# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""
This module provides a container class to store parameters for the
geometry of an ellipse.
"""
import math
import numpy as np
from astropy import log
__all__ = ['EllipseGeometry']
IN_MASK = [
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
]
OUT_MASK = [
[1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1],
[1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1],
[1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1],
[1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1],
[0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0],
[0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0],
[0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0],
[0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0],
[0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0],
[0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0],
[0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0],
[0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0],
[1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1],
[1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1],
[1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1],
[1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1],
]
def _area(sma, eps, phi, r):
"""
Compute elliptical sector area.
"""
aux = r * math.cos(phi) / sma
signal = aux / abs(aux)
if abs(aux) >= 1.0:
aux = signal
return abs(sma**2 * (1.0 - eps) / 2.0 * math.acos(aux))
[docs]
class EllipseGeometry:
r"""
Container class to store parameters for the geometry of an ellipse.
Parameters that describe the relationship of a given ellipse with
other associated ellipses are also encapsulated in this container.
These associated ellipses may include, e.g., the two (inner and
outer) bounding ellipses that are used to build sectors along the
elliptical path. These sectors are used as areas for integrating
pixel values, when the area integration mode (mean or median) is
used.
This class also keeps track of where in the ellipse we are when
performing an 'extract' operation. This is mostly relevant when
using an area integration mode (as opposed to a pixel integration
mode)
Parameters
----------
x0, y0 : float
The center pixel coordinate of the ellipse.
sma : float
The semimajor axis of the ellipse in pixels.
eps : ellipticity
The ellipticity of the ellipse.
pa : float
The position angle (in radians) of the semimajor axis in
relation to the positive x axis of the image array (rotating
towards the positive y axis). Position angles are defined in the
range :math:`0 < PA <= \pi`. Avoid using as starting position
angle of 0., since the fit algorithm may not work properly.
When the ellipses are such that position angles are near either
extreme of the range, noise can make the solution jump back and
forth between successive isophotes, by amounts close to 180
degrees.
astep : float, optional
The step value for growing/shrinking the semimajor axis. It can
be expressed either in pixels (when ``linear_growth=True``) or
as a relative value (when ``linear_growth=False``). The default
is 0.1.
linear_growth : bool, optional
The semimajor axis growing/shrinking mode. The default is
`False`.
fix_center : bool, optional
Keep center of ellipse fixed during fit? The default is False.
fix_pa : bool, optional
Keep position angle of semi-major axis of ellipse fixed during fit?
The default is False.
fix_eps : bool, optional
Keep ellipticity of ellipse fixed during fit? The default is False.
"""
def __init__(self, x0, y0, sma, eps, pa, astep=0.1, linear_growth=False,
fix_center=False, fix_pa=False, fix_eps=False):
self.x0 = x0
self.y0 = y0
self.sma = sma
self.eps = eps
self.pa = pa
self.astep = astep
self.linear_growth = linear_growth
# Fixed parameters are flagged in here. Note that the
# ordering must follow the same ordering used in the
# fitter._CORRECTORS list.
self.fix = np.array([fix_center, fix_center, fix_pa, fix_eps])
# limits for sector angular width
self._phi_min = 0.05
self._phi_max = 0.2
# variables used in the calculation of the sector angular width
sma1, sma2 = self.bounding_ellipses()
inner_sma = min((sma2 - sma1), 3.0)
self._area_factor = (sma2 - sma1) * inner_sma
# sma can eventually be zero!
if self.sma > 0.0:
self.sector_angular_width = max(min((inner_sma / self.sma),
self._phi_max), self._phi_min)
self.initial_polar_angle = self.sector_angular_width / 2.0
self.initial_polar_radius = self.radius(self.initial_polar_angle)
[docs]
def find_center(self, image, threshold=0.1, verbose=True):
"""
Find the center of a galaxy.
If the algorithm is successful the (x, y) coordinates in this
`~photutils.isophote.EllipseGeometry` (i.e., the ``x0`` and
``y0`` attributes) instance will be modified.
