Extension of Robust Principal Component Analysis for Incremental Face

Recognition

ıfa Nakouri and Mohamed Limam

Institut Sup

erieur de Gestion, LARODEC Laboratory

University of Tunis, Tunis, Tunisia

Keywords:

Image alignment, Robust Principal Component Analysis, Incremental RPCA.

Abstract:

Face recognition performance is highly affected by image corruption, shadowing and various face expressions.

In this paper, an efﬁcient incremental face recognition algorithm, robust to image occlusion, is proposed. This

algorithm is based on robust alignment by sparse and low-rank decomposition for linearly correlated images,

extended to be incrementally applied for large face data sets. Based on the latter, incremental robust principal

component analysis (PCA) is used to recover the intrinsic data of a sequence of images of one subject. A new

similarity metric is deﬁned for face recognition and classiﬁcation. Experiments on ﬁve databases, based on

four different criteria, illustrate the efﬁciency of the proposed method. We show that our method outperforms

other existing incremental PCA approaches such as incremental singular value decomposition, add block

singular value decomposition and candid covariance-free incremental PCA in terms of recognition rate under

occlusions, facial expressions and image perspectives.

1 INTRODUCTION

In the last two decades, face recognition has been an

active research area within the computer vision and

the pattern recognition communities. Since an orig-

inal input image space has a very high dimension,

dimensionality reduction techniques are usually per-

formed before classiﬁcation. Principal Component

Analysis (PCA) is one of the most popular representa-

tion methods for computer vision applications mainly

face recognition. Usually, PCA is performed in the

batch mode, where all training data are used to calcu-

late the PCA projection matrix. Once the training data

have been fully processed, the learning process stops.

In case we want to incorporate additional data into an

existing PCA projection matrix, the matrix has to be

retained with all training data. Therefore, such system

is hard to scale up. An incremental version for PCA is

a straightforward solution to overcome this limitation.

Incremental PCA (IPCA) has been studied for

more than two decades yielding many methods,

which are specially useful when not all observations

are simultaneously available. The aim of the IPCA

approach is to do not consider all available observa-

tions more than once even when new data are eventu-

ally upcoming. New data can be used to incrementally

update a previous computation. Such an approach

reduces storage requirements and large problems be-

come computationally feasible.

The performance of IPCA methods is evaluated

with face recognition standard databases (Hall et al.,

2000; Weng et al., 2003; Huang et al., 2009; Hall

et al., 2002). However, one of their major drawbacks

is that they cannot simultaneously handle large illu-

mination variations, image corruptions and partial oc-

clusions that often occur in real face data (e.g., self

shadowing, hats, sunglasses, scarf, incomplete face

data, etc), hence inducing important appearance vari-

ation. These image variations can be considered as

outliers or errors regarding the original face image

of one subject. Although classical PCA is effective

against the presence of small Gaussian noise in the

data, it is highly sensitive to even sparse errors of very

high magnitude.

On the other hand, it is known that well-aligned

face images of a person, under varying illumination,

lie very close to a low-dimensional linear subspace.

However, in practice, images deviate from this situ-

ation due to self shadowing, different angles and oc-

clusions. Thus, we have a set of coherent images cor-

rupted by essentially sparse errors. In order to efﬁ-

ciently extract low-rank face images from corrupted

and distorted ones, we should ﬁrst model those cor-

ruption factors and seek efﬁcient ways to eliminate

549

Nakouri H. and Mohamed L..

Extension of Robust Principal Component Analysis for Incremental Face Recognition.

DOI: 10.5220/0004288305490555

In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2013), pages 549-555

ISBN: 978-989-8565-47-1

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

them. The robust PCA (RPCA) (Wright et al., 2009)

is a powerful tool to get rid off such errors and retrieve

cleaner images potentially better suited for computer

vision application, namely face recognition.

In this paper, we propose an incremental method

for robust face recognition under various conditions

based on RPCA. The proposed method handles both

misalignment and occlusion problems on face images.

