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
