A Biquadratic Reflectance Model for Radiometric Image Analysis
Boxin Shi1∗ Ping Tan2 Yasuyuki Matsushita3 Katsushi Ikeuchi1
1
The University of Tokyo 2 National University of Singapore 3 Microsoft Research Asia

Abstract
Radiometric image analysis methods heavily rely on reflectance models. Due to the complexity of real materials,
methods based on simple models such as the Lambertianmodel often suffer from inaccuracy. On the other hand,more advanced models such as the Cook-Torrance model
severely complicate the analysis problem. We tackle thisdilemma by focusing on the low-frequency component of thereflectance. We propose a compact biquadratic reflectancemodel to represent the reflectance of a broad class of materials precisely in the low-frequency domain. We validate
our model by fitting to both existing parametric models andnon-parametric measured data, and show that our model
outperforms existing parametric diffuse models. We show
applications of reflectometry using general diffuse surfaces
and photometric stereo for general isotropic materials. Experimental results show the effectiveness of our biquadraticmodel and its usefulness in radiometric image analysis.

1. Introduction
General surface reflectance is described by 4-D bidirectional reflectance distribution functions (BRDFs). Radiometric image analysis methods, including reflectometry and photometric shape analysis, rely on accurate modeling of reflectance. There has been a tremendous effort
to build parametric BRDF models for realistic rendering(e.g., [9, 36, 18, 3]). While these models are successful ingraphics-rendering applications, they are seldom used in inverse problems in radiometric image analysis, because theyoften lead to highly nonlinear optimization problems. For
the sake of computational simplicity and stability, the Lambertian model is still most widely used for inverse problems
in spite of its limited accuracy in modeling real-world materials. To make the radiometric image analysis methods
more accurate and reliable, a general yet simple parametric
BRDF model is desired.
One of the key challenges in developing an analytic
BRDF model is accurately modeling specular reflections that exhibit high-frequency variations. Modeling high∗ Part of this work was done while the first author was visiting National

University of Singapore.

frequency specular reflections is crucial for photo-realistic
rendering in computer graphics. However, it is seldom necessary for inverse problems in computer vision like photometric stereo. Instead, the low-frequency component is more actively used in radiometric analysis, because its
dense observation provides a reliable source for the analysis problem1 . Motivated by this observation, we develop a
new analytic reflectance model for radiometric image analysis. This model relies on a biquadratic representation for
accurately modeling the low-frequency component whileneglecting the high-frequency component. Although it excludes strong specularities including specular spikes, it canrepresent broad and smooth specular lobes as well as the
diffuse component.
Our biquadratic representation uses a factorized form of
the bivariate BRDF model [29], i.e., the product of half- anddifference-angle terms as ρ(θh , θd ) ' ρ1 (θh )ρ2 (θd ). We
use a quadratic function for each term to achieve a compact parameterization. This representation is well suitedfor the low-frequency component of isotropic BRDFs as
we will see in the experimental results, and it is more precise than the Lambertian model, existing parametric models
with general diffuse terms [18, 3] and specular terms [9, 36].
In addition, because of its compactness, model estimation
becomes tractable for inverse problems in radiometric image analysis. As applications of the proposed model, weshow its usages in reflectometry for general diffuse surfaces
and photometric stereo for general isotropic materials.

2. Related Works
Various parametric BRDF models have been developed over the decades. These can be categorized into
physically-based and empirical models. The former group
includes microfacet-based models, such as the TorranceSparrow [35] and Cook-Torrance [9] models for specularsurfaces, and the Oren-Nayar model [25] for diffuse surfaces. On the other hand, empirical models, such as the
Phone [26], Blinn-Phong [4], and Lafortune models [18],use generic functions to express BRDFs. Some models
1 For example, strong specular spikes (high-frequency) are observed only when the surface normal coincides with the bisector of viewing andlighting directions.

