Camera Model

Camera Model Reference

Overview

Our camera system is implemented in JAX and follows OpenCV conventions. There are two layers:

1. Core library (multi_camera.analysis.camera) - Low-level functions operating in millimeters
2. Camera adapter (multiview_biomechanical_pose.geometry.camera_adapter) - High-level functions accepting/returning meters

Source files:

• MultiCameraTracking/multi_camera/analysis/camera.py - Core camera functions
• MultiviewBiomechanicalPose/multiview_biomechanical_pose/geometry/camera_adapter.py - Meter-safe wrappers

Camera Parameters Dict

All camera functions expect a single dict with parameters for ALL cameras:

camera_params = {
    "mtx":  array,  # (num_cameras, 4)  - [fx, fy, cx, cy] stored at 1/1000 scale
    "rvec": array,  # (num_cameras, 3)  - Rodrigues rotation vectors (dimensionless)
    "tvec": array,  # (num_cameras, 3)  - Translation vectors in METERS (stored at 1/1000 scale)
    "dist": array,  # (num_cameras, 5+) - Distortion coefficients [k1, k2, p1, p2, k3, ...]
}

The mtx and tvec fields are stored at 1/1000 scale (effectively meters). The library functions get_intrinsic() and get_extrinsic() multiply by 1000 internally to produce pixel-scale intrinsics and mm-scale extrinsics.

TFRecord format uses plural names (camera_intrinsics, camera_rvecs, camera_tvecs, camera_distortion) which the dataset pipeline converts to singular names automatically.

Unit Conventions (CRITICAL)

The Conversion Rule

When calling core library functions directly, always convert:

# Meters -> mm before projection
points_2d = cam_lib.project_distortion(camera_params, cam_idx, points_3d_m * 1000.0)

# mm -> meters after triangulation
points_3d_m = triangulate_point(camera_params, points_2d) / 1000.0

When using camera_adapter functions, no conversion is needed - they handle it internally.

OpenCV Extrinsic Convention

Our camera model follows the OpenCV world-to-camera convention:

P_camera = E @ P_world

• get_extrinsic() returns a 4x4 SE(3) matrix that transforms points FROM world coordinates TO camera coordinates
• This is NOT the camera position in world space
• The camera position in world space would be -R^T @ t (or E_inv[:3, 3])
• Rotation is stored as Rodrigues vectors, converted via SO3.exp(rvec)
• Uses jaxlie SE3/SO3 for all rigid body transformations

# get_extrinsic returns world-to-camera transform (units: mm)
E = get_extrinsic(camera_params, i)  # 4x4 matrix

# To get camera position in world (in mm):
camera_pos_mm = jnp.linalg.inv(E)[:3, 3]

# To get camera position in meters:
camera_pos_m = camera_pos_mm / 1000.0

COLMAP Conversion Warning

COLMAP stores camera-to-world transforms. When converting from COLMAP:

R_w2c = R_c2w.T
t_w2c = -R_w2c @ t_colmap

Core Library Functions (multi_camera.analysis.camera)

All core functions operate in millimeters. Import as:

from multi_camera.analysis import camera as cam_lib
from multi_camera.analysis.camera import get_intrinsic, get_extrinsic

get_intrinsic(camera_params, i) -> 3x3 K matrix

Returns the intrinsic matrix with focal lengths and principal point in pixel units:

K = get_intrinsic(camera_params, 0)
# [[fx,  0, cx],
#  [ 0, fy, cy],
#  [ 0,  0,  1]]

get_extrinsic(camera_params, i) -> 4x4 SE(3) matrix

Returns the world-to-camera transformation matrix in millimeters:

E = get_extrinsic(camera_params, 0)
# 4x4 homogeneous transform [R|t; 0 0 0 1]
# t is in MILLIMETERS
# This is NOT the camera position - it's a world-to-camera transform

get_projection(camera_params, i) -> 3x4 projection matrix

Returns K @ E[:3] - the full 3x4 projection matrix:

