Yanushkevich S.N., Wang P.S.P., Gavrilova M.L., Srihari S.N. (eds.) Image Pattern Recognition. Synthesis and Analysis in Biometrics
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198 Synthesis and Analysis in Biometrics
g(r) = grad(G(r) ∗ I(r)),
I(r) is the image and G(r)=e
|r|
2
2σ
2
is a Gaussian kernel.
The Marr-Hildreth operator, also known as Laplacian of Gaussian
(LoG), uses a Gaussian kernel which is optimal in Gaussian noise, whereas
the 1/r kernel is an artifact of its force field nature, and the intrinsic
smoothing it affords is merely a fortunate consequence of its formulation.
We must also stress that whilst the Marr-Hidlreth operator is linear,
convergence is non-linear because it is based on force direction rather
than force. This nonlinearity means that we are obliged to perform the
operations in the order shown; we cannot take the divergence of the
force and then divide by the force magnitude: div (grad/ | grad |) =
(div grad)/ | grad |. This is easily illustrated by a simple example using
the scalar field e
x
:
$
div(grad/|grad|)
∇·
∇e
x
|∇e
x
|
= ∇·
e
x
i
e
x
= ∇·i =0
%
=
(divgrad)/|grad|
∇·∇e
x
|∇e
x
|
=
e
x
e
x
=1
(7.12)
where i is a unit vector in the x direction. The convergence is zero because
we have a field of parallel unit magnitude vectors, whereas in the second
case the vectors are parallel but the magnitude changes, resulting in a net
outflow of flux at any point. This illustrates that even though convergence
looks very much like a Laplacian operator, it definitely is not.
Figure 7.9(b) shows the convergence field for an ear image, while
Fig. 7.9(a) shows the corresponding field lines. A magnified version of
a small section of the force direction field, depicted by a small rectangular
insert in Fig. 7.9(a), is shown in Fig. 7.9(c). In Fig. 7.9(b) the convergence
values have been adjusted to fall within the range 0 to 255, so that negative
convergence values corresponding to antichannels appear as dark bands,
and positive values corresponding to channels appear as white bands.
Notice that the antichannels are dominated by the channels, and that the
antichannels tend to lie within the confines of the channels. Also, notice
how wells appear as bright white spots.
“The convergence map provides more
information than the field lines, in the form
of negative versions of wells and channels”
The correspondence between the convergence function and the field line
features can be seen by observing the patterns in the force direction field
shown in Fig. 7.9(c). Notice the correspondence between the converging
arrows and white ridges, and between the diverging arrows and black ridges.
The convergence map provides more information than the field lines,
in the form of negative versions of wells and channels or antiwells and
April 2, 2007 14:42 World Scientific Review Volume - 9in x 6in Main˙WorldSc˙IPR˙SAB
Force Field Feature Extraction for Ear Biometrics 199
Field line features Conv ergence map Magnified insert force field
(a) (b) (c)
Fig. 7.9. Convergence field for an ear. (b) shows the convergence field whilst (a) shows
the corresponding field line feature extraction. (c) depicts a magnified portion of the
force field corresponding to the little rectangular insert show in (a).
antichannels. Although it should be possible to modify the field line
technique to extract this extra information by seeding test pixels on a
regular grid and reversing the direction of test pixel propagation.
Figure 7.10 shows the convergence field of an ear image with the
corresponding field lines superimposed. Figure 7.10(a) is the field line
map, and Fig. 7.10(b) is the convergence map, while Fig. 7.10(c) is the
superposition of one on the other. We can see clearly how channels coincide
with white ridges in the convergence map and that potential wells coincide
with the convergence peaks. Notice the extra information in the center
of the convergence map that is not in the field line map, illustrating an
advantage of convergence over field lines.
7.5. Ear Recognition
The technique was validated on a set of 252 ear images taken from 63
subjects selected from the XM2VTS face database
[
11
]
achieving almost
total correct recognition.
Multiplicative template matching of ternary thresholded convergence
maps was used where levels less than minus one standard deviation are
mapped to −1, whilst those greater than one standard deviation map to
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200 Synthesis and Analysis in Biometrics
Field lines Convergence map Superposition
(a) (b) (c)
Fig. 7.10. Correspondence between channels and convergence. (a) shows the field line
map whilst (b) shows the convergence map and (c) shows the superposition of these
two maps where we can see clearly that the channels coinccide with the ridges in the
convergence map.
