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.
23 November 2003
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