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lensmodels.py
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lensmodels.py
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from astropy import constants as const
from astropy.constants import c
from astropy import units as u
import astropy.units as units
import numpy as np
from skimage import measure
from scipy.ndimage import map_coordinates
import matplotlib.pyplot as plt
from matplotlib import cm
from astropy.modeling.functional_models import Sersic2D
from scipy.optimize import fsolve, newton,brenth, brentq
class genlen(object):
def __init__(self):
self.initialized=True
def setGrid(self,theta=None, thetax=None, thetay=None, compute_potential=False):
"""
set Grid covering the source plane, and compute lens maps such as convergence,
shear, and deflection angles
:param theta: a numpy array setting the pixel positions along one axis
:param compute_potential: if True, the map of the potential is computed. This is time and memory consuming,
therefore it is False by default.
It is neccessary to set this flag to True if you want to compute time delay surfaces.
:return: the function does not return any value, but the maps will be accessible as
properties:
example:
kappa=genlens.ka
g1=genlens.g1
g2=genlens.g2
a1=genlens.a1
a2=genlens.a2
pot=genlens.potential
"""
# construct a mesh:
if thetax is None or thetay is None:
self.theta1, self.theta2 = np.meshgrid(theta, theta)
self.thetax = theta
self.thetay = theta
else:
self.theta1, self.theta2 = np.meshgrid(thetax, thetay)
if np.round(thetax[1]-thetax[0], 6) == np.round(thetay[1]-thetay[0], 6):
self.thetax = thetax
self.thetay = thetay
else:
raise Exception('thetax and thetay must have the same pixel scale '
+ str(thetax[1]-thetax[0])
+ ', ' + str(thetay[1]-thetay[0]))
self.g1, self.g2 = self.gamma(self.theta1, self.theta2)
self.ka = self.kappa(self.theta1, self.theta2)
self.a1,self.a2=self.angle(self.theta1, self.theta2)
self.pot = self.potential(self.theta1, self.theta2)
self.size1 = np.max(self.thetax) - np.min(self.thetax)
self.size2 = np.max(self.thetay) - np.min(self.thetay)
self.nray1 = len(self.thetax)
self.nray2 = len(self.thetay)
self.pixel_scale = self.thetax[1]-self.thetax[0]
#self.conv_fact_time = \
# ((1. + self.zl) / c.to(u.km / u.s) *
# (self.dl * self.ds / self.dls).to(u.km)).to(u.d) * \
# (np.pi / 180.0 / 3600.) ** 2
def tancl(self):
lambdat=1.0-self.ka-np.sqrt(self.g1*self.g1+self.g2*self.g2)
cl = measure.find_contours(lambdat, 0.0)
for i in range(len(cl)):
cl[i][:,1] = cl[i][:,1]*self.pixel_scale - self.size1/2.0
cl[i][:,0] = cl[i][:,0]*self.pixel_scale - self.size2/2.0
return cl
def radcl(self):
lambdar=1.0-self.ka+np.sqrt(self.g1*self.g1+self.g2*self.g2)
cl = measure.find_contours(lambdar, 0.0)
for i in range(len(cl)):
cl[i][:,1] = cl[i][:,1]*self.pixel_scale - self.size1/2.0
cl[i][:,0] = cl[i][:,0]*self.pixel_scale - self.size2/2.0
return cl
def crit2cau(self,cl):
thetac2, thetac1 = np.array(cl[:,0]), np.array(cl[:,1])
#thetac1_ = ((thetac1+self.size1/2.0)/self.pixel_scale).astype(int)
#thetac2_ = ((thetac2+self.size1/2.0)/self.pixel_scale).astype(int)
#ac1 = map_coordinates(self.a1, [[thetac2_], [thetac1_]], order=1, prefilter=True)
#ac2 = map_coordinates(self.a2, [[thetac2_], [thetac1_]], order=1, prefilter=True)
a1, a2 = self.angle(thetac1,thetac2)
betac1 = thetac1 - a1#ac1[0]
betac2 = thetac2 - a2#ac2[0]
cau = np.zeros(np.array(cl).shape)
cau[:,0], cau[:,1] = betac2, betac1
return cau
def combinewith(self,gl):
if np.round(gl.pixel_scale,6) != np.round(self.pixel_scale,6)\
or gl.size1 != self.size1 \
or gl.size2 != self.size2:
raise Exception('Incompatible sizes of deflectors (conbinewith)')
self.ka=self.ka+gl.ka
self.g1=self.g1+gl.g1
self.g2=self.g2+gl.g2
self.a1=self.a1+gl.a1
self.a2=self.a2+gl.a2
class sie(genlen):
def __init__(self, co, **kwargs):
self.computed_potential = False
if ('zl' in kwargs):
self.zl = kwargs['zl']
else:
self.zl = 0.3
if ('zs' in kwargs):
self.zs = kwargs['zs']
else:
self.zs = 1.5
if ('theta_c' in kwargs):
self.theta_c = kwargs['theta_c']
else:
self.theta_c = 0.0
if ('pa' in kwargs):
self.pa = kwargs['pa'] - np.pi/2.
