import numpy as np
from .model_object import ComponentModel
from ..utils.decorators import ignore_numpy_warnings, combine_docstrings
from . import func
from ..param import forward
from ..backend_obj import backend, ArrayLike
__all__ = ["GaussianEllipsoid"]
[docs]
@combine_docstrings
class GaussianEllipsoid(ComponentModel):
"""Model that represents a galaxy as a 3D Gaussian ellipsoid.
The model is triaxial, meaning it has three different standard deviations
along the three axes. The orientation of the ellipsoid is defined by Euler
angles.
If all three Euler angles are set to zero, the ellipsoid is aligned with the
image axes meaning sigma_a gives the std along the x axis of the tangent
plane, sigma_b gives the std along the y axis of the tangent plane, and
sigma_z gives the std into the tangent plane. We use the ZXZ convention for
the Euler angles. This means that for a disk galaxy, one can naturally
consider sigma_c as the disk thickness and sigma_a=sigma_b as the disk
radius; setting the Euler angles to zero would leave the disk face-on in the
x-y tangent plane.
Note:
the model is highly degenerate, meaning that it is not possible to
uniquely determine the parameters from the data. The model is useful if
one already has a 3D model of the galaxy in mind and wants to produce
mock data. Alternately, if one applies some constraints on the
parameters, such as sigma_a = sigma_b and alpha=0, then the model will
be better determined. In that case, beta is related to the inclination
of the disk and gamma is related to the position angle of the disk. The
initialization for this model assumes exactly this interpretation with a
disk thickness of sigma_c = 0.2 *sigma_a.
:param sigma_a: Standard deviation of the Gaussian along the alpha axis in arcseconds.
:param sigma_b: Standard deviation of the Gaussian along the beta axis in arcseconds.
:param sigma_c: Standard deviation of the Gaussian along the gamma axis in arcseconds.
:param alpha: Euler angle representing the rotation around the alpha axis in radians.
:param beta: Euler angle representing the rotation around the beta axis in radians.
:param gamma: Euler angle representing the rotation around the gamma axis in radians.
:param flux: Total flux of the galaxy in arbitrary units.
"""
_model_type = "gaussianellipsoid"
_parameter_specs = {
"sigma_a": {
"units": "arcsec",
"valid": (0, None),
"shape": (),
"dynamic": True,
"description": "Standard deviation of the Gaussian along the alpha axis in arcseconds.",
},
"sigma_b": {
"units": "arcsec",
"valid": (0, None),
"shape": (),
"dynamic": True,
"description": "Standard deviation of the Gaussian along the beta axis in arcseconds.",
},
"sigma_c": {
"units": "arcsec",
"valid": (0, None),
"shape": (),
"dynamic": True,
"description": "Standard deviation of the Gaussian along the gamma axis in arcseconds.",
},
"alpha": {
"units": "radians",
"valid": (0, 2 * np.pi),
"cyclic": True,
"shape": (),
"dynamic": True,
"description": "Euler angle representing the rotation around the alpha axis in radians.",
},
"beta": {
"units": "radians",
"valid": (0, 2 * np.pi),
"cyclic": True,
"shape": (),
"dynamic": True,
"description": "Euler angle representing the rotation around the beta axis in radians.",
},
"gamma": {
"units": "radians",
"valid": (0, 2 * np.pi),
"cyclic": True,
"shape": (),
"dynamic": True,
"description": "Euler angle representing the rotation around the gamma axis in radians.",
},
"flux": {
"units": "flux",
"shape": (),
"dynamic": True,
"description": "Total flux of the galaxy in arbitrary units.",
},
}
usable = True
[docs]
@ignore_numpy_warnings
def initialize(self):
super().initialize()
if any(self[key].initialized for key in GaussianEllipsoid._parameter_specs):
return
self.sigma_b = self.sigma_a
self.sigma_c = lambda p: 0.2 * p.sigma_a.value
self.sigma_c.link(self.sigma_a)
self.alpha = 0.0
target_area = self.target[self.window]
dat = backend.to_numpy(target_area._data).copy()
mask = backend.to_numpy(target_area._mask).copy()
dat[mask] = np.median(dat[~mask])
edge = np.concatenate((dat[:, 0], dat[:, -1], dat[0, :], dat[-1, :]))
edge_average = np.nanmedian(edge)
dat -= edge_average
x, y = target_area.coordinate_center_meshgrid()
center = self.center.value
x = x - center[0]
y = y - center[1]
r = backend.to_numpy(self.radius_metric(x, y, params=()))
self.sigma_a.value = np.sqrt(np.sum((r * dat) ** 2) / np.sum(r**2))
x = backend.to_numpy(x)
y = backend.to_numpy(y)
mu20 = np.median(dat * np.abs(x))
mu02 = np.median(dat * np.abs(y))
mu11 = np.median(dat * x * y / np.sqrt(np.abs(x * y) + self.softening**2))
M = np.array([[mu20, mu11], [mu11, mu02]])
if np.any(np.iscomplex(M)) or np.any(~np.isfinite(M)):
PA = np.pi / 2
l = (0.7, 1.0)
else:
PA = (0.5 * np.arctan2(2 * mu11, mu20 - mu02) - np.pi / 2) % np.pi
l = np.sort(np.linalg.eigvals(M))
q = np.clip(np.sqrt(l[0] / l[1]), 0.1, 0.9)
self.beta.value = np.arccos(q)
self.gamma.value = PA
self.flux.value = np.sum(dat)
[docs]
@forward
def brightness(
self,
x: ArrayLike,
y: ArrayLike,
sigma_a: ArrayLike,
sigma_b: ArrayLike,
sigma_c: ArrayLike,
alpha: ArrayLike,
beta: ArrayLike,
gamma: ArrayLike,
flux: ArrayLike,
) -> ArrayLike:
"""Brightness of the Gaussian ellipsoid."""
D = backend.diag(backend.stack((sigma_a, sigma_b, sigma_c)) ** 2)
R = func.euler_rotation_matrix(alpha, beta, gamma)
Sigma = R @ D @ R.T
Sigma2D = Sigma[:2, :2]
inv_Sigma = backend.linalg.inv(Sigma2D)
v = backend.stack(self.transform_coordinates(x, y), dim=0).reshape(2, -1)
return (
flux
* backend.exp(-0.5 * backend.sum(v * (inv_Sigma @ v), dim=0))
/ (2 * np.pi * backend.sqrt(backend.linalg.det(Sigma2D)))
).reshape(x.shape)
