models with general diffuse terms, suchas the Lafortune and Ashikhmin2 models. In the Lafortune
2. We only consider the non-Lambertian diffuse terms of these two models.

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model [11], a rotationally-symmetric diffuse component DL
is written as
k > k
DL = Cd n> l
n v ,
(12)
where Cd and k are model parameters. The general diffuse
term of the Ashikhmin model [13] DA is defined as

5 !
5 !

n> v
n> l
1− 1−
, (13)DA = R 1 − 1 −22where R is the model parameter.
Several parametric models assume a Lambertian diffuse term, and use a microfacet-based specular component, such as the
Cook-Torrance and Ward models. According to Equation (8)and Equation (11), these models can be represented as

6

TABLE 1
BRDF fitting comparison.
The table shows average BRDF fitting errors (RMSE×10−4 ) over all materials for various reflectance models. The result of noise-free data is shown in thefirst row, and the other rows are the results with variant levels of additive noise
besides quantization. The two parameters, µ (×10−7 ) and λ (×10−4 ), are
the weights for signal-independent and signal-dependent noise, respectively.
µ/λNo Noise
0/0
1/5
5/5
5/10
5/30
10/5

Bicub.
6.71
16.04
21.00
24.43
49.06
174.88
26.27

Biquad.
7.18
16.79
21.82
25.57
50.13
177.57
26.92

Bilin.
9.01
18.47
22.98
26.04
51.94
187.40
30.57

Laf.
11.01
18.01
24.08
26.24
49.88
179.92
28.48

C.-T.
11.97
29.23
39.98
32.93
47.61
128.11
42.53

Lamb.
13.00
23.01
29.81
31.95
59.12
211.8436.37

kd+ ks S(n, l, v, m),(14)
πwhere kd and ks are model parameters representing thestrength of diffuse and specular terms, respectively; S isa nonlinear function with m as another model parameter.
In the Cook-Torrance model, m is encoded in the term D
in Equation (9).
We again use the MERL database for evaluation by setting
the threshold Tlow = 25% to extract the low-frequency
reflectances. Other experiment settings are the same as thoseused in model validation in Section 4.2. Fitting the Ashikhmin
model in Equation (13) is straightforward. For fitting theLafortune model in Equation (12), we take the logarithm atboth sides of the equation and estimate the log parameters
using linear least squares. For fitting the Cook-Torrance andWard models of Equation (14), we adopt a similar strategy
as [14] and use a Matlab function “lsqnonlin” to solve thenonlinear optimization.
The fitting errors are summarized in Figure 5, and meanerrors across all materials are listed in the first row of Table 13 .
We take the biquadratic case as an example to test both
the relaxed model of Equation (6) and the original modelof Equation (5). Their average fitting errors are 7.18×10−4 and7.97 × 10−4 , respectively, which suggests their approximatedequivalence in accuracy. Since the relaxed form is more
efficient in computation, for the bi-polynomial model we use
the relaxed form in the rest of the experiments.
Our model has a consistently smaller fitting error than thegeneral diffuse terms of the Lafortune and Ashikhmin models.
The Cook-Torrance and Ward models underperform our modelin representing the low-frequency reflectance, although specular terms are included in these models. This is mainly because
the Cook-Torrance and Ward models behave very similarly
as the Lambert’s model in low-frequency reflectance domain.
The bicubic model has the highest modeling accuracy for lowfrequency reflectance, while the bilinear model is less accurate
than the Lafortune model. In terms of modeling complexity,
the bi-polynomial model has simpler analytic forms than otherparametric models, and only a linear least squares fitting isrequired to estimate the reflectance parameters. We have also
tested a higher-order bi-polynomial model, i.e., the biquartic
model. However, the performance gain was rather limited andshowed an almost identical error curve with the bicubic model(which is omitted in Figure 5) with RMSE of 5.60 × 10−4 .
Therefore, we limit our discussion to polynomials up to thethird order.
Note that the BRDF fittings here are performed using only
low-frequency reflectances. The modeling accuracy of the
bi-polynomial model will be deteriorated at high-frequency
reflectances for materials with specularity. In such a case, themodels with the specular terms such as the Cook-Torrance
model may outperform our model. We refer the readers
to [14] for evaluations of various parametric BRDF models
in the complete BRDF domain (both low-frequency and highfrequency).

3. According to Figure 5, the result of the Ward model is very close to the
Cook-Torrance model, and error of the Ashikhmin model is much larger thanothers. Hence, these two models are omitted thereafter.

4. Usually, when dealing with the general BRDFs, HDR images are used
as done in [18], [39]. So we simulate 16-bit quantization instead of 8-bit LDR
images.

ρS =Evaluation using noisy data. The previous experiments are
performed using a carefully measured data (MERL BRDFs),
where a double precision is used for data storage. However,in practical scenarios, the image formation process involves
various types of noise and quantization errors. Therefore, we
simulate these factors and evaluate their influences on theperformances of different BRDF models. Here, we consider
the additive noise and 16-bit image quantization4 . We apply
both signal-independent and signal-dependent noise by adding√
them to the original signal as ỹ = y + (µ + λ y)X, which
is a commonly used noise model of an imaging sensor [53].
Here, ỹ and y are data with and without noise, µ and λ areweighting factors for signal-independent and signal-dependent
noise, respectively, and X ∼ N (0, 1) is a random variable
following a Gaussian distribution with mean and standard
deviation of 0 and 1. To apply quantization, we first cap
the data with a lower bound 10−6 and an upper bound 1.0.
The capped data are then uniformly quantized to 216 levels.
The original data are first corrupted by additive noise before
performing quantization. We vary µ and λ to change the noiselevels (µ = λ = 0 indicates the case where only quantizationnoise is applied).
The average BRDF fitting errors for noisy input are summa-

