__author__ = "sibirrer"
# description of the polar shapelets in potential space
import numpy as np
import scipy.special
import math
import lenstronomy.Util.param_util as param_util
from lenstronomy.LensModel.Profiles.base_profile import LensProfileBase
__all__ = ["PolarShapelets"]
[docs]
class PolarShapelets(LensProfileBase):
"""This class contains the function and the derivatives of the Singular Isothermal
Sphere."""
param_names = ["coeffs", "beta", "center_x", "center_y"]
lower_limit_default = {"coeffs": [0], "beta": 0, "center_x": -100, "center_y": -100}
upper_limit_default = {
"coeffs": [100],
"beta": 100,
"center_x": 100,
"center_y": 100,
}
[docs]
def __init__(self):
n = 10
self.poly = [[[] for i in range(n)] for j in range(n)]
for i in range(0, n):
for j in range(0, n):
self.poly[i][j] = scipy.special.genlaguerre(i, j)
super(PolarShapelets, self).__init__()
[docs]
def function(self, x, y, coeffs, beta, center_x=0, center_y=0):
shapelets = self._createShapelet(coeffs)
r, phi = param_util.cart2polar(x, y, center_x=center_x, center_y=center_y)
f_ = self._shapeletOutput(r, phi, beta, shapelets)
return f_
[docs]
def derivatives(self, x, y, coeffs, beta, center_x=0, center_y=0):
"""Returns df/dx and df/dy of the function."""
shapelets = self._createShapelet(coeffs)
r, phi = param_util.cart2polar(x, y, center_x=center_x, center_y=center_y)
alpha1_shapelets, alpha2_shapelets = self._alphaShapelets(shapelets, beta)
f_x = self._shapeletOutput(r, phi, beta, alpha1_shapelets)
f_y = self._shapeletOutput(r, phi, beta, alpha2_shapelets)
return f_x, f_y
[docs]
def hessian(self, x, y, coeffs, beta, center_x=0, center_y=0):
"""Returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx,
d^f/dy^2."""
shapelets = self._createShapelet(coeffs)
r, phi = param_util.cart2polar(x, y, center_x=center_x, center_y=center_y)
kappa_shapelets = self._kappaShapelets(shapelets, beta)
gamma1_shapelets, gamma2_shapelets = self._gammaShapelets(shapelets, beta)
kappa_value = self._shapeletOutput(r, phi, beta, kappa_shapelets)
gamma1_value = self._shapeletOutput(r, phi, beta, gamma1_shapelets)
gamma2_value = self._shapeletOutput(r, phi, beta, gamma2_shapelets)
f_xx = kappa_value + gamma1_value
f_xy = gamma2_value
f_yy = kappa_value - gamma1_value
return f_xx, f_xy, f_xy, f_yy
def _createShapelet(self, coeff):
"""Returns a shapelet array out of the coefficients *a, up to order l.
:param num_l: order of shapelets
:type num_l: int.
:param coeff: shapelet coefficients
:type coeff: floats
:returns: complex array
:raises: AttributeError, KeyError
"""
n_coeffs = len(coeff)
num_l = self._get_num_l(n_coeffs)
shapelets = np.zeros((num_l + 1, num_l + 1), "complex")
nl = 0
k = 0
i = 0
while i < len(coeff):
if i % 2 == 0:
shapelets[nl][k] += coeff[i] / 2.0
shapelets[k][nl] += coeff[i] / 2.0
if k == nl:
nl += 1
k = 0
i += 1
continue
else:
k += 1
i += 1
continue
else:
shapelets[nl][k] += 1j * coeff[i] / 2.0
shapelets[k][nl] -= 1j * coeff[i] / 2.0
i += 1
return shapelets
def _shapeletOutput(self, r, phi, beta, shapelets):
"""Returns the the numerical values of a set of shapelets at polar coordinates.
:param shapelets: set of shapelets [l=,r=,a_lr=]
:type shapelets: array of size (n,3)
:param coordPolar: set of coordinates in polar units
:type coordPolar: array of size (n,2)
:returns: array of same size with coords [r,phi]
:raises: AttributeError, KeyError
"""
if (
type(r) == float
or type(r) == int
or type(r) == type(np.float64(1))
or len(r) <= 1
):
values = 0.0
else:
values = np.zeros(len(r), "complex")
for nl in range(0, len(shapelets)): # sum over different shapelets
for nr in range(0, len(shapelets)):
value = shapelets[nl][nr] * self._chi_lr(r, phi, nl, nr, beta)
values += value
return values.real
def _chi_lr(self, r, phi, nl, nr, beta):
"""Computes the generalized polar basis function in the convention of
Massey&Refregier eqn 8.
:param nl: left basis
:type nl: int
:param nr: right basis
:type nr: int
:param beta: beta --the characteristic scale typically choosen to be close to
the size of the object.
:type beta: float.
:param coord: coordinates [r,phi]
:type coord: array(n,2)
:returns: values at positions of coordinates.
