稀疏表示在鲁棒人脸识别中的应用
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收藏 4.57MB PDF 举报 "这篇论文是关于基于稀疏表示的鲁棒人脸识别技术，它在图像处理领域具有重要影响。研究者提出将人脸识别问题转化为在多个线性回归模型之间的分类问题，并指出稀疏信号表示的新理论是解决这个问题的关键。通过使用ℓ1-最小化计算的稀疏表示，他们提出了一种用于图像对象识别的通用分类算法。这一新框架对人脸识别中的两个关键问题——特征提取和遮挡的鲁棒性——提供了新的理解。"
在本文中，作者John Wright等人探讨了如何在不同表情、光照、遮挡和伪装情况下自动识别人脸的问题。他们采用稀疏表示的方法来解决这些问题，这种方法在当前的图像处理领域备受关注。稀疏表示是一种数学技术，它可以将复杂的信号或数据表示为少量关键成分的线性组合，这些关键成分往往对应于信号的基础或重要模式。
论文的核心观点是，将人脸识别问题转化为多个线性回归模型的分类任务。在这种框架下，识别过程不再依赖特定的特征选择，而是关注特征的数量是否足够大以及是否能形成有效的稀疏表示。如果能正确利用识别问题中的稀疏性，特征选择的影响力将大大减弱。关键在于，特征集合应该足够丰富，以便捕捉到人脸的多样性和变化，同时保证稀疏性，即每个样本可以用相对较少的特征来表示。
稀疏表示通过ℓ1-最小化实现，这是一种优化技术，旨在找到一个最稀疏的解决方案，即在满足约束条件下，使得非零元素数量最少。这种技术在处理噪声和异常值时表现出良好的鲁棒性，因此在处理遮挡和伪装等复杂情况时非常有效。
此外，文章还强调了稀疏表示在特征提取和遮挡鲁棒性方面的贡献。对于特征提取，即使不精心设计特定的特征，只要稀疏性得到充分利用，也能实现良好的识别性能。对于遮挡问题，稀疏表示能够通过强调未被遮挡的部分，忽略或弱化遮挡区域的影响，从而提高识别的准确性。
这篇论文提供了一个基于稀疏表示的人脸识别新方法，它不仅简化了传统的人脸识别流程，而且通过引入稀疏性，增强了系统的鲁棒性和适应性，对于实际应用具有很高的价值。
MANUSCRIPT ACCEPTED BY IEEE TRANS. PAMI, MARCH 2008. 5
C. Classification Based on Sparse Representation
Given a new test sample y from one of the classes in the
training set, we first compute its sparse representation
ˆ
x
1
via
(6) or (10). Ideally, the nonzero entries in the estimate
ˆ
x
1
will
all be associated with the columns of A from a single object
class i, and we can easily assign the test sample y to that class.
However, noise and modeling error may lead to small nonzero
entries associated with multiple object classes (see Figure 3).
Based on the global, sparse representation, one can design many
possible classifiers to resolve this. For instance, we can simply
assign y to the object class with the single largest entry in
ˆ
x
1
.
However, such heuristics do not harness the subspace structure
associated with images in face recognition. To better harness such
linear structure, we instead classify y based on how well the
coefficients associated with all training samples of each object
reproduce y.
For each class i, let δ
i
: R
n
→ R
n
be the characteristic function
which selects the coefficients associated with the i-th class. For
x ∈ R
n
, δ
i
(x) ∈ R
n
is a new vector whose only nonzero entries
are the entries in x that are associated with class i. Using only the
coefficients associated with the i-th class, one can approximate
the given test sample y as
ˆ
y
i
= Aδ
i
(
ˆ
x
1
). We then classify y
based on these approximations by assigning it to the object class
that minimizes the residual between y and
ˆ
y
i
:
min
i
r
i
(y)
.
= ky − A δ
i
(
ˆ
x
1
)k
2
. (12)
Algorithm 1 below summarizes the complete recognition proce-
dure. Our implementation minimizes the `
1
-norm via a primal-
dual algorithm for linear programming based on [39], [40].
Algorithm 1 : Sparse Representation-based Classification
(SRC)
1: Input: a matrix of training samples A = [A
1
, A
2
, . . . , A
k
] ∈
R
m×n
for k classes, a test sample y ∈ R
m
, (and an optional
error tolerance ε > 0.)