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE

7

-3

5

x 10

Bicubic
Biquadratic
Bilinear

RMSE

4

Ward
Ashikhmin

Lafortune
Cook-Torrance
Lambert

3
2
1

0

10

20

30

40

50
BRDF index

60

70

80

90

100

VIOLET-RUBBERDARK-RED-PAINT
YELLOW-PAINT
WHITE-ACRYLIC
ORANGE-PAINT
YELLOW-MATTE-PLASTICPEARL-PAINT
BLUE-RUBBERGOLD-METALLIC-PAINT3
LIGHT-BROWN-FABRIC
RED-FABRIC2
GREEN-METALLIC-PAINT
GOLD-PAINT
SPECULAR-MAROON-PHENOLIC
WHITE-DIFFUSE-BBALL
LIGHT-RED-PAINT
DARK-SPECULAR-FABRIC
SPECULAR-ORANGE-PHENOLICMAROON-PLASTIC
GREEN-PLASTIC
PINK-FABRICSILVER-METALLIC-PAINT2SPECULAR-VIOLET-PHENOLICSPECULAR-RED-PHENOLICGRAY-PLASTICPURPLE-PAINT
SILVER-PAINT
SILVER-METALLIC-PAINT
SPECULAR-YELLOW-PHENOLIC
BLUE-ACRYLIC
GOLD-METALLIC-PAINT2
WHITE-FABRIC
WHITE-MARBLE
RED-METALLIC-PAINT
DARK-BLUE-PAINT
YELLOW-PHENOLICVIOLET-ACRYLIC
NATURAL-209
ALUMINA-OXIDETEFLON
TWO-LAYER-GOLDPOLYURETHANE-FOAM
NEOPRENE-RUBBERNREEN-FABRIC
COLOR-CHANGING-PAINT2
PURE-RUBBERAVENTURNINE
SPECULAR-BLUE-PHENOLICIPSWICH-PINE-221
BLUE-FABRIC
SPECULAR-GREEN-PHENOLICGREEN-METALLIC-PAINT2FRUITWOOD-241
BLACK-OXIDIZED-STEEL
WHITE-FABRIC2
BLUE-METALLIC-PAINT2COLONIAL-MAPLE-223
RED-PHENOLICPINK-FELTDELRIN
PVC
RED-SPECULAR-PLASTICCHERRY-235
SPECIAL-WALNUT-224
RED-FABRICPINK-JASPER
BLACK-SOFT-PLASTIC
GREEN-ACRYLIC
WHITE-PAINT
BLACK-FABRIC
GREEN-LATEXBEIGE-FABRICPINK-FABRIC2
HEMATITE
SPECULAR-BLACK-PHENOLICBLACK-PHENOLICTWO-LAYER-SILVER
ALUM-BRONZEBLACK-OBSIDIAN
GOLD-METALLIC-PAINT
SILICON-NITRADE
SS440
PINK-PLASTIC
BLUE-METALLIC-PAINT
RED-PLASTICPICKLED-OAK-260
COLOR-CHANGING-PAINT3
ALUMINIUM
YELLOW-PLASTIC
SPECULAR-WHITE-PHENOLICBRASSNICKEL
COLOR-CHANGING-PAINT1
CHROMETUNGSTEN-CARBIDENYLON
CHROME-STEELPOLYETHYLENESTEELGREASE-COVERED-STEEL

0

Fig. 5. BRDF fitting comparison for various reflectance models and all materials in the MERL database. The Y -axis
shows the RMSE values; the X-axis shows BRDF names ordered by the fitting errors of the biquadratic model, withsome renderer spheres below for a visualization purpose.

rized in the second through seventh rows of Table 1. Comparedwith the noise-free result in the first row, the errors becomelarger with the increasing noise level as anticipated. The
bicubic model shows the best performance in the majority of
the cases, unless the noise is too strong (fifth and sixth rows).

5

A PPLICATION TO R EFLECTOMETRY

Although the bi-polynomial model is designed to represent
low-frequency reflectances, it can also be used for reflectometry of materials without significant specular spikes. Using thelinear representation of bi-polynomial model in Equation (4),reflectometry under directional light sources only requires
solving a linear equation Ax = i, where i records radiancevalues. For each observation, we can calculate the matrix A
from n, l, and v when the shape and lighting are all calibrated.
The matrix A has 4, 9 and 16 columns for bilinear, biquadratic,
and bicubic models respectively according to Equation (4). In
the biquadratic case for instance, from p (p > 9) independent
samples, the matrix A ∈ Rp×9 and observations i ∈ Rp×1
are constructed. The model parameter x can be determined bysimply solving the linear system as x = (A> A)−1 A> i. If
only one image of a curved surface under a directional lightsource is available, the reflectance shows only variations along
the half angle,
reduces to a univariate function
Pkour model>ias ρ(θh ) 'i=0 Ai (n h) . In such a case, the estimatedreflectance only shows the variations along the half angle.
Reflectometry using measured BRDFs. To verify themethod, we select some materials from the MERL database(e.g., fabric, matte-paint, rubber, etc.) that do not containstrong specularities. We render a single image of a sphere
under a directional light source as input to estimate the
BRDF. We test this simple reflectometry method using various
parametric BRDF models (i.e., bilinear, biquadratic, bicubic,
Lafortune, Cook-Torrance, and Lambert). We then reconstruct
images under the same lighting and viewing directions using
the estimated reflectance parameters and evaluate the reconstruction errors. The reconstruction errors are defined as themean of pixel-wise absolute difference.
Here we show the B LUE - FABRIC and G REEN - LATEX results as two examples in top two rows of Figure 6. We show
three rendered spheres using the ground truth BRDF, fittings
to the biquadratic model, and to the Lambert’s model for eachexample. The ground truth and estimated BRDFs of various
models are visualized as 2-D polar plots. As summarized in thelegends of Figure 6, the reconstruction errors show a similar
performance ordering to the result in Figure 5 and Table 1,
where the bicubic model performs the best, and the Lambert’s
model performs the worst. From the plotted BRDF slices, we
can see that the bicubic and biquadratic models fit closely
to the measured data even though they only require a few
parameters. The biquadratic model is a little bit worse than thebicubic model, but outperforms all other parametric models.
The bilinear model shows some inaccuracy, and is worse thanthe Cook-Torrance model but better than the Lafortune model.
Notice that this experiment setting is slightly different fromthat in Figure 5. Here, we fit the models to the complete
BRDF domain for materials without strong highlight, while
in Figure 5 we fit the models only to the low-frequency
component extracted by Tlow .