:raises: AttributeError, KeyError
"""
m = int((nr - nl).real)
n = int((nr + nl).real)
p = int((n - abs(m)) / 2)
p2 = int((n + abs(m)) / 2)
q = int(abs(m))
if p % 2 == 0: # if p is even
prefac = 1
else:
prefac = -1
prefactor = (
prefac
/ beta ** (abs(m) + 1)
* np.sqrt(math.factorial(p) / (np.pi * math.factorial(p2)))
)
poly = self.poly[p][q]
return (
prefactor
* r**q
* poly((r / beta) ** 2)
* np.exp(-((r / beta) ** 2) / 2)
* np.exp(-1j * m * phi)
)
def _kappaShapelets(self, shapelets, beta):
"""Calculates the convergence kappa given lensing potential shapelet
coefficients (laplacian/2) :param shapelets: set of shapelets [l=,r=,a_lr=]
:type shapelets: array of size (n,3) :returns: set of kappa shapelets.
:raises: AttributeError, KeyError
"""
output = np.zeros((len(shapelets) + 1, len(shapelets) + 1), "complex")
for nl in range(0, len(shapelets)):
for nr in range(0, len(shapelets)):
a_lr = shapelets[nl][nr]
if nl > 0:
output[nl - 1][nr + 1] += a_lr * np.sqrt(nl * (nr + 1)) / 2
if nr > 0:
output[nl - 1][nr - 1] += a_lr * np.sqrt(nl * nr) / 2
output[nl + 1][nr + 1] += a_lr * np.sqrt((nl + 1) * (nr + 1)) / 2
if nr > 0:
output[nl + 1][nr - 1] += a_lr * np.sqrt((nl + 1) * nr) / 2
return output / beta**2
def _alphaShapelets(self, shapelets, beta):
"""Calculates the deflection angles given lensing potential shapelet
coefficients (laplacian/2) :param shapelets: set of shapelets [l=,r=,a_lr=]
:type shapelets: array of size (n,3) :returns: set of alpha shapelets.
:raises: AttributeError, KeyError
"""
output_x = np.zeros((len(shapelets) + 1, len(shapelets) + 1), "complex")
output_y = np.zeros((len(shapelets) + 1, len(shapelets) + 1), "complex")
for nl in range(0, len(shapelets)):
for nr in range(0, len(shapelets)):
a_lr = shapelets[nl][nr]
output_x[nl][nr + 1] -= a_lr * np.sqrt(nr + 1) / 2
output_y[nl][nr + 1] -= a_lr * np.sqrt(nr + 1) / 2 * 1j
output_x[nl + 1][nr] -= a_lr * np.sqrt(nl + 1) / 2
output_y[nl + 1][nr] += a_lr * np.sqrt(nl + 1) / 2 * 1j
if nl > 0:
output_x[nl - 1][nr] += a_lr * np.sqrt(nl) / 2
output_y[nl - 1][nr] -= a_lr * np.sqrt(nl) / 2 * 1j
if nr > 0:
output_x[nl][nr - 1] += a_lr * np.sqrt(nr) / 2
output_y[nl][nr - 1] += a_lr * np.sqrt(nr) / 2 * 1j
return output_x / beta, output_y / beta # attention complex numbers!!!!
def _gammaShapelets(self, shapelets, beta):
"""Calculates the shear gamma given lensing potential shapelet coefficients
:param shapelets: set of shapelets [l=,r=,a_lr=] :type shapelets: array of size
(n,3) :returns: set of alpha shapelets.
:raises: AttributeError, KeyError
"""
output_x = np.zeros((len(shapelets) + 2, len(shapelets) + 2), "complex")
output_y = np.zeros((len(shapelets) + 2, len(shapelets) + 2), "complex")
for nl in range(0, len(shapelets)):
for nr in range(0, len(shapelets)):
a_lr = shapelets[nl][nr]
output_x[nl + 2][nr] += a_lr * np.sqrt((nl + 1) * (nl + 2)) / 2
output_x[nl][nr + 2] += a_lr * np.sqrt((nr + 1) * (nr + 2)) / 2
output_x[nl][nr] += a_lr * (1 - (nr + 1) - (nl + 1))
if nl > 1:
output_x[nl - 2][nr] += a_lr * np.sqrt((nl) * (nl - 1)) / 2
if nr > 1:
output_x[nl][nr - 2] += a_lr * np.sqrt((nr) * (nr - 1)) / 2
output_y[nl + 2][nr] += a_lr * np.sqrt((nl + 1) * (nl + 2)) * 1j / 4
output_y[nl][nr + 2] -= a_lr * np.sqrt((nr + 1) * (nr + 2)) * 1j / 4
if nl > 0:
output_y[nl - 1][nr + 1] += a_lr * np.sqrt((nl) * (nr + 1)) * 1j / 2
if nr > 0:
output_y[nl + 1][nr - 1] -= a_lr * np.sqrt((nr) * (nl + 1)) * 1j / 2
if nl > 1:
output_y[nl - 2][nr] -= a_lr * np.sqrt((nl) * (nl - 1)) * 1j / 4
if nr > 1:
output_y[nl][nr - 2] += a_lr * np.sqrt((nr) * (nr - 1)) * 1j / 4
return (
output_x / beta**2,
output_y / beta**2,
) # attention complex numbers!!!!
@staticmethod
def _get_num_l(n_coeffs):
"""
:param n_coeffs: number of coeffs
:return: number of n_l of order of the shapelets
"""
num_l = int(round((math.sqrt(8 * n_coeffs + 9) - 3) / 2 + 0.499))
return num_l