else:
self.pa = 0.0 - np.pi/2.
if ('q' in kwargs):
self.q = kwargs['q']
else:
self.q = 1.0
if ('sigma0' in kwargs):
self.sigma0 = kwargs['sigma0']
else:
self.sigma0 = 200.0
if ('x1' in kwargs):
self.x1 = kwargs['x1']
else:
self.x1 = 0.0
if ('x2' in kwargs):
self.x2 = kwargs['x2']
else:
self.x2 = 0.0
if self.theta_c < 1e-4:
self.lens_type = 'SIE'
else:
self.lens_type = 'NIE'
self.co = co
self.dl = co.angular_diameter_distance(self.zl)
self.ds = co.angular_diameter_distance(self.zs)
self.dls = co.angular_diameter_distance_z1z2(self.zl, self.zs)
self.conv_fact_time = \
((1. + self.zl) / c.to(u.km / u.s) *
(self.dl * self.ds / self.dls).to(u.km)).to(u.d) * \
(np.pi / 180.0 / 3600.) ** 2
# @property
def bsie(self):
conv = 180.0 / np.pi * 3600.0 # radians to arcsec
return conv * 4.0 * np.pi * self.sigma0 ** 2 / const.c.to('km/s').value ** 2 * self.dls.value / self.ds.value / np.sqrt(
self.q)
def sfunc(self):
return (self.theta_c / np.sqrt(self.q))
def kappa(self, theta1__, theta2__):
theta1 = (theta1__ - self.x1)
theta2 = (theta2__ - self.x2)
theta1_ = theta1 * np.sin(self.pa) - theta2 * np.cos(self.pa)
theta2_ = theta1 * np.cos(self.pa) + theta2 * np.sin(self.pa)
kappa_ = self.bsie() / 2 * (1.0 / np.sqrt(self.sfunc() ** 2 + theta1_ ** 2 + theta2_ ** 2 / self.q ** 2))
return kappa_
def angle(self, theta1__, theta2__):
theta1 = theta1__ - self.x1
theta2 = theta2__ - self.x2
theta1_ = theta1 * np.sin(self.pa) - theta2 * np.cos(self.pa)
theta2_ = theta1 * np.cos(self.pa) + theta2 * np.sin(self.pa)
psi = np.sqrt(self.q ** 2 * (self.sfunc() ** 2 + theta1_ ** 2) + theta2_ ** 2)
if (self.q < 1):
alphax = self.bsie() * \
self.q / np.sqrt(1 - self.q ** 2) * \
np.arctan(np.sqrt(1 - self.q ** 2) * theta1_ / (psi + self.sfunc()))
alphay = self.bsie() * \
self.q / np.sqrt(1 - self.q ** 2) * \
np.arctanh(np.sqrt(1 - self.q ** 2) * theta2_ / (psi + self.q ** 2 * self.sfunc()))
elif (self.q == 1):
alphax = self.bsie() * theta1_ / (psi + self.sfunc())
alphay = self.bsie() * theta2_ / (psi + self.sfunc())
else:
print('q cannot be larger than 1')
alphax_ = alphax * np.sin(self.pa) + alphay * np.cos(self.pa)
alphay_ = -alphax * np.cos(self.pa) + alphay * np.sin(self.pa)
return (alphax_, alphay_)
def potential(self, theta1__, theta2__):
theta1 = theta1__ - self.x1
theta2 = theta2__ - self.x2
theta1_ = theta1 * np.sin(self.pa) - theta2 * np.cos(self.pa)
theta2_ = theta1 * np.cos(self.pa) + theta2 * np.sin(self.pa)
psi = np.sqrt(self.q ** 2 * (self.sfunc() ** 2 + theta1_ ** 2) + theta2_ ** 2)
if (self.q < 1):
alphax = self.bsie() * \
self.q / np.sqrt(1 - self.q ** 2) * \
np.arctan(np.sqrt(1 - self.q ** 2) * theta1_ / (psi + self.sfunc()))
alphay = self.bsie() * \
self.q / np.sqrt(1 - self.q ** 2) * \
np.arctanh(np.sqrt(1 - self.q ** 2) * theta2_ / (psi + self.q ** 2 * self.sfunc()))
