A
New Force Field Transform for Ear and Face Recognition
David J. Hurley, Mark S.
Nixon
and John
N.
Carter
Department
of
Electronics and Computer Science
University
of
Southampton, Southampton
SO1
7
1
BJ,
UK
[dj
h9
7r
I
msn
I
jncl
@ecs
.
sot
on.
a
c
.
uk
Abstract
The objective in defining feature space is to reduce the
dimension of the original pattem space yet maintaining
discriminatory power for classijkation
[l].
To meet this
objective in the context of ear and face biometrics
a
novel
force field transformation has been developed in which
the image is treated as an array of Gaussian attractors
that act as the source
of
a force field. The directional
properties of the force field are exploited to automatically
locate a small number of potential energy wells and
channels that form the basis
of
a characteristic feature
vector. Here, we generalise the analysis, and the stock of
applications.
1
Introduction
In vision-based biometrics we aim to extract a
compact description from an image which may
subsequently be used to identify or confirm the identity of
the owner [2]. Now in order to meet this objective we
have developed
an
invertible linear transformation which
we call the
forcefield transform
[3].
Our primary concern
has been automatic ear recognition
[3];
we now extend
our analysis and show that the transformation appears to
have similar potential for automatic face recognition.
The entire ear image is converted into a force field by
pretending that each pixel exerts an isotropic force on all
the other pixels that is proportional to pixel intensity and
inversely proportional to the square of the distance. There
is a potential energy surface associated with this force
field, which in the case of
an
ear can be likened to a small
mountain with a few peaks joined by ridges. We call
these peaks potential energy wells and the ridges joining
them potential energy channels. The directional property
of the force field is exploited to automatically locate these
potential wells and channels, which then form the basis of
the ear's signature.
An array of unit value exploratory mobile test pixels is
arranged in a closed loop formation surrounding the target
ear. Each test pixel is then allowed to follow the pull of
the force field
so
that its trajectory forms a field line and it
will continue moving until it reaches the center of a well
where no force is exerted and no further movement is
possible. Since the force field at a point is unique all field
lines which arrive at a given point will follow the same
path from that point onwards thus forming channels. This
process is illustrated in Figure
3
where an elliptic array
of
50
test pixels is placed in the force field and iterated to
produce the field lines shown in the center. The most
striking example of the channel formation process is seen
at the top of the ear where
14
field lines combine to form
a channel which flows rightwards following the contour
of the ear-rim to finally terminate
in
a well. The locations
of the wells are extracted by simply noting the
coordinates of the clusters of test pixels that eventually
form. These locations are shown on the right,
superimposed on force field magnitude.
The structure of the force field as described by the
field lines shows remarkable initialization invariance
[3]
in
the sense that if the radius
of
the ellipse is altered
or
if
its center is translated, the same channel and well
description will result.
Also
if
the image is scaled the
force field structure scales with the image. We have also
found that the process is very tolerant of noise, due to its
inherent averaging. In this paper we generalise our earlier
presentation to show more basic properties of the
technique and how it can be applied to other pattern
recognition problems.
2
Force Field Approach
The image is transformed by pretending that it consists
of an array of
N
Gaussian attractors, which act as the
source of a force field. Each pixel is considered to
generate a spherically symmetrical force field
so
that the
force
Fi(rj)
exerted on a pixel of unit intensity at the pixel
location with position vector
rj
by any other pixel with
position vector
ri
and pixel intensity
P(rJ
is given by
The units of pixel intensity, force, and distance are
arbitrary, as are the co-ordinates of the origin of the
vector field. The total force
F(rj)
exerted on a pixel of unit
intensity at the pixel location with position vector
rj
is the
vector sum of all the forces due to the other pixels in the
image and is given by,
25
0-7803-6297-7/00/$10.00
0
2000
IEEE
N-1
~-1
[
ri -rj3]
F(rj)
=
Fi(rj)
=
P(ri)
(2)
i=O,#
j
i=O,#
j
Iri
-
rj
I
In order to calculate the force field for the entire
image, this equation should be applied at every pixel
position in the image. Since this procedure is quadratic in
the number of pixels
N,
a more efficient approach is to
exploit the speed of the
Fast Fourier Transform
by
viewing the process as a convolution of the image with
the field associated with a
unit
value test pixel. A nine-
fold memory penalty is incurred since zero padding is
required for anti-aliasing. However the reward is a
computation complexity of order 9Mog(9N).
Associated with the force field generated by each
pixel there is a spherically symmetrical scalar potential
energy field where
Ei(rj)
is the potential energy imparted
to a pixel of unit intensity at the pixel location with
position vector
rj
by the energy field of any other pixel
with position vector
ri
and pixel intensity
P(c),
and is
given by
(3)
The defining equation is simpler than the force field
equation but the concept is less intuitive. If an
exploratory unit test pixel is moved around in the force
field generated by a given pixel, energy will be exchanged
if the net effect is to change the distance of the test pixel
from the given pixel. Thus the field consists of concentric
rings of equal potential energy known as equipotentials.
If
the
test pixel moves to a different location on the same
equipotential ring, no energy is exchanged. If it moves to
a different equipotential, an amount of energy will be
exchanged equal to the difference in potential energy
between the two rings. The potential energy function of a
single isolated pixel looks like an inverted vortex as
shown in Figure 1
I
Figure
1
Isolated pixel potential energy function
Now to find the total potential energy at a particular
pixel location in the image, the scalar sum is taken of the
values of the overlapping potential energy functions of all
the image pixels at that precise location and is given by
This summation is then carried out at each pixel
location to generate a potential energy surface, which is a
smoothly varying surface due to the fact that the
underlying inverted vortices have smooth surfaces.