bridge these two categories by partly using physically motivated terms, e.g., the Ward [36] and Ashikhmin models [3].
An experimental evaluation of various models can be foundin [23]. Generally, parametric models are compact and easy
to use for forward problems, but they are often designed fora limited class of materials.
A BRDF can also be naturally represented in a nonparametric form by a 4-D discrete table of densely measured data. This can be simplified to a 3-D table if thematerials are isotropic [11, 20, 21]. It can be further reduced to a 2-D table for a wide range of isotropic reflectances [27, 2, 28]. Although high-quality rendering canbe achieved using measured data, the non-parametric form is generally unsuitable for inverse problems because its
large number of parameters complicate the problem. There
are recent efforts that efficiently represent the BRDFs using non-parametric [40], semi-parametric [7], or parametric [24] forms to achieve high modeling accuracy by employing different basis functions. Our goal shares a similar spirit, but we focus on modeling the low-frequency reflectance using a simpler parametric form with an emphasison radiometric image analysis.
Radiometric image analysis. Radiometric image analysis seeks to recover scene properties, such as reflectance or
shape, from recorded scene radiance. Here we briefly review related works in reflectometry and photometric stereo.
Most of the works on reflectometry are based on parametric reflectance models, such as [39, 5, 14], with exceptions of [27, 28], which assume non-parametric bivariate BRDFs and adopt a discrete table representation. All
these methods focus on reflectance estimation assuming aknown shape with different input: multiple input images
under a moving point light source [39, 14], a single input
image with known [5, 27] or unknown environment illumination [28]. A successful reflectometry requires both anaccurate reflectance model and a reliable model estimation
method.
In the shape estimation context, early photometric stereo works [37, 32] are based on the Lambertian reflectancemodel. To handle real surfaces that are often nonLambertian, some algorithms attempt to discard the nonLambertian reflectance as outliers [38]. The others employ parametric reflectance models, such as the TorranceSparrow model [12], the Ward model [8, 13], and othermulti-lobe models [33].
There are more recent approaches that attempt to solve
the photometric stereo problem with reflectance symmetries, such as isotropy and reciprocity [1, 2, 34, 6], without using a strong assumption on reflectance. Photometric stereo can also be applied to surfaces with general reflectance by using thousands of images [16], or considering some consensus properties [15]. We study photometricstereo based on our biquadratic reflectance model. Since

n
l

v

h
θh
θd

θd
o

Figure 1. The definitions of θh and θd .

our reflectance model is compact yet accurate for many real materials, our method requires simple data capturing andoptimization.

3. Biquadratic Reflectance Model
We denote n as a unit surface normal, h as the bisectorof lighting direction l and viewing direction v, i.e., h = (l+v)/kl + vk. Following [29], we use θh to denote the angle
between n and h, and θd for the angle between l (or v) andh as illustrated in Fig. 1. Hence, the following relationships
hold: nT h = cos θh and lT h = cos θd .
Our reflectance model is based on the bivariate BRDF
model [29], which shows that most of the isotropic BRDFs
can be represented as a bivariate function ρ(θh , θd ). This
representation is evaluated by [27] with a large number ofmeasured BRDFs [21] in the development of passive reflectometry. It is further discussed in [29] that any isotropic
BRDF based on microfacet theory should consist of a univariate function of θh , and its Fresnel term should be a univariate function of θd . As shown by [3], the masking andshadowing terms in a microfacet-based BRDF model varysmoothly 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 [19] to assist material capturing
and editing.
To obtain a compact parametric model suitable for inverse problems, we make a further simplification by representing the factorized terms as quadratic functions ofcos θh and cos θd . As a result, our BRDF model becomesa biquadratic function. As we will see later, the proposed biquadratic model can represent a wide class of lowfrequency BRDFs and has a computationally tractable form
in radiometric image analysis.
Based on these discussions, we define a biquadratic reflectance model for isotropic low-frequency reflectance as
the following:ρ(θh , θd ) 'ρ1 (θh )ρ2 (θd ) = ρ˜1 (nT h)ρ˜2 (lT h)= A1 (nT h)2 + B1 (nT h) + C1A2 (lT h)2 + B2 (lT h) + C2 .

(1)BRDF fitting error
Normal: thetaN 2 : 2: 90, phiN 10 : 10 : 360
Light: Random 100
Cook-Torrance model

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

1

Dark-redpaint

0.5

Natural209

0.3

0.1

Figure 2. BRDF fitting errors of our model to the Cook-Torrance
model. The columns vary with Tlow and the rows vary with m in
the Cook-Torrance model. Some rendered images are displayedon the left side for reference.