elif (self.q == 1):
alphax = self.bsie() * theta1_ / (psi + self.sfunc())
alphay = self.bsie() * theta2_ / (psi + self.sfunc())
else:
print('q cannot be larger than 1')
if (np.abs(self.theta_c)>0.0):
pot = theta1_*alphax+ \
theta2_*alphay+ \
self.bsie()*self.q*self.sfunc()*np.log((1.+self.q)*self.sfunc()/
np.sqrt((psi+self.sfunc())**2+(1.-self.q**2)*
theta1_**2))
else:
pot = theta1_*alphax+theta2_*alphay
return pot
def psi11(self, theta1_, theta2_):
psi = np.sqrt(self.q ** 2 * (self.sfunc() ** 2 + theta1_ ** 2) + theta2_ ** 2)
den = (1.0 + self.q ** 2) * self.sfunc() ** 2 + 2.0 * psi * self.sfunc() + theta1_ ** 2 + theta2_ ** 2
psi11_ = self.bsie() * self.q / psi * (
self.q ** 2 * self.sfunc() ** 2 + theta2_ ** 2 + self.sfunc() * psi) / den
return (psi11_)
def psi22(self, theta1_, theta2_):
psi = np.sqrt(self.q ** 2 * (self.sfunc() ** 2 + theta1_ ** 2) + theta2_ ** 2)
den = (1.0 + self.q ** 2) * self.sfunc() ** 2 + 2.0 * psi * self.sfunc() + theta1_ ** 2 + theta2_ ** 2
psi22_ = self.bsie() * self.q / psi * (self.sfunc() ** 2 + theta1_ ** 2 + self.sfunc() * psi) / den
return (psi22_)
def psi12(self, theta1_, theta2_):
psi = np.sqrt(self.q ** 2 * (self.sfunc() ** 2 + theta1_ ** 2) + theta2_ ** 2)
den = (1.0 + self.q ** 2) * self.sfunc() ** 2 + 2.0 * psi * self.sfunc() + theta1_ ** 2 + theta2_ ** 2
psi12_ = -self.bsie() * self.q / psi * (theta1_ * theta2_) / den
return (psi12_)
def gamma(self, theta1__, theta2__):
theta1 = theta1__ - self.x1
theta2 = theta2__ - self.x2
theta1_ = theta1 * np.sin(self.pa) - theta2 * np.cos(self.pa)
theta2_ = theta1 * np.cos(self.pa) + theta2 * np.sin(self.pa)
psi11 = self.psi11(theta1_, theta2_)
psi22 = self.psi22(theta1_, theta2_)
psi12 = self.psi12(theta1_, theta2_)
psi11_ = psi11 * np.sin(self.pa) ** 2 + 2.0 * psi12 * np.sin(self.pa) * np.cos(self.pa) + psi22 * np.cos(
self.pa) ** 2
psi22_ = psi11 * np.cos(self.pa) ** 2 - 2.0 * psi12 * np.sin(self.pa) * np.cos(self.pa) + \
psi22 * np.sin(self.pa) ** 2
psi12_ = -psi11 * np.sin(self.pa) * np.cos(self.pa) + psi12 * (np.sin(self.pa) ** 2 - np.cos(self.pa) ** 2) + \
psi22 * np.sin(self.pa) * np.cos(self.pa)
gammax = 0.5 * (psi11_ - psi22_)
gammay = psi12_
return (gammax, gammay)
def mu(self, theta1__, theta2__):
kappa = self.kappa(theta1__, theta2__)
gammax,gammay = self.gamma(theta1__, theta2__)
gamma=np.sqrt(gammax**2+gammay**2)
mu = 1.0/abs((1-kappa)**2-gamma**2)
return mu
class piemd(genlen):
def __init__(self, co, **kwargs):
self.computed_potential = False
self.lens_type = 'PIEMD'
if ('zl' in kwargs):
self.zl = kwargs['zl']
else:
self.zl = 0.3
if ('zs' in kwargs):
self.zs = kwargs['zs']
else:
self.zs = 1.5
self.co=co
self.dl = co.angular_diameter_distance(self.zl)
self.ds = co.angular_diameter_distance(self.zs)