The isophote fit algorithm requires an initial guess for the
galaxy center (x, y) coordinates and these coordinates must be
close to the actual galaxy center for the isophote fit to work.
This method provides can provide an initial guess for the galaxy
center coordinates. See the **Notes** section below for more
details.
Parameters
----------
image : 2D `~numpy.ndarray`
The image array. Masked arrays are not recognized here. This
assumes that centering should always be done on valid pixels.
threshold : float, optional
The centerer threshold. To turn off the centerer, set this
to a large value (i.e., >> 1). The default is 0.1.
verbose : bool, optional
Whether to print object centering information. The default is
`True`.
Notes
-----
The centerer function scans a 10x10 window centered on the (x,
y) coordinates in the `~photutils.isophote.EllipseGeometry`
instance passed to the constructor of the
`~photutils.isophote.Ellipse` class. If any of the
`~photutils.isophote.EllipseGeometry` (x, y) coordinates are
`None`, the center of the input image frame is used. If the
center acquisition is successful, the
`~photutils.isophote.EllipseGeometry` instance is modified in
place to reflect the solution of the object centerer algorithm.
In some cases the object centerer algorithm may fail even though
there is enough signal-to-noise to start a fit (e.g., objects
with very high ellipticity). In those cases the sensitivity
of the algorithm can be decreased by decreasing the value of
the object centerer threshold parameter. The centerer works by
looking where a quantity akin to a signal-to-noise ratio is
maximized within the 10x10 window. The centerer can thus be shut
off entirely by setting the threshold to a large value (i.e.,
>> 1; meaning no location inside the search window will achieve
that signal-to-noise ratio).
"""
self._centerer_mask_half_size = len(IN_MASK) / 2
self.centerer_threshold = threshold
# number of pixels in each mask
sz = len(IN_MASK)
self._centerer_ones_in = np.ma.masked_array(np.ones(shape=(sz, sz)),
mask=IN_MASK)
self._centerer_ones_out = np.ma.masked_array(np.ones(shape=(sz, sz)),
mask=OUT_MASK)
self._centerer_in_mask_npix = np.sum(self._centerer_ones_in)
self._centerer_out_mask_npix = np.sum(self._centerer_ones_out)
# Check if center coordinates point to somewhere inside the frame.
# If not, set then to frame center.
shape = image.shape
_x0 = self.x0
_y0 = self.y0
if (_x0 is None or _x0 < 0 or _x0 >= shape[1] or _y0 is None
or _y0 < 0 or _y0 >= shape[0]):
_x0 = shape[1] / 2
_y0 = shape[0] / 2
max_fom = 0.0
max_i = 0
max_j = 0
# scan all positions inside window
window_half_size = 5
for i in range(int(_x0 - window_half_size),
int(_x0 + window_half_size) + 1):
for j in range(int(_y0 - window_half_size),
int(_y0 + window_half_size) + 1):
# ensure that it stays inside image frame
i1 = int(max(0, i - self._centerer_mask_half_size))
j1 = int(max(0, j - self._centerer_mask_half_size))
i2 = int(min(shape[1] - 1, i + self._centerer_mask_half_size))
j2 = int(min(shape[0] - 1, j + self._centerer_mask_half_size))
window = image[j1:j2, i1:i2]
# averages in inner and outer regions.
inner = np.ma.masked_array(window, mask=IN_MASK)
outer = np.ma.masked_array(window, mask=OUT_MASK)
inner_avg = np.sum(inner) / self._centerer_in_mask_npix
outer_avg = np.sum(outer) / self._centerer_out_mask_npix
# standard deviation and figure of merit
inner_std = np.std(inner)
outer_std = np.std(outer)
stddev = np.sqrt(inner_std**2 + outer_std**2)
fom = (inner_avg - outer_avg) / stddev
if fom > max_fom:
max_fom = fom
max_i = i
max_j = j
# figure of merit > threshold: update geometry with new coordinates.
if max_fom > threshold:
self.x0 = float(max_i)
self.y0 = float(max_j)
if verbose:
log.info(f'Found center at x0 = {self.x0:5.1f}, '
f'y0 = {self.y0:5.1f}')
else:
if verbose:
log.info('Result is below the threshold -- keeping the '
'original coordinates.')
