如何将坐标系从世界转换为相机（3D 到 2D 转换）？

Question

I am working on a project involving 3D world to 2D film conversion, and I am having trouble obtaining the correct output.

I am using the information as located on https://www.scratchapixel.com/lessons/3d-basic-rendering/computing-pixel-coordinates-of-3d-point/mathematics-computing-2d-coordinates-of-3d-points to figure out the code, and this is part of what I have so far:

# Camera's Focal Point f is known, and predefined.

# Define camera's position in terms of world coordinates (Fixed)
cx = 281.771;
cy = 6109.4;
cz = 1141.63;

# Each of the world's axes is described in terms of camera coordinates

r11, r21, r31 = 0.0026, -0.9999, 0.0038; #World X-Axis in Cam Coords
r12, r22, r32 = -0.0043, -0.0038, -0.9998; #World Y-Axis in Cam Coords
r13, r23, r33 = -0.9999, 0.0157, 0.0042; #World Z-Axis in Cam Coords


r_matrix = np.array([[r11, r21, r31, 0],
                [r12, r22, r32, 0],
                [r13, r23, r33, 0],
                [cx, cy, cz, 1]]); #Rotation Matrix

# Inverse of this since we are doing world-to-local
global r_mat

r_mat = inv(r_matrix); # This will also be passed to function since it is fixed for each camera
print(r_mat);
r_mat = r_mat.astype(float)

world = np.array([])
world = np.array([U, V, W, 1.0]); #World Coordinate Matrix (x,y,z,1)
world = world.astype(float)
local = world.dot(r_mat)
print(local)
localx = local[0]
localy = local[1]
localz = local[2]

camx = -1 * f * (localx/localz)
camy = -1 * f * (localy/localz)

This gives me incorrect values for the 2D coordinates. Could someone please please help me out with this? It would be a huge help!

(Extra: Does the orientation of the camera-axes and the world-axes have to be the same? Or does the rotation matrix take care of this?)

Answer 1

first of all i would like to have commented instead but my reputation does not allow this at the moment.

So you got your camera at some coordniantes in the 3D space and a point in 3D space which you want to project into the 2D space of the camera.

You need to describe the point in camera coordinates which corespondes to a translation and a rotation, which can be described in a 4x4 matrix. The extra dimension is needed to make wokring with the matrix easier.

Lets define some variables

pc = point in camera coordinates (3D)

pw = point in wolrd coordinates (3D)

Cwc = rotation of the camera ( matrix which describes the rotation of the camera)

Tw = describes the translation

Twc = combination of translation and rotation

Therefore pc =Twc * pw so you now have described the point in the camera coordinates. Now you need to project the 3D point into the 2D camera plain.

p2d = point in camera coordinates (2D)

Z = projection matrix (involves focal length)

p2d = Z*pc

For example:

p2d = [[f, 0, 0, 0], [0, f, 0, 0], [0, 0, 1, 0]]*[pc1, pc2, pc3, 1]

Thats it for the theory, what is missing is the actual calculation of the Cwc, the matrix which describes the rotation in of the camera plain. Unfortunatley i dont know how to do this.