+1, and those remaining map to 0. A threshold level of one standard
deviation was chosen experimentally resulting in the template channel
thickness shown in Fig. 7.11. This figure also shows a rectangular exclusion
zone centered on the convergence magnitude centroid; the centroid of the
convergence tends to be stable with respect to the ear features and this
approach has the added advantage of removing unwanted outliers such as
bright spots caused by spectacles. The size of the rectangle was chosen
as 71 × 51 pixels by adjusting its proportions to give a good fit for the
majority of the convergence maps. Notice how for image 000-2 which is
slightly lower than the other three, that the centroid-centered rectangle
has correctly tracked the template downwards.
The inherent automatic extraction advantage was demonstrated by
deliberately not accurately extracting or registering the ears in the sense
that the database consists of 141×101 pixel images where the ears have only
an average size of 111×73 and are only roughly located by eye in the center
of these images. This can be seen clearly in the top row of Fig. 7.11 where
we see a marked variation both in vertical and horizontal ear-location, and
also that there is a generous margin surrounding the ears.
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Force Field Feature Extraction for Ear Biometrics 201
000-1-L1 000-2-L1 000-3-L1 000-4-L1
Fig. 7.11. Feature extraction for subject 000 at 141× 101 pixels. The top row shows the
four sample ear images for the subject, whilst the middle row shows the corresponding
convergence maps. The bottom row shows the maps after ternary thresholding and also
depicts a rectangle centered on the convergence centroid. Notice that the ear image for
000-2 is lower than the others but that the rectangle has correctly tracked it downwards.
7.5.1. Experiment Results and Analysis
The force field technique gives a correct classification rate of 99.2% on this
set. Running PCA
[
19
]
on the same set gives a result of only 62.4%, but
when the ears are accurately extracted by cropping to the average ear size
of 111 ×73, running PCA then gives a result of 98.4%, thus demonstrating
the inherent extraction advantage. The first image of the four samples from
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202 Synthesis and Analysis in Biometrics
Fig. 7.12. This figure shows the first 4 of the 63 eigenvectors for the 111 × 73-pixel
images. The first image of the four samples from each of the 63 subjects was used in
forming the PCA covariance matrix.
each of the 63 subjects was used in forming the PCA covariance matrix.
Figure 7.12 shows the first 4 eigenvectors for the 111 × 73-pixel images.
The effect of brightness change by addition was also tested where we see
that in the worst case where every odd image is subjected to an addition of 3
standard deviations the force field results only change by 2%. whereas those
for PCA under the same conditions fall by 36%, or by 16% for normalized
intensity PCA, thus confirming that the technique is robust under variable
lighting conditions.
These results are presented in Table 7.1. We also include the decidability
index after Daugman
[
17
]
which combines the mean and standard deviation
of the intra-class and inter-class measurement distributions giving a good
single indication of the nature of the results. This index d
measures
how well separated the distributions are, since recognition errors are
caused by their overlap. The measure aims to give the highest scores
to distributions with the widest separation between means, and smallest
standard deviations. If the two means are µ
1
and µ
2
and the two standard
deviations are σ
1
and σ
2
then d
is defined as
d
=
| µ
1
− µ
2
|
(σ
2
1
+ σ
2
2
)/2
(7.13)
Notice that the best case index for PCA is slightly higher than the value
of 3.43 obtained for the 141×101 images but this could be attributed to the
reduction in data set size from 252 to 189 also to the fact that the images
have been more fully extracted for PCA.
We have also included noise performance figures where noise has been
modeled as additive noise with a zero mean Gaussian distribution. The
signal to noise ratios of 6dB, 12dB, and18dB used are calculated as
20log
10
(S/N). We see that the technique enjoys excellent noise tolerance
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Force Field Feature Extraction for Ear Biometrics 203
where even for extreme noise of 6dB the performance only falls by about
3.6%. Interestingly at a ratio of 18dB the recognition rate actually improves
over the noiseless recognition rate, but this must be put down to the
combination of small changes and the random nature of the noise process.
For reference we have also included the corresponding noise results for PCA
under the same conditions, where we see that PCA also performs well under
noisy conditions but not quite as well as FFE at 6dB where the fall is about
4.8%.
Table 7.1. Comparison of force field and PCA recognition results.
