稀疏表示在人脸识别中的应用

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"本文探讨了基于稀疏表示的人脸识别方法，提出了一种新的分类算法，该算法在处理面部表情、光照变化、遮挡和伪装等情况下效果极佳，适用于人脸检测。" 在计算机视觉领域，人脸检测和识别是一项重要的任务，它广泛应用于安全监控、身份验证和人机交互等多个场景。稀疏表示是近年来在图像处理和模式识别中发展起来的一种有效技术，尤其在人脸识别中表现出了强大的潜力。文章"Robust Face Recognition via Sparse Representation"由John Wright等人撰写，他们提出将人脸识别问题转化为多个线性回归模型的分类问题，并指出稀疏信号表示的新理论是解决这个问题的关键。传统的面部识别方法通常依赖于特定的特征提取，如局部二值模式(LBP)或主成分分析(PCA)，这些方法对光照、表情和遮挡等因素敏感。而稀疏表示通过最小化‘1范数来计算，能够从原始图像中找到一个简洁的、非零元素较少的表示，即稀疏编码。这种方法能够容忍一定程度的面部变化，因为关键特征即使在噪声或部分遮挡下也能被识别出来。在特征提取方面，文章指出，如果正确利用稀疏性，选择什么样的特征就变得不那么重要了。关键在于特征的数量是否足够大，以及能否正确计算出稀疏表示。当特征数量足够且稀疏表示准确时，算法能够从大量可能的特征中挑选出对识别最有贡献的部分，从而提高了识别的鲁棒性。对于遮挡问题，稀疏表示方法通过只关注那些对识别至关重要的特征，能够在部分面部被遮挡的情况下仍能进行有效的识别。这为解决实际应用中的复杂情况提供了新的思路。此外，该方法还能处理伪装识别，因为即使人脸的外观发生了变化，其内在的结构特征（如眼睛、鼻子和嘴巴的位置关系）仍然保持不变，这些关键特征可以通过稀疏表示被有效地提取出来。 "Robust Face Recognition via Sparse Representation"这篇文章提出了基于稀疏表示的新型人脸识别框架，该框架不仅对遮挡和伪装具有较强的鲁棒性，而且在特征选择上提供了新的见解，降低了对预定义特征的依赖，提高了人脸识别的准确性和实用性。这一方法在人脸检测领域具有极大的应用价值，对于研究者和实践者来说，都是一种值得试验和探索的技术。

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

WRIGHT ET AL.: ROBUST FACE RECOGNITION VIA SPARSE REPRESENTATION 213

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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