FACE HALLUCINATION USING PCA IN WAVELET DOMAIN
Abdu Rahiman V.
Department of Electronics and Communication Engineering, College of Engineering, Trivandrum, University of Kerala, India
Jiji C. V.
Department of Electronics and Communication Engineering, College of Engineering, Trivandrum, University of Kerala, India
Keywords:
Face Hallucination, Principal Component Analysis, Wavelets, Eigen Images.
Abstract:
The term face hallucination stands for recognition based super resolution of face images to improve the spatial
resolution. In this paper, we propose two face hallucination algorithms based on principal component analysis
(PCA) in the wavelet transform domain. In the spatial domain, the PCA based super resolution algorithm; a
low resolution (LR) observation is represented as the linear combination of LR images in an image database.
Super resolved image is obtained as the linear combination of the corresponding high resolution (HR) images
in the database. In the first approach proposed in this paper, PCA based hallucination algorithm is applied to
the wavelet coefficients of face image. The hallucinated face image is reconstructed from the super resolved
wavelet coefficients. In second method, face image is split in to four sub images and the first method is
separately applied to three textured regions. Fourth region, which is relatively smooth, is interpolated using
standard interpolation techniques. We compare the performance of the two proposed algorithms with their
spatial domain counter parts. The proposed method shows significant improvement over the spatial domain
approaches.
1 INTRODUCTION
Image acquisition is the front end of image process-
ing. Resolution of digital images is limited, because
of the fixed dimensions of sensor elements used for
image acquisition. Higher resolution requires smaller
sensor elements arranged at higher density which ob-
viously increases the cost of the device. Even if the
cost is affordable, sensor dimension cannot be min-
imized beyond a physical limit which has already
reached. HR images contain more details than the
corresponding LR image. Many image processing
applications need HR images for better machine in-
terpretation. Image super resolution is the process
of acquiring a HR image from one or more LR im-
ages, using signal processing means, by synthesizing
the missing high frequency details (Sung et al., 2003).
As it is a purely computational process, it will not in-
crease the cost of sensor. There are mainly two tech-
niques for image super resolution. In the first method,
multiple LR observations of a scene are used to re-
construct the HR image. This method is called multi-
frame super resolution. The second method called
single frame super resolution uses only a single LR
observation of the scene to reconstruct an HR image.
The super resolution process may also introduce
some unwanted high frequency components. In the
case of highly textured images like grass, leaves, etc.
the errors in super resolved image due to these spuri-
ous frequencies may not be significant. But in face
images, the effect due to spurious frequencies may
be very serious. Hence it is necessary that the face
super resolution process should not introduce many
unwanted details (Liu et al., 2007). The face super
resolution problem has wide applications in the ar-
eas of detection, recognition, surveillance and mon-
itoring. Face image super resolution based on recog-
nition is termed as face hallucination (Baker and
Kanade, 1999). Algorithm proposed by Baker and
Kanade(1999) uses a single LR observation to synthe-
size an HR face image, which makes use of a training
set of HR face images. High frequency details of the
HR face image is learned by identifying local features
from the training set.
180
Rahiman V. A. and C. V. J. (2008).
FACE HALLUCINATION USING PCA IN WAVELET DOMAIN.
In Proceedings of the Third International Conference on Computer Vision Theory and Applications, pages 180-187
DOI: 10.5220/0001083001800187
Copyright
c
 SciTePress

In the above hallucination approach, a Gaussian
image pyramid is formed for every image in the train-
ing set as well as for the LR observation. A set of
features are computed for every image in the image
pyramid. A gradient prior is learned using gradients
and second gradients of the images as features. Hallu-
cinated face is estimated using maximum a posteriori
(MAP) framework, which uses leaned prior in its cost
function.
In the eigen transformation based hallucination
algorithm proposed by Wang and Tang (2005), first
an HR database and corresponding LR face image
database is prepared. LR observation image is then
represented as the linear combination of LR database
images. The coefficients are determined from the
PCA coefficients. The super resolution is achieved by
finding the linear combination of the HR images with
the same coefficients. To avoid abnormalities in the
image, a regularization is done with respect to eigen
values.