In order to improve the recognition process and based

on RPCA (Wright et al., 2009), we eliminate cor-

ruptions and occlusion in original face images. Be-

sides, the incremental aspect of our face recognition

method handles the memory constraint and computa-

tional cost of a large data set. To measure the similar-

ity between a query image and a sequence of images

of one person, we deﬁne a new similarity metric. To

evaluate the performance of our method, experiments

on the AR (Martinez and Benavente, 1998), ORL

(Samaria and Harter, 1994), PIE (Sim et al., 2002),

YALE (Belhumeur et al., 1997) and FERET (Phillips

et al., 1998) databases and a comparison with other

incremental PCA methods namely incremental singu-

lar value decomposition (SVD) (Hall et al., 2002), add

block SVD (Brand, 2006) and candid covariance-free

incremental PCA (Weng et al., 2003) are conducted.

We also compare our method to a face recognition

method based on batch robust PCA, denoted by face

recognition RPCA (FRPCA) (Wang and Xie, 2010).

This paper is organized as follows. In Section 2,

we introduce the RPCA method, incremental RPCA

(IRPCA) and our face Recognition method, denoted

by new incremental RPCA (NIRPCA). Finally, in

Section 3, we present our experimental results.

2 FACE RECOGNITION BASED

ON IRPCA

2.1 Robust Image Alignment by Sparse

and Low-rank Decomposition

Peng et al., (Peng et al., 2010) proposed robust align-

ment by sparse and low-rank decomposition for lin-

early correlated images (RASL). It is a scalable op-

timization technique for batch linearly correlated im-

age alignment. One of its objectives is to robustly

align a dataset of human faces based on the fact that

if faces are well-aligned, they show efﬁcient low-rank

structure up to some sparse corruptions. Even per-

fectly aligned images may not be identical, but at least

they lie near a low-dimensional subspace (Basri and

Jacobs, 2003). To the best of our knowledge, RASL

is the ﬁrst method that uses a trade-off between rank

minimization and alignment of image data. Hence,

the idea is to search for a set of transformations τ such

that the rank of the transformed images becomes as

small as possible and at the same time the sparse er-

rors are compensated. Generally, the applied transfor-

mation is the 2D afﬁne transform, where we implicitly

assume that the face of a person is approximately on

a plane in 3D-space.

2.2 Incremental Robust Principal

Component Analysis (IRPCA)

RPCA algorithm is aimed to recover the low-rank ma-

trix A from the corrupted observations D = A + E,

where corrupted entries E are unknown and the er-

rors can be arbitrarily large but assumed to be sparse.

More speciﬁcally, in face recognition, E is a sparse

matrix because it is assumed that only a small fraction

of image pixels are corrupted by large errors (e.g., oc-

clusions). Hence, being able to correctly identify and

recover the low structure A could be very interesting

for many computer vision applications namely face

recognition.

We assume that we have m subjects and each one

has n face images. Although RASL can give a very

accurate alignment for faces (Peng et al., 2010), it

is not applicable when the total number of images

m × n denoted by l is very large. Wu et al., (Wu

et al., 2011) proposed an extension to RASL from l

to L where L >> l, by reformulating the problem us-

ing a ”one-by-one” alignment approach. This incre-

mental alignment can be summarized in three steps.

First, l frames are selected to be aligned with batch

RASL method producing a low-rank summary A

∗

. In

the second step, the (l + 1)

image is aligned with

∗

which contains the information of the previously

aligned l images. Finally, the second step is repeated

for the remaining images, regardless of the size of the

data set.

We denote by I

, A

, E

the corrupted ,observed,

face image, the original face image and the error of

the j

image of the i

subject, respectively. Then,

we have I

= A

, where i denotes the subject and

j its corresponding image such that i = 1, . . . , m, j =

1, . . . , n. Let:

vec : R

w×h

→ R

(w×h)×1

, (1)

be a function which transforms a w × h image matrix

into a (w × h) × 1 vector by stacking its columns to

have vec(I

) = vec(A

) + vec(E

). Assuming that we

have m subjects and each one has n images, we deﬁne

for the i

subject:

:= [vec(I

)|. . . |vec(I

)] = A

+ E

(2)

