The reader of this paper
should first imagine a plate, as large as a continent. We
can place on the borders of this plate instruments, that are
sensitive to vibrations (earth- or platequakes), that can
occure somewhere on the plate. And now we will try to
imagine, that such earthquake happens on our plate (for
example in the middle of it).
On a drawing it can be shown in following way:
Arrows are showing the places, where our instruments are
situated. How we can obtain the information, where was the
source of earthquake (epicenter). It is probably very
popular knowledge, that this can be calculated, if the time
of flight of the signal form the epicenter to our instrument
is known and readings from at least two instruments are
available. Because we normaly do not known the time of
flight, but only the moment, when the signal from earthquake
was measured, three instruments and three measurements are
necessary.
In the case of the earthquake it is naturally more
complicated, because it is necessary to know the speed of
the wave on the way to the receiver and this can change on
the way. But it is true, that for a not very exact
calculation of the loaction of the earthquake source a
relativ simple calculation can be enough.
Now we can imagine a more realistic situation: If we take a
thic metal (or glass) plate and place on their side
microphones, that can receive sounds, propagating in the
plate, we can make following experiments:
If we generate a sound somewhere in the medium of the plate
for example knocking on it, our microphones will receive -
after some time, necessary for its propagation - this sound,
and we can calculate, at which place this sound was
generated.
If we generate a sound wave in the middle of plate, that is
going from their bottom to the upper wall, our microphones
will not hear anything, because the sound wave will be
reflected back and for the between both walls. But the
experiments shows, that in the case, if in the region, where
this sound wave will be reflected, something is lying (in
contact with the surface of the plate) part of the sound
wave will be scattered in different directions, and along
the plate too. Our microphones will receive the sound and we
can calculate where our object is lying on the plate.
This effect can be called as a contact scattering and is
probably easy to imagine, what causes it: If nothing is
disturbing the surface of the plate, the wave will be
reflected and propagates toward the source on the opposite
wall. But if the surface is disturbed by something lying on
it, the reflection is not normal and part of the wave energy
is taking another way. We are now not able to tell exactly
and describe theoretically, what happens in this case (especially
depending on materials in contact, pressure, etc.), butthis
can be easily shown.
We can assume, that the point of contact becomes a kind of
secondary sound sorce, that our microphones can "hear".
And it is easy to imagine, that the situation is about the
same as in the case of investigating of earthquakes, but may
be simpler, because we have here a homogenous plate.
It is easy to imagine, that the detection of small amount of
sources can be made with a simple mathematics (signals are
easy to separate). Especiall easy is to imagine, what
happens, when we have one source and many detectors. This
situation can be shown on the following picture, and was
detected with the setup, where the plate was round and the
source of pulse not central. Timeof flight (coming of pulse)
detected by each detector (y-axis) is different (x-axis),
but the differences are smooth and shows a sinus line on the
picture:
After
calculation (reconstruction), the location of the sound
source can be shown:
It is more
complicated to explaine, how to calculate the signals in the
situation, where many sources are existing. But even this is
not as complicated as it seems to be. And it could be
interesting for the reader, that procedures, that can be
used here was first proposed more than two hundred years ago
for earthquakes localisation and earth structureresearch
with sound waves, that was artificially generated.
This procedures was later used for tomography, and it is
possible to show, that the description of the situation in
the case, when we have to do with projection (as it is in
tomography) is similar to the situation, where we have a
source (or more of them) and will find its location.
For tomography and other cases, where so called inverse
problems must be solved many algorithms was proposed, that
has caused, that the people are now able to investigate and
detect objects, that are "hidden" and can be
investigated only due to changes, that they are causing in
the wave or field coming through them, or due to radiation
caused by them.
In the case of our camera for finger recognition we have a
bit special situation: Fingerprint is a two-dimensional
structure, and this means, that it can be reconstructed in
about the same way as an tomographic picture (or earthquake
source), but this is measured using scattering of the wave
and measuring its wave field with all information, its
contains. And this is nothing else as holography. Thus: it
is easy to understand, that in this case we have a point,
where tomography procedures can be directly used for
holography and it can be also shown, that the whole
information about the object can be collected using
receivers placed only on the circle.
The situation will be more complicated, if we are interested
in inestigating 3-D objects, because in this case it will be
not possible to use such simple methods and it will be
necessary to propose other methods, but even in this case,
we can use a part of experience, that was collected for
investigation of earthquakes, because the happen never on
the earth surface but in some depth. In many cases in large
depth.
And the reader, that knows something about the construction
of our actual prototype can believe, that even in the case
of volume waves, the situation will be - from the point of
view - almost exactly the same as in the case of the wave in
a plate. The only existing difference is the projection,
that must be made here - the angle of wave inclination is
different.
Wieslaw Bicz
23 November 2003