0-4IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE

Ground truth
Bicubic:4.93
Biquadratic:5.17
Bilinear:8.37
Lafortune:9.76
Cook-Torrance:6.83
Lambert:10.86

Ground truth

0-4

Biquadratic

0.02
Ground truth
Bicubic:14.56
Biquadratic:14.82
Bilinear:22.75
Lafortune:27.09
Cook-Torrance:18.58
Lambert:28.63

Ground truth

0.04

Biquadratic

Ground truth
Bicubic:2.08
Biquadratic:2.09
Bilinear:2.58
Lafortune:2.29
Cook-Torrance:2.15
Lambert:4.61

Lambertimages captured by a fixed camera under varying lightings.
We assume an orthographic camera and directional lightings.
The camera-centered coordinate system is chosen such thatv = [0, 0, 1]> . To use the bi-polynomial model for photometricstereo problems, we fit the reflectance model at each pixel
independently. Like previous methods that deal with spatiallyvarying BRDFs [18], our approach determines the BRDF foreach pixel from its intensity observations, lighting directions,and estimated surface normals. Therefore, our method is able
to handle spatially varying BRDFs.
6.1

0.04

0.02

10-2

Lambert8

Ground truth

Biquadratic

0.06
LambertAn iterative normal estimation method

The bilinear, biquadratic, and bicubic models can be used in
the same manner for solving photometric stereo. Here we use
the biquadratic model as an example. From the photometricstereo images, we observe multiple radiance intensities at
each pixel. For each pixel, we first use a very small intensity threshold (10−6 in our experiment for synthetic data)
to neglect shadows. Then we sort remaining observationsin an ascending order and keep only those ranked below
the percentage Tlow , which is empirically determined within[15%, 50%]. We use ilow to denote the concatenated vectorof these remaining observations and stack their corresponding
lighting directions to form a matrix Llow . We finally obtainthe following equation:
ilow = ρ̄(n, Llow ) ◦ (n> Llow ),

0.2

0.4

Fig. 6. Reflectometry results. Top two rows are fromsynthetic data, and the bottom row is from real data.
BRDF plots for all models and the rendered spheres using
measured data, the biquadratic model, and the Lambert’s
model are shown. The reconstruction errors (×10−4 for
top two examples, ×10−2 for the bottom example) for eachmodel are summarized in the legends. Each BRDF isvisualized as a 2-D curve, which is a polar plot with angle
as the elevation angle of surface normal and radius as
the reflectance magnitude. The surface normal with zeroazimuth angle is selected for a visualization in 2-D.

Reflectometry using real data. We also test the BRDF estimation capability using real data. A sphere under a directional
lighting is recorded. We show the result in the bottom rowof Figure 6, which is consistent with the synthetic test. The
bicubic model performs the best with a slight advantage overthe biquadratic model; modeling accuracy of the biquadraticmodel is much better than the bilinear model, and is close to
the bicubic model. Higher-order bi-polynomial models, suchas bicubic and biquartic, show only negligible improvementin our experiments, hence the biquadratic model is a goodtrade-off in terms of modeling accuracy and simplicity.

6

A PPLICATION TO P HOTOMETRIC S TEREOIn this section, we apply the bi-polynomial reflectance modelto photometric stereo for estimating surface normals from(15)

where “◦” indicates element-wise multiplication. ρ̄ encodes
the reflectance parameter x in the same manner as ρ̃in Equation (6), but operates on each n and l. The reflectance parameter x represents the 9 polynomial coefficients[C22 , C21 , · · · , C00 ]> for the biquadratic model, as defined in
Equation (7). We use the relaxed biquadratic model of Equation (6) for computational efficiency. Surface normal n
and the BRDF parameter x can be determined by iteratively
optimizing the following objective function:(n∗ , x∗ ) = argmin |ρ̄(n, Llow ) ◦ (n> Llow ) − ilow |2 .

(16)

n,x

At each iteration, we first fix the normal direction n
and refine x by computing a linear least squares. We thensubstitute x and n to determine ρ̄. Once ρ̄ is calculated, weupdate n again by a linear least squares. A normalization
to unit-norm is performed immediately after each time n issolved. To initialize this iterative optimization, we first apply
Lambertian photometric stereo [26] using ilow and Llow toestimate the initial normal. The iterative optimization stops
when the residual of Equation (16) does not change. In our
implementation, we stop it when the change becomes less than
10−7 or a maximum iteration of 100 times is exceeded. The
normal estimation algorithm is summarized as Algorithm 1.
Note that we take the biquadratic model as an example in
Algorithm 1, and the ρ̄ can be replaced by other parametricmodels. Theoretically, to optimize n with fixed x, minimizing
a multivariate polynomial system using Gröbner basis [54]is one of the solutions. But through experimentation, we
find our simpler approach outlined above produces equivalent
convergence with much lighter computation.

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE

10

Bicubic
Biquadratic
BilinearAngular Error in Degrees

89

Lafortune
Cook-Torrance
Lambert

6
4
2

0

10

20

30

40

50
BRDF index (reordered)

60

70

80

90

100

ORANGE-PAINT
WHITE-DIFFUSE-BBALLYELLOW-PAINT
LIGHT-RED-PAINTPEARL-PAINT
SILVER-PAINT
NATURAL-209
WHITE-ACRYLIC
VIOLET-RUBBERDARK-RED-PAINT
BLUE-RUBBERDARK-BLUE-PAINT
DARK-SPECULAR-FABRICPINK-FABRICTEFLONVIOLET-ACRYLIC
SPECULAR-ORANGE-PHENOLIC
PURE-RUBBERYELLOW-MATTE-PLASTIC
SPECULAR-VIOLET-PHENOLIC
LIGHT-BROWN-FABRIC
RED-FABRIC
WHITE-FABRIC
GOLD-PAINT
WHITE-FABRIC2POLYURETHANE-FOAM
SILVER-METALLIC-PAINT2GREEN-PLASTIC
RED-FABRIC2
GRAY-PLASTIC
GREEN-METALLIC-PAINT
PVC
SPECULAR-MAROON-PHENOLICPINK-FELTPURPLE-PAINT
ALUMINA-OXIDEGOLD-METALLIC-PAINT3
SPECIAL-WALNUT-224
NEOPRENE-RUBBER
RED-METALLIC-PAINT
COLOR-CHANGING-PAINT2AVENTURNINE
GOLD-METALLIC-PAINT2GREEN-FABRIC
SPECULAR-YELLOW-PHENOLIC
BLUE-METALLIC-PAINT2IPSWICH-PINE-221
DELRIN
BLUE-FABRICMAROON-PLASTIC
SPECULAR-RED-PHENOLIC
WHITE-PAINT
BLACK-SOFT-PLASTIC
TWO-LAYER-SILVER
BLUE-ACRYLIC
GREEN-LATEXNYLONRED-PHENOLICFRUITWOOD-241
SILVER-METALLIC-PAINTPINK-PLASTIC
GREEN-ACRYLIC
SPECULAR-GREEN-PHENOLICYELLOW-PLASTIC
RED-PLASTIC
BLACK-PHENOLIC
ALUM-BRONZEGREEN-METALLIC-PAINT2BLACK-OXIDIZED-STEEL
SPECULAR-BLUE-PHENOLICPOLYETHYLENEBLACK-FABRIC
BLUE-METALLIC-PAINT
RED-SPECULAR-PLASTIC
WHITE-MARBLE
COLONIAL-MAPLE-223
TWO-LAYER-GOLD
GOLD-METALLIC-PAINT
CLOR-CHANGING-PAINT3SPECULAR-BLACK-PHENOLICBLACK-OBSIDIANSS440
HEMATITE
PINK-FABRIC2PINK-JASPER
CHROME-STEELYELLOW-PHENOLIC
CHERRY-235
BEIGE-FABRIC
CHROMECOLOR-CHANGING-PAINT1BRASSSPECULAR-WHITE-PHENOLICSILICON-NITRADE
ALUMINIUMTUNGSTEN-CARBIDESTEEL
NICKELPICKLED-OAK-260
GREASE-COVERED-STEEL