In practice, to simplify model fitting, the biquadratic form of Eq. (1) can be further relaxed asρ̃(x, y) =Ax2 y 2 + Bx2 y + Cx2 + Dxy 2 + Exy+F x + Gy 2 + Hy + I,(2)

where x = nT h, y = lT h are defined for notation simplicity. There are 9 parameters in total, and we denote them ina vector form as x = [A, B, · · · , I]T ∈ R9×1 . Note thatthe conversion from Eq. (1) to Eq. (2) is unique, but theother direction is not, and Eq. (2) may not always have theproduct form of Eq. (1). Eq. (2) is a linear function of its
parameters A to I, while Eq. (1) is a bilinear function ofthese parameters. Hence, the relaxed model is much easier
to fit.

3.1. Model Validation
Intuitively, our model can accurately represent the lowfrequency part of conventional dichromatic reflectancemodels [31], which represent reflectances as a summationof Lambertian diffuse and specular terms. Since the specular term is mostly concentrated in the high-frequency part,the low-frequency Lambertian term can be well represented
by our model with A1 = B1 = A2 = B2 = 0. Our model
can also represent the low-frequency component of BRDF models that rely only on nT h, such as the Blinn-Phongmodel [4]. The low-frequency part of these models can beapproximated by our model with A2 = B2 = 0.
In the following, we assess the modeling accuracy ofour biquadratic reflectance model by fitting existing parametric models and measured data. We use a simple intensity thresholding to separate low-frequency and highfrequency reflectances, since high-frequency (strong specular) reflectances generally show high intensity. We further compare our biquadratic model with existing parametric models.

96
(25)

67
(46)

Alumbronze

28(1)

Nickel

1(48)

Figure 3. BRDF fitting errors of our model to the 100 materials in*“a(b)” a = sorted BRDF ID in the right plot, (b) = original ID in the database
the MERL
database. The columns vary with Tlow , and the rows
represent different BRDFs with indices ordered by the mean fitting
errors over columns. Some rendered images are displayed on theleft for reference, with reordered BRDF indices (the numbers inbrackets are their original indices in the database) aside.

Fitting to parametric BRDFs. The Cook-Torrance model [9] is widely used to represent surface reflectance. It consists of a Lambertian diffuse and a specular term. The specular component is modeled as a multiplication of microfacet
distribution D, the Fresnel term F and the geometrical attenuation term G. The microfacet distribution D is represented as



1
1
1
D=
exp
1
−
,
(3)
4m2 (nT h)4m2(nT h)2
where m indicates the surface roughness. D is clearly afunction of θh . F is often simplified by the Schlick’s approximation [30, 3], denoted as
F = ρs + (1 − ρs )(1 − lT h)5 ,(4)where ρs is a constant. Hence, F is a function of θd . Although the term G is relatively complicated, it varies smoothly and is close to constant over a large range of exitantangles as evaluated in [22, 3]. Therefore, ρ1 (θh )ρ2 (θd ) isa good approximation for the Cook-Torrance model. Other models based on the microfacet theory, such as theWard [36] model, have similar expressions.
For an experimental validation, we fit our biquadraticmodel to synthetic data generated by the Cook-Torrance
model with varying surface roughness. The relative strengthof diffuse and specular components is fixed to 1. We fix theviewing direction at (0, 0, 1) and sample 1620 normal directions from the visible hemisphere by uniformly choosing 36
longitudes and 45 altitudes. We then randomly sample 100
light directions to calculate the reflected radiance. For eachnormal direction under varying lightings, we sort the radiance intensities in an increasing order and select only thosenon-shadowed samples that are ranked below a predefinedpercentage Tlow . We then fit our biquadratic model to theseselected samples in a least-squares manner. Specifically, we
fit Eq. (2) by a linear least-squares and use its results as initial values to fit Eq. (1) in an iterative manner. At each iteration, we fix the parameters of one quadratic function andoptimize the other parameters by a linear least-squares. It
usually takes no more than 10 iterations to converge. This
process is repeated with different surface roughness m anddifferent Tlow .
The color encoded mean absolute errors (MAE) of thefitting are summarized in Fig. 2 in a matrix form withvarying m and Tlow . Note that reflectance with a large
roughness value has a broad specular lobe, which is mixed
with the Lambertian diffuse reflectance to form the lowfrequency reflectance. From the region within the light blue
rectangle in Fig. 2, we conclude that our model well fits the
low-frequency reflectance of the Cook-Torrance model withdifferent roughnesses.
Fitting to measured data. We also assess our biquadraticmodel by fitting to synthetic data generated from measured
MERL BRDF database [21]. The input samples are generated in the same way as the previous experiment, except that
we use the measured BRDFs instead of the Cook-Torrance
model. The MAE values are shown in Fig. 3. Note that therows indicate different BRDFs in the database now. Similar to Fig. 2, our model fits well to the low-frequency reflectance of different materials, and our model has smaller
fitting errors for materials with broader specular lobes.

