diffusionsfm/utils/geometry.py

import numpy as np
import torch
from pytorch3d.renderer import FoVPerspectiveCameras
from pytorch3d.transforms import quaternion_to_matrix


def generate_random_rotations(N=1, device="cpu"):
    q = torch.randn(N, 4, device=device)
    q = q / q.norm(dim=-1, keepdim=True)
    return quaternion_to_matrix(q)


def symmetric_orthogonalization(x):
    """Maps 9D input vectors onto SO(3) via symmetric orthogonalization.

    x: should have size [batch_size, 9]

    Output has size [batch_size, 3, 3], where each inner 3x3 matrix is in SO(3).
    """
    m = x.view(-1, 3, 3)
    u, s, v = torch.svd(m)
    vt = torch.transpose(v, 1, 2)
    det = torch.det(torch.matmul(u, vt))
    det = det.view(-1, 1, 1)
    vt = torch.cat((vt[:, :2, :], vt[:, -1:, :] * det), 1)
    r = torch.matmul(u, vt)
    return r


def get_permutations(num_images):
    permutations = []
    for i in range(0, num_images):
        for j in range(0, num_images):
            if i != j:
                permutations.append((j, i))

    return permutations


def n_to_np_rotations(num_frames, n_rots):
    R_pred_rel = []
    permutations = get_permutations(num_frames)
    for i, j in permutations:
        R_pred_rel.append(n_rots[i].T @ n_rots[j])
    R_pred_rel = torch.stack(R_pred_rel)

    return R_pred_rel


def compute_angular_error_batch(rotation1, rotation2):
    R_rel = np.einsum("Bij,Bjk ->Bik", rotation2, rotation1.transpose(0, 2, 1))
    t = (np.trace(R_rel, axis1=1, axis2=2) - 1) / 2
    theta = np.arccos(np.clip(t, -1, 1))
    return theta * 180 / np.pi


# A should be GT, B should be predicted
def compute_optimal_alignment(A, B):
    """
    Compute the optimal scale s, rotation R, and translation t that minimizes:
    || A - (s * B @ R + T) || ^ 2

    Reference: Umeyama (TPAMI 91)

    Args:
        A (torch.Tensor): (N, 3).
        B (torch.Tensor): (N, 3).

    Returns:
        s (float): scale.
        R (torch.Tensor): rotation matrix (3, 3).
        t (torch.Tensor): translation (3,).
    """
    A_bar = A.mean(0)
    B_bar = B.mean(0)
    # normally with R @ B, this would be A @ B.T
    H = (B - B_bar).T @ (A - A_bar)
    U, S, Vh = torch.linalg.svd(H, full_matrices=True)
    s = torch.linalg.det(U @ Vh)
    S_prime = torch.diag(torch.tensor([1, 1, torch.sign(s)], device=A.device))
    variance = torch.sum((B - B_bar) ** 2)
    scale = 1 / variance * torch.trace(torch.diag(S) @ S_prime)
    R = U @ S_prime @ Vh
    t = A_bar - scale * B_bar @ R

    A_hat = scale * B @ R + t
    return A_hat, scale, R, t


def compute_optimal_translation_alignment(T_A, T_B, R_B):
    """
    Assuming right-multiplied rotation matrices.

    E.g., for world2cam R and T, a world coordinate is transformed to camera coordinate
    system using X_cam = X_world.T @ R + T = R.T @ X_world + T

    Finds s, t that minimizes || T_A - (s * T_B + R_B.T @ t) ||^2

    Args:
        T_A (torch.Tensor): Target translation (N, 3).
        T_B (torch.Tensor): Initial translation (N, 3).
        R_B (torch.Tensor): Initial rotation (N, 3, 3).

    Returns:
        T_A_hat (torch.Tensor): s * T_B + t @ R_B (N, 3).
        scale s (torch.Tensor): (1,).
        translation t (torch.Tensor): (1, 3).
    """
    n = len(T_A)

    T_A = T_A.unsqueeze(2)
    T_B = T_B.unsqueeze(2)

    A = torch.sum(T_B * T_A)
    B = (T_B.transpose(1, 2) @ R_B.transpose(1, 2)).sum(0) @ (R_B @ T_A).sum(0) / n
    C = torch.sum(T_B * T_B)
    D = (T_B.transpose(1, 2) @ R_B.transpose(1, 2)).sum(0)
    E = (D * D).sum() / n

    s = (A - B.sum()) / (C - E.sum())

    t = (R_B @ (T_A - s * T_B)).sum(0) / n

    T_A_hat = s * T_B + R_B.transpose(1, 2) @ t

    return T_A_hat.squeeze(2), s, t.transpose(1, 0)


def get_error(predict_rotations, R_pred, T_pred, R_gt, T_gt, gt_scene_scale):
    if predict_rotations:
        cameras_gt = FoVPerspectiveCameras(R=R_gt, T=T_gt)
        cc_gt = cameras_gt.get_camera_center()
        cameras_pred = FoVPerspectiveCameras(R=R_pred, T=T_pred)
        cc_pred = cameras_pred.get_camera_center()

        A_hat, _, _, _ = compute_optimal_alignment(cc_gt, cc_pred)
        norm = torch.linalg.norm(cc_gt - A_hat, dim=1) / gt_scene_scale

        norms = np.ndarray.tolist(norm.detach().cpu().numpy())
        return norms, A_hat
    else:
        T_A_hat, _, _ = compute_optimal_translation_alignment(T_gt, T_pred, R_pred)
        norm = torch.linalg.norm(T_gt - T_A_hat, dim=1) / gt_scene_scale
        norms = np.ndarray.tolist(norm.detach().cpu().numpy())
        return norms, T_A_hat