crosstyan
032eb684ec
- Updated play notebook to include new tracking and clustering functionalities. - Introduced `distortion` and `project` functions for applying distortion to 2D points and projecting 3D points to 2D, respectively. - Enhanced `CameraParams` and `Camera` classes with methods for distortion and projection, improving usability. - Cleaned up execution counts in the notebook for better organization.
555 lines
17 KiB
Python
555 lines
17 KiB
Python
from collections import OrderedDict, defaultdict
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from dataclasses import dataclass
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from datetime import datetime
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from typing import Any, TypeAlias, TypedDict, Optional
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from beartype import beartype
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from jax import Array
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from jax import numpy as jnp
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from jaxtyping import Num, jaxtyped
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from typing_extensions import NotRequired
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CameraID: TypeAlias = str # pylint: disable=invalid-name
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@jaxtyped(typechecker=beartype)
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def distortion(
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points_2d: Num[Array, "N 2"],
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K: Num[Array, "3 3"],
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dist_coeffs: Num[Array, "5"],
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) -> Num[Array, "N 2"]:
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"""
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Apply distortion to 2D points
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Args:
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points_2d: 2D points in image coordinates
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K: Camera intrinsic matrix
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dist_coeffs: Distortion coefficients [k1, k2, p1, p2, k3]
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Returns:
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Distorted 2D points
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"""
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k1, k2, p1, p2, k3 = dist_coeffs
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# Get principal point and focal length
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cx, cy = K[0, 2], K[1, 2]
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fx, fy = K[0, 0], K[1, 1]
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# Convert to normalized coordinates
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x = (points_2d[:, 0] - cx) / fx
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y = (points_2d[:, 1] - cy) / fy
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r2 = x * x + y * y
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# Radial distortion
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xdistort = x * (1 + k1 * r2 + k2 * r2 * r2 + k3 * r2 * r2 * r2)
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ydistort = y * (1 + k1 * r2 + k2 * r2 * r2 + k3 * r2 * r2 * r2)
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# Tangential distortion
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xdistort = xdistort + 2 * p1 * x * y + p2 * (r2 + 2 * x * x)
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ydistort = ydistort + p1 * (r2 + 2 * y * y) + 2 * p2 * x * y
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# Back to absolute coordinates
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xdistort = xdistort * fx + cx
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ydistort = ydistort * fy + cy
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# Combine distorted coordinates
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return jnp.stack([xdistort, ydistort], axis=1)
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@jaxtyped(typechecker=beartype)
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def project(
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points_3d: Num[Array, "N 3"],
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projection_matrix: Num[Array, "3 4"],
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K: Num[Array, "3 3"],
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dist_coeffs: Num[Array, "5"],
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) -> Num[Array, "N 2"]:
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"""
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Project 3D points to 2D points
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Args:
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points_3d: 3D points in world coordinates
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projection_matrix: pre-computed projection matrix (K @ Rt[:3, :])
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K: Camera intrinsic matrix
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dist_coeffs: Distortion coefficients
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Returns:
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2D points in image coordinates
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"""
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P = projection_matrix
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p3d_homogeneous = jnp.hstack(
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(points_3d, jnp.ones((points_3d.shape[0], 1), dtype=points_3d.dtype))
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)
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# Project points
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p2d_homogeneous = p3d_homogeneous @ P.T
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# Perspective division
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p2d = p2d_homogeneous[:, 0:2] / p2d_homogeneous[:, 2:3]
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# Apply distortion if needed
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if dist_coeffs is not None:
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# Check if valid points (between 0 and 1)
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valid = jnp.all(p2d > 0, axis=1) & jnp.all(p2d < 1, axis=1)
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# Only apply distortion if there are valid points
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if jnp.any(valid):
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# Only distort valid points
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valid_p2d = p2d[valid]
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distorted_valid = distortion(valid_p2d, K, dist_coeffs)
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p2d = p2d.at[valid].set(distorted_valid)
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return jnp.squeeze(p2d)
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@jaxtyped(typechecker=beartype)
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@dataclass(frozen=True)
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class CameraParams:
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"""
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Camera parameters: intrinsic matrix, extrinsic matrix, and distortion coefficients
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"""
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K: Num[Array, "3 3"]
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"""
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intrinsic matrix
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"""
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Rt: Num[Array, "4 4"]
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"""
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[R|t] extrinsic matrix
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R and t are the rotation and translation that describe the change of
