Errors in pointcloud-to-image reprojection

[jaime-spencer-oxb]

Hi, thanks for creating this dataset, it seems very interesting! 😄

I've been looking at projecting the lidar points onto the images to generate depth maps, but have been seeing some errors while following the tutorial.

When projecting points in frames where the vehicle is static the quality of the reprojection is really high, which shows that the lidar-image calibration is very accurate.

However, when the car is in motion, I have noticed that the reprojections are incorrect (particularly on the left image border). Below I've attached a couple of examples. This seems happen even when the vehicle is travelling forward in an open street (i.e. no urban canyon).

Do you think this is due to incorrect pose estimates? Or could it be related to the pointcloud motion compensation?

As you mention in the paper, using the relative pose instead of the global pose would likely be more accurate for short timescales. Are those poses available in the dataset? As far as I can tell, there is only the raw IMU data.

Thank you!

[keenan-burnett]

It could be the motion compensation. The way I remove motion distortion in the tutorial is a bit "lazy".

The ground truth provides poses like T_enu_applanix at around 200Hz. If you want the best motion compensation for the lidar pointcloud, you need to interpolate the ground truth for each lidar point timestamp t and do p_enu = T_enu_applanix(t) T_applanix_lidar p_lidar(t).

You can use SLERP interpolation for the rotation and linear interpolation on position for T_enu_applanix.

[cppcute-pm]

@keenan-burnett Hello, I've done the exactly things you suggest, and the problem still exists. After I gain all the p_enu for one point cloud, I use the lidar_frame.pose to transform the point cloud back to the LiDAR coordinates, the whole part of the code is below:

def get_gt_data_for_traversal(root): """Retrieves ground truth applanix data for a given sensor frame Args: root (str): path to the sequence root Returns: gt (list): A list of ground truth values from the applanix sensor_poses.scv """ posepath = os.path.join(root, "gps_post_process.csv") pose_list = [] with open(posepath, "r") as f: f.readline() # header for line in f: pose_list.append([float(x) for x in line.split(",")]) pose_ndarray = np.array(pose_list) return pose_ndarray

def is_sorted_along_axis(arr, axis):

对数组沿给定轴方向上的差值进行计算

diff = np.diff(arr, axis=axis)
# 如果所有差值都是正数或零，则数组在该维度上是升序的
return np.all(diff >= 0)

def so3_to_quaternion(so3): """Converts an SO3 rotation matrix to a quaternion

Args:
    so3: 3x3 rotation matrix

Returns:
    numpy.ndarray: quaternion [w, x, y, z]

Raises:
    ValueError: if so3 is not 3x3
"""
if so3.shape != (3, 3):
    raise ValueError("SO3 matrix must be 3x3")

R_xx = so3[0, 0]
R_xy = so3[0, 1]
R_xz = so3[0, 2]
R_yx = so3[1, 0]
R_yy = so3[1, 1]
R_yz = so3[1, 2]
R_zx = so3[2, 0]
R_zy = so3[2, 1]
R_zz = so3[2, 2]

try:
    w = sqrt(so3.trace() + 1) / 2
except(ValueError):
    # w is non-real
    w = 0

# Due to numerical precision the value passed to `sqrt` may be a negative of the order 1e-15.
# To avoid this error we clip these values to a minimum value of 0.
x = sqrt(max(1 + R_xx - R_yy - R_zz, 0)) / 2
y = sqrt(max(1 + R_yy - R_xx - R_zz, 0)) / 2
z = sqrt(max(1 + R_zz - R_yy - R_xx, 0)) / 2

max_index = max(range(4), key=[w, x, y, z].__getitem__)

if max_index == 0:
    x = (R_zy - R_yz) / (4 * w)
    y = (R_xz - R_zx) / (4 * w)
    z = (R_yx - R_xy) / (4 * w)
elif max_index == 1:
    w = (R_zy - R_yz) / (4 * x)
    y = (R_xy + R_yx) / (4 * x)
    z = (R_zx + R_xz) / (4 * x)
elif max_index == 2:
    w = (R_xz - R_zx) / (4 * y)
    x = (R_xy + R_yx) / (4 * y)
    z = (R_yz + R_zy) / (4 * y)
elif max_index == 3:
    w = (R_yx - R_xy) / (4 * z)
    x = (R_zx + R_xz) / (4 * z)
    y = (R_yz + R_zy) / (4 * z)