3.2. Comparison with Other Parametric Models
Models with general diffuse terms. There are existing
parametric models with general diffuse terms, such as the
Lafortune and the Ashikhmin2 models.
In the Lafortune model [18], a general rotationally symmetric diffuse component ρL is written as
k T  kρL = Cd nT l
n v ,
(5)
where Cd and k are model parameters. The general diffuse
term of Ashikhmin model [3] ρA is defined as
5 !

5 !

nT v
nT l
1− 1−
, (6)
ρA = R 1 − 1 −22where R is the model parameter.
Models with specular terms. Several parametric models assume a Lambertian diffuse term and use a microfacetbased specular component, such as the Cook-Torrance andWard models. These models can be represented as
ρS =kd+ ks S(n, l, v, m),π(7)

2 In this paper, we only consider the general diffuse terms of these twomodels.

where kd and ks are model parameters representing thestrength of diffuse and specular terms respectively; S is anonlinear function with m as another model parameter. In
the Cook-Torrance model, m is encoded in the D term in Eq. (3) and the Ward model has a similar expression.
Fitting comparison. We again use the MERL BRDF
database for evaluation by setting the threshold Tlow =25% to extract the low-frequency reflectance. Fitting Eq. (6) is straightforward. For fitting Eq. (5), we take logarithm of both sides of the equation and estimate the logparameters by a linear least-squares. For fitting Eq. (7), we
adopt the similar strategies as [23], and use the Matlab function “lsqnonlin” to solve the nonlinear optimization. The
fitting errors are shown in Fig. 4, where “biquadratic (relax.)” and “biquadratic (iter.)” indicate the relaxed modelin Eq. (2) and the original model in Eq. (1), respectively.
Compare to existing general diffuse terms, our model has
a smaller fitting error than the Lafortune and Ashikhmin
models. Though with specular components included, theCook-Torrance and Ward models cannot outperform ourmodel in fitting the low-frequency reflectance. Besides, ourmodel has a simpler analytic form, which makes it easier
to use in inverse problems. The errors of “biquadratic (relax.)” and “biquadratic (iter.)” are very similar. It suggests
the relaxed model is a good replacement for the original
biquadratic model when computational efficiency is important.

3.3. Why Biquadratic?
ρ˜1 (x) and ρ˜2 (y) might also be represented by polynomials of any orders, e.g., linear or cubic. We choose to represent both of them by quadratic functions to achieve a goodtrade-off between modeling accuracy and model complexity. We compare the accuracy of bilinear, biquadratic, andbicubic models in Fig. 4. The bilinear model is less accurate
than the Lafortune model. On the other hand, the bicubicmodel has the smallest fitting error, because its higher degrees of freedom. However, as we will see in the photometric stereo application, its cubic function of n complicatesthe normal estimation. Therefore, we choose the biquadratic model.

4. Application to Reflectometry
Although our biquadratic model mainly intends to model
the low-frequency reflectance, it can also be used as an empirical reflectance model for materials without significant
specular spikes. With our model in Eq. (2), reflectometry
becomes a simple linear problem Ax = i, where i records
radiance values. For each observation, we can calculate the(x, y) terms (a 9-D row vector) according to Eq. (2) fromn, l, and v when the shape and lighting are all calibrated.
From p samples (p > 9) that are linearly independent, thematrix A ∈ Rp×9 and observations i ∈ Rp×1 are obtained.

-3

7
6

MAE

5
4

x 10

Bicubic: 5.75/2.38
Biquadratic(relax.): 6.32/2.77
Biquadratic(iter.): 6.89/3.08
Lafortune: 7.15/3.87
Bilinear: 7.25/3.61
Cook-Torrance: 7.62/4.53
Ward: 7.90/4.70
Ashikhmin: 24.83/12.81

3
2
1
0

10

20

30

40

50
BRDF index

60

70

80

90

100

Figure 4. BRDF fitting errors. The Y-axis shows the MAE values. The X-axis shows BRDF indices ordered by the fitting errors of“biquadratic (relax.)” for visualization purpose. The mean/median MAEs (×10−4 ) over the BRDF index are shown in the legend.
Rendering results

