It is well-known that such problem does not have a general solution at all, at least, not a solution in the traditional mathematical sense of this word.

There is a branch of mathematics devoted to some classes of so called

*ill-posed problems*. Most typically, ill-posed problems appears from the formulation of different kinds of

*inverse problems*:

http://en.wikipedia.org/wiki/Ill-posed_problem[^].

The model problem of this class, closest to the problem which OP is interested in, is the problem of

*tomography*:

http://en.wikipedia.org/wiki/Tomography[^],

http://www2.imm.dtu.dk/~pcha/Tomo.html[^].

Even though the problem of tomography looks like and extreme hi-tech (and it actually is), the seemingly easy problem of reconstruction of the 3D model by 2D projections is not easier at all.

The formulation for a wide class of ill-posed problem was formulated by Jacques Hadamard and the approach to such problems was developed by A.N.Tikhonov. According to this approach, the ideas of traditional "solution" is given up, and the concept of "

*quasi-solution*" is introduced. For example, if some quasi-solution is found and describes some 3D shape which really produces the correct set of projections used as the input data of the problem, some other 3D shape can be found, which produces exactly the same set of projections. None of such solution can be called a "real" solution, because an ambiguous is normally required. In practical applications of the quasi-solution, some preexisting knowledge of the expected shape is used. For example, a set of tomographic images of a brain can be resolved in a set of different 3D images, but the software produces only the 3D model of the brain which conforms to preexisting knowledge of how a human brain is generally shaped.

You can get better ideal on this extremely difficult topic if you read some books/articles. Please see:

http://books.google.com/books?id=b1GSCZCZk60C&pg=PA25&lpg=PA25&dq=ill-posed+tomography&source=bl&ots=aMLv63cW13&sig=NS1S82fBOlMEKWZet8OuFRYUXo4&hl=en&sa=X&ei=ciQQUbeDI5Go0AGSw4GQAw&ved=0CFYQ6AEwBA[^].

Most article needs special academic or personal subscription, so they are no easy to get.

Try to find some: http://bit.ly/11KWV51[^].

Yes, this is an adequate search criteria. If you try to find something using your formulation of the problem, you won't find anything.

~~geeks~~members here will be pleased with that. Cheers.—SA

Especially taking into account that such problem is

theoretically proven unsolvablein general case, and where it is "solvable", it does not mean "solution" in traditional mathematical sense of this word.This is a very interesting topic. I've put some information in my answer. Please take a look, this is really interesting... :-)

—SA