频率域不变量：球谐身份理论在BRDF照明转换与图像一致性中的应用
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"A theory of frequency domain invariants spherical harmonic identities for BRDF lighting transfer and image consistency"
本文探讨了频域不变量的理论，特别是在计算机视觉中的应用。作者Dhruv Mahajan、Ravi Ramamoorthi和Brian Curless，都是IEEE的成员，他们提出了一种新颖的基于球谐函数的身份验证方法，这些方法是角度频率域中类似于空间域中常见的反射率比率的不变量。球谐函数在计算机视觉领域常用于描述三维形状和光照交互。
在频域不变量理论中，研究重点在于球谐卷积框架，这是处理曲面反射的一种方法。通过这种方法，他们推导出一系列适用于多种情况的等式，包括单个或多个在同一光照条件下的物体图像，以及不同光照环境下的两个不同光泽物体的图像。对于后者，他们建立了一个新颖的独立于特定照明配置或BRDF（双向反射分布函数）的等式，这个等式允许在有三个图像的情况下直接估计第四个图像。
此外，这个等式还可用作图像一致性检测的不变量，有助于检测图像是否被篡改。虽然本文主要关注理论发展，但其潜在价值在于为光照传输、图像一致性以及可能的图像验证算法提供了坚实的理论基础。这将对计算机图形学、图像处理和数字取证等领域产生深远影响，进一步推动相关技术的发展和应用。通过深入理解和应用这些频域不变量球谐身份，可以优化光照条件下的物体建模，提高图像合成的真实感，以及增强数字图像的可信度和安全。
sources. This paper takes an important first step in laying the
theoretical foundations for this new research direction by
deriving a new class of identities which can be checked to
detect tampering and consistency of lighting and shading in a
complex lighting environment. A limitation of our approach
is that our identities require the knowledge of 3D model/
geometry of the object, though such geometry could be
available through prior acquisition or estimated from the
images, for example, based on known shape distributions [3].
The rest of this paper is organized as follows: Section 2
briefly explains the spherical convolution and signal proces-
sing framework. Section 3 demonstrates the use of deconvo-
lution to estimate lighting. In Sections 4 and 5, we introduce
identities for the simple case of a single image of an object.
Section 6 derives more identities for the case of multiple
images. In Section 7, we discuss the implications of our theory
and its relation to spatial domain invariants. Section 8 gives
experimental validation of our theory and shows potential
applications. Finally, we conclude our discussion in Section 9
and talk about the future research directions that this work
makes possible. This paper is an extended and detailed
version of a paper that was presented at European Con-
ference on Computer Vision (ECCV ’06) [12].
2BACKGROUND
We now briefly introduce the spherical convolution and
signal-processing framework [2], [19] needed for our later
derivations. We start with the Lambertian case
BðnÞ¼
Z
S
2
Lð!Þmaxðn !; 0Þ d!; ð1Þ
where BðnÞ denotes the reflected light as a function of the
surface normal. B is proportional to the irradiance (we omit
the albedo for simplicity), and Lð!Þ is the incident illumina-
tion. The integral is over the sphere S
2
, and the second term in
the integrand is the half-cosine function. The equations in this
paper do not explicitly consider color; the (R,G,B) channels
are simply computed independently. A similar mathematical
form holds for other radially symmetric BRDFs such as the
Phong model for specular materials. In the specular
2
case, we
reparameterize by the reflected direction R (the reflection of
the viewing ray about the surface normal), which takes the
place of the surface normal. For the Phong model, the
reflection equation becomes:
BðRÞ¼
s þ 1
2
Z
S
2
Lð!ÞmaxðR !; 0Þ
s
d!; ð2Þ
where s is the Phong exponent, and the BRDF is normalized
(by ðs þ 1Þ=2).
If we expand in spherical harmonics Y
lm
ð; Þ, using
spherical coordinates ! ¼ð; Þ, n or R ¼ð; Þ, and ðÞ
for the (radially symmetric) BRDF kernel, we obtain
Lð; Þ¼
X
1
l¼0
X
l
m¼l
L
lm
Y
lm
ð; Þ
Bð; Þ¼
X
1
l¼0
X
l
m¼l
B
lm
Y
lm
ð; Þ ðÞ¼
X
1
l¼0
l
Y
l0
ðÞ:
ð3Þ
It is a lso possible to derive analyti c forms and good
approximations for common BRDF filters . For the Lamber-
tian case, almost all of the energy is captured by l 2. For
Phong and Torrance-Sparrow models of specular reflection,
good approximations [19] are Gaussians: exp½l
2
=2s for
Phong and exp½ðlÞ
2
for Torrance-Sparrow, where is the
surface roughness parameter in the Torrance-Sparrow
model, and s is the Phong exponent.
In the angular (versus angular frequency) domain, (1) and
(2) represent rotational convolution of lighting with BRDF.
The BRDF can be thought of as the filter, whereas the lighting
is the input signal. This allows us to relate them multi-
plicatively in the angular frequency domain (convolution
theorem). In the frequency domain, the reflected light B is
given by a simple product formula or spherical convolution
(see [2], [19] for the derivation and an analysis of this
convolution)
B
lm
¼
l
l
L
lm
¼ A
l
L
lm
;
ð4Þ
where for convenience, we define the normalization
constant
l
as
l
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi
4
2l þ 1
r
A
l
¼
l
l
: ð5Þ
It is also possible to extend these results to nonradially
symmetric general isotropic BRDFs [19]. For this case, we
must consider the entire 4D light field, expressed as a function
of both orientation and outgoing direction
B
lmpq
¼
l
lq;pq
L
lm
; ð6Þ
where the reflected light field is now expanded in a mixed
basis of representation matrices and spherical harmonics and
has four indices because it is a 4D quantity. The 3D isotropic
BRDF involves an expansion over both incoming and
outgoing directions. The new indices p and q correspond to
the spherical harmonic in dices for the expansion over
outgoing angles (analogous to the indices l and m used for
the lighting).
The remainder of this paper derives new identities and
formulas from (4), B
lm
¼ A
l
L
lm
. Most glossy BRDFs (such as
Torrance-Sparrow) are approximately radially symmetric,
especially for nongrazing angles of reflection [19], [20]. Most
of the theory in this paper also carries over to general isotropic
materials, as per (6), if we consider the entire light field.
Another reason to focus on (4), is that it is simple and allows
practical spherical harmonic computations from only a single
image—a single view of a sufficiently curved object (assuming
a distant viewer) sees all reflected directions.
3
3KNOWN BRDF: DECONVOLUTION TO ESTIMATE
LIGHTING
Lighting estimation is a specific example of the general
inverse rendering problem. Given a single image and BRDF
of known geometry and homogenous material, we want to
estimate the directional distribution of the incident light.
This information can then be used to insert new objects in
the scene, alter the lighting of the object or check lighting
MAHAJAN ET AL.: A THEORY OF FREQUENCY DOMAIN INVARIANTS: SPHERICAL HARMONIC IDENTITIES FOR BRDF/LIGHTING TRANSFER... 3
2. “Specular” will always be used to mean generally glossy, including
but not restricted to mirror-like.
3. In case we do not have the full range of normals, we can use multiple
cameras. As we move the camera (viewer), the same point on the object now
corresponds to a different reflected direction. Hence, we can get all the
reflected directions even if the object has only a partial set of normals by the
careful placement of cameras.
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