Is there an algorithm for solving such projection reconstruction geometric problem?

Question

We have a grid with red squares on it. Meaning we have an array of 3 squares (with angles == 90 deg) which as we know have same size, lying on the same plane and with same rotation relative to the plane they are lying on, and are not situated on same line on plane.

We have a projection of the space which contains the plane with squares.

We want to turn our plane projection with squares so that we would see it like it's facing us, in general we need a formula for turning each point of that original plane projection so that it would be facing us like on the image below.

What formulas can be used for solving such problem, how to solve it, has any one faced something like this before?

Answer 1

This is a special case of finding mappings between quadrilaterals that preserve straight lines. These are generally called homographic transforms. Here, one of the quads is a square, so this is a popular special case. You can google these terms ("quad to quad", etc) to find explanations and code, but here are some for you.

Perspective Transform Estimation

a gaming forum discussion

extracting a quadrilateral image to a rectangle

Projective Warping & Mapping

ProjectiveMappings for ImageWarping by Paul Heckbert.

The math isn't particularly pleasant, but it isn't that hard either. You can also find some code from one of the above links.