self.dls = co.angular_diameter_distance_z1z2(self.zl, self.zs)
self.conv_fact_time = \
((1. + self.zl) / c.to(u.km / u.s) *
(self.dl * self.ds / self.dls).to(u.km)).to(u.d) * \
(np.pi / 180.0 / 3600.) ** 2
if ('theta_c' in kwargs):
self.theta_c = kwargs['theta_c']
else:
self.theta_c = 0.0
if ('pa' in kwargs):
self.pa = kwargs['pa']
else:
self.pa = 0.0
if ('q' in kwargs):
self.q = kwargs['q']
else:
self.q = 1.0
if ('sigma0' in kwargs):
self.sigma0 = kwargs['sigma0']
else:
self.sigma0 = 200.0
if ('x1' in kwargs):
self.x1 = kwargs['x1']
else:
self.x1 = 0.0
if ('x2' in kwargs):
self.x2 = kwargs['x2']
else:
self.x2 = 0.0
if ('theta_t' in kwargs):
self.theta_t = kwargs['theta_t']
else:
self.theta_t = self.rt()
self.theta_t = self.theta_t* 1e-3 / self.dl.value * \
180.0 / np.pi * 3600.0
kwargs1 = {'zl': self.zl,
'zs': self.zs,
'sigma0': self.sigma0,
'q': self.q,
'theta_c': self.theta_c,
'pa': self.pa,
'x1': self.x1,
'x2': self.x2}
kwargs2 = {'zl': self.zl,
'zs': self.zs,
'sigma0': self.sigma0,
'q': self.q,
'theta_c': self.theta_t,
'pa': self.pa,
'x1': self.x1,
'x2': self.x2}
self.s1 = sie(co, **kwargs1)
self.s2 = sie(co, **kwargs2)
def kappa(self, theta1, theta2):
#isel = (theta1 - self.x1)**2+(theta2 - self.x2)**2 < 5.0*self.theta_t
kappa_ = self.s1.kappa(theta1, theta2) - self.s2.kappa(theta1, theta2)
return kappa_
def potential(self, theta1, theta2):
pot = self.s1.potential(theta1, theta2) - self.s2.potential(theta1, theta2)
return pot
def angle(self, theta1, theta2):
alphax_1, alphay_1 = self.s1.angle(theta1, theta2)
alphax_2, alphay_2 = self.s2.angle(theta1, theta2)
return (alphax_1 - alphax_2, alphay_1 - alphay_2)
def gamma(self, theta1, theta2):
gammax_1, gammay_1 = self.s1.gamma(theta1, theta2)
gammax_2, gammay_2 = self.s2.gamma(theta1, theta2)
gammax = gammax_1 - gammax_2
gammay = gammay_1 - gammay_2
return (gammax, gammay)
def mass(self):
GG=const.G.to(units.km * units.km / units.s / units.s * units.Mpc / units.Msun).value
return(np.pi*self.sigma0**2/GG *self.theta_t*self.dl.value*np.pi/180.0/3600)
def m2Dr(self,r):
GG = const.G.to(units.km * units.km / units.s / units.s * units.Mpc / units.Msun).value
rcut=self.theta_t * self.dl.value * np.pi / 180.0 / 3600
rcore=self.theta_c * self.dl.value * np.pi / 180.0 / 3600
return (np.pi * self.sigma0 ** 2 / GG * rcut / (rcut-rcore) *
(np.sqrt(r**2 + rcore**2) - rcore -
np.sqrt(r**2 + rcut**2) + rcut))
def m3Dr(self,r):
GG = const.G.to(units.km * units.km / units.s / units.s * units.Mpc / units.Msun).value
rcut=self.theta_t * self.dl.value * np.pi / 180.0 / 3600
rcore=self.theta_c * self.dl.value * np.pi / 180.0 / 3600
return(2 * self.sigma0**2 / GG * rcut / (rcut-rcore) *
(rcut*np.arctan(r / rcut) - rcore*np.arctan(r / rcore)))
def vcirc(self,r):
GG = const.G.to(units.km * units.km / units.s / units.s * units.Mpc / units.Msun).value