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coordinates from world to camera coordinate systems (or camera frame)
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Rt is expected to be World-to-Camera (W2C) transformation matrix,
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which is the result of `solvePnP` in OpenCV. (but converted to homogeneous coordinates)
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World-to-Camera (W2C): Transforms points from world coordinates to camera coordinates
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- The world origin is transformed to camera space
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- Used for projecting 3D world points onto the camera's image plane
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- Required for rendering/projection
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"""
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dist_coeffs: Num[Array, "5"]
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"""
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An array of distortion coefficients of the form
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[k1, k2, [p1, p2, [k3]]], where ki is the ith
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radial distortion coefficient and pi is the ith
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tangential distortion coeff.
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"""
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image_size: Num[Array, "2"]
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"""
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The size of image plane (width, height)
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"""
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@property
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def pose_matrix(self) -> Num[Array, "4 4"]:
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"""
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The inversion of the extrinsic matrix, which gives Camera-to-World (C2W) transformation matrix.
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Camera-to-World (C2W): Transforms points from camera coordinates to world coordinates
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- The camera is the origin in camera space
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- This transformation tells where the camera is positioned in world space
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- Often used for camera positioning/orientation
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The result is cached on first access. (lazy evaluation)
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"""
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t = getattr(self, "_pose", None)
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if t is None:
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t = jnp.linalg.inv(self.Rt)
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object.__setattr__(self, "_pose", t)
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return t
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@property
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def projection_matrix(self) -> Num[Array, "3 4"]:
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"""
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Returns the 3x4 projection matrix K @ [R|t].
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The result is cached on first access. (lazy evaluation)
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"""
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pm = getattr(self, "_proj", None)
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if pm is None:
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pm = self.K @ self.Rt[:3, :]
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# object.__setattr__ bypasses the frozen‐dataclass blocker
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object.__setattr__(self, "_proj", pm)
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return pm
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@jaxtyped(typechecker=beartype)
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@dataclass(frozen=True)
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class Camera:
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"""
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a description of a camera
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"""
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id: CameraID
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"""
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Camera ID
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"""
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params: CameraParams
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"""
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Camera parameters
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"""
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def project(self, points_3d: Num[Array, "N 3"]) -> Num[Array, "N 2"]:
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"""
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Project 3D points to 2D points using this camera's parameters
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Args:
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points_3d: 3D points in world coordinates
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Returns:
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2D points in image coordinates
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"""
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return project(
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points_3d=points_3d,
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K=self.params.K,
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dist_coeffs=self.params.dist_coeffs,
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projection_matrix=self.params.projection_matrix,
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)
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def distortion(self, points_2d: Num[Array, "N 2"]) -> Num[Array, "N 2"]:
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"""
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Apply distortion to 2D points using this camera's parameters
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Args:
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points_2d: 2D points in image coordinates
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Returns:
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Distorted 2D points
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"""
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return distortion(
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points_2d=points_2d,
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K=self.params.K,
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dist_coeffs=self.params.dist_coeffs,
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)
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@jaxtyped(typechecker=beartype)
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@dataclass(frozen=True)
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class Detection:
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"""
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One detection from a camera
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"""
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keypoints: Num[Array, "N 2"]
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"""
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Keypoints
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"""
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confidences: Num[Array, "N"]
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"""
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Confidences
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"""
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camera: Camera
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"""