Image

which do not show strong specularities from the MERLBRDF database. We synthesize a single image of a sphere
under a point light source as input and fit our model to it.
All the tested materials show the similar results with anaverage MAE about 8 × 10−4 . Here we show the BLUE RUBBER example in Fig. 5. The left most is the original
input image. The others are images rendered according to
the reflectance model estimated from the input image. We
also show the comparison with the bivariate model [27] andthe Lambertian model. The bivariate model is generated
similarly as [27] by taking one slice of the 3-D discrete
BRDF table (ρ(θh , θd , φd )) and averaging over φd (the rotation of difference-angle). Although our model performs slightly worse than the bivariate model, we have significantly
fewer parameters. In contrast, the simple Lambertian model generates the largest error. Generally, our model bears
relatively larger errors when high-frequency reflectance appears (e.g., center of the sphere), since it only models lowfrequency reflectance.
We also use real data to evaluate our method. The input
image is captured from a ping pong ball. For visual validation, we show a sphere rendered according to the biquadratic model, the Lafortune model and the Lambertian model
estimated from this input image in the lower part of Fig. 5.
The result here is consistent with the BRDF fitting resultsin Fig. 4. Our model can better describe the low-frequency
reflectance than the Lafortune and Lambertian models.

Biquadratic

Lambert0

5.25×10-4

6.82 ×10-4

11.67 ×10-4

Ground truth

Biquadratic

Lafortune

Lambert

0.018

0.031

0.035

0.01

0.2

DifferenceBLUE - RUBBER , DARK - RED - PAINT, YELLOW- PAINT, etc.),

Bivariate

Image

To verify the method, we pick some materials (e.g.,

Difference

Ground truth

The model parameters x can be determined by simply solving the linear system as x = (AT A)−1 AT i. For a curved
surface with a single material, a single image under a point
light source provides sufficient information for reflectometry.

0

Figure 5. A synthetic and a real-world results of reflectometry. Below the rendered (captured) spheres we show the difference maps
w.r.t. the ground truth and their MAEs.

5. Application to Photometric Stereo
In this section, we apply our biquadratic reflectancemodel to photometric stereo to recover surface normals
from images captured from a fixed camera under varying
lightings. We assume an orthographic camera and directional lightings. The camera-centered coordinate system ischosen such that v = (0, 0, 1)T . We fit our biquadratic
reflectance model at each pixel independently, thus this approach can handle spatially varying BRDFs.

5.1. Algorithm Detail
From the photometric stereo images, we observe multiple radiance intensities at each pixel. For each pixel, we firstuse a very small intensity threshold to skip shadows. And

Ground truth

Biquadratic

Bicubic

Cook-Torrance

Ward

Lambert (low)

Lafortune

Lambert (all)

45

2.65

3.30

3.95

3.99

4.04

5.61

15.94

0

Figure 6. Photometric stereo with synthetic data. One of the input images is given under the ground truth normal map. Below other normal
maps are angular difference maps w.r.t. the ground truth. The numbers on the difference maps show mean angular errors (degree) of normal
estimates.

then we sort remaining observations in increasing order andonly keep those ranked below a percentage Tlow , which isempirically determined within [20%, 40%]. We use ilow todenote the concatenated vector of these remaining observations and stack their corresponding lighting directions tofrom a matrix Llow . We then obtain the following equation:
3.11

4.01

ilow = ρ̄(n, Llow ) ◦ (nT Llow ),

5.18

(8)

where ‘◦’ indicates element-wise multiplication. ρ̄(·, ·) encodes the reflectance parameter x in the same manner asρ̃(·, ·) in Eq. (2), but operates on each n and l. We use the relaxed biquadratic model of Eq. (2) for better computational
efficiency. The surface normal n and the BRDF parametersx can be determined by iteratively optimizing the following
objective function:(n∗ , x∗ ) = argmin kρ̄(n, Llow ) ◦ (nT Llow ) − ilow k2 . (9)n,xAt each iteration, we first fix the normal direction n andrefine x by a linear least-squares. We then substitute x andn to determine ρ̄(·, ·). Once ρ̄(·, ·) is calculated, we update n again by a linear least-squares. To initialize this iterative optimization, we first apply Lambertian photometricstereo [37] with ilow and Llow to estimate an initial normal.
The iterative optimization stops when the residual of Eq. (9)does not change. In our implementation, we stop it when thechange becomes less than 10−7 .
This same approach can be applied for other analytic
BRDF models, such as the Lafortune, Cook-Torrance andWard models. However, when it is used, the estimation of
their BRDF parameters by fixing n becomes highly nonlinear. Similarly, though we might use higher-order polynomials such as cubic functions to represent ρ˜1 (x) and ρ˜2 (y),high-order polynomials are prone to different instabilities(numerical errors, Runge’s phenomenon [10], etc.). Hence,
the biquadratic representation is a good trade-off between
model complexity and accuracy.