return(np.sqrt(GG*self.m3Dr(r)/r))
def rt(self):
# best fit sigma-rt relation from Bergamini et al. 2019
r_t = 32.01 * (np.sqrt(3.0 / 2.0) * self.sigma0 / 350.0) ** 2.42
return r_t
def density(self,r_in):
GG = const.G.to(units.km * units.km / units.s / units.s * units.Mpc / units.Msun).value
rcut = self.theta_t * self.dl.value * np.pi / 180.0 / 3600
rcore = self.theta_c * self.dl.value * np.pi / 180.0 / 3600
r = r_in * self.dl.value * np.pi / 180.0 / 3600
den = self.sigma0**2.0 / 2.0 / np.pi / GG * (rcut + rcore) / (rcore ** 2 * rcut) / \
(1.0 + (r/rcore)**2)/(1.0+(r/rcut)**2)
return den
def surf_density(self,r_in):
GG = const.G.to(units.km * units.km / units.s / units.s * units.Mpc / units.Msun).value
rcut = self.theta_t * self.dl.value * np.pi / 180.0 / 3600
rcore = self.theta_c * self.dl.value * np.pi / 180.0 / 3600
r = r_in * self.dl.value * np.pi / 180.0 / 3600
den = self.sigma0**2.0 / 2.0 / GG * rcut/(rcut - rcore) * \
(1.0/(rcore**2 + r**2)-1.0/(rcut**2+r**2))
return den
class gensrc(object):
def __init__(self) -> object:
self.initialized=True
def ray_trace(self):
px = self.df.pixel_scale
x1pix = (self.x1 - self.df.thetax[0]) / px
x2pix = (self.x2 - self.df.thetay[0]) / px
if len(x1pix.shape) > 1:
x1pix[:, -1] = np.round(x1pix[:, -1], 0)
x2pix[-1, :] = np.round(x2pix[-1, :], 0)
x1pix[:, 0] = np.round(x1pix[:, 0], 0)
x2pix[0, :] = np.round(x2pix[0, :], 0)
else:
x1pix[-1] = np.round(x1pix[-1], 0)
x2pix[-1] = np.round(x2pix[-1], 0)
x1pix[0] = np.round(x1pix[0], 0)
x2pix[0] = np.round(x2pix[0], 0)
a1 = map_coordinates(self.df.a1,
[x2pix, x1pix], order=1, prefilter=True)
a2 = map_coordinates(self.df.a2,
[x2pix, x1pix], order=1, prefilter=True)
y1 = (self.x1 - a1 * self.rescf) # y1 coordinates on the source plane
y2 = (self.x2 - a2 * self.rescf) # y2 coordinates on the source plane
return (y1, y2)
class pointsrc(gensrc):
def __init__(self, size=100.0, sizex=None, sizey=None, Npix=100, gl=None,
save_unlensed=False, fast=False, **kwargs):
if ('ys1' in kwargs):
self.ys1 = kwargs['ys1']
else:
self.ys1 = 0.0
if ('ys2' in kwargs):
self.ys2 = kwargs['ys2']
else:
self.ys2 = 0.0
if ('flux' in kwargs):
self.flux = kwargs['flux']
else:
self.flux = 100.0
if ('zs' in kwargs):
self.zs = kwargs['zs']
else:
self.zs = 1.0
self.rescf = 1.0
if gl != None:
if self.zs != gl.zs:
if self.zs > gl.zl:
ds = gl.co.angular_diameter_distance(self.zs).value
dls = gl.co.angular_diameter_distance_z1z2(gl.zl,self.zs).value
self.rescf=dls/ds*gl.ds/gl.dls
else:
self.rescf = 0.0
self.N = Npix
self.df = gl
if not fast:
# define the pixel coordinates
if sizex == None or sizey == None:
self.size = float(size)
pcx = np.linspace(-self.size / 2.0, self.size / 2.0, self.N)
pcy = np.linspace(-self.size / 2.0, self.size / 2.0, self.N)
self.center_frame = [0.,0.]
