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Inertial 3D Body Scanner

Three Dimensional matter Imaging by Moment of Inertia De-Convolution

The idea starts out with a simple puzzle and takes the solution and runs with it. This takes some explaining, so here goes...

Supposing you have a puzzle where there are two balls. One of them is solid aluminium, and the other is a shell of aluminium with a shell of lead (Pb) inside, and then the middle is hollow. Both of the spheres have been constructed to be the same size and also the same weight. So, how do you tell which is which?

Well let's assume that you're not allowed to x-ray them, or to open them up, etc. How can you tell which is which without much fuss? Bear in mind the spheres weigh the same and are the same diameter, and the outside of each is aluminium. So, this leaves some people stumped.

The answer is: Roll the balls down a slope and see how they accelerate. It's a fact that the solid aluminium ball will roll just how you'd expect, with a moment of inertia of two-fifths m R-squared, but the other ball which has a heavy internal shell but hollow in the middle, will gain rotational speed slower as it has a higher moment of inertia.

OK, so in effect, by consideration of the moment of inertia of objects it's possible to "see into them"? How far can this be carried, though?

Supposing you have a cube of polystyrene about half a metre along each edge, and you know that within it there are two relatively small (5cm diameter) spheres made of metal. Without using some sort of beam probing method, how could you find where the spheres are within the cube? By measuring the moment of inertia about different lines, you could find out where they are. With two small spheres inside a polystyrene block, I believe you could determine approximately where they are by waving the cube about and feeling how it moves. Of course the axis containing both spheres would be the axis of least rotational inertia. However, the precise mass and location of the objects could be evaluated by a process in which multiple results are put together and then de-convoluted.

Convolution is what happens when multiple things are put together and mixed up in a definite way. The opposite, deconvolution, is possible by a computer algorithm similar to Fourier Transforms.

What about three solid heavy objects within an opaque lightweight polystyrene block? I suggest that there is a way of measuring the moment of inertia for different axes of rotation and then putting the data through a system to work out the locations of the objects.

Such de-convolution and momentum based analysis could be conducted for any solid object, the result being a three dimensional density map of the internal structure of the object.

There is a refinement to the process, which is necessary because it's inconvenient to rotate an object through all possible axes of rotation in three dimensions. After an initial set of rotations, the next set of choices of axes through which to rotate the object could be determined from the results of the first set, so making a more discerning set of results. The type of idea is similar to that seen in playing the logical game of "Mastermind" where the codemaker and the codebreaker contend with each other in a game of intelligent cleverness.

To make sure I am right about this theory and not completely talking out of my hat, let's take a simplistic example. Supposing we simplify it to two dimensions and have a chessboard on a vinyl gramophone deck, and put weights strategically on some of the squares. Now, if someone has remote control of the deck, and can spin the deck with precise motor control and can check its rotational speed at a distance, then without seeing the chessboard, can they work out where the weights are? (The board can be moved so the centre of the deck corresponds to any position on the board). The total moment of inertia is the sum of all of the moments of inertia of all of the weights, which for each is its mass times the square of its distance from the centre of rotation.

It's clear to see that this would work for the analysis of weights on squares of a chessboard. I believe it can also be extended into 3D and into an arbitrary level of precision of measurement.

The idea of using this as a body scanner may make some people feel they should be a bit queasy of it, as they'll be put in a tube like in an MRI scanner and then spun around in all kinds of directions. It's the "pukomatic"! But no, it may not be anything like as emetic. Fine measurement and convolution will advance, and also for non-rigid bodies it may be possible to vibrate them and analyse the inertial movement. This becomes practical once the level of computer power exceeds some function of the accuracy required. Computers seem to get more powerful at a remarkable geometric rate, so sooner or later this will be practical, even if it currently seems fanciful.

To sum it up:

* It's possible to see a 3D map of density of the inside of an opaque object.

* The multiple moments of inertia about different axes are the key to this. Deconvolution finds the answers.