[docs]
def radius(self, angle):
"""
Calculate the polar radius for a given polar angle.
Parameters
----------
angle : float
The polar angle (radians).
Returns
-------
radius : float
The polar radius (pixels).
"""
return (self.sma * (1.0 - self.eps)
/ np.sqrt(((1.0 - self.eps) * np.cos(angle))**2
+ (np.sin(angle))**2))
[docs]
def initialize_sector_geometry(self, phi):
"""
Initialize geometry attributes associated with an elliptical
sector at the given polar angle ``phi``.
This function computes:
* the four vertices that define the elliptical sector on the
pixel array.
* the sector area (saved in the ``sector_area`` attribute)
* the sector angular width (saved in ``sector_angular_width``
attribute)
Parameters
----------
phi : float
The polar angle (radians) where the sector is located.
Returns
-------
x, y : 1D `~numpy.ndarray`
The x and y coordinates of each vertex as 1D arrays.
"""
# These polar radii bound the region between the inner
# and outer ellipses that define the sector.
sma1, sma2 = self.bounding_ellipses()
eps_ = 1.0 - self.eps
# polar vector at one side of the elliptical sector
self._phi1 = phi - self.sector_angular_width / 2.0
r1 = (sma1 * eps_ / math.sqrt((eps_ * math.cos(self._phi1))**2
+ (math.sin(self._phi1))**2))
r2 = (sma2 * eps_ / math.sqrt((eps_ * math.cos(self._phi1))**2
+ (math.sin(self._phi1))**2))
# polar vector at the other side of the elliptical sector
self._phi2 = phi + self.sector_angular_width / 2.0
r3 = (sma2 * eps_ / math.sqrt((eps_ * math.cos(self._phi2))**2
+ (math.sin(self._phi2))**2))
r4 = (sma1 * eps_ / math.sqrt((eps_ * math.cos(self._phi2))**2
+ (math.sin(self._phi2))**2))
# sector area
sa1 = _area(sma1, self.eps, self._phi1, r1)
sa2 = _area(sma2, self.eps, self._phi1, r2)
sa3 = _area(sma2, self.eps, self._phi2, r3)
sa4 = _area(sma1, self.eps, self._phi2, r4)
self.sector_area = abs((sa3 - sa2) - (sa4 - sa1))
# angular width of sector. It is calculated such that the sectors
# come out with roughly constant area along the ellipse.
self.sector_angular_width = max(min((self._area_factor / (r3 - r4)
/ r4), self._phi_max),
self._phi_min)
# compute the 4 vertices that define the elliptical sector.
vertex_x = np.zeros(shape=4, dtype=float)
vertex_y = np.zeros(shape=4, dtype=float)
# vertices are labelled in counterclockwise sequence
vertex_x[0:2] = np.array([r1, r2]) * math.cos(self._phi1 + self.pa)
vertex_x[2:4] = np.array([r4, r3]) * math.cos(self._phi2 + self.pa)
vertex_y[0:2] = np.array([r1, r2]) * math.sin(self._phi1 + self.pa)
vertex_y[2:4] = np.array([r4, r3]) * math.sin(self._phi2 + self.pa)
vertex_x += self.x0
vertex_y += self.y0
return vertex_x, vertex_y
[docs]
def bounding_ellipses(self):
"""
Compute the semimajor axis of the two ellipses that bound the
annulus where integrations take place.
Returns
-------
sma1, sma2 : float
The smaller and larger values of semimajor axis length that
define the annulus bounding ellipses.
"""
if self.linear_growth:
a1 = self.sma - self.astep / 2.0
a2 = self.sma + self.astep / 2.0
else:
a1 = self.sma * (1.0 - self.astep / 2.0)
a2 = self.sma * (1.0 + self.astep / 2.0)
return a1, a2
[docs]
def polar_angle_sector_limits(self):
"""
Return the two polar angles that bound the sector.