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Camera
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"""
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timestamp: datetime
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"""
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Timestamp of the detection
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"""
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def classify_by_camera(
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detections: list[Detection],
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) -> OrderedDict[CameraID, list[Detection]]:
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"""
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Classify detections by camera
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"""
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# or use setdefault
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camera_wise_split: dict[CameraID, list[Detection]] = defaultdict(list)
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for entry in detections:
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camera_id = entry.camera.id
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camera_wise_split[camera_id].append(entry)
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return OrderedDict(camera_wise_split)
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@jaxtyped(typechecker=beartype)
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def to_homogeneous(points: Num[Array, "N 2"] | Num[Array, "N 3"]) -> Num[Array, "N 3"]:
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"""
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Convert points to homogeneous coordinates
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"""
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if points.shape[-1] == 2:
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return jnp.hstack((points, jnp.ones((points.shape[0], 1))))
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elif points.shape[-1] == 3:
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return points
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else:
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raise ValueError(f"Invalid shape for points: {points.shape}")
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@jaxtyped(typechecker=beartype)
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def point_line_distance(
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points: Num[Array, "N 3"] | Num[Array, "N 2"],
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line: Num[Array, "N 3"],
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eps: float = 1e-9,
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):
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"""
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Calculate the distance from a point to a line
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Args:
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point: (possibly homogeneous) points :math:`(N, 2 or 3)`.
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line: lines coefficients :math:`(a, b, c)` with shape :math:`(N, 3)`, where :math:`ax + by + c = 0`.
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eps: Small constant for safe sqrt.
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Returns:
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the computed distance with shape :math:`(N)`.
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See also:
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https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line
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"""
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numerator = abs(line[:, 0] * points[:, 0] + line[:, 1] * points[:, 1] + line[:, 2])
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denominator = jnp.sqrt(line[:, 0] * line[:, 0] + line[:, 1] * line[:, 1])
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return numerator / (denominator + eps)
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@jaxtyped(typechecker=beartype)
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def left_to_right_epipolar_distance(
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left: Num[Array, "N 3"],
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right: Num[Array, "N 3"],
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fundamental_matrix: Num[Array, "3 3"],
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):
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"""
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Return one-sided epipolar distance for correspondences given the fundamental matrix.
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Args:
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left: points in the left image (homogeneous) :math:`(N, 3)`
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right: points in the right image (homogeneous) :math:`(N, 3)`
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fundamental_matrix: fundamental matrix :math:`(3, 3)`
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Returns:
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the computed distance with shape :math:`(N)`.
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See also:
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https://en.wikipedia.org/wiki/Fundamental_matrix_%28computer_vision%29
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$$x^{\\prime T}Fx = 0$$
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"""
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F_t = fundamental_matrix.transpose()
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line1_in_2 = jnp.matmul(left, F_t)
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return point_line_distance(right, line1_in_2)
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@jaxtyped(typechecker=beartype)
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def right_to_left_epipolar_distance(
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left: Num[Array, "N 3"],
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right: Num[Array, "N 3"],
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fundamental_matrix: Num[Array, "3 3"],
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):
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"""
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Return one-sided epipolar distance for correspondences given the fundamental matrix.
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Args:
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left: points in the left image (homogeneous) :math:`(N, 3)`
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right: points in the right image (homogeneous) :math:`(N, 3)`
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fundamental_matrix: fundamental matrix :math:`(3, 3)`
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Returns:
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the computed distance with shape :math:`(N)`.
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See also:
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https://en.wikipedia.org/wiki/Fundamental_matrix_%28computer_vision%29
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$$x^{\\prime T}Fx = 0$$
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"""
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line2_in_1 = jnp.matmul(right, fundamental_matrix)
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return point_line_distance(left, line2_in_1)
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def distance_between_epipolar_lines(
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x1: Num[Array, "N 2"] | Num[Array, "N 3"],
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x2: Num[Array, "N 2"] | Num[Array, "N 3"],
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fundamental_matrix: Num[Array, "3 3"],
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):
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"""
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Calculate the total epipolar line distance between x1 and x2.