return np.array([w, x, y, z])

def my_interpolate_poses(pose_timestamps, requested_timestamps, abs_quaternions, abs_positions):

upper_indices = np.minimum(np.searchsorted(pose_timestamps, requested_timestamps, side='left'), len(pose_timestamps) - 1)
lower_indices = np.maximum(upper_indices - 1, 0)

assert np.all(pose_timestamps[upper_indices] - pose_timestamps[lower_indices] > 0.0)

fractions = (requested_timestamps - pose_timestamps[lower_indices]) / \
            (pose_timestamps[upper_indices] - pose_timestamps[lower_indices])

quaternions_lower = abs_quaternions[:, lower_indices]
quaternions_upper = abs_quaternions[:, upper_indices]

d_array = (quaternions_lower * quaternions_upper).sum(0)

linear_interp_indices = np.nonzero(d_array >= 1)
sin_interp_indices = np.nonzero(d_array < 1)

scale0_array = np.zeros(d_array.shape)
scale1_array = np.zeros(d_array.shape)

scale0_array[linear_interp_indices] = 1 - fractions[linear_interp_indices]
scale1_array[linear_interp_indices] = fractions[linear_interp_indices]

theta_array = np.arccos(np.abs(d_array[sin_interp_indices]))

scale0_array[sin_interp_indices] = \
    np.sin((1 - fractions[sin_interp_indices]) * theta_array) / np.sin(theta_array)
scale1_array[sin_interp_indices] = \
    np.sin(fractions[sin_interp_indices] * theta_array) / np.sin(theta_array)

negative_d_indices = np.nonzero(d_array < 0)
scale1_array[negative_d_indices] = -scale1_array[negative_d_indices]

quaternions_interp = np.tile(scale0_array, (4, 1)) * quaternions_lower \
                     + np.tile(scale1_array, (4, 1)) * quaternions_upper

positions_lower = abs_positions[:, lower_indices]
positions_upper = abs_positions[:, upper_indices]

positions_interp = np.multiply(np.tile((1 - fractions), (3, 1)), positions_lower) \
                   + np.multiply(np.tile(fractions, (3, 1)), positions_upper)

poses_mat = np.zeros((4, 4 * len(requested_timestamps)))

poses_mat[0, 0::4] = 1 - 2 * np.square(quaternions_interp[2, :]) - \
                     2 * np.square(quaternions_interp[3, :])
poses_mat[0, 1::4] = 2 * np.multiply(quaternions_interp[1, :], quaternions_interp[2, :]) - \
                     2 * np.multiply(quaternions_interp[3, :], quaternions_interp[0, :])
poses_mat[0, 2::4] = 2 * np.multiply(quaternions_interp[1, :], quaternions_interp[3, :]) + \
                     2 * np.multiply(quaternions_interp[2, :], quaternions_interp[0, :])

poses_mat[1, 0::4] = 2 * np.multiply(quaternions_interp[1, :], quaternions_interp[2, :]) \
                     + 2 * np.multiply(quaternions_interp[3, :], quaternions_interp[0, :])
poses_mat[1, 1::4] = 1 - 2 * np.square(quaternions_interp[1, :]) \
                     - 2 * np.square(quaternions_interp[3, :])
poses_mat[1, 2::4] = 2 * np.multiply(quaternions_interp[2, :], quaternions_interp[3, :]) - \
                     2 * np.multiply(quaternions_interp[1, :], quaternions_interp[0, :])

poses_mat[2, 0::4] = 2 * np.multiply(quaternions_interp[1, :], quaternions_interp[3, :]) - \
                     2 * np.multiply(quaternions_interp[2, :], quaternions_interp[0, :])
poses_mat[2, 1::4] = 2 * np.multiply(quaternions_interp[2, :], quaternions_interp[3, :]) + \
                     2 * np.multiply(quaternions_interp[1, :], quaternions_interp[0, :])
poses_mat[2, 2::4] = 1 - 2 * np.square(quaternions_interp[1, :]) - \
                     2 * np.square(quaternions_interp[2, :])

poses_mat[0:3, 3::4] = positions_interp
poses_mat[3, 3::4] = 1

poses_out = poses_mat.reshape((4, len(requested_timestamps), 4))
poses_out = np.transpose(poses_out, (1, 0, 2))