Jiji et al.(2004) proposed a wavelet based single
frame super resolution method, for general images,
making use of an HR image database. Observed im-
age as well as the data base images are decomposed
using wavelet transform. Wavelet coefficients of the
super resolved image is learned from the coefficients
of images in the database. The problem is solved
under a regularization frame work using the learned
wavelet prior. An edge preserving smoothness con-
straint is used to maintain the continuity of edges in
super resoved image. Capel and Zisserman (2001) di-
vided the face image in to six regions or subspaces
and then PCA based super resolution is applied on
the respective portions independantly. The subspaces
are combined and a global regularization is done to
minimize artifacts at the boundaries of the regions.
In this paper, we propose two methods for face
hallucination in wavelet domain. In both the meth-
ods, face hallucination based on eigen transformation
is applied on the wavelet coefficients. In the first
method, face image is treated as a single image and
in the second method, it is split into several regions
and then the super resolution techniques are applied.
Eigen transformation uses PCA coefficients for im-
plementing the super resolution. Therefore the hallu-
cination algorithms proposed in this paper are refered
as face hallucination using PCA in wavelet domain
and face hallucination using subspace PCA in wavelet
domain.
The remaining part of this paper is organized as
below. The background for Wavelets and PCA rela-
vant to the problem of face hallucination are discussed
in subsections 2.1 and 2.2 respectively. The signif-
icance of super resolution using PCA and subspace
models in spatial domain as well as in the wavelet do-
main are given in subsections from 2.3 to 2.6. Section
3 consists of the details of simulation work and results
and the paper concludes in section 4.
2 HALLUCINATION WITH PCA
IN WAVELET DOMAIN
In this paper two methods are proposed for face hal-
lucination, using PCA in wavelet domain. Both the
methods use, a training set of face images consisting
of HR and corresponding LR images. In the first al-
gorithm, wavelet coefficients of images in the training
set are determined. Wavelet coefficients of LR obser-
vation is also determined. Eigen transformation based
hallucination is applied individually on all the four
wavelet coefficients, to get the super resolved wavelet
coefficients. The hallucinated face is obatained by
computing the inverse wavelet transform of super re-
solved wavelet coefficients. In the second method,
face image is split in to four regions out of which three
regions are textured and are more significant. The
fourth region is relatively flat. Textured regions are
super resolved seperately using the method described
above. Flat regions are interpolated using standard in-
terpolation techniques. In the following subsections
we discuss briefly on wavelets and PCA followed by
the usage of eigen vectors for super resolution.
2.1 Discrete Wavelet Transform
The Discrete Wavelet Transforms (DWT) splits the
image into four spectral bands (Daubechies, 1992).
Each of the subbands is one quarter in size of the orig-
inal image and these coefficients (subbands) preserve
the locality of spatial and spectral details in the im-
age. This property of spectral and spatial localization
is useful in problemslike image analysis, especially in
super resolution. Wavelet coefficients of an image are
determined using filters arranged as shown in figure 1.
g(n) and h(n) are the half band high pass and low pass
filters respectively. Resulting wavelet coefficients are
LL (low-low), LH (low-high), HL (high-low) and HH
(high-high). Perfect reconstruction of the image is
possible from four wavelet coefficients, using inverse
DWT (IDWT). There are many types of wavelets de-
pending on the type of filters used for g(n) and h(n).
Wayo et al (2006) shows that Symlets givesbetter per-
formance in PCA based face recognition techniques,
over other types of wavelet. In this paper, proposed
algorithms are tested with Daubechies, Symlet and
Coiflet wavelets. Figure 2 shows the typical single
level wavelet decomposition of a face image.
FACE HALLUCINATION USING PCA IN WAVELET DOMAIN
181

Figure 1: Filter structure for the wavelet decomposition of
an image.
Figure 2: Single level wavelet decomposition of face image.
