Is there any algorithm for determining 3d position in such case? (images below)

Question

So first of all I have such image (and ofcourse I have all points coordinates in 2d so I can regenerate lines and check where they cross each other)

_{(source: narod.ru)}

But hey, I have another Image of same lines (I know thay are same) and new coords of my points like on this image
_{(source: narod.ru)}

So... now Having points (coords) on first image, How can I determin plane rotation and Z depth on second image (asuming first one's center was in point (0,0,0) with no rotation)?

Answer 1

What you're trying to find is called a projection matrix. Determining precise inverse projection usually requires that you have firmly established coordinates in both source and destination vectors, which the images above aren't going to give you. You can approximate using pixel positions, however.

This thread will give you a basic walkthrough of the techniques you need to use.

Answer 2

Let me say this up front: this problem is hard. There is a reason Dan Story's linked question has not been answered. Let provide an explanation for people who want to take a stab at it. I hope I'm wrong about how hard it is, though.

I will assume that the 2D screen coordinates and projection/perspective matrix is known to you. You need to know at least this much (if you don't know the projection matrix, essentially you are using a different camera to look at the world). Let's call each pair of 2D screen coordinates (a_i, b_i), and I will assume the projection matrix is of the form

P = [ px  0  0  0 ]
    [ 0   py 0  0 ]
    [ 0   0  pz pw]
    [ 0   0  s  0 ], s = +/-1

Almost any reasonable projection has this form. Working through the rendering pipeline, you find that

a_i = px x_i / (s z_i)
b_i = py y_i / (s z_i)

where (x_i, y_i, z_i) are the original 3D coordinates of the point.

Now, let's assume you know your shape in a set of canonical coordinates (whatever you want), so that the vertices is (x0_i, y0_i, z0_i). We can arrange these as columns of a matrix C. The actual coordinates of the shape are a rigid transformation of these coordinates. Let's similarly organize the actual coordinates as columns of a matrix V. Then these are related by

V = R C + v 1^T             (*)

where 1^T is a row vector of ones with the right length, R is an orthogonal rotation matrix of the rigid transformation, and v is the offset vector of the transformation.

Now, you have an expression for each column of V from above: the first column is { s a_1 z_1 / px, s b_1 z_1 / py, z_1 } and so on.

You must solve the set of equations (*) for the set of scalars z_i, and the rigid transformation defined R and v.

Difficulties

• The equation is nonlinear in the unknowns, involving quotients of R and z_i
• We have assumed up to now that you know which 2D coordinates correspond to which vertices of the original shape (if your shape is a square, this is slightly less of a problem).
• We assume there is even a solution at all; if there are errors in the 2D data, then it's hard to say how well equation (*) will be satisfied; the transformation will be nonrigid or nonlinear.

Answer 3

It's called (digital) photogrammetry. Start Googling.

Answer 4

If you are really interested in this kind of problems (which are common in computer vision, tracking objects with cameras etc.), the following book contains a detailed treatment:

Ma, Soatto, Kosecka, Sastry, An Invitation to 3-D Vision, Springer 2004.

Beware: this is an advanced engineering text, and uses many techniques which are mathematical in nature. Skim through the sample chapters featured on the book's web page to get an idea.