Figure
2
Potential energy surface
for
an ear
3
Invertible Linear Transform
In the appendix we show that the force field transform
is a linear transformation and this is confirmed here by
giving its matrix representation
[4].
The form of the
matrix is illustrated for a trivial
2
x
2
pixel image. It is
easily verified that this represents the application of
Equation
2
at each of the four pixel locations. This
equation multiplies a column vector of pixel intensities
(Pi)
by a matrix
of
inverse square displacement vectors
d,
to give a column vector of forces
(Fi).
We have,
ri -r.
0
dl0 dzo d30
P,
'do,
0
dzl
d3~
I:]=
[
i]
where
d,
=
dm
dl2
0
42
14
-rjl
pm
dl3
dz3
0
4
This
is
a skew-symmetric matrix: the leading diagonal
of zeros reflects the fact that no pixel attracts itself and
the skew symmetry is accounted for by the fact that we
are dealing with a fully connected network but with a pair
of directed edges connecting every pair of nodes.
There is a corresponding representation for the
potential energy transformation since the vector force
field and scalar potential energy fields are related by the
fact that the force at a given point is equal to the additive
inverse of the gradient of the potential energy surface at
that point,
F(r)
=
-grud(E(r))=
-VE(r)
(5)
Since the representation matrices are square it is of
theoretical interest whether they are invertible or not. If
they are invertible then the original image can
be
recovered for example from the potential energy surface.
This implies that all the information in the original image
is conserved by the transformation, which is an important
result. In practice the representation matrices for images
of
even
modest size are very large, for example
a
10x10
image has a matrix with 10,000 elements. However we
26
have tested the potential energy representation matrices
for all square images up to 32x32 pixels and all non-
square images up to
7x8
pixels and have found them to be
invertible. These results suggest that the potential energy
transform is indeed invertible for most image sizes and
aspect ratios.
Figure
3
Force field feature extraction for an ear
Figure
4
Different descriptions for different ears
Even if there are some particular combinations of aspect
ratio and size that yield singular matrices, this should not
detract from the overall conclusion that all of the image
information is conserved by the transformation.
In Figure 4 we show results of the same extraction
process for different ears. Clearly the new transformation
leads not only to a different channel description for each
ear, but also to different wells
4
Force
Field
Faces
Here we demonstrate that the technique may also be
employed in face recognition. We find generally that
there are fewer potential wells in the results for faces
when compared with those for ears,
so
that greater
emphasis and reliance must be placed on potential
~
Figure
5
Force field feature extraction applied
to
faces
channel descriptions.
As
shown in Figure
5
the potential
channels are quite unique for each of the four faces.
Notice that only one potential well is located for the last
face in the set, roughly in the middle of the nose. We see
however that a rich set of distinctive channels leads into
this single well that can be used to provide
a
reliable
27
description. Notice how two channels grow downwards
to surround the mouth and a single channel runs up along
the length of the nose which then splits into two further
channels running parallel to the eyebrows. In all cases
there are also channels forming along the cheekbones.
5
Conclusions
We have developed a new feature extraction
technique, targeted primarily at ear biometrics but which
readily extends to include face recognition. The
technique is robust and reliable with remarkable
invariance to initialization and possessing excellent noise
tolerance. The beauty of this technique is that an explicit
description of the target topology is not necessary.
Extracting the potential well description merely involves
following the force field lines and observing eventual
clustering of coordinates. Taking account of the channel
shape and ultimately the underlying shape of the energy
surface can increase the level of detail in the description
to meet any demand.
We are currently investigating how linearity can be
exploited to perform force field tracking in dynamic
images. A dynamic force field can be updated by simply
adding the force field associated with the difference
between sequential images, often a sparse array and
therefore easily computed. We are also investigating how
information conservation, which is a consequence of the
transform invertiblilty, can be exploited to perform image
compression.
A
very important aspect of the transformation is the
fact that it simulates a natural process, namely the
formation
of
electric fields in the vicinity of electric
charge distributions.
For
example the image formed on a
charge-coupled device will result in a charge distribution
which will have an associated electric force field. This
holds out the prospect of a solid state device with direct
image to force field conversion in real time.
References
[
1
J
H. C. Andrews,
Introduction to Mathematical Techniques in
Pattern Recognition,
John Wiley
&
Sons Inc., p 9, 1972
[2] M. Burge, and
W.
Burger, Ear Recognition,
In:
A.
K.
Jain,
R. Bolle and
S.
Pankanti Eds.,
Biometrics: Personal
Identzjication in Networked Society,
pp. 273-286, Kluwer
Academic Publishing, 1998
[3] D.J. Hurley, M.S. Nixon, J.N. Carter, “Force Field Energy
Functionals for Image Feature Extraction”,
Proc. British
Machine Vision
Con$
BMVA
Press,
11,
pp.
604-613, 1999
linear transformations, M203 Handbook, p29, Open University
Press, 1999
Appendix
Proof of Linearity
Let
A
and
B
be elements of the vector space
V
of
MxN
matrices whose elements
A,
and
Bi,,
are real numbers
representing pixel intensities. Also let
W
be the vector
space of
MxN
matrices where the elements are forces as
defined by the relation:
(4
n)
=
(i,
j)
Now, to check that the defining relation for a linear
transform is satisfied, we need to show that
t(a,A+a,B)
=
a,t(A)+a,t(B),
foralla,,a,
E
%,A,BE
V
(6)
That is, we need to show that the force field of the sum of
two separately scaled images is the same as the sum of the
scaled force fields associated with the individual images.
Accordingly, for the sum of the force fields associated
with the two images we have,
Hence, since we have shown that the defining relation is
satisfied, the force field transform is indeed a linear
transformation.
[4]
M203 Introduction to Pure Mathematics, Linear Algebra
Block:
Unit
4
Linear Transformations, Section 2: Matrices of
28