0

Fig. 7. Photometric stereo results comparison for various reflectance models and all materials in the MERL database.
The Y -axis shows mean angular errors (degree); the X-axis shows BRDF names ordered by the mean angular errorsof the biquadratic model, with some selected rendered spheres below for a visualization purpose.
Ground truth

Bicubic

Biquadratic

Bilinear

Lafortune

Cook-Torrance

Lambert (low)Lambert (all)

45

4.01

3.11

4.61

6.77

5.18

5.27

22.00

0

Fig. 8. Photometric stereo results using synthetic data. One of the input images is shown under the ground truth
normal map. Normal map estimates using different BRDF models are shown in the top row; the bottom row shows
angular difference maps w.r.t. the ground truth. The numbers on the difference maps show mean angular errors(degree).

6.2

Surface normal estimation results

Using the above iterative solution method, we perform surface
normal estimation using the bi-polynomial model in comparison with other parametric BRDF models. For other parametric
BRDF models, such as the Cook-Torrance and Lafortune
models, we use the same iterative solution method for deriving
the surface normal (simply replacing ρ̄ with a designatedmodel). We use the same dataset as used in Section 4.2
and Tlow is set to 25%. Note that although the intensity
thresholding is applied in the same way, for photometric stereo
the BRDF fitting is performed for each pixel (intensity profile),
while for the experiment in Section 4.2 the BRDF fitting is
performed for all pixels with their corresponding lightings.

When the Cook-Torrance and Lafortune models are used, theestimation of their BRDF parameters by fixing n becomes
highly nonlinear. This causes some numerical instability. A
similar issue arises when a high-order bi-polynomial model isused (empirically, higher than cubic).
The results for all 100 materials are summarized in Figure 7
with mean errors listed in the first row of Table 2 as aquantitative evaluation. Among all the tested models, thebiquadratic model performs the best on the average over 100
materials. The mean error from the bilinear model is larger
than the biquadratic case due to the model’s poor accuracy, andthe bicubic case also has larger errors than the biquadratic casedue to the instability caused by high-order polynomial fitting.

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE

10

Algorithm 1 Normal estimation
INPUT: Scene radiance values i, lighting directions L,
threshold Tlow .
for each pixel doExtract ilow and Llow using Tlow ;Solve for initial n with ρ̄ as constant (Equation (15));
while resid. Equation (16) > 10−7 OR #iter. ≤ 100 do
Update ρ̄ by fixing n (e.g., Equation (6));
Update n by fixing ρ̄ (Equation (15));
end while
end forOUTPUT: Estimated surface normal n for all pixels.

reflectance models by applying different noise levels in thesecond through seventh rows of Table 2. The results are
generally consistent with the noise-free case (first row). Whenthe signal-dependent noise level is very strong (sixth row), thebilinear model performs best due to its simpler form. As shownin the seventh row when the magnitude of signal-independent
noise reaches the shadow threshold (10−6 ), all models show
worse performances than the initial results (Lambert). Thisis because shadowed observations are mis-classified as lowfrequency reflectances and results in unpredictable errors.
Increasing the shadow threshold might solve this problem, but
how to choose the optimal threshold remains difficult.

TABLE 2
Photometric stereo result comparison.
The table shows the mean angular errors (degree) of surface normal estimatesover all materials for various reflectance models. The result of noise-freedata is shown in the first row, and the other rows are the results with variant
levels of additive noise besides quantization. The two parameters, µ (×10−7 )and λ (×10−4 ), are the weights for signal-independent and signal-dependent
noise respectively.
µ/λNo Noise
0/0
1/5
5/5
5/10
5/30
10/5

Bicub.
1.25
1.42
1.48
1.85
1.98
2.79
6.38

Biquad.
1.12
1.34
1.39
1.81
1.94
2.68
6.57

Bilin.
1.37
1.58
1.60
1.98
2.03
2.35
6.56

Laf.
4.07
4.17
4.17
4.30
4.33
4.55
6.47

C.-T.
2.13
3.18
3.19
3.40
3.43
3.69
6.13

Lamb.
2.14
2.28
2.28
2.46
2.49
2.73
5.04

For the Cook-Torrance and Lafortune models, their errors arelarger than the bi-polynomial model partly due to their lower
modeling accuracy of low-frequency reflectances. With only
having a Lambertian diffuse term, the Cook-Torrance model
hardly improves the accuracy from the initial result of theLambert’s model. In addition, these two nonlinear models
cause optimization difficulties, which also explains the larger
error of the Lafortune model.
Our model allows a simple alternating optimization whileother reflectance models require a more sophisticated optimization technique due to their highly nonlinear nature. In
our experiments, the bilinear, biquadratic, and bicubic models
usually converge in a similar manner with fewer than 10
iterations. The Cook-Torrance model shows either a quick
convergence after one or two iterations, or shows no decrease
from the initial value, since its low-frequency term is thesame as the Lambert’s model. The Lafortune model shows
instability during the iterations, which is caused by boththe nonlinearity of the model and the simple optimization
technique that is employed. Though there is no theoreticalguarantee for the convergence in our optimization method, we
empirically find that this simple optimization technique shows
better convergence for our model than for other models.
Evaluation using noisy data. We evaluate the effect of
noisy input to photometric stereo estimates across different
parametric BRDF models. The noise is applied in the same
way as in Section 4.3. We summarize the mean angularerrors of photometric stereo across all materials for various

