基于稀疏表示的鲁棒人脸识别算法

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"《PAMI-Robust Face Recognition via Sparse Representation》一文探讨了在人脸识别领域的一个前沿方法，针对人脸从不同角度、表情、光照变化以及遮挡和伪装的情况下进行自动识别。论文将人脸识别问题重新定义为对多个线性回归模型的分类，并强调了稀疏信号表示理论在解决这个问题中的关键作用。作者提出了一种基于'1-范数最小化'计算得到的稀疏表示的通用分类算法，旨在处理基于图像的对象识别任务。这个新框架对人脸识别的两个关键问题提供了新的见解：特征提取和对遮挡的鲁棒性。通过稀疏表示，作者指出，如果能够有效利用问题的稀疏特性，选择特定的特征（如Eigenfaces或Laplacianfaces）就不再是决定因素，重要的是特征空间的维度是否足够大，以及稀疏表示是否正确构建。即使非传统特征如降采样图像和随机投影也能与传统特征达到同样效果，只要特征维度超过了由稀疏表示理论预测的阈值。对于遮挡的处理，该框架利用了错误在标准像素基下的稀疏性，能够统一地应对因遮挡造成的误差。稀疏表示理论不仅帮助预测算法能处理多少程度的遮挡，还指导如何选择训练数据来最大化对遮挡的鲁棒性。作者通过大量公开数据库的实验验证了所提算法的有效性，并证实了他们的理论主张。这篇论文揭示了稀疏表示在人脸识别中的强大潜力，它不仅改变了我们看待特征选择的方式，也为设计更为鲁棒的面部识别系统提供了创新的策略和技术。"

As the entries of the vector xx

encode the identity of the

test sample yy, it is tempting to attempt to obtain it by

solving the linear system of equations yy ¼ Axx. Notice,

though, that using the entire training set to solve for xx

represents a significant departure from one sample or one

class at a time methods such as NN and NS. We will later

argue that one can obtain a more discriminative classifier

from such a global representation. We will demonstrate its

superiority over these local methods (NN or NS) both for

identifying objects represented in the training set and for

rejecting outlying samples that do not arise from any of the

classes present in the training set. These advantages can

come without an increase in the order of growth of the

computation: As we will see, the complexity remains linear

in the size of training set.

Obviously, if m>n, the system of equations yy ¼ Axx is

overdetermined, and the correct xx

can usually be found as

its unique solution. We will see in Section 3, however, that

in robust face recognition, the system yy ¼ Axx is typically

underdetermined, and so, its solu tion is not unique.

Conventionally, this difficulty is resolved by choosing the

minimum ‘

-norm solution:

ð‘

Þ :

¼ arg min kxxk

subject to Axx ¼ yy: ð4Þ

While this optimization problem can be easily solved (via

the pseudoinverse of A), the solution

is not especially

informative for recognizing the test sample yy. As shown in

Example 1, ^xx

is generally dense, with large nonzero entries

corresponding to training samples from many different

classes. To resolve this difficulty, we instead exploit the

following simple observation: A valid test sample yy can be

sufficiently represented using only the training samples

from the same class. This representation is naturally sparse if

the number of object classes k is reasonably large. For

instance, if k ¼ 20, only 5 percent of the entries of the

desired xx

should be nonzero. The more sparse the

recovered xx

is, the easier will it be to accurately determine

the identity of the test sample yy.

This motivates us to seek the sparsest solution to yy ¼ Axx,

solving the following optimization problem:

ð‘

Þ :

¼ arg min kxxk

subject to Axx ¼ yy; ð5Þ

where kk

denotes the ‘

-norm, which counts the number

of nonzero entries in a vector. In fact, if the columns of A are

in general position, then whenever yy ¼ Axx for some xx with

less than m=2 nonzeros, xx is the unique sparsest solution:

¼ xx [33]. However, the problem of finding the sparsest

solution of an underdetermined system of linear equations is

NP-hard and difficult even to approximate [13]: that is, in the

general case, no known procedure for finding the sparsest

solution is significantly more efficient than exhausting all

subsets of the entries for xx.

2.2 Sparse Solution via ‘

-Minimization

Recent development in the emerging theory of sparse

representation and compressed sensing [9], [10], [11] reveals

that if the solution xx

sought is sparse enough, the solution of

the ‘

-minimization problem (5) is equal to the solution to

the following ‘

-minimization problem:

ð‘

Þ :

¼ arg min kxxk

subject to Axx ¼ yy: ð6Þ

This problem can be solved in polynomial time by standard

linear programming methods [34]. Even more efficient

methods are available when the solution is known to be

very sparse. For example, homotopy algorithms recover

solutions with t nonzeros in Oðt

þ nÞ time, linear in the size

of the training set [35].

2.2.1 Geometric Interpretation

Fig. 2 gives a geometric interpretation (essentially due to

[36]) of why minimizing the ‘

-norm correctly recovers

sufficiently sparse solutions. Let P



denote the ‘

-ball (or

crosspolytope) of radius :



fxx : kxxk

 gIR

: ð7Þ

In Fig. 2, the unit ‘

-ball P

is mapped to the polytope

P¼

AðP

ÞIR

, consisting of all yy that satisfy yy ¼ Axx for

some xx whose ‘

-norm is  1.

The geometric relationship between P



and the polytope

AðP



Þ is invariant to scaling. That is, if we scale P



, its

image under multiplication by A is also scaled by the same

amount. Geometrically, finding the minimum ‘

-norm

solution

to (6) is equivalent to expanding the ‘

-ball P



until the polytope AðP



Þ first touches yy. The value of  at

which this occurs is exactly k

Now, suppose that yy ¼ Axx

for some sparse xx

. We wish

to know when solving (6) correctly recovers xx

.This

question is easily resolved from the geometry of that in

Fig. 2: Since

is found by expanding both P



and AðP



until a point of AðP



Þ touches yy, the ‘

-minimizer

must

generate a point A

on the boundary of P .

Thus,

¼ xx

if and only if the point Aðxx

=kxx

Þ lies on

the boundary of the polytope P. For the example shown in

Fig. 2, it is easy to see that the ‘

-minimization recovers all

with only one nonzero entry. This equivalence holds

because all of the vertices of P

map to points on the

boundary of P.

In general, if A maps all t-dimensional facets of P

facets of P , the polytope P is referred to as (centrally)

t-neighborly [36]. From the above, we see that the

‘

-minimization (6) correctly recovers all xx

with  t þ 1

nonzeros if and only if P is t-neighborly, in which case, it is

4 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 31, NO. 2, FEBRUARY 2009

Fig. 2. Geometry of sparse representation via ‘

-minimization. The

‘

-minimization determines which facet (of the lowest dimension) of the

polytope AðP



Þ, the point yy=kyyk

lies in. The test sample vector yy is

represented as a linear combination of just the vertices of that facet, with

coefficients xx

6. Furthermore, even in the overdetermined case, such a linear equation

maynotbeperfectlysatisfiedinthepresenceofdatanoise(see

Section 2.2.2).

7. This intuition holds only when the size of the database is fixed. For

example, if we are allowed to append additional irrelevant columns to A,

we can make the solution xx

have a smaller fraction of nonzeros, but this

does not make xx

more informative for recognition.

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