VISAPP2013-InternationalConferenceonComputerVisionTheoryandApplications

550

where

:= [vec(A

)|. . . |vec(A

)] ∈ R

(w×h)×n

(3)

and

:= [vec(E

)|. . . |vec(E

)] ∈ R

(w×h)×n

, (4)

with i = 1, . . . , m. D

is formed by stacking the n im-

age vectors of the i

subject, A

and E

are the cor-

responding original images matrix and the error ma-

trix, respectively. Since all images of the same per-

son are approximately linearly correlated, A

is re-

garded as a low-rank matrix and E

is a large ma-

trix but sparse. As proved in (Peng et al., 2010),

the original face A

can be efﬁciently recovered from

the corrupted face image D

. It is well known that if

images are well-aligned, they should present a low-

rank structure up to some sparse errors (e.g., occlu-

sions) (Peng et al., 2010). Therefore, we search for

a set of transformations, τ = τ

, . . . , τ

, such that the

rank of the transformed images becomes as small as

possible, and simultaneously images become as well-

aligned as possible. Many works (Peng et al., 2010;

Cand

es et al., 2011) prove that practical misalignment

can be modeled as a certain transformation τ

−1

∈ G

acting on the two-dimensional domain of an image I.

G is assumed to be a ﬁnite dimensional group that

has a parametric representation, such as the similarity

group SE(2) × G

or the 2D afﬁne group A f f (2). In

this paper we assume that G is the afﬁne group.

Consider that after performing a batch image

alignment using the RASL method, we obtain the set

of transformation τ, the low rank matrix A and the

error matrix E. The IRASL algorithm is given in Al-

gorithm 1.

2.3 The Proposed Face Recognition

Algorithm based on IRPCA

In this section, we introduce a new face recognition

algorithm called NIRPCA based on the one-by-one

RASL method discussed in Section 2.2. We also de-

ﬁne a new similarity metric, which is used for mea-

suring the similarity between a query image and a se-

quence of images, and later used for our main face

recognition application.

Given, m different subjects where each one has n

training images I

, i = 1, . . . , m; j = 1, . . . , n, we need

to classify a query image I

n+1

. The basic idea of the

proposed algorithm is to recover the sparse error E

n+1

of the test image I

n+1

using approximated A

n+1

. Once

n+1

is recovered, we use it to compute a similarity

metric for our face recognition algorithm NIRPCA.

Let D be the observation matrix of each subject in

the training set, and A its recovered low-rank matrix

Algorithm 1: Incremental robust alignment by sparse and

low-rank decomposition.

• INPUT: Images I

n+1

∈ R

w×h

, initial transformations

n+1

in certain parametric group G, weight µ > 0

• WHILE not converged DO

step1: compute Jacobian matrix w.r.t transformation

n+1

J ←

∂

∂ζ



vec(I

n+1

◦ζ)

kvec(I

n+1

◦ζ)k



ζ=τ

n+1

;

step2: wrap and normalize the image:

n+1

◦ τ

n+1

←

vec(I

n+1

◦τ

n+1

)

kvec(I

n+1

◦τ

n+1

;

step3: solve the linearized convex optimization:

∗

, ∆τ

∗

n+1

) ←

arg min

x,∆τ

n+1

k I

n+1

◦ τ

n+1

+ J∆τ

n+1

−

Ax k

+µ k x k

;

step4: update transformation :

n+1

← τ

n+1

+ ∆τ

∗

n+1

;

• END WHILE

• OUTPUT: solution x, τ

n+1

and E the error matrix. For the i

subject in the train-

ing set, i = 1, . . . , m, we have D

, A

and E

as given in

Equations (2), (3) and (4).

