December 10, 2007 / Vol. 5, No. 12 / CHINESE OPTICS LETTERS 683
A polynomial hybrid reflection model and measurement of
its parameters based on images of sample
Lei Yang (
) and Jiuqiang Han (
)
School of Electronics and Information Engineering, Xi’an Jiaotong University, Xi’an 710049
Received April 27, 2007
Reflectance model is a basic concept in computer vision. Some existing mo dels combining the classical
diffuse reflectance model and those for surfaces containing specular components can approximately describe
real reflectance. But the ratio of diffuse and specular reflection decided manually has no clear meaning.
We propose a new polynomial hybrid reflectance model. The reflectance map equation with a known shape
(for example cylinder) as a sample is used to estimate parameters of the proposed reflectance model by least
square regression algorithm. Then the reflectance parameters for surfaces of the same class of materials
can be determined. Experiments are performed for a metal surface. The synthesis images produced by the
proposed method and existing ones are compared with the real acquired image, and the results show that
the proposed reflectance model is suitable for describing real reflectance.
OCIS codes: 100.2960, 120.5820, 160.4760, 290.5820, 290.5880.
Reflectance model is an important concept in computer
vision
[1]
. Using the conception of reflectance model, the
procedure of imaging can be simulated. The bright-
ness of image pixel points is associated with proper-
ties of light source, attributes of camera, the orienta-
tion (shape) of object, and reflectance properties of the
surface
[2]
. Physical and geometrical optics are two ap-
proaches to the study of reflectance
[3]
.Geometrical
models process simpler mathematical forms than phys-
ical ones, which makes them more useful. In general,
geometrical models are applicable only when the wave-
length of incident light is much smaller compared with
the dimensions of the surface. A few papers investi-
gate the unified reflectance models from both physical
and geometrical optics
[4−6]
. At early time, Lambertian
model was adopted by computer vision due to its sim-
ple form
[7]
. Real materials surfaces show heavy specu-
lar component namely highlight, so specular reflectance
models are used
[8]
. But real surface contains both diffuse
and specular components. So lots of hybrid reflectance
models were proposed
[5,9,10]
, most of which used a lin-
ear combination of diffuse and specular reflectance mod-
els and isotropic property. However, there is no general
reflectance model containing all reflectance properties.
On the other hand, the measurement of the reflectance
models is still a problem. Gonioreflectometer was used to
measure the parameters
[5]
. But the apparatus for mea-
suring the parameters of reflectance are relative expen-
sive. The measuring methodology based on images is usu-
ally chosen
[11,12]
. Recently, linear subspace was used to
investigate Lambertian reflectance and image formation
was considered as the analog of convolution
[13]
.Howto
express reflectance function was investigated
[14]
,adata-
driven reflectance model was also brought forward
[15]
.
We think that the diffuse and specular reflectance
properties are distinguished but associated with each
other for an object. The image intensity based on diffuse
reflectance model is the inner multiplication of vectors
of the light source direction and surface normal. The
specular image intensity is exponential (or n-order poly-
nomial) of the inner multiplication of the specular direc-
tion and surface normal. Using principal direction deter-
mined by light direction and specular direction, a math-
ematical refined m-lobed reflection model was derived in
Ref. [9]. The value of weighting factor was gained empir-
ically for the linear combination hybrid model of diffuse
and specular models
[2]
. Based on the reflectance mor-
phology of rough surface
[3,9]
, and inspired by the form
of diffuse and specular reflectance models, a new gen-
eral polynomial hybrid reflectance model is proposed in
this paper. The coefficients of the polynomial reflectance
model can be estimated using image with known shape by
least square regression algorithm. Then the reflectance
models for a class of surfaces of the same material can be
acquired. Additionally, the proposed method has the fol-
lowing characteristics. 1) Polynomial reflectance models
contain wide surface from diffuse to high specular ones
(the former is lower-order item and the latter is higher-
order item). 2) The reflectance map does not need to
be normalized because the irradiance is merged in the
coefficients of reflectance model.
Intensities of image are determined by several param-
eters: location and properties of light, surface properties
and orientation n of objective in scene, location of cam-
era, and laws of image formation. In shape-from-shading
case, we generally assume that the light is a point light
source located in infinite position
[16,17]
, its direction is n
i
in image center coordinates, and the light strength is E.
Observingcameraislocatedinn
o
=(0, 0, −1). The ra-
diance of a distant light source can be described by Dirac
function as
L
i
(n)=Eδ(n − n
o
). (1)
The reflected light depends on the radiance of the light
and the bidirectional reflection function (BRDF) defined
as
f(θ
i
,φ
i
,θ
o
,φ
o
)=
∂L
o
(θ
o
,φ
o
)
∂L
i
(θ
i
,φ
i
)
, (2)
where (θ
i
,φ
i
)and(θ
o
,φ
o
) denote directions of irradiance
of incident light L
i
and reflected radiance L
o
in polar
1671-7694/2007/120683-04
c
2007 Chinese Optics Letters
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