return poses_out

def process_sequence(seq_list): seq_ID = seq_list[0] lidar_idxs = seq_list[1] print(f"enter {seq_ID}") curr_seq = dataset.get_seq_from_ID(seq_ID) target_seq_path = os.path.join(target_path, str(seq_ID)) os.makedirs(target_seq_path, exist_ok=True) target_seq_lidar_path = os.path.join(target_seq_path, "lidar") os.makedirs(target_seq_lidar_path, exist_ok=True) calib_root = os.path.join(dataset_path, str(seq_ID), 'calib') calib = Calib(calib_root) gt_file_ndarray = get_gt_data_for_traversal(curr_seq.applanix_root) # (gt_num, 18) assert is_sorted_along_axis(gt_file_ndarray[:, 0], 0) gt_trans_matrix = np.zeros((gt_file_ndarray.shape[0], 4, 4)) gt_positions = np.zeros((3, gt_file_ndarray.shape[0])) gt_quaternions = np.zeros((4, gt_file_ndarray.shape[0])) for i in range(gt_file_ndarray.shape[0]): gt_trans_matrix[i] = get_transform(gt_file_ndarray[i]) gt_quaternions[:, i] = so3_to_quaternion(gt_trans_matrix[i][0:3, 0:3]) gt_positions[:, i] = np.ravel(gt_trans_matrix[i][0:3, 3]) gt_file_timestamp = gt_file_ndarray[:, 0]

for lidar_idx in lidar_idxs:
    t1 = time.perf_counter()
    curr_lidar_frame = curr_seq.lidar_frames[int(lidar_idx)]
    filename = curr_lidar_frame.path.split("/")[-1].split(".")[0]
    save_path = os.path.join(target_seq_lidar_path, filename + ".npy")

    if os.path.exists(save_path):
        print(f"pass the {lidar_idx} in {seq_ID}")
        continue

    curr_lidar_frame.load_data()

    # curr_lidar_frame.remove_motion(curr_lidar_frame.body_rate)
    curr_lidar_frame_points_time = curr_lidar_frame.points[:, -1]
    curr_lidar_frame_points_coords = curr_lidar_frame.points[:, :3] # (N, 3)

    vis_pc_1 = curr_lidar_frame_points_coords

    curr_lidar_frame_points_poses = my_interpolate_poses(gt_file_timestamp, curr_lidar_frame_points_time, gt_quaternions, gt_positions) # (N, 4, 4)
    T_applanix_lidar = calib.T_applanix_lidar.astype(np.float32) # (4, 4)
    curr_lidar_frame_points_coords_to_mul = np.concatenate([curr_lidar_frame_points_coords, np.ones_like(curr_lidar_frame_points_coords[:, :1])], axis=1) # (N, 4)
    curr_lidar_frame_points_coords_in_applanix = np.matmul(curr_lidar_frame_points_coords_to_mul, T_applanix_lidar.T) # (N, 4)
    curr_lidar_frame_points_coords_in_enu = np.matmul(np.expand_dims(curr_lidar_frame_points_coords_in_applanix, axis=1), curr_lidar_frame_points_poses.transpose(0, 2, 1)) # (N, 1, 4)
    curr_lidar_frame_points_coords_in_enu = curr_lidar_frame_points_coords_in_enu.squeeze(1) # (N, 4)

    vis_pc_2 = curr_lidar_frame_points_coords_in_enu[:, :3]

    T_enu_lidar = curr_lidar_frame.pose.astype(np.float32) # (4, 4)
    T_lidar_enu = np.linalg.inv(T_enu_lidar)
    curr_lidar_frame_points_coords_after_mc = np.matmul(curr_lidar_frame_points_coords_in_enu, T_lidar_enu.transpose(1, 0)) # (N, 4)
    curr_lidar_frame_points_coords_after_mc = curr_lidar_frame_points_coords_after_mc[:, :3] # (N, 3)

    image_id = lidar2image[seq_ID][str(lidar_idx)][0]
    curr_image_frame = curr_seq.camera_frames[image_id]
    curr_image_frame.load_data()
    T_enu_camera = curr_image_frame.pose
    T_enu_lidar = curr_lidar_frame.pose
    T_camera_lidar = np.matmul(get_inverse_tf(T_enu_camera), T_enu_lidar)

    lidar_curr_frame = copy.deepcopy(curr_lidar_frame)
    lidar_curr_frame.remove_motion(lidar_curr_frame.body_rate)
    lidar_curr_frame.transform(T_camera_lidar)