2.2 Principal Component Analysis
PCA is a powerful tool for analyzing data by per-
forming dimensionality reduction in which the orig-
inal data or image is projected on to a lower dimen-
sional space. An image in a collection of images can
be represented as the linear combination of some ba-
sis images. Let there be M images with N pixels each
in a collection, the basis function to represent these
images can be found by using Eigen vectors (Moon
and Stirling, 2005). Let x
i
be the individual image
vectors and x be the mean image vector, then the mean
removed image is given by
l
i
= x
i
− x (1)
All images in the database are arranged in to column
vectors by scanning them in raster scan order. All
the mean removed images are arranged in columns to
form the matrix
L = [l
1
,l
2
,...,l
M
] (2)
Covariance matrix of L can be found as
C = L × L
T
(3)
Let E be the matrix of eigen vectors; e
1
,e
2
,...be the
eigen vectors and S be the eigen values of C. Then
C, S and E will satisfy the following equation.
E = [e
1
,e
2
,...,e
N
] (4)
C× E = E × S (5)
where S = diag(λ
1
,λ
2
,...,λ
N
) and λ
1
,λ
2
,...,λ
N
are
the eigen values. The eigen vectors are arranged in
such a way that respective eigen values are in decreas-
ing order. The images can be projected on to these
eigen vectors and the coefficients w
l
thus obtained are
called Principal Component Analysis (PCA) coeffi-
cients.
w
l
= E
T
× (x
i
− x) = E
T
× l
i
(6)
The mean removed image can be reconstructed using
the following relation.
l
i
= E × w
l
(7)
Adding the mean image vector to l
i
gives the actual
image vector. In the discussions followed, image is
considered as the mean removed image unless other-
wise mentioned. An important fact about PCA co-
efficients is that the image can be reconstructed with
minimum mean square error, using the first few coef-
ficients alone. This feature is used in image compres-
sion and noise reduction applications.
2.3 Super Resolution with Eigen Images
PCA can be used for super resolution by slightly mod-
ifying the equations to find Eigen vectors and PCA
coefficients. The covariance matrix C will have large
dimensions. Therefore finding eigen vectors will be
difficult while the number of significant eigen vectors
will be much less. Therefore an alternate scheme is
used to determine the significant Eigen vectors (Turk
and Pentland, 1991). Define the matrix K as
K = L
T
× L (8)
Let V be the matrix containing eigen vectors of K.
V = [v
1
,v
2
,...,v
M
] (9)
Most significant M eigen vectors of C can be deter-
mined by
e
i
= L×
v
i
kL× v
i
k
for i = 1 to M (10)
The reconstructed image
ˆ
l is obtained as
ˆ
l = E × w
l
= L× c (11)
where
c =
v
i
kL× v
i
k
× w
l
(12)
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182

In super resolution process, we use databases of
registered HR images and corresponding LR images.
Let H be the matrix of mean removed image vectors
of HR images in database, corresponding to the ma-
trix L. The LR image is represented as the linear com-
bination of the image vectors as shown in equation
(11) (Wang and Tang, 2003). Hallucinated face im-
age can be determined by using the same coefficients
but by using the HR image vectors H
h
SR
= H × c (13)
where h
SR
is the hallucinated face image. It simply
means that if LR image is the linear combination of
image vectors in the LR face images, then the cor-
responding HR image will be linear combination of
the respective HR image vectors while keeping the
same coefficients. As the input resolution decreases
the hallucinated image will have artefacts. These arte-
facts are minimised by applying a constraint based on
eigen values. Let Q× Q be the resolution magnifica-
tion to be done and α a positive constant. To apply
the constraint, super resolved image is projected onto
the HR Eigen vectors E
h
, to get the coefficients w
h
.
Constrained PCA coefficients, ˆw
h
are obtained as
ˆw
h
=

w
h
(i) if |w
h
(i)| < λ
1/2
i
α/Q
2
sign(w
h
(i))λ
1/2
i
otherwise
(14)
where the λ
i
are the eigen values corresponding to E
h
.