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"""
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if x1.shape[0] != x2.shape[0]:
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raise ValueError(
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f"x1 and x2 must have the same number of points: {x1.shape[0]} != {x2.shape[0]}"
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)
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if x1.shape[-1] == 2:
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points1 = to_homogeneous(x1)
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elif x1.shape[-1] == 3:
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points1 = x1
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else:
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raise ValueError(f"Invalid shape for correspondence1: {x1.shape}")
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if x2.shape[-1] == 2:
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points2 = to_homogeneous(x2)
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elif x2.shape[-1] == 3:
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points2 = x2
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else:
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raise ValueError(f"Invalid shape for correspondence2: {x2.shape}")
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if fundamental_matrix.shape != (3, 3):
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raise ValueError(
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f"Invalid shape for fundamental_matrix: {fundamental_matrix.shape}"
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)
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# points 1 and 2 are unnormalized points
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dist_1 = jnp.mean(
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right_to_left_epipolar_distance(points1, points2, fundamental_matrix)
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)
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dist_2 = jnp.mean(
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left_to_right_epipolar_distance(points1, points2, fundamental_matrix)
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)
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distance = dist_1 + dist_2
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return distance
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@jaxtyped(typechecker=beartype)
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def calculate_fundamental_matrix(
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camera_left: Camera, camera_right: Camera
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) -> Num[Array, "3 3"]:
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"""
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Calculate the fundamental matrix for the given cameras.
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"""
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# Intrinsics
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K1 = camera_left.params.K
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K2 = camera_right.params.K
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# Extrinsics (World to Camera transforms)
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Rt1 = camera_left.params.Rt
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Rt2 = camera_right.params.Rt
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# Convert to Camera to World (Inverse)
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T2: Array = jnp.linalg.inv(Rt2)
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# Relative transform from Left to Right
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T_rel = T2 @ Rt1
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R = T_rel[:3, :3]
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t = T_rel[:3, 3:]
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# Skew-symmetric matrix for cross product
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def skew(v: Num[Array, "3"]) -> Num[Array, "3 3"]:
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return jnp.array(
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[
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[0, -v[2], v[1]],
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[v[2], 0, -v[0]],
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[-v[1], v[0], 0],
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]
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)
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t_skew = skew(t.reshape(-1))
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# Essential Matrix
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E = t_skew @ R
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# Fundamental Matrix
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F = jnp.linalg.inv(K2).T @ E @ jnp.linalg.inv(K1)
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return F
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def compute_affinity_epipolar_constraint_with_pairs(
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left: Detection, right: Detection, alpha_2d: float
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):
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"""
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Compute the affinity between two groups of detections by epipolar constraint,
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where camera parameters are included in the detections.
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Note:
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Originally, alpha_2d comes from the paper as a scaling factor for epipolar error affinity.
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Its role is mainly to normalize error into [0,1] range, but it could lead to negative affinity.
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An alternative approach is using normalized epipolar error relative to image size, with soft cutoff,
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like exp(-error / threshold), for better interpretability and stability.
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"""
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fundamental_matrix = calculate_fundamental_matrix(left.camera, right.camera)
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d = distance_between_epipolar_lines(
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left.keypoints, right.keypoints, fundamental_matrix
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)
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return 1 - (d / alpha_2d)
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def calculate_affinity_matrix_by_epipolar_constraint(
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detections: list[Detection] | dict[CameraID, list[Detection]],
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alpha_2d: float,
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) -> tuple[list[Detection], Num[Array, "N N"]]:
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"""
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Calculate the affinity matrix by epipolar constraint
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This function evaluates the geometric consistency of every pair of detections
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across different cameras using the fundamental matrix. It assumes that
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detections from the same camera are not comparable and should have zero affinity.
