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"""
Useful geometric operations, e.g. Orthographic projection and a differentiable Rodrigues formula
Parts of the code are taken from https://github.com/MandyMo/pytorch_HMR
"""
import torch
def rodrigues(theta):
"""Convert axis-angle representation to rotation matrix.
Args:
theta: size = [B, 3]
Returns:
Rotation matrix corresponding to the quaternion -- size = [B, 3, 3]
"""
l1norm = torch.norm(theta + 1e-8, p = 2, dim = 1)
angle = torch.unsqueeze(l1norm, -1)
normalized = torch.div(theta, angle)
angle = angle * 0.5
v_cos = torch.cos(angle)
v_sin = torch.sin(angle)
quat = torch.cat([v_cos, v_sin * normalized], dim = 1)
return quat2mat(quat)
def quat2mat(quat):
"""Convert quaternion coefficients to rotation matrix.
Args:
quat: size = [B, 4] 4 <===>(w, x, y, z)
Returns:
Rotation matrix corresponding to the quaternion -- size = [B, 3, 3]
"""
norm_quat = quat
norm_quat = norm_quat/norm_quat.norm(p=2, dim=1, keepdim=True)
w, x, y, z = norm_quat[:,0], norm_quat[:,1], norm_quat[:,2], norm_quat[:,3]
B = quat.size(0)
w2, x2, y2, z2 = w.pow(2), x.pow(2), y.pow(2), z.pow(2)
wx, wy, wz = w*x, w*y, w*z
xy, xz, yz = x*y, x*z, y*z
rotMat = torch.stack([w2 + x2 - y2 - z2, 2*xy - 2*wz, 2*wy + 2*xz,
2*wz + 2*xy, w2 - x2 + y2 - z2, 2*yz - 2*wx,
2*xz - 2*wy, 2*wx + 2*yz, w2 - x2 - y2 + z2], dim=1).view(B, 3, 3)
return rotMat
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
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