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