Results on spatially varying BRDFs. We also generate synthetic images with spatially varying BRDFs to show the photometric stereo results. We use two different BRDFs, namely, theB LUE - METALLIC - PAINT and GREEN - METALLIC - PAINT materials from the MERL database on a C AESAR model, and render
100 images under varying illumination as input. The estimatednormal maps and their angular errors are shown in Figure 8.
We linearly map x, y, z components of surface normal to theR, G, B color channels, as indicated by a reference sphereshown in the ground truth normal map. We add the resultsof the traditional Lambertian photometric stereo [26] with all
images (“Lambert (all)”) for observing its failure mode on
those materials. By only using the low-frequency reflectance,
photometric stereo works much better on general reflectanceseven using the Lambert’s model (“Lambert (low)”), thus it
can provide reasonable initial normals used for fitting allother parametric BRDF models. The performance ranking ofmodels by mean angular errors here is consistent with Figure 7
and Table 2; the biquadratic model yields the most accurate
normal estimates.
6.3

Effect of varying numbers of lightings

We perform a similar experiment as Section 6.2 with varying
numbers of lighting directions (images) to observe its effect on
the accuracy. We vary the number of lighting directions from25 to 250 with a step of 25, and perform photometric stereo
using various BRDF models. The threshold Tlow is fixed to
25% in this experiment. The mean angular errors over 100
materials of MERL database are plotted in Figure 95 . Here
we do not compare with the Lafortune model because of its
unstable convergence. Generally, a larger number of images
improves the results. Compared with the Cook-Torrance model, the bi-polynomial model shows greater improvement on
accuracy with the increasing number of light directions, but
the Cook-Torrance model still shows a similar performance tothe Lambert’s model. When the number of lighting directions
becomes greater, the Cook-Torrance model even shows aslightly worse result than the Lambert’s model due to its
numerical instability during the optimization. Note that in our
experiment, the fitting is performed using the low-frequency
reflectance data.
5. The biquadratic and bicubic models need at least 9 and 16 equations forfitting, therefore their curves start from using 50 and 75 images, respectively,
when Tlow = 25%.

Angu

1
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25

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE

2
1

Angular Error in Degrees

Cook-Torrance:4.56
Lambert:5.53
Biquad./noise:5.17

With more input images, the low-frequency reflectances can
2
be more reliably extracted using Tlow . Therefore, regardless of0
the ordersof 30
polynomials,
results generally
10 20
40 50 60 the
70 bi-polynomial
80 90 100(%)number of input images increases.
become better as TthelowEmpirically, about 100 images are sufficient for photometricstereo with the biquadratic model to produce accurate results
for general isotropic reflectance. On average, the angular error
becomes about 1◦ for the materials in the MERL database. The
major reason for requiring many images is that the method
involves the estimation of BRDFs, and it requires a sufficient
sampling resolution in the angular domain. Similar numbersof images have been used for other state-of-the-art techniquessuch as [18], [36], [42]. When there is noise in the input data
(curve “Biquad./noise” in Figure 9), the angular errors increase
in all settings of different numbers of input lightings, but ingeneral more lights yield higher accuracy.
6.4

10

11

Analysis on intensity threshold Tlow

Extracting the low-frequency reflectance observations plays animportant role in our BRDF modeling. We now analyze theperformance variation using various Tlow to see its influenceon the photometric stereo results.
Performance variation with diverse Tlow . Again, we perform
similar experiments as Section 6.2 and Section 6.3 with thenumber of lighting directions fixed to 100. We plot the mean
angular errors of 100 materials with varying Tlow from 5% to100% with a step of 5% in Figure 106 . The overall tendency isthat all models show larger errors with increasing Tlow . Thisshows that as more observations of high-frequency reflectances
are involved, the problem of photometric stereo becomes moredifficult. This observation agrees well with our motivation of
focusing only on the low-frequency reflectance. Degradation of
the Lambertian photometric stereo results with increasing Tlow
also influences the results of all other models, because they
all rely on the Lambertian photometric stereo for initialization.
For the Cook-Torrance model, we find that it performs very
similar to the Lambert’s model when Tlow is smaller than
30%, but it outperforms the bi-polynomial model when Tlow
is larger than 75%. For the latter case, the specularity becomes
6. The biquadratic and bicubic models need at least 9 and 16 equations
for fitting, therefore their curves start from using Tlow = 10% and 20%,
respectively, when 100 images are given as input.

Bicubic:4.65
Biquadratic:4.16
Bilinear:4.76
Cook-Torrance:4.56
Lambert:5.53
Biquad./noise:5.17

8
6
4
2
0

50 75 100 125 150 175 200 225 250
Number of different lighting directions

Fig.12 9. Angular error (degree) varying with number oflightings.
The numbers in the legend are the mean values
10
Bicubic:4.65 to theover X-axis. The “Biquad./noise” case corresponds
8
same noise level as the sixth row of Table 2.Biquadratic:4.16
Bilinear:4.76
4Angular Error in Degrees

Angular Error in Degrees

Bicubic:1.37
Biquadratic:1.09
Bilinear:1.49
Cook-Torrance:2.31
Lambert:2.30
Biquad./noise:2.69

3

6

50 75 100 125 150 175 200 225 250
Number of different lighting directions

12

4

0
25

Biquad./noise:2.69

10

20

30

40

50 60
Tlow (%)

70

80

90

100

Fig. 10. Angular error (degree) varying with Tlow . The
numbers in the legend are the mean values over X-axis.
The “Biquad./noise” case corresponds to the same noiselevel as the sixth row of Table 2.

significant, and it becomes necessary to model high-frequency
components with explicit specular terms.
As for the difference between the bilinear, biquadratic,
and bicubic models, we find that the biquadratic model still
performs best on average across varying Tlow . However, when
Tlow is larger than 55%, the bicubic model improves due to its
greater representation ability. When Tlow is very small (below20%), the bilinear model can be more stably estimated due toits simpler form, and the reason here is similar to the increased
noise cases in the bottom three rows of Table 2. According
to Figure 10, a Tlow around 20% is a good choice for the
bi-polynomial model with ideal data.
When there is noise in the input data (curve “Biquad./noise”
in Figure 10), the angular errors increase accordingly. To
handle the input data with such large noise, a larger Tlow isdesired for obtaining the optimal performance. Fortunately, as
discussed above, our method is not sensitive to the choice ofTlow in the range of [20%, 50%], therefore we can pick a Tlow
that could be larger than necessary.
What materials are (in)sensitive to Tlow ? From Figure 10 wecan see that a Tlow below 50% is safe to use for various BRDFs. However, to determine the best Tlow for each material is noteasy in our empirical model. If we plot and check the curve ofangular errors varying with Tlow for each material, we can findsome material-related properties. We again take the biquadraticmodel as an example and plot the curves in Figure 11. We
plot thin curves for all materials and use a thick curve for
their average in the same figure. Based on the shape of thesecurves, we find that the materials can be categorized into twogroups according to the sensitivity of normal estimation error
to Tlow . The sensitive materials show a sudden increase oferrors when Tlow is greater than 50%, while the insensitive
materials show almost constant error curves. To check the typeof materials in each group, we look at the point of Tlow = 70%
and the corresponding materials with large and small errors.
We list the best nine and worst nine materials in Figure 11
with rendered spheres and material names. Interestingly, theerrors of metal-like materials are sensitive to Tlow and that of
fabric-like materials are insensitive. Our method is not suitablefor dealing with materials having relatively broad and strong
highlight, like some metallic paint. But when the highlightis very focused or it is very weak, our method works well Sensitive: very large error (~20)
 Insensitive: very small error (<0.7)