Let I

n+1

be a new occluded face image that we

need to classify. According to RASL method (Peng

et al., 2010; Cand

es et al., 2011), this new observation

can be decomposed as:

n+1

◦ τ

n+1

= A

n+1

+ E

n+1

, (5)

where, τ

n+1

is the transformation applied to corrupted

image I

n+1

to resolve the image misalignment. A

n+1

is the occlusion-free image and E

n+1

is the error im-

age representing the occlusion. Our objective is to

estimate τ

n+1

, A

n+1

and E

n+1

. To solve Equation (5),

we propose to use the low rank matrix A

∗

generated

by the RASL method. As indicated in Algorithm 1,

the one-by-one alignment approach proposed by (Wu

et al., 2011) computes τ

n+1

and x having the low rank

matrix A

∗

and the corrupted face image I

n+1

. Since

∗

is a low-rank matrix, let

A denote the summary

of low rank data resulting from batch RASL, such

that

A ∈ R

m×rank(A

∗

)

, where the columns are equal to

rank(A

∗

), i.e., the independent columns of A

∗

. The

vector x of dimension rank(A

∗

) is represented as an

approximation of the coefﬁcients of the linear com-

bination of

A and A

n+1

. Hence, an approximation of

n+1

is obtained by the following equation

n+1

A ∗ x. (6)

ExtensionofRobustPrincipalComponentAnalysisforIncrementalFaceRecognition

551

Once we estimate A

n+1

and by using Equation (5), we

estimate the error vector E

n+1

(standing for the occlu-

sion on the I

n+1

test image) and use it to compare the

similarity between the test image and the stored im-

ages. Let the similarity metric be

=k E

− E

n+1

, i = 1, . . . , n (7)

where M

measures the similarity between the input

corrupted test image I

n+1

and the a class of images of

the i

subject. M

is the Euclidean distance between

the input image I

n+1

and a class of images belong-

ing to the i

subject. If the query image I

n+1

belongs

to the i

subject, D

contains the image of the same

subject so that the assumption that A

is linearly cor-

related and the low-rank condition can be satisﬁed. In

this case, the parameters of E

are small, and then M

should be small, otherwise, the value of M

is rela-

tively large. For face recognition, the test image I

n+1

is recognized as the subject which has the smallest

value of M

. If the similarity M

is greater than a given

threshold α, the face image I

n+1

is not recognized. It

will be considered as a new subject and a new class

subject A

n+1

will be created and added to summary

A. Otherwise, M

≤ α, and I

n+1

is recognized as be-

longing to the i

subject and we just should update

the class of the i

subject in the

A summary with

n+1

. The classiﬁcation criterion α is set to be 0.5.

We should also set at most 15 face images per sub-

ject to be stored in the

A summary in order to keep a

sort of balance between the training faces. The NIR-

PCA algorithm is summarized in Algorithm 2. Figure

1 illustrates the recovered original images A and error

images E of two different subjects using RPCA (see

Section 2.1). In our case, RPCA is used to recover

original (A) and error (E) images of the training set.

As for the test set, we start using our NIRPCA method

deﬁned in Algorithm 2. Figure 2 shows an example

of face recognition using NIRPCA method.

3 EXPERIMENT RESULTS AND

DISCUSSION

In this section, we evaluate the performance of the

proposed face algorithm NIRPCA based on the AR

(Martinez and Benavente, 1998), ORL (Samaria and

Harter, 1994), PIE (Sim et al., 2002), YALE (Bel-

humeur et al., 1997) and FERET (Phillips et al., 1998)

databases. All testing images are grayscale and nor-

malized. We precisely use the canonical size of the

images rather than the original one, based on the eye

corner locations. Table 1 provides information about

these databases.

Algorithm 2 : Face Recognition based on IRASL (NI-

RASL).

• INPUT: Images I

n+1

∈ R

w×h

, initial transformations

n+1

, in certain parametric group G,

A summary,

weight µ > 0

• 1. (τ

n+1

,x) = IRASL (

A, τ

n+1

) (Algorithm 1)

• 2. A

n+1

A ∗ x

• 3. E

n+1

= I

n+1

◦τ

n+1

−A

n+1

(Equation (5))

• 4. FOR each subject I

=k E

− E

n+1

, i = 1, . . . , m (Equation (7))

END FOR

• 5. M

min

← min(M

), i = 1, . . . , m

• 6. IF M

min

> α THEN

A ← (

A|A

n+1

)

m ← m + 1

n ← n + 1

ELSE

A ← (

A|A

n+1

)

n ← n + 1

END IF

• OUTPUT: Subject class of face image I

n+1

Table 1: Face databases used for experiments.