These new coefficients, ˆw
h
is used to reconstruct the
super resolved images from HR eigen vectors. Super
resolved image x
h
is given by
x
h
= E
h
× ˆw
h
+ x
h
(15)
x
h
is the mean of HR images in the database. As the
value of α increase, super resolved image may have
more high frequency details as well as spurious high
frequency components. On the other hand, when α
is reduced, the super resolved image tends towards
mean face image.
Figure 3: Mean face image and first few Eigen faces.
2.4 Super Resolution using Eigen
Subspaces
In the case of a normal face image, some of the por-
tions like eyes, nose, etc. are highly textured and more
significant, so it needs more attention during super
resolution. Bicubic interpolation will be sufficient at
the smooth regions like forehead, cheeks etc. In the
second algorithm proposed in this work, face image
is split in to left eye, right eye, mouth with nose and
the remaining area as shown in figure 4. These re-
gions are the subspaces of the entire face space. The
dimension of such subspaces will be much less com-
pared to the dimension of entire face space.
PCA based super resolution techniques out per-
forms other hallucination methods if the test image
is closely similar to the images in database. Other-
wise, it gives better results if number of images in the
database is high. If sub images are used for super res-
olution, the number of images required in the database
is less compared to the case of whole face image for
a given reconstruction error. PCA based hallucination
is applied on all the subimages separately and the re-
sulting super resolved regions are combined with the
interpolated version of remaining area. The computa-
tional cost associated with this method is much less
because it is easy to compute the Eigen vectors of
smaller subspace images.
Figure 4: Face image divided into subspaces. (1) Entire face
image with regions marked, (2 a, b, c) Textured regions,
left eye, right eye and mouth with nose respectively. (3)
Remaining smooth region.
Face image has a specific structure and this prior
information is utilized in face hallucination algo-
rithms. In a specific class of properly aligned face im-
ages, contours, patterns and such facial features will
be closely aligned. DWT decomposition of face im-
age splits the image into four spectral bands without
losing spatial details. Details in respective subbands
will be more similar for different face images. There-
fore, using very less number of eigen images we will
be able to capture the finer details accurately. Hence
a PCA based super resolution scheme in wavelet
domain will be more efficient and computationally
FACE HALLUCINATION USING PCA IN WAVELET DOMAIN
183

less expensive. Another importance of such trans-
form domain approach is that, all images are stored
in compressed format and most of the popular im-
age compression techniques are in transform domain.
Wavelet based compression is used in JPEG2000,
MPEG4/H.264 and in many other standard image and
video compression techniques. Therefore the pro-
posed algorithm can be directly applied on a com-
pressed image after properly rearranging the com-
pressed coefficients. So there is no need for decom-
pressing the image which will considerably reduce the
computational cost.
2.5 Super Resolution with Eigen Images
in Wavelet Domain
In the first method proposed for face hallucination in
wavelet domain, HR and LR face images in database
are decomposed using DWT to form LR and HR
wavelet coefficient database
[L
xx
] = DWT(L) (16)
[H
xx
] = DWT(H) (17)
where xx stands for LL, LH, HL and HH wavelet sub-
bands. Test image also is decomposed with DWT and
the PCA based super resolution method described in
section 2.3 is applied on these wavelet coefficients
separately. Resulting wavelet coefficients,h
SR−xx
, are
given by
h
SR−xx
= H
xx
× c
xx
(18)
where c
xx
is the coefficient for linear combination
in different subbands, calculated using equation (12).
The constraint based on eigen value, as given in equa-
tion (14), is applied individually on all the wavelet
coefficients to give
ˆ
h
SR−xx
. Super resolved face im-
age h
SR
is computed by determining the IDWT of the
coefficients
ˆ
h
SR−xx
.
h
SR
= IDWT(
ˆ
h
SR−xx
) (19)
The complete algorithm for face hallucination us-
ing PCA in wavelet domain is summarized below.
Step 1: Prepare the HR and LR image databases and
compute the wavelet subbands of all the images in
the databases.
Step 2: For all the wavelet coefficients, find the vec-
tors L, the matrix K and the eigen vectors V as in
section 2.3.