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE

12

25

Angular Error in Degrees

PINK-FABRIC

ORANGE-PAINT
GREEN-PLASTIC

20

RED-METALLIC-PAINT
CHROME
BLUE-METALLIC-PAINT2

15

WHITE-ACRYLIC
SPECULAR-MAROON-PHENOLICSPECULAR-VIOLET-PHENOLIC

10

BLUE-METALLIC-PAINT
COLOR-CHANGING-PAINT3COLOR-CHANGING-PAINT1

5

0
10

20

30

40

50
60
Tlow (%)

70

80

90

100 SPECULAR-ORANGE-PHENOLIC SPECULAR-RED-PHENOLIC
BLUE-RUBBERSILVER-METALLIC-PAINT
CHROME-STEEL

STEEL

Fig. 11. Angular error (degree) varying with Tlow for the biquadratic model. Each thin curve represents the result of
one material; the thick red curve is the average of 100 materials. The materials in dark green and red frames areinsensitive and sensitive material examples to Tlow .
with relatively arbitrary selections of Tlow . This is consistent
with the above discussions, since wide and strong specularities
contain significant high-frequency reflectances which cannot
be easily discarded by Tlow , while spiky specular lobes areeasily separable.
6.5 Results using real-world data
We show the results of real-world data in Figure 12. We use
the biquadratic model, since it is found to be the optimal
one according to the synthetic test. We compare with themethod from Alldrin et al. [18] using their datasets in the
top three rows. We refer to these scenes as (from top tobottom row) GOURD 1 (102), GOURD 2 (98), and APPLE (112),
with the number of input images in the paranthesis. Since
we do not have the ground truth of these data, a quantitativeevaluation cannot be performed. For a qualitative evaluation,
we reconstruct the surfaces using the estimated normals andthe method in [55]. In Figure 12, the left column shows oneof the input images to the photometric stereo algorithm and areference image (not used in calculation) of rendered resultsfrom [18]. The middle and right columns show the estimatedsurface normals and recovered surfaces using our method andthe Lambertian photometric stereo. The recovered surfaces arealigned to the reference views, and our reconstruction agrees
closely with the result of [18]. The data in the bottom tworows (named POST (91) and TEAPOT (73)) were captured witha Sony XCD-X710CR camera with a linear response function.
While we did not carefully control exposures to avoid saturation, our method can naturally skip undesired strong specularand saturation regions by setting Tlow . The consistency of
the reconstructed surfaces using estimated normals with thepictures of the objects taken from another viewpoint indicatesthe effectiveness of the proposed approach.

7

C ONCLUSIONSWe present the bi-polynomial reflectance model for representing low-frequency reflectances of isotropic surfaces. The

proposed reflectance model accurately captures the general
diffuse reflectance that spans in the low-frequency domain
in comparison with other parametric BRDF models and isuseful for inverse problems such as reflectometry and surface
normal recovery. We make comparisons with existing parametric models and demonstrate the usefulness of the proposed
BRDF model. We also discuss the choices of different ordersof polynomials, and conclude that the biquadratic model isgenerally the most suitable. The current model is limited toisotropic materials. In future work, we hope to analyze thecharacteristics of the low-frequency component of anisotropic
reflectance and extend our bi-polynomial model to a wider
variety of materials.

ACKNOWLEDGEMENTS
The authors would like to thank Yinqiang Zheng for helpful
discussion. P. Tan was supported by the Singapore ASTARPSF grant R-263-000-698-305.

R EFERENCES
[1]

[2]
[3]
[4]
[5]
[6]
[7]

L. Wu, A. Ganesh, B. Shi, Y. Matsushita, Y. Wang, and Y. Ma, “Robust
photometric stereo via low-rank matrix completion and recovery,” inProc. of Asian Conference on Computer Vision (ACCV), 2010, pp. 703–
717.
S. Ikehata, D. Wipf, Y. Matsushita, and K. Aizawa, “Robust photometricstereo using sparse regression,” in Proc. of IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2012, pp. 318–325.
S. Rusinkiewicz, “A new change of variables for efficient BRDF representation,” in Rendering Techniques (Proc. of Eurographics Workshop
on Rendering), 1998, pp. 11–22.
F. Romeiro, Y. Vasilyev, and T. Zickler, “Passive reflectometry,” in Proc.
of European Conference on Computer Vision (ECCV), 2008, pp. 859–
872.
B. Shi, P. Tan, Y. Matsushita, and K. Ikeuchi, “A biquadratic reflectancemodel for radiometric image analysis,” in Proc. of IEEE Conference on
Computer Vision and Pattern Recognition (CVPR), 2012, pp. 230–237.
K. Torrance and E. Sparrow, “Theory for off-specular reflection fromroughened surfaces,” Journal of the Optical Society of America, vol. 57,
no. 9, pp. 1105–1114, 1967.
R. Cook and K. Torrance, “A reflectance model for computer graphics,”
ACM Trans. on Graph., vol. 1, no. 1, pp. 7–24, 1982.

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE

Image (input/reference)

13

Biquadratic (normal/surface)

Lambert (all) (normal/surface)Fig. 12. Photometric stereo results using real-world data in comparison with the results of [18] and Lambertian
photometric stereo. The top three rows of data are courtesy of Alldrin et al. [18].