Database AR ORL PIE YALE FERET

Original

size

64 × 64 92 × 112 640 × 486 320 × 243 80 × 80

Canonical

size

50 × 50 90 × 90 110 × 130 120 × 140 80 × 80

Number

subjects

70 40 68 15 47

Number

of total

images

420 200 680 162 465

3.1 Comparison with Standard

Incremental Face Recognition

Methods

In this section, our method is tested on ﬁve different

databases as shown in Table 2. For each database,

of the images are used for the training set and the

remaining

of the images are randomly selected for

the test set. In these databases, face images are cap-

tured under varied conditions such as illumination

and shadowing levels, facial expression, different face

perspectives and with/without occlusion (sunglasses,

scarf, etc.). Besides, our method is compared to other

well-known incremental face recognition methods,

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552

(a) (b) (c) (d) (e) (f) (a') (b') (c') (d') (e') (f')

Figure 1: Images recovered by Robust PCA (RPCA), (a) - (f) are six images of two different subjects. D are original subjects,

A and E are recovered low-rank and error faces. Sunglasses, scarf and face expressions are successfully removed. These

images correspond to the training face images of the corresponding subjects.

(a)(a) (b) (c) (d) (a') (b') (c') (d')

Figure 2: Face recognition using NIRPCA method for two different subjects. (a), (a’): test Face (on the canonical frame). (b),

(b’): approximated low-rank matrix A using IRPCA. (c), (c’): recovered error matrix E. (d), (d’): the reconstructed face. The

classiﬁcation measure M

computed for these test faces is less than α, when α = 0.5. M

= 0.406, 0.428, 0.274 respectively

for each test face of subject (a) and M

= 0.364, 0.394, 0.383 respectively for each test face of subject (a’). Hence both test

faces are correctly classiﬁed.

i.e., ISVD (Hall et al., 2002), ABSVD (Brand, 2006)

and CCIPCA (Weng et al., 2003). In this section, oc-

cluded images in the AR database (with sunglasses

or scarfs) are omitted in the evaluation. Performance

on images with occlusions is considered in Section

3.3 and the accuracy rate is given in Table 2. These

Table 2: Accuracy rate of different incremental face recog-

nition algorithms.

% IPCA ABSVD CCIPCA NIRPCA

AR 79.71 73.44 77.43 86.42

ORL 72.68 72.23 76.82 77.39

PIE 72.17 75.89 79.64 88.72

YALE 78.12 80.27 85.47 90

FERET 76.09 80.63 85.44 89.58

results indicate that our method achieves the best per-

formance, speciﬁcally on images under different fa-

cial expressions, head positions and shadow levels.

This can be explained by the application of IRPCA

which recovers original face and removes shape and

expression variations better than other algorithms. In

fact, various angles of a face image or different head

positions of a subject can be reduced to an image dis-

tortion problem. Our algorithm can solve this prob-

lem since IRPCA approximates the original image

matrix A

n+1

, the sparse error matrix E

n+1

and a trans-

formation τ

n+1

at the same time and for each input im-

age I

n+1

. Thus applying the recovered afﬁne transfor-

mation τ

n+1

to the input image will generate a well-

aligned and distortion-free face image.

3.2 Incremental Face Recognition with

Different Numbers of Training

Images

In this section, we evaluate the performance when the

number of training images per subject varies. Based

on the AR database, K images are randomly selected

as the training set, and the remaining constitute the

test set, where K = 1, . . . , 10. Figure 3 shows the vari-

ation trend of different face recognition algorithms

and NIRPCA, in terms of recognition rate. This ex-

periment shows that beyond 3 images per subject

NIRPCA achieves the best performance. In fact, with

less than 4 images per subject, the D matrix, presented

in Section 2.1, is itself a low-rank matrix, hence it

cannot generate an exact and efﬁcient error matrix

E. Accordingly, approximated E

n+1

and similarity

measure are both inaccurate, which proves the slow

growth of NIRPCA’s accuracy rate with few images

per subject.

