多模态人脸识别人工智能：MMD方法与图像集应用

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本文发表于2012年的IEEE Transactions on Image Processing，主题是"Manifold-Manifold Distance and Its Application to Face Recognition with Image Sets"，由Ruiping Wang、Shiguang Shan、Xilin Chen、Qionghai Dai和Wen Gao等作者共同完成。文章关注的是人脸识别领域中的一个重要问题——如何有效地对包含同一主题但变化巨大的图像集进行分类。人脸识别的传统方法通常处理单张图片，但本文则探讨了如何将每个图像集视为一个嵌入高维空间的数据集合，即“流形”，从而转化为计算两个流形之间的距离，即所谓的“流形-流形距离”（Manifold-Manifold Distance, MMD）。在实际应用中，人脸图像集可能包含点（像素级特征）、子空间（局部特征或特征向量的集合）和整个流形三个层次的变化。作者系统地研究了这三者之间的距离，并提出了一个多层级MMD框架。他们采用局部线性模型来表示流形，将MMD转换为计算涉及的两个流形中子空间间的距离。理论分析部分探讨了MMD的不同配置及其有效性，包括不同层次的特征融合策略和子空间距离度量方法的选择。实验部分展示了这种多层级MMD方法在实际人脸识别任务中的优越性能，比如对抗图像质量变化、光照变化以及姿势和表情变化时的鲁棒性。通过与传统方法的比较，文章强调了MMD在处理大规模、复杂变化人脸图像集时的优势，为后续的研究提供了新的视角和技术支持。总结来说，这篇文章不仅深化了人脸识别技术的理解，还引入了一种创新的流形建模和距离度量方法，为提高人脸识别系统的准确性和适应性提供了有价值的理论基础和实践指导。对于从事计算机视觉、机器学习和人工智能领域的研究人员和工程师来说，这篇论文具有很高的参考价值。

4468 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 10, OCTOBER 2012

Fig. 2. Hierarchical structure formed by three pattern levels (i.e., point,

subspace, and manifold) in face recognition.

by a large number of samples). See Fig. 2 for an illustration.

In some sense, the core of pattern classiﬁcation is the distance

computation among these representations. The distances over

point and subspace have been well studied in the literature;

whereas very few studies have been done on the distance

related to manifold.

A. Distances Over Point and Subspace

Hereinafter we always denote points by x

, y

, subspaces

by S

, and manifolds by M

. The distances over point and

subspace include the following three ones:

1) Point to Point Distance (PPD): denote by d(x

, x

) the

distance from point x

to x

. The most commonly used PPD

is the Euclidean distance as follows:

d(x

, x

) =



− x



. (1)

2) Point to Subspace Distance (PSD): denote by d(x, S) the

distance from point x to subspace S. It is generally deﬁned as

the so-called L2-Hausdorff distance:

d(x, S) = min

y∈S



x − y



x − x





. (2)

In fact, x



is the projection of x in the subspace S,alsothe

nearest point to x in S. Thus, the PSD is actually the PPD

from x to its projection x



in S. It is also known as “distance-

from-feature-space” (DFFS) in [23].

3) Subspace to Subspace Distance (SSD): denote by

d(S

, S

) the distance between two subspaces S

and S

While there is not a uniﬁed deﬁnition yet to measure the SSD,

the concept of principal angles [18], [19] is perhaps the most

commonly exploited one due to its favorable performance.

Recently, another SSD is proposed in [24], which can be

regarded as utilizing the sum of DFFS between the bases of

two subspaces.

As known in linear algebra, the single point x

spans a

special linear subspace, i.e., the trivial zero subspace L{0},

which is centered on x

and of zero dimensional. In this sense,

both PPD and PSD are special cases of SSD.

B. Distances Over Manifold

Our main motivation arises from the fact that local lin-

earity holds everywhere on a globally nonlinear manifold.

Thus, a manifold can be modeled by a collection of local

linear models, each depicted by a subspace [25]. In general,

manifold can be viewed as extending subspace to account

for more general and complex data variations. The distances

associated with manifold are then related to those deﬁned on

subspace. Formally, we denote the i-th component subspace

of a manifold M by C

, and express M as a set containing all

the C

M ={C

: i = 1, 2,...,m}={C

, C

,...,C

}. (3)

where m is the number of local linear subspaces.

1) Point to Manifold Distance (PMD): denote by d(x, M)

the distance from point x to manifold M. Similar to PSD, one

can deﬁne this distance by ﬁnding the closest point to x in M

as follows:

d(x, M)= min

∈M

d(x, C

)= min

∈M

min

y∈C



x−y



x−x





. (4)

In analogy to x



in the PSD, here we call x



the projection of

x in the manifold M.

2) Subspace to Manifold Distance (SMD): denote by

d(S, M) the distance from subspace S to manifold M. It can

be deﬁned by seeking the closest subspace to S in manifold

d(S, M) = min

∈M

d(S, C

). (5)

It comes that SMD is reduced to SSD in a simple manner

similar to that from PSD to PPD.

3) Manifold to Manifold Distance (MMD): denote by

d(M

, M

) the distance between two manifolds M

and

. With the local linear model representation in (3), MMD

can be converted to integrating the distances between pair of

subspaces respectively from one of the involved manifolds.

See Fig. 3 for a conceptual illustration.

Formally, given two manifolds M

={C

: i =

1, 2,...,m}, M

={C



: j = 1, 2,...,n}, we formulate

MMD as follows:

d(M

, M

) =



i=1



j=1



, C





s.t.



i=1



j=1

= 1, f

≥ 0. (6)

In this general formulation, MMD comes in the form of a

weighted average of pairwise SSDs, i.e., d(C

, C



It has been ﬁgured out that point is a special case of

subspace. Similarly, subspace can be viewed as a special case

of manifold under the formulation in (3). Therefore, the three

pattern levels form a hierarchical structure and all the six

distances can be formulated in a general multi-level MMD

framework.

III. M

ANIFOLD–MANIFOLD DISTANCE

From Fig. 3 and (6), one can ﬁnd that there are three key

ingredients in MMD: (i) local linear model construction, i.e.,

the component subspaces C

, C



, (ii) local model distance

measure, i.e., the SSD d(C

, C



), and (iii) global integration

of local distances, i.e., the choice of the weights f

.Inthis

section, we present details of these ingredients and extensive

investigations on their various conﬁgurations.

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多模态人脸识别人工智能：MMD方法与图像集应用

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