[8]M. Oren and S. Nayar, “Diffuse reflectance from rough surfaces,” inProc. of IEEE Conference on Computer Vision and Pattern Recognition(CVPR), 1993, pp. 763–764.
[9] B. Phong, “Illumination for computer generated pictures,” in Proc. of
SIGGRAPH, 1975, pp. 117–126.
[10] J. Blinn, “Models of light reflection for computer synthesized pictures,”in Proc. of SIGGRAPH, 1977, pp. 192–198.
[11] E. Lafortune, S. Foo, K. Torrance, and D. Greenberg, “Non-linear
approximation of reflectance functions,” in Proc. of SIGGRAPH, 1997.
[12] G. Ward, “Measuring and modeling anisotropic reflection,” Computer
Graphics, vol. 26, no. 2, pp. 265–272, 1992.
[13] M. Ashikhmin and P. Shirley, “An anisotropic Phong BRDF model,”Journal of Graphics Tools, vol. 5, no. 2, pp. 25–32, 2000.
[14] A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of BRDF
models,” in Proc. of Eurographics Symposium on Rendering (EGSR),
2005, pp. 117–126.
[15] K. Dana, B. Ginneken, S. Nayar, and J. Koenderink, “Reflectance andtexture of real world surfaces,” ACM Trans. on Graph., vol. 18, no. 1,
pp. 1–34, 1999.
[16] S. Marschner, S. Westin, E. Lafortune, K. Torrance, and D. Greenberg,“Image-based BRDF measurement including human skin,” in Proc. ofEurographics Symposium on Rendering (EGSR), 1999, pp. 139–152.
[17] W. Matusik, H. Pfister, M. Brand, and L. McMillan, “A data-driven
reflectance model,” in Proc. of SIGGRAPH (ACM Trans. on Graph.),2003, pp. 759–769.
[18] N. Alldrin, T. Zickler, and D. Kriegman, “Photometric stereo with
non-parametric and spatially-varying reflectance,” in Proc. of IEEE
Conference on Computer Vision and Pattern Recognition (CVPR), 2008.
[19] F. Romeiro and T. Zickler, “Blind reflectometry,” in Proc. of European
Conference on Computer Vision (ECCV), 2010, pp. 45–58.
[20] T. Zickler, R. Ramamoorthi, S. Enrique, and P. Belhumeur, “Reflectancesharing: Predicting appearance from a sparse set of images,” IEEE Trans.
Pattern Anal. Mach. Intell., vol. 28, no. 8, pp. 1287–1302, 2006.
[21] K. Nishino, “Directional statistics BRDF model,” in Proc. of International Conference on Computer Vision (ICCV), 2009, pp. 476–483.
[22] M. Chandraker and R. Ramamoorthi, “What an image reveals aboutmaterial reflectance,” in Proc. of International Conference on Computer
Vision (ICCV), 2011, pp. 1076–1083.

[23] Y. Yu, P. Debevec, J. Malik, and T. Hawkins, “Inverse global illumination: Recovering reflectance models of real scenes from photographs,”
in Proc. of SIGGRAPH, 1999, pp. 215–224.
[24] S. Boivin and A. Gagalowicz, “Image-based rendering of diffuse, specular and glossy surfaces from a single image,” in Proc. of SIGGRAPH,
2001, pp. 107–116.
[25] K. Hara, K. Nishino, and K. Ikeuchi, “Mixture of spherical distributions
for single-view relighting,” IEEE Trans. Pattern Anal. Mach. Intell.,vol. 30, no. 1, pp. 25–35, 2008.
[26] R. Woodham, “Photometric method for determining surface orientationfrom multiple images,” Optical Engineering, vol. 19, no. 1, pp. 139–144,
1980.
[27] W. Silver, “Determining shape and reflectance using multiple images,”
Master’s thesis, MIT, 1980.
[28] E. Coleman and R. Jain, “Obtaining 3-dimensional shape of textured andspecular surfaces using four-source photometry,” Computer Graphics
and Image Processing, vol. 18, no. 4, pp. 309–328, 1982.
[29] F. Solomon and K. Ikeuchi, “Extracting the shape and roughness ofspecular lobe objects using four light photometric stereo,” IEEE Trans.
Pattern Anal. Mach. Intell., vol. 18, no. 4, pp. 449–454, 1996.
[30] G. Kay and T. Caelly, “Estimating the parameters of an illumination
model using photometric stereo,” Graphical Models and Image Processing, vol. 57, no. 5, pp. 365–388, 1995.
[31] A. Georghiades, “Incorporating the Torrance and Sparrow model ofreflectance in uncalibrated photometric stereo,” in Proc. of International
Conference on Computer Vision (ICCV), 2003, pp. 816–823.
[32] H. Chung and J. Jia, “Efficient photometric stereo on glossy surfaces
with wide specular lobes,” in Proc. of IEEE Conference on Computer
Vision and Pattern Recognition (CVPR), 2008.
[33] D. Goldman, B. Curless, A. Hertzmann, and S. Seitz, “Shape andspatially-varying BRDFs from photometric stereo,” IEEE Trans. PatternAnal. Mach. Intell., vol. 32, no. 6, pp. 1060–1071, 2010.
[34] P. Tan, S. Lin, and L. Quan, “Subpixel photometric stereo,” IEEE Trans.
Pattern Anal. Mach. Intell., vol. 30, no. 8, pp. 1460–1471, 2008.
[35] N. Alldrin and D. Kriegman, “Toward reconstructing surfaces with
arbitrary isotropic reflectance: A stratified photometric stereo approach,”in Proc. of International Conference on Computer Vision (ICCV), 2007.
[36] B. Shi, P. Tan, Y. Matsushita, and K. Ikeuchi, “Elevation angle fromIEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE

14

reflectance monotonicity: Photometric stereo for general isotropic reflectances,” in Proc. of European Conference on Computer Vision(ECCV), 2012, pp. 455–468.
[37] P. Tan, L. Quan, and T. Zickler, “The geometry of reflectance symmetries,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 33, no. 12, pp.
2506–2520, 2011.
[38] M. Chandraker, J. Bai, and R. Ramamoorthi, “A theory of photometricreconstruction for unknown isotropic reflectances,” in Proc. of IEEE
Conference on Computer Vision and Pattern Recognition (CVPR), 2011,
pp. 2505–2512.
[39] Z. Zhou, Z. Wu, and P. Tan, “Multi-view photometric stereo withspatially varying isotropic materials,” in Proc. of IEEE Conference on
Computer Vision and Pattern Recognition (CVPR), 2013.
[40] T. Higo, Y. Matsushita, and K. Ikeuchi, “Consensus photometric stereo,”in Proc. of IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2010, pp. 1157–1164.
[41] I. Sato, T. Okabe, Q. Yu, and Y. Sato, “Shape reconstruction based on
similarity in radiance changes under varying illumination,” in Proc. ofInternational Conference on Computer Vision (ICCV), 2007.
[42] F. Lu, Y. Matsushita, I. Sato, T. Okabe, and Y. Sato, “Uncalibrated
photometric stereo for unknown isotropic reflectances,” in Proc. of IEEE
Conference on Computer Vision and Pattern Recognition (CVPR), 2013.
[43] T. Okabe, I. Sato, and Y. Sato, “Attached shadow coding: Estimating
surface normals from shadows under unknown reflectance and lighting
conditions,” in Proc. of International Conference on Computer Vision(ICCV), 2009, pp. 1693–1700.
[44] M. Holroyd, J. Lawrence, G. Humphreys, and T. Zickler, “A photometricapproach for estimating normals and tangents,” in Proc. of SIGGRAPH
Asia (ACM Trans. on Graph.), 2008, pp. 133–141.
[45] T. Malzbender, D. Gelb, and H. Wolters, “Polynomial texture maps,” inProc. of SIGGRAPH, 2001, pp. 519–528.
[46] M. Drew, Y. Hel-Or, T. Malzbender, and N. Hajari, “Robust estimation of
surface properties and interpolation of shadow/specularity components,”
Image and Vision Computing, vol. 30, no. 4-5, pp. 317–331, 2012.
[47] E. Reinhard, M. Stark, P. Shirley, and J. Ferwerda, “Photographics tone
reproduction for digital images,” in Proc. of SIGGRAPH (ACM Trans.
on Graph.), 2002, pp. 267–276.
[48] R. Ramamoorthi and P. Hanrahan, “Frequency space environment map
rendering,” in Proc. of SIGGRAPH (ACM Trans. on Graph.), 2002, pp.
517–526.
[49] S. Nayar, K. Ikeuchi, and T. Kanade, “Surface reflection: Physicaland geometrical perspectives,” IEEE Trans. Pattern Anal. Mach. Intell.,vol. 13, no. 7, pp. 611–634, 1991.
[50] J. Lawrence, A. Ben-Artzi, C. DeCoro, W. Matusik, H. Pfister, R. Ramamoorthi, and S. Rusinkiewicz, “Inverse shade trees for non-parametricmaterial representation and editing,” in Proc. of SIGGRAPH (ACM
Trans. on Graph.), 2006, pp. 735–745.
[51] S. Shafer, “Using color to separate reflection components,” Color
Research, vol. 10, no. 4, pp. 210–218, 1985.
[52] C. Schlick, “An inexpensive BRDF model for physically-based rendering,” Computer Graphics Forum, vol. 13, no. 3, pp. 233–246, 1994.
[53] N. Ratner and Y. Schechner, “Illumination multiplexing within fundamental limits,” in Proc. of IEEE Conference on Computer Vision and
Pattern Recognition (CVPR), 2007.
[54] Z. Kukelova, M. Bujnak, and T. Pajdla, “Automatic generator of minimal
problem solvers,” in Proc. of European Conference on Computer Vision(ECCV), 2008, pp. 302–315.
[55] P. Kovesi, “Shapelets correlated with surface normals produce surfaces,”in Proc. of International Conference on Computer Vision (ICCV), 2005,
pp. 994–1001.

Pin Tan received the BS degree in Applied Mathematics from the Shanghai Jiao Tong University
in China in 2000 and the PhD degree in Computer Science and Engineering from the Hong Kong
University of Science and Technology in 2007.
He joined the Department of Electrical and Computer Engineering at the National University ofSingapore as an assistant professor in 2007. Hereceived the inaugural MIT TR35@Singapore
award in 2012, and the Image and Vision Computing Outstanding Young Researcher Honorable Mention Award in 2012. He is an editorial board member of theInternational Journal of Computer Vision (IJCV), Machine Vision andApplications (MVA). His research interests include computer vision andcomputer graphics. He is a member of the IEEE and ACM.

Boxin Shi received the B.E. degree from Beijing
University of Posts and Telecommunications in2007 and M.E. degree from Peking University in 2010. He joined the graduate school of
information science and technology at the University of Tokyo in 2010, where currently heis a Ph.D. candidate under the supervision ofProf. Katsushi Ikeuchi. His research interests
include photometric methods in computer vision,reflectance and illumination modeling, and 3D
reconstruction. He is a student member of IEEE.

Yasuyuki Matsushita received his B.S., M.S.,
and Ph.D. degrees in EECS from the Universityof Tokyo in 1998, 2000, and 2003, respectively.
He joined Microsoft Research Asia in April 2003.
He is a Lead Researcher in Visual Computing
Group. His areas of research are computer vision (photometric techniques, such as radiometric calibration, photometric stereo, shape-fromshading), computer graphics (image relighting,video analysis and synthesis). He is on the editorial board member of IEEE Transactions on
Pattern Analysis and Machine Intelligence (TPAMI), International Journal of Computer Vision (IJCV), IPSJ Journal of Computer Vision andApplications (CVA), The Visual Computer Journal, and Encyclopedia of
Computer Vision. He served/is serving as a Program Co-Chair of PSIVT
2010, 3DIMPVT 2011, and ACCV 2012 and a General Co-Chair for
ACCV 2014. He is appointed as a Guest Associate Professor at Osaka
University (April 2010-), Visiting Associate Professor at National Instituteof Informatics (April 2011-) and Tohoku University (April 2012-), Japan.
He is a senior member of IEEE.

Katsushi Ikeuchi received the BE degree from
Kyoto University in 1973 and the PhD degreefrom the University of Tokyo in 1978. After working at the Massachusetts Institute of Technology(MIT) Artificial Intelligence (AI) Laboratory forthree years, ETL for five years, and Carnegie
Mellon University (CMU) Robotics Institute for 10
years, he joined the University of Tokyo in 1996
and is currently a full professor. His researchinterest spans computer vision, robotics, andcomputer graphics. In these research fields, he
has received several awards, including the David Marr Prize in computational vision for the paper “Shape from Interreflection”, and IEEE
Robotics and Automation Society K. S. Fu memorial best transaction
paper award for the paper “Toward Automatic Robot Instruction from
Perception-Mapping Human Grasps to Manipulator Grasps”. In addition,in 1992, his paper “Numerical Shape from Shading and Occluding
Boundaries” was selected as one of the most influential papers to haveappeared in the Artificial Intelligence Journal within the past 10 years.
His IEEE activities include general chair of IROS ’95, ITSC ’00, IV ’01;
program chair of the CVPR ’96 and ICCV ’03; associate editor of IEEE
Transactions on Robotics and Automation, IEEE Transactions on PatternAnalysis and Machine Intelligence; and a distinguished lecturer of the
Signal Processing Society in 2000-2002 and Robotics and Automation
Society in 2004-2006. He was elected an IEEE fellow in 1998. He is
the editor-in-chief of the International Journal of Computer Vision. He
received the Computer Vision Significant Researcher Award in ICCV’11.