ExtensionofRobustPrincipalComponentAnalysisforIncrementalFaceRecognition

553

(e)

(a) (b) (c) (d)

(f) (g) (h)

(S1) (S2)

Figure 4: Some corrupted images with occlusions in the AR database of two subjects S

and S

. Images (a), (b), (e) and (f)

are occluded images with sunglasses. Images (c), (d), (g) and (h) are occluded images with a scarf.

1 2 3 4 5 6 7 8 9 10

60%

65%

70%

75%

80%

85%

90%

95%

100%

ISVD

BlockISVD

CCIPCA

NIRPCA

Figure 3: Accuracy rate of face recognition methods with

different number of training images per subject.

3.3 Performance on Images with

Occlusions

In practical face recognition applications, occlusions

(e.g., sunglasses or scarfs on faces) could not be

avoided. Robust and efﬁcient face recognition algo-

rithms should achieve good performance when faces

are practically occluded. We use the occluded face

images in the AR database to test the performance of

different algorithms. In this experiment, 70 subjects

are selected for the dataset, 5 occluded images and 6

non-occluded ones for each one. Some corrupted im-

ages with occlusions are shown in Figure 4.

These results show us that occluded images con-

siderably reduce the performance of IPCA, ABSVD

and CCIPCA, where the best recognition rate is only

about 20%. Whereas a high recognition rate is still

maintained by our algorithm NIRPCA as shown in

Table 3. This is due to the robustness of our method

regarding occlusions while other incremental algo-

rithms cannot efﬁciently remove disruption caused by

corruption on images.

Table 3: Accuracy rate of different incremental face recog-

nition algorithms with occluded test faces.

% IPCA ABSVD CCIPCA NIRPCA

AR 28.89 24.36 22.27 58.67

3.4 Comparison with Batch FRPCA

Wang and Xie (Wang and Xie, 2010) presented a face

recognition method, FRPCA, based on batch RPCA

(Peng et al., 2010) presented in Section 2.1. Fig-

ure 5 shows the variation trend between the FRPCA

1 2 3 4 5 6 7 8 9 10

65%

70%

75%

80%

85%

90%

95%

100%

FRPCA

NIRPCA

Figure 5: Accuracy rate variation of FRPCA and NIRPCA

with different numbers of training images per subject.

method and NIRPCA with reference to various num-

ber of training sets. Results show that the accuracy

of our method with different numbers of training face

images is very close to that of FRPCA. This experi-

ment shows high accuracy of the recovered low rank

face image A

n+1

approximated by Equation 6.

3.5 Runtime Comparison

In this section, we compare the runtime of NIR-

PCA with other IPCA-based face recognition meth-

ods namely ISVD, ABSVD and CCIPCA. This ex-

periment is carried on 70 different subjects, 13 images

for each subject from the AR database. Table 4 sum-

marizes the runtime results. Although the recogni-

Table 4: Runtime of different incremental face recognition

algorithms.

% IPCA ABSVD CCIPCA NIRPCA

Runtime (s) 96.89 135.41 26.53 60.29

tion rate of NIRPCA is the best, its runtime is slower

than that of CCIPCA due to iterative linearization of

the convex optimization using the splitting Bregman

method (Goldstein and Osher, 2009) (step 3 in Algo-

rithm 1). On the other hand, the runtime of NIRPCA

is faster than those of face recognition methods based

on ISVD or ABSVD. When we decrease the number

of principal components, the runtimes of ISVD and

ABSVD are similar to that of NIRPCA , but their face

recognition ratio decreases.

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554

4 CONCLUSIONS

In this paper, we proposed a new face incremental

recognition method based on one-by-one RASL. Our

method is robust to sparse corruptions on face images

and performed experiments on different databases

show its efﬁciency. The advantages of our method

is that it handles many aspects of image variations

such as face expression, image shadowing and various

angles. Above all, unlike other existing incremental

face recognition methods, the proposed method han-

dles efﬁciently corrupted images, mainly the occluded

ones. Experiment based on ﬁve different databases

show that our proposed method has better accuracy

rates. Further experiments can be extended to video

images, so that it could be used in real face recogni-

tion applications. However, for video images, impor-

tant image preprocessing work, such as face detec-

tion, should be done before the recognition step.

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