Step 3: Determine the significant eigen vectors of C
using equation (10).
Step 4: Find the PCA coefficients w
l
of the test image
from equation (6).
Figure 5: Eigen images of wavelet coefficients. First few
Eigen images of LL, LH, HL and HH coefficients.
Step 5: Using w
l
compute the linear combination co-
efficients c.
Step 6: The super resolved coefficients are obtained
from equation (13).
Step 7: Reconstruct the coefficients with equation
(15) after applying the constraints in equation
(14).
Step 8: Reconstruct the super resolved subimages by
finding the IDWT of super resolved wavelet coef-
ficients.
2.6 Subspaces in Wavelet Domain
The second method proposed for face hallucination is
an extension of the first algorithm. In this method HR
and LR face images in database as well as the LR test
image are split in to four regions as shown in figure 4.
Then the three textured regions are individually super
resolved using the algorithm explained in section 2.5.
The fourth region is interpolated with bicubic interpo-
lation technique and the three super resolved regions
are combined with the fourth interpolated region to
form the hallucinated face image. This subspace tech-
nique in wavelet domain for super resolution, reduces
computational cost considerably, because the size of
regions are small and therefore the computation re-
quired to determine wavelet coefficients are very less.
PCA in wavelet domain further reduces the memory
required for implementation. These favourable fea-
tures along with the acceptable performance make it
suitable for smaller systems which use input images
with sufficient resolution. This method is not suitable
where input image resolution is very less, because it
is not feasible to split and align test image into subim-
ages when the input image resolution is very less. The
steps involved in the hallucination with subspace PCA
in wavelet domain, are listed below:
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Step 1: Split all the face images in the HR and LR
databases into mouth with nose left eye, right eye
and remaining area.
Step 2: Determine the wavelet coefficients of eyes
and mouth with nose.
Step 3: Repeat steps 2 to 8 of the algorithm de-
scribed in section 2.5 on all the three textured por-
tions.
Step 4: Combine the super resolved regions with in-
terpolated version of remaining part to form the
hallucinated face image.
3 EXPERIMENTAL RESULTS
Face image database for the experiments is prepared
from BioID face database, PIE database and Yale
database. Front facial images are selected and man-
ually cropped to uniform size of 128× 96 after align-
ing with three control points, centres of left and right
eyes and the centre of the lips. The distance between
the control points are predefined. Distance between
eye centres is fixed to 50 pixels. 103 images are pre-
pared as described above to form the HR face image
database. LR image database is prepared by down
sampling the HR images. Performance of the pro-
posed algorithms is analysed by using input images
with different resolutions. To obtain a LR test im-
age and in order to be able to quantify the improve-
ments during super resolution, we consider a HR im-
age which does not belong to the training set and
downsample it by required magnification factor Q.
Figure 6: Hallucinated faces with the method proposed in
section 2.5. Input, original, Bicubic interpolated and hal-
lucinated images. For magnification factors of four (top),
eight (middle) and eleven (bottom).
Figure 6 shows the hallucination result of the
method proposed in section 2.5 with magnification
factors 4, 8 and 11. The hallucinated result is much
better when the images in database precisely repre-
sent the features of the test face. But the result seems
to be noisy when the test face has significant compo-
nents in the space orthogonal to the space spanned by
Eigen vectors. As it can be observed from figure 6, the
hallucinated result is much better than the bicubic in-
terpolation, for higher values of Q. But if the number
of pixels in the input image is very less, the proposed
PCA based method fails to find the super resolvedim-
age. In our experiment, the resolution of HR image is
128 × 96 pixels. Therefore, it is observed that when
the value of Q is above 11, size of input image size
will be less than 11× 9 pixels and the algorithm fails
to produce correct result. Figure 7 show the result of
the proposed algorithm, when the test image not sim-
ilar to the images in the database.
Figure 7: Hallucination result with a test image not similar
to the database images. Input, original, Bicubic interpolated
and hallucinated face images.
