The Grids Methods
By Douglas Moreman
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This rather skimpy introduction is placed on the website on 26 July 2013.
The Grids Methods were introduced in US Patent "Echo scope".

Suppose there is a suitable notion of distance defined on 6-Space and Er() is a function such that for points X and Y in 6-Space, Er(X,Y) is the distance from X to Y.

A Grid is superimposed on the space in front of Device. Before being used for imaging, for each sensor M and for each gridpoint P is pre-computed GridTime(P,M), the sum of the times-of-travel of a fang from Clicker to P and from P to M. So, for every grid-point P and every scopion S there is GridKey(P,S) which is a six-term sequence t = (t1, t2, ...,t6) such that tK = GridTime(P,SK) for K= 1 to 6. The sequence t is a point in the above-mentioned 6-Space.

In a click event, for every scopion S, if T1 is a toa on S1, T2 is a toa on S2,..., and T6 is a toa on S6 then the six-term sequence (T1,T2,...,T6) is an “EchoKey” of S for that click-event.

The distance-function Er( ) can be used like this: in case, for some scopion S, each of X and Y is an EchoKey of S then Er(X,Y) can be regarded as the error in taking Y to represent X.

Suppose a click-event has produced recorded echoes on all the sensors of Device and FeatureDetector has sent, for every sensor M, a list of toas of fangs on M to HitDetector.

HitDetector does this: For every Scopion S, for every EchoKey T of S (computed by a subroutine EchoKeyDetector), Grid is searched for a “grid-hitpoint” P that minimizes Er(T,GridKey(P,S)).

The set of all the grid-hitpoints P could used to compute an image, but, there can be this problem: it may be, in terms of computational resources, impossible to make the Grid fine enough for acceptable quality of such images.

So, a second searching procedure moves P off of the Grid and into uncharted space so that Er(T, P) is reduced further.

If the error fails to fall below some maxError2, then T is classified as a phantom. Earlier, a similar check against some maxError1, used in the search on the Grid could have, in some cases, reached the same conclusion and saved the time of the second search.

In simulations, a Grids method seemed, by 2005, adequate for low-resolution applications such as “fish-finding.”

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