P = get_projection(camera_params, 0)  # 3x4

project_distortion(camera_params, i, points) -> 2D pixels

Projects 3D points to 2D image coordinates with radial distortion:

# CRITICAL: points must be in MILLIMETERS
points_2d = cam_lib.project_distortion(camera_params, cam_idx, points_3d_m * 1000.0)

Currently uses simplified radial model (k1 only). Handles behind-camera points by shrinking them toward image center.

undistort_points(points, K, dist, num_iters=5) -> undistorted 2D

Iteratively removes lens distortion from 2D image points. Supports full OpenCV model (k1-k6, p1-p2, s1-s4, tau_x, tau_y):

undistorted = cam_lib.undistort_points(
    points_2d[None, ...],   # (1, N, 2) - needs batch dim
    K[None, ...],           # (1, 3, 3)
    dist_coeffs[None, :],   # (1, n_coeffs)
)[0]                        # remove batch dim

distort_3d(camera_params, i, points) -> distorted 3D in camera coords

Computes equivalent 3D camera-frame points that would project to the same 2D coordinates under an ideal pinhole camera. Used for rendering 3D models (e.g., SMPL) on distorted images:

# points in mm, returns camera-frame coords in mm
vertices_distorted = cam_lib.distort_3d(camera_params, i, vertices_mm)

Uses full radial+tangential model (k1, k2, k3, p1, p2).

triangulate_point(camera_params, points2d, return_confidence=False) -> 3D mm

DLT triangulation from multiple 2D observations. Handles undistortion internally:

# points2d: (num_cameras, T, J, 3) with [u, v, confidence]
# Returns: (T, J, 3) or (T, J, 4) with confidence, in MILLIMETERS
points_3d_mm = cam_lib.triangulate_point(camera_params, points2d, return_confidence=True)
points_3d_m = points_3d_mm[..., :3] / 1000.0

robust_triangulate_points(camera_params, points2d, sigma=150, threshold=0.5) -> 3D mm

Robust triangulation using geometric median of pairwise triangulations (Roy et al. 2022). Automatically weights cameras by agreement:

points_3d, camera_weights = cam_lib.robust_triangulate_points(
    camera_params, points2d, return_weights=True
)
# points_3d: (T, J, 4) in mm
# camera_weights: (num_cameras, T, J)

reprojection_error(camera_params, points2d, points3d) -> residual in pixels

Computes difference between observed and projected 2D points:

# points3d must be in mm (same space as projection)
residual = cam_lib.reprojection_error(camera_params, points2d, points3d_mm)

Camera Adapter Functions (Meter-Safe API)

These functions accept and return meters. Prefer these over direct core library calls when possible.

from multiview_biomechanical_pose.geometry.camera_adapter import (
    transform_points_world_to_camera,
    transform_points_camera_to_world,
    transform_points_between_cameras,
    project_camera_coordinates,
    unproject_from_image_with_distortion,
    unproject_and_transform_to_world,
    transform_pose_se3_world_to_camera,
    transform_pose_se3_camera_to_world,
    compute_relative_extrinsics,
)

Point Transformations

# World -> Camera (meters in, meters out)
pts_cam = transform_points_world_to_camera(pts_world_m, camera_params, cam_idx)

# Camera -> World (meters in, meters out)
pts_world = transform_points_camera_to_world(pts_cam_m, camera_params, cam_idx)

# Camera A -> Camera B (meters in, meters out)
pts_target = transform_points_between_cameras(pts_source_m, camera_params, source_idx, target_idx)

Projection from Camera Coordinates

# Projects points already in camera frame to 2D (meters in, pixels out)
pts_2d = project_camera_coordinates(pts_cam_m, camera_params, cam_idx)

Uses identity extrinsics internally to avoid double-transformation.

