Before I start
with the description of the idea, I must tell the story,
that began about 10 years ago. This explains much. It is
very difficult for me to tell, if 10 years ago, somebody
else has already had the idea, that it would be possible to
make a full mathematical description of fingerprints. In
this time I have found no paper, describing such idea.
During this time Optel worked mainly on the project, having
the goal of creating a device, that would be able to
recognize people measuring their fingers with the help of acoustical
holography. We have already created first working
prototypes of such devices and spend many days on
discussions how to find a best way to compare the results
and recognize fingerprints. We have studied papers, already
published and discussed our ideas with a specialist, working
since some years at the fingerprint recognition department
of the local police, that was a member of our team during
this time. We have also tried to create our own software for
fingerprint recognition, that have mostly used classical
approach (based on methods, proposed in this time). I
suppose, that during this years many teams in the whole
world have worked on such software. Our team was probably
not much more skilled, than other teams, but there was one
large difference: Nobody else in the world have tried to use
acoustical holography for measurement of fingerprints. And
although the picture of fingerprint obtained by this method
is not significantly different, than picture, obtained by
other methods (optical, capacitive chips, etc.), the first
step - the information, that is directly "seen" by
such device is significantly different. It is a hologram,
that can be presented in the form of Fourier transform or
pulse answer. And because we have used waves with the length,
corresponding to the distance between the ridges of
fingerprints, the structure of such pulse answer has some
similarity to fingerprint itself. This is especially good to
see on the following picture:
Our
fingerprint technician (Adam Nowak) has even proposed to use
direct comparison of such pulse answers and not to use
reconstructed pictures. He has tried to find a method for
this comparison, but was not able to find it. The work with
such pictures has caused, that I have had the idea, that
fingerprint can be treated as a kind of structure, that may
be considered as the result of interference of some waves.
Optel team I have tried to find a method of generation of
such pattern using simulations with different wave sources.
We was able to produce very interesting patterns, some of
them very similar to fingerprints. This was surely not a
proper working idea, but probably not a bad way. From other
point of view, it is obviously true, that fingerprint seems
to be not far away from wave pattern, that could be
generated using different methods and for example found in
pictures, published in books about physics.
Although I have had no evidence, that this can be possible,
after such numerical experiments I was sure, that it must be
possible to find an algorithm, that would allow to describe
fingerprint using small amount of parameters. This was the
reason, why I have tried to convince all people working on
this problem together with me. We have discussed many ideas
together and with Mieczyslaw Pluta, who began to work on
finger recognition at Optel and also on reconstruction of
pictures, that we have obtained from ultrasonic data. The
work on this topic and our discussions have produced one
very significant result: algorithm, that allows mathematical
description of minutia. He has assumed, that fingerprint can
be seen as a wave pattern (periodical sinusoidal structure
on the surface), that can be described using following
function:
F(x,y)=cos(j(x,y))
(1)
Where j(x,y))
is a function, that describes the phase of the "wave
structure", that is "frozen" in the form of
fingerprint. This "phase describing" function has
normally two parts:
- Slowly
changing part, that describe the shape of ridges,
- Quick changing part, that causes so called minutias.
This can be
mathematically formulated in following way:
j(x,y)
= jg(x,y)
+ jo(x,y)
(2)
where the
second part describes quick changes.
Because fingerprint can be locally interpreted as a kind of
a grid, constructed from parallel lines (it is a very good
known approach), the first part of its function can be also
interpreted as information about the direction of this grid
and its density.
It is probably a relatively simple job for a good
mathematician to construct a function, that describes a
shape, that is similar to the shapes found in fingerprints.
This is the reason, why I will not describe functions, that
produce such shapes.
Much more interesting is the description of the second part
of fingerprint function. To do it, we will define special
function Yqxz(x,y), giving each point (x,y) the
value of angle position in relation to a specially chosen
middle point (x,z). For each x and y, the function Yqxz(x,y) should be monotonically
changing from 0 to 2p around the point (x,z) and have a characteristic jump
from 0 to 2p for a chosen angle q. With certain assumptions,
all this is given for the following function:
Yqxz(x,y) = arctg ((x-x)
/ (y-z) )
(3)
If this function will be implemented in the finger
describing function, it will cause in the point (x,z) a phase jump, that will be
observed as minutia. Two kinds of minutias can be observed,
depending on the sign (subtraction or addition) of the
second part in the formula (2). Minutia is also defined
using angle q (starting point of the phase of formula (3)).
