Weak-perspective camera coordinates

[vladsterz]

Hello, great job.
I've a question about the estimated perspective camera position. I assume that the SMPL model goes as (x,y,z) and after watching the values at the estimated camera I noticed that the first coefficient was big ( x? ) , but I expected the last coordinate to be big ( z )

[nkolot]

If you look at how we are computing the orthographic projection:

def orthographic_projection(X, camera):
    """Perform orthographic projection of 3D points X using the camera parameters
    Args:
        X: size = [B, N, 3]
        camera: size = [B, 3]
    Returns:
        Projected 2D points -- size = [B, N, 2]
    """ 
    camera = camera.view(-1, 1, 3)
    X_trans = X[:, :, :2] + camera[:, :, 1:]
    shape = X_trans.shape
    X_2d = (camera[:, :, 0] * X_trans.view(shape[0], -1)).view(shape)
    return X_2d

You can see that the first coefficient in the camera is the scaling factor whereas the second and third the translation in the x and y direction. So more formally, the transformation we are doing is

[X,Y,Z] -> camera[0] * [X+camera[1], Y+camera[2]]

I hope this helps

[vladsterz]

Thanks for the fast reply.
This in fact helps alot.

[vladsterz]

Hello, I can't figure how to use the predicted camera parameters in order to project the predicted points of the model onto the image.

Actually I want to make my intention clear. Given this camera parameters and the predicted SMPL model, is it possible for me to project every point of the mesh to the image? How is it done?
As I understand, this is handled by OpenDR.
Thanks in advance.

[nkolot]

The orthographic projection function that I mentioned above projects the mesh in the normalized image plane. If you want to do the projection in pixel coordinates, you have to convert the projected coordinates from [-1,1] to [0,224] by a simple translation and scaling.

You can also do the perspective projection from scratch as K * [R,t] * [X,Y,Z] as we show in demo.py. Assuming a fixed focal length, first we get the camera translation

camera_translation = torch.stack([pred_camera[:,1], pred_camera[:,2], 2*FOCAL_LENGTH/(224 * pred_camera[:,0] +1e-9)],dim=-1)

This is t in the above projection equation. R will be the identity matrix and K will be

        FOCAL_LENGTH                0                  224/2
K =          0                 FOCAL_LENGTH.           224/2
             0                      0                    1

[vladsterz]

Thanks once again for the helpful response. Closing it, for now...

[RohanChacko]

Hi,
Can you tell me how you arrived at the tz term in the translation expression
camera_translation = torch.stack([pred_camera[:,1], pred_camera[:,2], 2*FOCAL_LENGTH/(224 * pred_camera[:,0] +1e-9)],dim=-1)? It would be helpful if you could explain the reasoning.

[Ironbrotherstyle]

Same question. If you have any mathematical solution, please let me know, thank you.