stereo vision 3d point calculation with known intrinsic and extrinsic matrix

I have successfully calculated Rotation, Translation with the intrinsic camera matrix of two cameras. I also got rectified images from the left and right cameras. Now, I wonder how I calculate the 3D coordinate of a point, just one point in an image. Here, please see the green points. I have a look at the equation, but it requires baseline which I don't know how to calculate. Could you show me the process of calculating the 3d coordinate of the green point with the given information (R, T, and intrinsic matrix)?

FYI 1. I also have a Fundamental matrix and Essential matrix, just in case we need them. 2. Original image size is 960 x 720. Rectified ones are 925 x 669 3. The green point from the left image: (562, 185), from the right image: (542, 185)

The term "baseline" usually just means translation. Since you already have your rotation, translation and intrinsics matrices (let's not them R, T and K). you can triangulate and don't need either the Fundamental or Essential matrices (they could be used to extract R, T etc but you already have them). You don't really need your images to be rectified either, since it doesn't change the triangulation process that much. There are many ways to triangulate, each with their pros and cons, and many libraries that implement them. So, all I can do here is give you and overview of the problem and potential solutions, as well as pointers to resources that you can either use as their are or as a source of inspiration to write your own code.

• Formalization and solution outlines. Let's formalize what we are after here. You have a 3d point X, with two observations x_1 and x_2 respectively in the left and right images. If you backproject them, you obtain two rays:

  ray_1=K^{1}x_1
  rat_2=R*K^{-1}x_2+T  //I'm assuming that [R|T] is the pose of the second camera expressed in the referential of the first camera 
  

  Ideally, you'd want those two rays to meet at point X. Since in practice we always have some noise (discretization noise, rounding errors and so on) the two rays wont meet at X, so the best answer would be a point Q such that

  Q=argmin_X {d(X,ray_1)^2+d(X,ray_2)^2}
  

  where d(.) denotes the Euclidian distance between a line and a point. You can solve this problem as a regular least squares problem, or you can just take the geometric approach (called midpoint) of considering the line segment l that is perpendicular to both ray_1 and ray_2, and take its middle as your solution. Another quick and dirty way is to use the DLT. Basically, you re-write the constrains (i.e. X should be as close as possible to both rays) as a linear system AX=0 and solve it with SVD.

  Usually, the geometric (midpoint) method is the less precise. The DLT based one, while not the most stable numerically, usually produces acceptable results.

• Ressources that present in depth formalization

  Hartley-Zisserman's book of course! Chapter 12. A simple DLT-based method, which is the one used in opencv (both in the calibration and sfm modules) is explained on page 312. It is very easy to implement, it shouldn't take more that 10 minutes in any language.

  Szeliski'st book. It has an intersting discussion on triangulation in the chapter on SFM, but is not as straight-forward or in depth as Hartley-Zisserman's.

• Code. You can use the triangulation methods from opencv, either from the calib3d module, or from the contribs/sfm module. Both use the DLT, but the code from the SFM module is more easily understandable (the calib3d code has a lot of old-school C code which is not very pleasant to read). There is also another lib, called openGV, which has a few interesting methods for triangulation.

  cv::triangulatePoints

  cv::sfm::triangulatePoints

  OpenGV

  The openGV git repo doesn't seem very active, and I'm not a big fan of the design of the library, but if I remember correctly (feel free to tell me otherwise) it offers methods other that the DLT for triangulations.

  Naturally, those are all written in C++, but if you use other languages, finding wrappers or similar libraries wont be difficult (with python you still have opencv wrappers, and MATLAB has a bundle module, etc.).