METHOD PASSES NOISE CCR BRIGHT DECIDABILITY
141 x 101 image with deliberately p oor extraction and registration
FFE 250/252 Nil 99.2% 0 3.432
FFE 251/252 18 dB 99.6% 0 3.488
FFE 249/252 12 dB 98.8% 0 3.089
FFE 241/252 6 dB 95.6% 0 1.886
FFE 250/252 Nil 99.2% 1 3.384
FFE 247/252 Nil 98.0% 2 3.137
FFE 245/252 Nil 97.2% 3 2.846
PCA 118/189 Nil 62.4% 0 1.945
111 x 73 image with accurate extraction and extraction and registration
PCA 186/189 Nil 98.4% 0 3.774
PCA 186/189 18 dB 98.4% 0 3.743
PCA 186/189 12 dB 98.4% 0 3.685
PCA 177/189 6 dB 93.6% 0 3.606
PCA 130/189 Nil 68.8% 1 1.694
PCA 120/189 Nil 63.6% 2 0.878
PCA 118/189 Nil 62.4% 3 0.476
PCA 181/189 Nil 95.6% 1
∗
3.171
PCA 172/189 Nil 91.0% 2
∗
1.910
PCA 166/189 Nil 82.5% 3
∗
1.140
Note: The table is divided between images which have deliberately poor registration in
the upper half and those which have accurate registration in the lower half. The first
column indicates whether force field feature extraction (FFE) or PCA extraction has
been used. The second column gives the number of passes as a fraction of the total
number of trials and this has been converted to a correct classification rated (CCR) in
column four. An asterisk indicates that the PCA has been normalized with respect to
image intensity.
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204 Synthesis and Analysis in Biometrics
7.6. Conclusions
In conclusion we may say that in the context of ear biometrics we
have developed a new linear transform that transforms an ear image,
with very powerful smoothing and without loss of information, into a
smooth dome shaped surface whose special shape facilitates a new form
of feature extraction that extracts the essential ear signature without the
need for explicit ear extraction; and in the process we have also verified
the recognition potential of the human ear for biometrics. We have
also described the convergence operator and shown that it is a valuable
alternative form of the field line feature extraction. We have validated the
technique by experiment and in the process we have contributed to the
mounting evidence that ears are indeed viable as a biometric. In our future
work we will continue to promote the case for ears as a biometric and to
extend our work in new directions. For example, this chapter has focused
on ear biometrics from an analysis point of view, but we believe that ear
synthesis could play an important role in future and some work has already
beguninthisdirection.
Acknowledgments
The author would like to pay tribute to Dr. Vlad P. Shmerko for his
generous help in preparing the LaTeX version of this manuscript and for
his valuable advice and suggestions regarding its content.
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Functionals for Image Feature Extraction. Proc. 10th British Machine Vision
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206 Synthesis and Analysis in Biometrics
Appendix: Force and energy field computation in Mathcad
Fig. A.1. Mathcad code for the force field by convolution in which “j” denotes
the complex operator and “cfft” and “icfft” denote the Fourier and inverse Fourier
transforms, respectively, and “usr” denotes the unit sample response for a force field.
You may need to type “1j” to tell Mathcad that “j” is not a Mathcad variable.
Fig. A.2. Mathcad code for the energy field by convolution. Note that the code fragment
“usr” or unit sample response is the only difference from the code for the force field shown
in Fig. A.1. The “usr” in this case corresponds to the potential energy function shown
in Fig. 7.4.
April 2, 2007 14:42 World Scientific Review Volume - 9in x 6in Main˙WorldSc˙IPR˙SAB
Chapter 8
Nontensor-Product-Wavelet-Based Facial
Feature Representation
Xinge You
∗,†,§
,QiuhuiChen
∗
,PatrickWang
¶
, and Dan Zhang
∗
∗
Faculty of Mathematics and Computer Science, Hubei University,
WuHan, HuBei, 430062, China
†
Department of Computer Science,
Hong Kong Baptist University, Hong Kong
‡
Image Processing Group, College of Computer and Information Science,
Northeastern University, Boston, MA 02115, USA
§
xyou@hubu.edu.cn
¶
pwang@ccs.neu.edu
Biometrics are automated methods of recognizing a person based on
a physiological or behavioural characteristic. Among the features
measured are: face, fingerprints, hand geometry, handwriting, iris,
retinal, vein, and voice. Facial Recognition technologies are becoming
one of the foundation of an extensive array of highly secure identification
and personal verification solutions. Meanwhile, a smart environment is
one that is able to identify people, interpret their actions, and react
appropriately. one of the most important building blocks of smart
environments is a person identification system. Face recognition devices
are ideal for such systems, since they have recently become fast, cheap,
unobtrusive, and, when combined with voice identification, are very
robust against changes in the environment.
As with all biometrics, facial recognition follows the same four steps
below: sample capture, feature extraction and representation, template
comparison, and matching. Feature representation of facial image play
a key role in facial recognition system. In other words, the performance
of facial recognition technology is very closely tied to the quality of the
representation of facial image. In this chapter, we develop a new method
for facial feature representation by using a nontensor product bivariate
wavelet transform. A new nontensor product bivariate wavelet filter
banks with linear phase are constructed from the centrally symmetric
matrices. Our investigations demonstrate that these filter banks have
a matrix factorization and they are capable of describing the features
of face image. The implementations of our algorithm are made of
207