Unprojection

# 2D pixels + depth -> 3D camera coords (pixels+meters in, meters out)
pts_cam = unproject_from_image_with_distortion(pts_2d, depth_m, camera_params, cam_idx)

# 2D pixels + depth -> 3D world coords (pixels+meters in, meters out)
pt_world = unproject_and_transform_to_world(image_point, depth_m, camera_params, cam_idx)

SE(3) Pose Transformations

# Transform full pose (position + quaternion) between frames
pos_cam, quat_cam = transform_pose_se3_world_to_camera(pos_world_m, quat_wxyz, camera_params, cam_idx)
pos_world, quat_world = transform_pose_se3_camera_to_world(pos_cam_m, quat_wxyz, camera_params, cam_idx)

Quaternions use wxyz format (jaxlie convention).

Common Workflows

Project 3D Skeleton to Image

from multi_camera.analysis import camera as cam_lib

# joints_3d: (T, J, 3) in meters
# Project to camera i
joints_2d = cam_lib.project_distortion(camera_params, i, joints_3d * 1000.0)
# joints_2d: (T, J, 2) in pixels

Triangulate from Multi-View 2D Keypoints

# keypoints_2d: (num_cameras, T, J, 3) with [u, v, confidence]
points_3d = cam_lib.triangulate_point(camera_params, keypoints_2d, return_confidence=True)
# points_3d: (T, J, 4) in mm - convert to meters:
points_3d_m = points_3d[..., :3] / 1000.0
confidence = points_3d[..., 3]

Render SMPL Mesh on Distorted Image

from multi_camera.analysis.camera import get_intrinsic, distort_3d

# vertices: (N, 3) in meters
K = np.array(get_intrinsic(camera_params, i))
vertices_distorted_mm = np.array(distort_3d(camera_params, i, vertices * 1000.0))
vertices_distorted_m = vertices_distorted_mm / 1000.0

# Render with ideal pinhole (K, identity R/T) since distortion is baked into 3D

Compute Reprojection Error

from multi_camera.analysis.fit_quality import reprojection_quality

# keypoints3d: (T, J, 3) in mm
# keypoints2d: (N_cams, T, J, 3) with [u, v, confidence]
metrics = reprojection_quality(keypoints3d_mm, camera_params, keypoints2d)

Convert Between Aniposelib and Our Format

# Aniposelib -> our format (mm -> meter storage)
from aniposelib.cameras import CameraGroup
cgroup = CameraGroup.from_names(cam_names)
cam_dicts = cgroup.get_dicts()
camera_params = {
    "mtx": np.array([[c["matrix"][0][0], c["matrix"][1][1],
                       c["matrix"][0][2], c["matrix"][1][2]]
                      for c in cam_dicts]) / 1000.0,
    "dist": np.array([c["distortions"] for c in cam_dicts]),
    "rvec": np.array([c["rotation"] for c in cam_dicts]),
    "tvec": np.array([c["translation"] for c in cam_dicts]) / 1000.0,
}

# Our format -> Aniposelib (meter storage -> mm)
from aniposelib.cameras import Camera, CameraGroup
cameras = []
for i, name in enumerate(cam_names):
    K = np.array([[params["mtx"][i,0]*1000, 0, params["mtx"][i,2]*1000],
                  [0, params["mtx"][i,1]*1000, params["mtx"][i,3]*1000],
                  [0, 0, 1]])
    cam = Camera(name=name, matrix=K, dist=params["dist"][i],
                 rvec=params["rvec"][i], tvec=params["tvec"][i]*1000)
    cameras.append(cam)
cgroup = CameraGroup(cameras)

Distortion Model

Our cameras use the OpenCV radial-tangential distortion model:

• Projection (project_distortion): Simplified model using k1 only (tangential terms commented out)
• Unprojection (undistort_points): Full model with k1-k6, p1-p2, s1-s4, tau
• 3D distortion (distort_3d): Full model with k1, k2, k3, p1, p2

Distortion coefficients are stored in OpenCV order: [k1, k2, p1, p2, k3, ...]

For EgoHumans data: images are pre-undistorted and distortion is set to [0, 0, 0, 0, 0].

