utils/geo_opensfm.py

"""Copied from opensfm.geo to minimize hard dependencies."""
import numpy as np
from numpy import ndarray
from typing import Tuple

WGS84_a = 6378137.0
WGS84_b = 6356752.314245


def ecef_from_lla(lat, lon, alt: float) -> Tuple[float, ...]:
    """
    Compute ECEF XYZ from latitude, longitude and altitude.

    All using the WGS84 model.
    Altitude is the distance to the WGS84 ellipsoid.
    Check results here http://www.oc.nps.edu/oc2902w/coord/llhxyz.htm

    >>> lat, lon, alt = 10, 20, 30
    >>> x, y, z = ecef_from_lla(lat, lon, alt)
    >>> np.allclose(lla_from_ecef(x,y,z), [lat, lon, alt])
    True
    """
    a2 = WGS84_a**2
    b2 = WGS84_b**2
    lat = np.radians(lat)
    lon = np.radians(lon)
    L = 1.0 / np.sqrt(a2 * np.cos(lat) ** 2 + b2 * np.sin(lat) ** 2)
    x = (a2 * L + alt) * np.cos(lat) * np.cos(lon)
    y = (a2 * L + alt) * np.cos(lat) * np.sin(lon)
    z = (b2 * L + alt) * np.sin(lat)
    return x, y, z


def lla_from_ecef(x, y, z):
    """
    Compute latitude, longitude and altitude from ECEF XYZ.

    All using the WGS84 model.
    Altitude is the distance to the WGS84 ellipsoid.
    """
    a = WGS84_a
    b = WGS84_b
    ea = np.sqrt((a**2 - b**2) / a**2)
    eb = np.sqrt((a**2 - b**2) / b**2)
    p = np.sqrt(x**2 + y**2)
    theta = np.arctan2(z * a, p * b)
    lon = np.arctan2(y, x)
    lat = np.arctan2(
        z + eb**2 * b * np.sin(theta) ** 3, p - ea**2 * a * np.cos(theta) ** 3
    )
    N = a / np.sqrt(1 - ea**2 * np.sin(lat) ** 2)
    alt = p / np.cos(lat) - N
    return np.degrees(lat), np.degrees(lon), alt


def ecef_from_topocentric_transform(lat, lon, alt: float) -> ndarray:
    """
    Transformation from a topocentric frame at reference position to ECEF.

    The topocentric reference frame is a metric one with the origin
    at the given (lat, lon, alt) position, with the X axis heading east,
    the Y axis heading north and the Z axis vertical to the ellipsoid.
    >>> a = ecef_from_topocentric_transform(30, 20, 10)
    >>> b = ecef_from_topocentric_transform_finite_diff(30, 20, 10)
    >>> np.allclose(a, b)
    True
    """
    x, y, z = ecef_from_lla(lat, lon, alt)
    sa = np.sin(np.radians(lat))
    ca = np.cos(np.radians(lat))
    so = np.sin(np.radians(lon))
    co = np.cos(np.radians(lon))
    return np.array(
        [
            [-so, -sa * co, ca * co, x],
            [co, -sa * so, ca * so, y],
            [0, ca, sa, z],
            [0, 0, 0, 1],
        ]
    )


def ecef_from_topocentric_transform_finite_diff(lat, lon, alt: float) -> ndarray:
    """
    Transformation from a topocentric frame at reference position to ECEF.

    The topocentric reference frame is a metric one with the origin
    at the given (lat, lon, alt) position, with the X axis heading east,
    the Y axis heading north and the Z axis vertical to the ellipsoid.
    """
    eps = 1e-2
    x, y, z = ecef_from_lla(lat, lon, alt)
    v1 = (
        (
            np.array(ecef_from_lla(lat, lon + eps, alt))
            - np.array(ecef_from_lla(lat, lon - eps, alt))
        )
        / 2
        / eps
    )
    v2 = (
        (
            np.array(ecef_from_lla(lat + eps, lon, alt))
            - np.array(ecef_from_lla(lat - eps, lon, alt))
        )
        / 2
        / eps
    )
    v3 = (
        (
            np.array(ecef_from_lla(lat, lon, alt + eps))
            - np.array(ecef_from_lla(lat, lon, alt - eps))
        )
        / 2
        / eps
    )
    v1 /= np.linalg.norm(v1)
    v2 /= np.linalg.norm(v2)
    v3 /= np.linalg.norm(v3)
    return np.array(
        [
            [v1[0], v2[0], v3[0], x],
            [v1[1], v2[1], v3[1], y],
            [v1[2], v2[2], v3[2], z],
            [0, 0, 0, 1],
        ]
    )


def topocentric_from_lla(lat, lon, alt: float, reflat, reflon, refalt: float):
    """
    Transform from lat, lon, alt to topocentric XYZ.

    >>> lat, lon, alt = -10, 20, 100
    >>> np.allclose(topocentric_from_lla(lat, lon, alt, lat, lon, alt),
    ...     [0,0,0])
    True
    >>> x, y, z = topocentric_from_lla(lat, lon, alt, 0, 0, 0)
    >>> np.allclose(lla_from_topocentric(x, y, z, 0, 0, 0),
    ...     [lat, lon, alt])
    True
    """
    T = np.linalg.inv(ecef_from_topocentric_transform(reflat, reflon, refalt))
    x, y, z = ecef_from_lla(lat, lon, alt)
    tx = T[0, 0] * x + T[0, 1] * y + T[0, 2] * z + T[0, 3]
    ty = T[1, 0] * x + T[1, 1] * y + T[1, 2] * z + T[1, 3]
    tz = T[2, 0] * x + T[2, 1] * y + T[2, 2] * z + T[2, 3]
    return tx, ty, tz


def lla_from_topocentric(x, y, z, reflat, reflon, refalt: float):
    """
    Transform from topocentric XYZ to lat, lon, alt.
    """
    T = ecef_from_topocentric_transform(reflat, reflon, refalt)
    ex = T[0, 0] * x + T[0, 1] * y + T[0, 2] * z + T[0, 3]
    ey = T[1, 0] * x + T[1, 1] * y + T[1, 2] * z + T[1, 3]
    ez = T[2, 0] * x + T[2, 1] * y + T[2, 2] * z + T[2, 3]
    return lla_from_ecef(ex, ey, ez)


class TopocentricConverter(object):
    """Convert to and from a topocentric reference frame."""

    def __init__(self, reflat, reflon, refalt):
        """Init the converter given the reference origin."""
        self.lat = reflat
        self.lon = reflon
        self.alt = refalt

    def to_topocentric(self, lat, lon, alt):
        """Convert lat, lon, alt to topocentric x, y, z."""
        return topocentric_from_lla(lat, lon, alt, self.lat, self.lon, self.alt)

    def to_lla(self, x, y, z):
        """Convert topocentric x, y, z to lat, lon, alt."""
        return lla_from_topocentric(x, y, z, self.lat, self.lon, self.alt)

    def __eq__(self, o):
        return np.allclose([self.lat, self.lon, self.alt], (o.lat, o.lon, o.alt))