    # Remove points outside our region of interest
    lidar_curr_frame.passthrough([-75, 75, -20, 10, 0, 40])  # xmin, xmax, ymin, ymax, zmin, zmax

    # Project lidar points onto the camera image, using the projection matrix, P0.
    uv, colors, _ = lidar_curr_frame.project_onto_image(calib.P0)

    # Draw the projection
    fig = plt.figure(figsize=(24.48, 20.48), dpi=100)
    ax = fig.add_subplot()
    ax.imshow(curr_image_frame.img)
    ax.set_xlim(0, 2448)
    ax.set_ylim(2048, 0)
    ax.scatter(uv[:, 0], uv[:, 1], c=colors, marker=',', s=3, edgecolors='none', alpha=0.7, cmap='jet')
    ax.set_axis_off()
    plt.savefig(f'../vis1020/pc_image_projection_before_mc_{seq_ID}_{int(lidar_idx)}.jpg', bbox_inches='tight', pad_inches=0, dpi=200)
    print('look look')
    plt.close()

    lidar_curr_frame_mc = copy.deepcopy(curr_lidar_frame)
    lidar_curr_frame_mc.points[:, :3] = curr_lidar_frame_points_coords_after_mc
    lidar_curr_frame_mc.transform(T_camera_lidar)

    # Remove points outside our region of interest
    lidar_curr_frame_mc.passthrough([-75, 75, -20, 10, 0, 40])  # xmin, xmax, ymin, ymax, zmin, zmax

    # Project lidar points onto the camera image, using the projection matrix, P0.
    uv_mc, colors_mv, _ = lidar_curr_frame_mc.project_onto_image(calib.P0)

    # Draw the projection
    fig = plt.figure(figsize=(24.48, 20.48), dpi=100)
    ax = fig.add_subplot()
    ax.imshow(curr_image_frame.img)
    ax.set_xlim(0, 2448)
    ax.set_ylim(2048, 0)
    ax.scatter(uv_mc[:, 0], uv_mc[:, 1], c=colors_mv, marker=',', s=3, edgecolors='none', alpha=0.7, cmap='jet')
    ax.set_axis_off()
    plt.savefig(f'../vis1020/pc_image_projection_after_mc_{seq_ID}_{int(lidar_idx)}.jpg', bbox_inches='tight', pad_inches=0, dpi=200)
    print('look look')
    plt.close()

    t2 = time.perf_counter()
    print(f"cost {t2 - t1} seconds")

    # np.save(save_path, curr_lidar_frame.points[:, :3].astype(np.float32))
    # print(f"saved {lidar_idx} in {seq_ID}")

Would you mind to check the code above for me?

[keenan-burnett]

Your code looks fine, can you post some examples of what the projection looks like using SLERP / linear interp? Unless there's a bug in your code, the remaining projection error could come from:

• The camera timestamps are slightly inaccurate (dataset problem). In this case, the 3D pose assigned to the camera image may be a pose that is slightly off as well. You could try adding a small constant offset to the camera image timestamps and then interpolating for a new camera pose for each camera image from the ground truth. For example, add +5ms vs. -5ms to each camera image timestamp, get a new pose for each, see what that does to the alignment.
• Camera intrinsics are slightly off: the focal length / intrinsic calibration of the camera may be slightly off.

[cppcute-pm]

Below are several example pairs of projection, the first image takes the motion compensation you provided, and the second one takes my version. The red circle is where I think exists slightly difference between two images, which can prove that the projections don't follow the same procedure:

example pair 1

example pair 2

example pair 3

From above, you can see that two kinds of projection is almostly the same, even where the projection errors lie

[keenan-burnett]

There's not much I can do to fix this at this point. As I mentioned above, you try adjusting the camera-lidar temporal synchronization to get better poses for the camera or you can try to adjust the camera's intrinsics.

[keenan-burnett]

These projections look nearly identical, you would have to pick an example where the car is driving faster to make the difference more noticeable.