Figure 8: Hallucinated face with noisy input image. (a)
Input image with Gaussian noise, (b) Original image, (c)
Bicubic interpolated image and (d) hallucinated image.
Noise variance σ = 0.001 (top), σ = 0.01 (middle) and
σ = 0.1 (bottom).
FACE HALLUCINATION USING PCA IN WAVELET DOMAIN
185

Next we perform the experiment with noisy test
image. Gaussian noise is added to the test image. In
this case, the test image and its correspondingHR ver-
sion is included in the LR and HR databases respec-
tively and the corresponding result is shown in fig-
ure 8. This result shows the recognition performance
of the algorithm with noisy observations. Figure 9
gives a comparison between the results of the pro-
posed method with the corresponding spatial domain
approach.
Figure 9: Hallucinated faces; Input image , Original image,
Bicubic interpolated image, hallucinated image with pro-
posed method and hallucinated result with spatial domain
approach.
The proposed algorithm is then tested for the
variation in performance with the different types of
wavelets. In this particular case, algorithm increases
the resolution by a factor of two horizontally and ver-
tically (magnification factor is two, Q = 2). Table 1
give the change in performance with wavelet types.
Figure 10: Textured regions reconstructed using the algo-
rithm proposed in section 2.6. Hallucinated, original and
bicubic interpolated images.
Now we show the result using the second algo-
rithm proposed in section 2.6. Super resolved subim-
ages are separately shown in figure 10 along with their
original and bicubic interpolated versions. These re-
sults are for a magnification factor of two (Q = 2).
Figure 11 shows the final hallucinated face. Eyes,
nose, lips etc are sharper than the bicubic interpolated
version. Boundaries of the subimages are barely vis-
ible in this image, but it will become more visible as
the magnification factor increases.
Figure 11: Hallucinated face image with subspace PCA in
wavelet domain. Hallucinated face, Original face and bicu-
bic interpolated face image.
Table 1: Change in PSNR with different types of wavelets.
Wavelet PCA in Subspace PCA
Type Wavelet Domain in Wavelet domain
Symlet2 28.777 24.630
Symlet5 28.773 24.046
Symlet7 28.706 25.048
Symlet9 28.794 24.891
Coif3 28.641 25.048
Coif4 28.719 25.029
Coif5 28.765 24.961
Daub2 28.777 24.630
Daub3 28.671 24.802
Daub7 28.684 24.875
Table 2: Comparison of performance of different PCA
based hallucination algorithms.
PCA in PCA in Subspace Subspace
Bicubic spatial Wavelet PCA in PCA in
domain domain spatial Wavelet
domain domain
When test image is not included in database
24.898 26.590 26.664 24.130 24.230
Test image within database and only 10 images
in database (Recognition performance)
29.367 32.655 44.924 30.503 30.542
The experiments are repeated with different types
of wavelet function. Table 1 give the change in PSNR
with different types of wavelets. Though the PSNR
seems to be low, hallucinated images are sharp at
eyes, nose, etc. compared to the bicubic interpolated
image. In both the cases presented here, result im-
proves considerably when the test image is accurately
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186

represented as the linear combination of images in
database. Table 2 compares the PCA based halluci-
nation algorithms in spatial domain and wavelet do-
main. Wavelet domain approach gives better results
while providing the advantages like lower memory re-
quirement computational efficiency.
4 CONCLUSIONS
In this paper two algorithms are proposed for face hal-
lucination using PCA in wavelet domain. First algo-
rithm uses the face image as a single image where
as the second method splits the image into textured
and flat regions. In both the cases wavelet subbands
of the images are determined and PCA based hallu-
cination technique is applied on these subbands inde-
pendently. The results are compared with the eigen
transformation based hallucination in spatial domain.
An advantage of PCA based methods discussed here,
over many other hallucination methods, is that it does
not require optimization of the result. Simulation re-
sults show significant improvement in the super res-
olution performance of proposed algorithms. Over-
all performance can be improved by using multi level
wavelet decomposition. Currently we are trying to ap-
ply the proposed methods in other transform domains
like curvelet, contourlet etc.
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