In the following picture, where the basic structure is build
form concentric rings, the minutia is caused by the setting (x,z)=(30,20)
If the simple function (3) will be replaced by a sum of such
functions:
N
jo(x,y) = S sk Yk (x,y) , where sk = signum (Yk ) (4)
k=1
it will be possible to describe a structure with many
minutias. If the basis function of the finger is good
described, this formula allows to implement all known
minutias in any desired place. In the following picture it
can be seen, how many such points are working on a simple
arch structure:
If the simple
function (3) will be replaced by a sum of such functions:
(4)
it will be
possible to describe a structure with many minutias. If the
basis function of the finger is good described, this formula
allows to implement all known minutias in any desired place.
In the following picture it can be seen, how many such
points are working on a simple arch structure:
The idea
described before was also used for the software, that
everybody can download from our internet page, called "fingerprint
creator". It was also used as the basis for finger
recognition software. This software uses the idea of
minutiae description in different direction - from
fingerprint data to the set of information, describing
fingerprint pattern. This software has some advantages (it
can for example work very good with relatively bad
fingerprints), but it would be surely necessary to continue
the work on this project to achieve very good performance.
This was not done, because the financing of this project was
stopped.
"Fingerprint
creator" should be treated as a demonstration, that
the idea of mathematical description can work at least in
some cases. And the limit of this idea is easily to see, too:
To describe all possible fingerprints it is necessary to
find a mathematical description of the basic structure of
fingerprint, not only of minutias. We have tried to find
such description and this considerations has caused, that I
would assume, that the required function should have
probably the same structure, as the basic function of the
fingerprint: one function describing the basic shape and the
second one, describing such structures as delta end points,
centers of circular lines etc. The function of this kind
would have two stages, having similar structure. But this is
only my assumption.
Existing finger creator software can produce very large
amount of fingerprint pattern, using only a very small
amount of data (exactly 10 bytes). If it would be possible (and
this is what I believe, but cannot exactly prove now), the
realistic chance of storing large amounts of fingerprints in
a very small storage space (I would assume, that not more,
than 10 bytes will be necessary for the full description of
fingerprint) will occur. This is especially interesting for
people storing large amounts of such data, but has a very
important advantage for all other: it could cause, that
comparison of fingerprints will work much quicker, that with
existing algorithms, because such description allows the
creation of a kind of catalogue of fingerprint (each
fingerprint has a defined place in such catalogue, that has
a number, but this number is also a parameter for an
algorithm, that allows to create its pattern).
We was not able to continue the work on this project (I am
searching new partners for it, and it seems to be realistic,
that I will find them soon), but may be the publication of
this paper will cause, that more people will think about the
possibility of mathematical description of fingerprints and
propose some new ideas.
I assume, that mainly following work should be done if the
project of mathematical description of fingerprint should be
continued:
1. Description
of the jg(x,y)
function from the
formula (2).
2. Finding a method of analysis of fingerprint pictures in
the way, that they can be described using mathematical
algorithm.
3. Finding a method of sorting (finding a place in a
catalogue) of fingerprints in the way, that the comparison
can be possible in the simplest way.
I have decided
to publish this paper mainly because I think, that the idea
of mathematical description of fingerprints can be
interesting for many people working in this area and our
achievements can be used to achieve the goal of the
description of all fingerprints and - in this way - for the
creation of a very concise catalog of all existing
fingerprints, that can take only a relatively small amount
of place (for example a collection of all fingerprints of
today living people together with their names could be
placed on a hard disk, that is easily available today -
about 200 Gbytes).
I am sure, that the project, having the goal of creation of
such catalogue can be advantageous for all people, if they
have no reason for making their identification difficult. It
can bring the large advantage of secure identification with
any kind of automatic fingerprint reader.
I am interested in cooperation with people, interested in
developing such algorithm and hope to find in this way
contact to them.
Wieslaw Bicz,
written in May
2003
