反向投影2D点转换成3D [英] Reverse-projection 2D points into 3D
问题描述
假设我们有一个三维空间上有一个平面，一个arbitary方程：AX +的+ CZ + D = 0 现在假设我们挑选那架飞机上3个随机点：（X0，Y0，Z0）（X1，Y1，Z1）（X1，Y1，Z1）
Suppose we have a 3d Space with a plane on it with an arbitary equation : ax+by+cz+d=0 now suppose that we pick 3 random points on that plane: (x0,y0,z0) (x1,y1,z1) (x1,y1,z1)
现在我有一个不同的角度（摄像头）的这架飞机。我的意思是我有一个不同的摄像头，将着眼于这架飞机从不同的角度来看。但从该摄像机点，这些点的位置不同。例如（X0，Y0，Z0）将是（X0'，Y0'） 和（x1，Y1，Z1）将是从视图中新相机点（X1'，Y1'）和（X2，Y2，Z2）将是（X2'，Y2'）。
now i have a different point of view(camera) for this plane. i mean i have a different camera that will look at this plane from a different point of view. From that camera point of view these points have different locations. for example (x0,y0,z0) will be (x0',y0') and (x1,y1,z1) will be (x1',y1') and (x2,y2,z2) will be (x2',y2') from the new camera point of view.
因此，这里是我的一个小问题很难回答！我要挑但从新相机点，例如一个点（X，Y），并说出它会在那架飞机。所有我知道的是，3分，他们在三维空间的位置和新的摄像机视图的投影位置。
So here is my a little hard question! I want to pick a point for example (X,Y) from the new camera point of view and tell where it will be on that plane. All i know is that 3 points and their locations on 3d space and their projection locations on the new camera view.
你知道平面方程的系数和摄像头的位置（以及投影），或者你只是有六分？ - 尼尔斯
Do you know the coefficients of the plane-equation and the camera positions (along with the projection), or do you only have the six points? - Nils
我所知道的第一个3点的位置。因此，我们可以计算出平面的系数。所以我们知道确切位置的平面距离（0,0,0）的观点。然后我们有摄像头，只能看到点！使摄像头看到的仅仅是3分，这也是他们知道在三维空间中的位置（和肯定他们对2D摄影机视图平面位置）。毕竟我想看看相机视图，选择一个点（例如（X1，Y1）），并告诉哪里是上飞机了这一点。 （为确保本（X，Y，Z）点应该适合于平面方程）。此外，我对此事一无所知的摄像头位置。
i know the location of first 3 points. therefore we can calculate the coefficients of the plane. so we know exactly where the plane is from (0,0,0) point of view. and then we have the camera that can only see the points! So the only thing that camera sees is 3 points and also it knows their locations in 3d space (and for sure their locations on 2d camera view plane). and after all i want to look at camera view, pick a point (for example (x1,y1)) and tell where is that point on that plane. (for sure this (X,Y,Z) point should fit on the plane equation). Also i know nothing about the camera location.
推荐答案
有不可能给出一个明确的解决这个问题。然而，这里就是我会抽取不同的解决方案：
It is not possible to give an unambiguous solution to this problem. However, here's how I would extract the different solutions:
1）解决了使用P3P（透视-3点），从原来的 RANSAC纸，这四个可能的可行的解决办法放弃（与分镜头前）。
1) Solve for the camera position and direction using the P3P (Perspective-3-Point) algorithm from the original RANSAC paper, which give up to four possible feasible solutions (with the points in front of the camera).
2）项目的光线与具有照相机位置作为原点（X，Y），为投影在摄像机并计算其与平面相交。
2) Project a ray with the camera position as origin having (X,Y) as projection in the camera and calculate its intersection with the plane.
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