2: Normalize the columns of A to have unit `
2
-norm.
3: Solve the `
1
-minimization problem:
ˆ
x
1
= arg min
x
kxk
1
subject to Ax = y. (13)
(Or alternatively, solve
ˆ
x
1
= arg min
x
kxk
1
subject to kAx − yk
2
≤ ε.)
4: Compute the residuals r
i
(y) = ky − A δ
i
(
ˆ
x
1
)k
2
for i = 1, . . . , k.
5: Output: identity(y) = arg min
i
r
i
(y).
Example 1 (`
1
-Minimization versus `
2
-Minimization): To
illustrate how Algorithm 1 works, we randomly select half of the
2, 414 images in the Extended Yale B database as the training set,
and the rest for testing. In this example, we subsample the images
from the original 192 × 168 to size 12 × 10. The pixel values of
the downsampled image are used as 120-D features – stacked as
columns of the matrix A in the algorithm. Hence matrix A has
size 120 × 1207, and the system y = Ax is underdetermined.
Figure 3 left illustrates the sparse coefficients recovered by
Algorithm 1 for a test image from the first subject. The figure
also shows the features and the original images that correspond
to the two largest coefficients. The two largest coefficients are
both associated with training samples from subject 1. Figure
3 right shows the residuals w.r.t. the 38 projected coefficients
δ
i
(
ˆ
x
1
), i = 1, 2, . . . , 38. With 12 × 10 downsampled images
as features, Algorithm 1 achieves an overall recognition rate
of 92.1% across the Extended Yale B database. (See Section
IV for details and performance with other features such as
Eigenfaces and Fisherfaces, as well as comparison with other
methods.) Whereas the more conventional minimum `
2
-norm
solution to the underdetermined system y = Ax is typically
quite dense, minimizing the `
1
-norm favors sparse solutions,
and provably recovers the sparsest solution when this solution is
sufficiently sparse. To illustrate this contrast, Figure 4 left shows
the coefficients of the same test image given by the conventional
`
2
-minimization (4), and Figure 4 right shows the corresponding
residuals w.r.t. the 38 subjects. The coefficients are much less
sparse than those given by `
1
-minimization (in Figure 3), and
the dominant coefficients are not associated with subject 1. As a
result, the smallest residual in Figure 4 does not correspond to
the correct subject (subject 1).
D. Validation Based on Sparse Representation
Before classifying a given test sample, we must first decide if
it is a valid sample from one of the classes in the dataset. The
ability to detect and then reject invalid test samples, or “outliers,”
is crucial for recognition systems to work in real-world situations.
A face recognition system, for example, could be given a face
image of a subject that is not in the database, or an image that is
not a face at all.
Systems based on conventional classifiers such as nearest
neighbor (NN) or nearest subspace (NS), often use the residuals
r
i
(y) for validation, in addition to identification. That is, the
algorithm accepts or rejects a test sample based on how small the
smallest residual is. However, each residual r
i
(y) is computed
without any knowledge of images of other object classes in the
training dataset and only measures similarity between the test
sample and each individual class.
In the sparse representation paradigm, the coefficients
ˆ
x
1
are
computed globally, in terms of images of all classes. In a sense, it
can harness the joint distribution of all classes for validation. We
contend that the coefficients
ˆ
x are better statistics for validation
than the residuals. Let us first see this through an example.
Example 2 (Concentration of Sparse Coefficients): We
randomly select an irrelevant image from Google, and
downsample it to 12 × 10. We then compute the sparse
representation of the image against the same Extended Yale
B training data as in Example 1. Figure 5 left plots the
obtained coefficients, and right plots the corresponding residuals.
Compared to the coefficients of a valid test image in Figure 3,
notice that the coefficients
ˆ
x here are not concentrated on any
one subject and instead spread widely across the entire training
set. Thus, the distribution of the estimated sparse coefficients
ˆ
x contains important information about the validity of the test
image: A valid test image should have a sparse representation
whose nonzero entries concentrate mostly on one subject,
whereas an invalid image has sparse coefficients spread widely
among multiple subjects.
To quantify this observation, we define the following measure
of how concentrated the coefficients are on a single class in the
dataset:
Definition 1 (Sparsity Concentration Index): The sparsity
concentration index (SCI) of a coefficient vector x ∈ R
n
is
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