Anti-Patterns (NEVER DO THESE)

1. Never manually access camera parameter arrays

# WRONG - bypasses scaling and formatting
fx = camera_params["mtx"][0, 0]
R = cv2.Rodrigues(camera_params["rvec"][0])[0]
t = camera_params["tvec"][0]

# CORRECT - use library functions
K = get_intrinsic(camera_params, 0)        # Properly scaled 3x3
E = get_extrinsic(camera_params, 0)        # Properly scaled 4x4 SE(3)

2. Never create single-camera parameter dicts

# WRONG - breaks the API contract
cam0_params = {
    "mtx": camera_params["mtx"][0:1],
    "rvec": camera_params["rvec"][0:1],
    "tvec": camera_params["tvec"][0:1],
    "dist": camera_params["dist"][0:1],
}

# CORRECT - pass full dict with camera index
result = cam_lib.project_distortion(camera_params, 0, points)

3. Never forget the mm conversion for core library functions

# WRONG - passing meters to mm-expecting function
projected = cam_lib.project_distortion(camera_params, i, points_3d_m)

# CORRECT
projected = cam_lib.project_distortion(camera_params, i, points_3d_m * 1000.0)

4. Never reimplement geometric transforms

# WRONG - manual projection ignoring distortion
x_2d = fx * (x_3d / z_3d) + cx
y_2d = fy * (y_3d / z_3d) + cy

# CORRECT - use library with distortion
from multiview_biomechanical_pose.geometry.camera_adapter import project_camera_coordinates
pts_2d = project_camera_coordinates(pts_cam_m, camera_params, cam_idx)

5. Never manually unproject without distortion correction

# WRONG - ignores lens distortion
x_3d = (x_2d - cx) * depth / fx

# CORRECT
from multiview_biomechanical_pose.geometry.camera_adapter import unproject_from_image_with_distortion
pts_3d = unproject_from_image_with_distortion(pts_2d, depth_m, camera_params, cam_idx)

6. Never confuse extrinsic matrix with camera position

# WRONG - treating tvec as camera position in world
camera_position = camera_params["tvec"][i]

# CORRECT - extrinsic is world-to-camera, invert to get camera position
E = get_extrinsic(camera_params, i)
camera_pos_mm = jnp.linalg.inv(E)[:3, 3]
camera_pos_m = camera_pos_mm / 1000.0

Quick Reference: Import Paths

# Core camera library
from multi_camera.analysis import camera as cam_lib
from multi_camera.analysis.camera import (
    get_intrinsic,          # camera_params, i -> 3x3 K (pixels)
    get_extrinsic,          # camera_params, i -> 4x4 E (mm, world-to-camera)
    get_projection,         # camera_params, i -> 3x4 P (mm)
    project_distortion,     # camera_params, i, pts_mm -> 2D pixels
    undistort_points,       # pts_px, K, dist -> undistorted_px
    distort_3d,             # camera_params, i, pts_mm -> distorted 3D mm
    triangulate_point,      # camera_params, pts_2d -> 3D mm
    robust_triangulate_points,  # camera_params, pts_2d -> 3D mm + weights
    reprojection_error,     # camera_params, pts_2d, pts_3d_mm -> residual px
)

# High-level meter-safe adapter
from multiview_biomechanical_pose.geometry.camera_adapter import (
    transform_points_world_to_camera,       # pts_m -> pts_m (camera frame)
    transform_points_camera_to_world,       # pts_m -> pts_m (world frame)
    transform_points_between_cameras,       # pts_m -> pts_m (target camera)
    project_camera_coordinates,             # pts_cam_m -> 2D pixels
    unproject_from_image_with_distortion,   # 2D px + depth_m -> 3D cam_m
    unproject_and_transform_to_world,       # 2D px + depth_m -> 3D world_m
    transform_pose_se3_world_to_camera,     # pos_m, quat -> pos_m, quat
    transform_pose_se3_camera_to_world,     # pos_m, quat -> pos_m, quat
    compute_relative_extrinsics,            # -> 4x4 relative transform (mm)
)

# Fit quality metrics
from multi_camera.analysis.fit_quality import reprojection_quality

Common Mistakes