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The affinity is computed by:
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1. Calculating the fundamental matrix between the two cameras.
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2. Measuring the average point-to-epipolar-line distance for all keypoints.
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3. Mapping the distance to affinity with the formula: 1 - (distance / alpha_2d).
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Args:
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detections: Either a flat list of Detection or a dict grouping Detection by CameraID.
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alpha_2d: Image resolution-dependent threshold controlling affinity scaling.
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Typically relates to expected pixel displacement rate.
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Returns:
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sorted_detections: Flattened list of detections sorted by camera order.
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affinity_matrix: Array of shape (N, N), where N is the number of detections.
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Notes:
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- Detections from the same camera always have affinity = 0.
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- Affinity decays linearly with epipolar error until 0 (or potentially negative).
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- Consider switching to exp(-error / scale) style for non-negative affinity.
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- alpha_2d should be adjusted based on image resolution or empirical observation.
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Affinity Matrix layout:
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assuming we have 3 cameras
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C0 has 3 detections: D0_C0, D1_C0, D2_C0
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C1 has 2 detections: D0_C1, D1_C1
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C2 has 2 detections: D0_C2, D1_C2
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D0_C0(0), D1_C0(1), D2_C0(2), D0_C1(3), D1_C1(4), D0_C2(5), D1_C2(6)
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D0_C0(0) 0 0 0 a_03 a_04 a_05 a_06
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D1_C0(1) 0 0 0 a_13 a_14 a_15 a_16
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...
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D0_C1(3) a_30 a_31 a_32 0 0 a_35 a_36
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...
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D1_C2(6) a_60 a_61 a_62 a_63 a_64 0 0
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"""
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if isinstance(detections, dict):
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camera_wise_split = detections
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else:
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camera_wise_split = classify_by_camera(detections)
|
||
num_entries = sum(len(entries) for entries in camera_wise_split.values())
|
||
affinity_matrix = jnp.ones((num_entries, num_entries), dtype=jnp.float32) * -jnp.inf
|
||
affinity_matrix_mask = jnp.zeros((num_entries, num_entries), dtype=jnp.bool_)
|
||
|
||
acc = 0
|
||
total_indices = set(range(num_entries))
|
||
camera_id_index_map: dict[CameraID, set[int]] = defaultdict(set)
|
||
camera_id_index_map_inverse: dict[CameraID, set[int]] = defaultdict(set)
|
||
# sorted by [D0_C0, D1_C0, D2_C0, D0_C1, D1_C1, D0_C2, D1_C2...]
|
||
sorted_detections: list[Detection] = []
|
||
for camera_id, entries in camera_wise_split.items():
|
||
for i, _ in enumerate(entries):
|
||
camera_id_index_map[camera_id].add(acc)
|
||
sorted_detections.append(entries[i])
|
||
acc += 1
|
||
camera_id_index_map_inverse[camera_id] = (
|
||
total_indices - camera_id_index_map[camera_id]
|
||
)
|
||
|
||
# ignore self-affinity
|
||
# ignore same-camera affinity
|
||
# assuming commutative
|
||
for i, det in enumerate(sorted_detections):
|
||
other_indices = camera_id_index_map_inverse[det.camera.id]
|
||
for j in other_indices:
|
||
if i == j:
|
||
continue
|
||
if affinity_matrix_mask[i, j] or affinity_matrix_mask[j, i]:
|
||
continue
|
||
a = compute_affinity_epipolar_constraint_with_pairs(
|
||
det, sorted_detections[j], alpha_2d
|
||
)
|
||
affinity_matrix = affinity_matrix.at[i, j].set(a)
|
||
affinity_matrix = affinity_matrix.at[j, i].set(a)
|
||
affinity_matrix_mask = affinity_matrix_mask.at[i, j].set(True)
|
||
affinity_matrix_mask = affinity_matrix_mask.at[j, i].set(True)
|
||
|
||
return sorted_detections, affinity_matrix
|