The two bounding polar angles become available only after
calling the
:meth:`~photutils.isophote.EllipseGeometry.initialize_sector_geometry`
method.
Returns
-------
phi1, phi2 : float
The smaller and larger values of polar angle that bound the
current sector.
"""
return self._phi1, self._phi2
[docs]
def to_polar(self, x, y):
r"""
Return the radius and polar angle in the ellipse coordinate
system given (x, y) pixel image coordinates.
This function takes care of the different definitions for
position angle (PA) and polar angle (phi):
.. math::
-\pi < PA < \pi
0 < phi < 2 \pi
Note that radius can be anything. The solution is not tied to
the semimajor axis length, but to the center position and tilt
angle.
Parameters
----------
x, y : float
The (x, y) image coordinates.
Returns
-------
radius, angle : float
The ellipse radius and polar angle.
"""
# We split in between a scalar version and a
# vectorized version. This is necessary for
# now so we don't pay a heavy speed penalty
# that is incurred when using vectorized code.
# The split in two separate functions helps in
# the profiling analysis: most of the time is
# spent in the scalar function.
if isinstance(x, (int, float)):
return self._to_polar_scalar(x, y)
else:
return self._to_polar_vectorized(x, y)
def _to_polar_scalar(self, x, y):
x1 = x - self.x0
y1 = y - self.y0
radius = x1**2 + y1**2
if radius > 0.0:
radius = math.sqrt(radius)
angle = math.asin(abs(y1) / radius)
else:
radius = 0.0
angle = 1.0
if x1 >= 0.0 and y1 < 0.0:
angle = 2 * np.pi - angle
elif x1 < 0.0 and y1 >= 0.0:
angle = np.pi - angle
elif x1 < 0.0 and y1 < 0.0:
angle = np.pi + angle
pa1 = self.pa
if self.pa < 0.0:
pa1 = self.pa + 2 * np.pi
angle = angle - pa1
if angle < 0.0:
angle = angle + 2 * np.pi
return radius, angle
def _to_polar_vectorized(self, x, y):
x1 = np.atleast_2d(x) - self.x0
y1 = np.atleast_2d(y) - self.y0
radius = x1**2 + y1**2
angle = np.ones(radius.shape)
imask = (radius > 0.0)
radius[imask] = np.sqrt(radius[imask])
angle[imask] = np.arcsin(np.abs(y1[imask]) / radius[imask])
radius[~imask] = 0.0
angle[~imask] = 1.0
idx = (x1 >= 0.0) & (y1 < 0.0)
angle[idx] = 2 * np.pi - angle[idx]
idx = (x1 < 0.0) & (y1 >= 0.0)
angle[idx] = np.pi - angle[idx]
idx = (x1 < 0.0) & (y1 < 0.0)
angle[idx] = np.pi + angle[idx]
pa1 = self.pa
if self.pa < 0.0:
pa1 = self.pa + 2 * np.pi
angle = angle - pa1
angle[angle < 0] += 2 * np.pi
return radius, angle
[docs]
def update_sma(self, step):
"""
Calculate an updated value for the semimajor axis, given the
current value and the step value.
The step value must be managed by the caller to support both
modes: grow outwards and shrink inwards.
Parameters
----------
step : float
The step value.
Returns
-------
sma : float
The new semimajor axis length.
"""
if self.linear_growth:
sma = self.sma + step
else:
sma = self.sma * (1.0 + step)
return sma
[docs]
def reset_sma(self, step):
"""
Change the direction of semimajor axis growth, from outwards to
inwards.
Parameters
----------
step : float
The current step value.
Returns
-------
sma, new_step : float
The new semimajor axis length and the new step value to
initiate the shrinking of the semimajor axis length. This is
the step value that should be used when calling the
:meth:`~photutils.isophote.EllipseGeometry.update_sma`
method.
"""
if self.linear_growth:
sma = self.sma - step
step = -step
else:
aux = 1.0 / (1.0 + step)
sma = self.sma * aux
step = aux - 1.0
return sma, step