变量光照下人脸识别的线性子空间获取方法

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"该资源是一篇关于在变化光照下进行人脸识别的学术论文，主要探讨了如何获取线性子空间以适应不同光照条件下的面部识别问题。作者包括Kuang-Chih Lee、Jeffrey Ho和David Kriegman，他们都是IEEE的成员。文章分析了在多光源和阴影环境下，人脸图像的变化可以由低维线性空间有效建模，并介绍了三种获取这些空间基图像的方法：通过大量不同光照条件下的人脸图像进行主成分分析(PCA)；利用3D模型生成合成图像，再用PCA估计子空间；以及基于球谐函数的漫射光照渲染的3D模型图像直接作为基图像。论文提出了一个新的方法，即通过特定的物理光照设置，使得获取的每个对象图像可以直接作为低维线性空间的基向量，且这个子空间接近于传统的那些方法得到的空间。" 本文的核心知识点包括： 1. **线性子空间与人脸识别**：在人脸识别领域，线性子空间是处理光照变化的关键工具，它能捕获人脸图像在不同光照条件下的主要变化模式。 2. **主成分分析(PCA)**：PCA是一种常用的数据降维技术，可以用来从大量的样本图像中提取关键特征，构建低维空间，用于表示图像变化。 3. **3D模型重建与合成图像**：通过3D模型重建可以从不同角度和光照条件下生成合成图像，然后使用PCA估计出代表图像变化的子空间。 4. **基于球谐函数的漫射光照渲染**：这是一种模拟真实世界光照条件的技术，通过球谐函数可以生成更贴近实际的3D模型渲染图像，这些图像可以直接作为线性子空间的基。 5. **新型光照安排方法**：论文提出的新方法旨在通过特定的物理光照设置，使得采集到的图像直接作为低维线性空间的基向量，减少了中间步骤，可能提高识别效率和准确性。 6. **光照条件对人脸识别的影响**：光照条件的变化是人脸识别中的一个挑战，如多光源和阴影会增加识别难度，而有效的线性子空间建模有助于解决这一问题。 7. **线性空间的逼近**：新方法得到的子空间与传统方法（如PCA）得到的子空间相近，表明新方法在理论和实际应用上都具有可行性。这篇论文对于理解在复杂光照环境下的面部识别技术有重要意义，特别是对于优化数据采集和预处理过程，提高识别系统的鲁棒性和准确性提供了新的思路。

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2PRELIMINARIES

2.1 Illumination Cone

Let x 2 IR

denote an image with n pixels of a convex object

with a Lambertian reflectance function illuminated by a

single point source at infinity, represented by a vector s 2

such that its magnitude jsj represents the intensity of the

source and the unit normal s=jsj represents the direction.

Let B 2 IR

n3

be a matrix where each row b in B is the

product of the albedo with unit normal for a point on the

surface projecting to a particular pixel in the image. Under

the Lambertian assumption, x is given by

x ¼ maxðBs; 0Þ; ð1Þ

where maxðBs; 0Þ sets to zero all negative components of

the vector Bs. If the object is illuminated by k light sources

at infinity, then the image is given by the superposition of

the images that would have been produced by the

individual light sources, i.e.,

x ¼

i¼1

maxðBs

; 0Þ: ð2Þ

Due to this superposition, the set of all possible images C of

a convex Lambertian surface created by varying the

direction and strength of an arbitrary number of point light

sources at infinity is a convex cone. Furthermore, any image

in the illumination cone C (including the boundary) can be

determined as a convex combination of extreme rays

(images) given by

¼ maxðBs

; 0Þ; ð3Þ

where s

¼ b

 b

are rows of B with i 6¼ j. It is clear that

there are at most mðm  1Þ extreme rays for m  n distinct

surface normals [7].

In computer vision, it has been a customary practice to

treat the human face as a Lambertian surface. Although

human faces are not convex, the degree of nonconvexity is

not serious enough to render the concept of the illumination

cone inapplicable [7]. The only difference between the

illumination cone of a human face and a convex object is

that (3) no longer accounts for all the extreme rays and there

are extreme rays that are the result of cast shadows.

Therefore, the formula for the upper bound on the number

of extreme rays is generally more complicated than the

quadratic expression mðm  1Þ above. This poses a formid-

able difficulty for computing the exact illumination cone

(i.e., specifying all the extreme rays). Instead, a subset of the

illumination cone can be computed by sampling lighting

directions on the unit sphere, and (1) is accompanied by ray

tracing to account for the cast shadows.

2.2 Lambertian Reflection and Spherical Harmonics

In this section, we briefly summarize the recent work

presented in [2], [14], [ 15], [19]. Consider a convex

Lambertian object with uniform albedo illuminated by

distant isotropic light sources, and p is a point on the

surface of the object. Pick a loca l ðx; y; zÞ coordinates

system F

centered at p such that the z-axis coincides with

the surface normal at p, and let ð; Þ

denote the spherical

coordinates centered at p. Under the assumption of distant

and isotropic light sources, the configuration of lights that

illuminate the object can be expressed as a nonnegative

function Lð; Þ. The reflected radiance at p is given by

rðpÞ¼

kðÞLð; ÞdA

¼ 

2



kðÞLð; Þsindd;

ð4Þ

where  is the albedo, and kðÞ¼maxðcos ; 0Þ is called the

Lambertian kernel. A similar integral can be formed for any

other point q on the surface to compute the reflected

radiance rðqÞ. The only difference between the integrals at p

and q is the lighting function L: At each point, L is

expressed in a local coordinate system (or coordinate frame

) at that point. Therefore, considered as a function on the

unit sphere, L

and L

differ by a rotation given by

ð; Þ¼L

ðgð; ÞÞ, where gð; Þ rotates the directions

ð; Þ in the L

to L

frame.

The spherical harmonics are a set of functions that form

an orthonormal basis for the set of all square-integrable (L

)

functions defined on the unit sphere. They are the analogue

on the sphere to the Fourier basis on the line or circle. The

spherical harmonics, Y

, are indexed by two integers l and

m obeying l  0 and l  m  l:

ð; Þ¼

jmj

ðcosÞcosðjmjÞ if m>0;

jmj

ðcosÞ if m ¼ 0;

jmj

ðcosÞsinðjmjÞ if m<0;

ð5Þ

where N

is a normalization factor guaranteeing that the

integral of Y

 Y

¼ 



, and P

jmj

is the associated

LEE ET AL.: ACQUIRING LINEAR SUBSPACES FOR FACE RECOGNITION UNDER VARIABLE LIGHTING 3

1. To conform with the notation used in spherical harmonics literature, 

denotes the elevation angle and  denotes the azimuth angle. In the next

section, however, we will switch the roles of  and .

TABLE 1

Summary of Notation Used in this Paper

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