photometric stereo method. We mix ALUM - BRONZE and
CHERRY-235 from the MERL BRDF database on a BUN 45 set
NY model, and render 100 images as input. We still
Tlow = 25% for this synthetic data. We use the same solution method as Sec. 5.1, and replace ρ̄(·, ·) as bicubic, Lafortune, Cook-Torrance and Ward models respectively to make3
a5.23
comparison. 5.27
The estimated6.77
normal maps22.00
and their0 angular errors are shown in Fig. 6.
The traditional Lambertian photometric stereo with all
images (“Lambert (all)”) [37] fails on those materials withcomplicated reflectances (e.g., the texture boundaries on
BUNNY is clearly visible in the error map). By only focusing on the low-frequency reflectance, photometric stereo
can work reasonably well on general reflectance even with
the Lambertian model (“Lambert (low)”). Further by fitting
the low-frequency reflectance with our biquadratic model,
the estimated normals become even more accurate. Compared with other BRDF models, the mean error from thebicubic case is larger than the biquadratic case due to thehigh-order polynomial fitting, and the errors from the CookTorrance, Ward and Lafortune models are larger than ourmodel partly due to their lower modeling ability for lowfrequency reflectance. Besides, the nonlinear reflectancemight cause the iterative optimization performs poor, whichmakes the Lafortune case has a larger error than the “Lambert (low)” case.

5.2. Experiments

Real-world data. We show the results of real-world data
in Fig. 7. For an easy reference, we refer to these examples
as (from top to bottom row) GOURD 1 (102), GOURD 2 (98),
APPLE (112), POST (91) and TEAPOT (73), with the numberof input images in the bracket. The first three datasets arerecorded by Alldrin et al. [2]. Others are captured by ourselves with a Sony XCD-X710CR camera. We do not carefully control exposure to avoid saturation, since our method
can naturally skip undesired strong specular and saturation
regions by setting Tlow . These objects include various challenging materials for photometric stereo, such as porcelain

Spatially varying BRDFs. We generate synthetic images
with spatially varying BRDFs to quantitatively verify our

3 x, y, z components of surface normal are linearly mapped to theR, G, B channels.

and shiny plastic.
In Fig. 7, the left column shows one of the input images to the photometric stereo algorithm and a reference
image (not used in calculation) of the same object under natural lighting and a new view point4 . The middle and rightcolumns show the estimated surface normal and integrated
surfaces using our method and the Lambertian photometricstereo. The integrated surfaces are produced by the shapelet
surface reconstruction method [17]. The recovered surfaces
are aligned to the reference views for making qualitative evaluation, and our reconstruction agrees well with them.

6. Conclusion
We present a biquadratic reflectance model for lowfrequency reflectance of isotropic surfaces. Our model canrepresent a broad class of real materials at high precision
while retaining sufficient simplicity for radiometric image
analysis. Extensive experiments using both synthetic andmeasured data demonstrate its effectiveness in modeling
low-frequency BRDF components. We further show thatit can be used for reflectometry to capture general diffuse
reflectance. We also propose a novel photometric stereo solution to general isotropic materials using the proposed reflectance model. These applications show the applicabilityof our model in radiometric image analysis.
The current model is limited to isotropic materials. We
also hope to analyze the characteristics of the low-frequency
component of anisotropic reflectance and extend our biquadratic model to a wider variety of reflectances. At thesame time, currently the low-frequency reflectance is identified by a simple intensity threshold. How to detect it in amore principled manner is also left as our future work.

Acknowledgement
B. Shi and K. Ikeuchi was partially supported by the Digital Museum Project, Ministry of Education, Culture, Sports, Science & Technology in Japan. P. Tan was supported by
the Singapore ASTAR grant R-263-000-592-305 and MOE
grant R-263-000-555-112.

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