else:
pcx = np.linspace(sizex[0], sizex[1], self.N)
pcy = np.linspace(sizey[0], sizey[1], self.N)
self.size=sizex[1]-sizex[0]
self.center_frame = [(sizex[0] + sizex[1]) / 2.0, (sizey[0] + sizey[1]) / 2.0]
self.x1, self.x2 = np.meshgrid(pcx, pcy)
self.pixel = self.size / self.N
if self.df == None: # NO LENS
self.image = self.brightness(self.ys1,self.ys2)
if(save_unlensed):
self.image_unlensed = self.image
else: # LENS
if(save_unlensed):
self.image_unlensed = self.brightness(self.ys1,self.ys2)
self.image = np.zeros((self.N, self.N))
self.xi1, self.xi2, self.mui, self.tdi = self.find_images()
for i in range(len(self.xi1)):
image_tmp = self.brightness(self.xi1[i],self.xi2[i], self.mui[i])
self.image=self.image+image_tmp.copy()
else:
self.xi1, self.xi2, self.mui, self.tdi = self.find_images()
def brightness(self,ys1,ys2,mu=1.0):
# convert image positions with respect to the deflector center into
# image positions with respect to the image center
#print (self.center_frame)
ys1 = ys1 - self.center_frame[0]
ys2 = ys2 - self.center_frame[1]
pix = self.size / self.N # LB: this is already defined above as self.pixel...
px = int(ys1/pix + self.N/2.)
py = int(ys2/pix + self.N/2.)
brightness=np.zeros((self.N,self.N))
if ((px >= 0) & (px<self.N) & (py >= 0) & (py < self.N)):
brightness[py,px]=self.flux*mu
return (brightness)
'''
Find the images of a source at (ys1,ys2) by mapping triangles on the lens plane into
triangles in the source plane. Then search for the triangles which contain the source.
The image position is then computed by weighing with the distance from the vertices of the
triangle on the lens plane
'''
def find_images(self):
if (self.df.lens_type == 'SIE'):
"""
If the lens is singular, use a faster method to find the multiple images than triangle mapping
"""
x,phi = self.phi_ima(checkplot=False,verbose=False)
xi=x*np.cos(phi)
yi=x*np.sin(phi)
mui = self.df.mu(xi,yi)
poti = self.df.potential(xi,yi)
tdi = (0.5*((self.ys1-xi)**2+(self.ys2-yi)**2)-poti)*self.df.conv_fact_time.value
tdi = tdi-tdi.min()
return xi, yi, mui, tdi
# map the source position in pixels onto the deflector grid
y1s = self.ys1/self.df.pixel_scale + len(self.df.thetax) / 2.0
y2s = self.ys2/self.df.pixel_scale + len(self.df.thetay) / 2.0
# ray-trace the deflector grid onto the source plane
y1 = self.df.theta1 - self.df.a1*self.rescf
y2 = self.df.theta2 - self.df.a2*self.rescf
# convert to pixel units
xray = y1.copy() / self.df.pixel_scale + len(self.df.thetax) / 2.0
yray = y2.copy() / self.df.pixel_scale + len(self.df.thetay) / 2.0
# shift the maps by one pixel
xray1 = np.roll(xray, 1, axis=1)
xray2 = np.roll(xray1, 1, axis=0)
xray3 = np.roll(xray2, -1, axis=1)
yray1 = np.roll(yray, 1, axis=1)
yray2 = np.roll(yray1, 1, axis=0)
yray3 = np.roll(yray2, -1, axis=1)
"""
For each pixel on the LENS plane, build two triangles. By means of
ray-tracing these are mapped onto the source plane into other two
triangles. Compute the distances of the vertices of the triangles on
the SOURCE plane from the source and check using cross-products if the
source is inside one of the two triangles.
"""
# l1=((yray1-yray2)*(ys1-xray2)+(xray2-xray1)*(ys2-yray2))/((yray1-yray2)*(xray-xray2)+(xray2-xray1)*(yray-yray2))
x1 = y1s - xray
y1 = y2s - yray
x2 = y1s - xray1
y2 = y2s - yray1
x3 = y1s - xray2
y3 = y2s - yray2
x4 = y1s - xray3
y4 = y2s - yray3
prod12 = x1 * y2 - x2 * y1
prod23 = x2 * y3 - x3 * y2
prod31 = x3 * y1 - x1 * y3
prod13 = -prod31
prod34 = x3 * y4 - x4 * y3
prod41 = x4 * y1 - x1 * y4
image = np.zeros(xray.shape)
image[((np.sign(prod12) == np.sign(prod23)) & (np.sign(prod23) == np.sign(prod31)))] = 1
image[((np.sign(prod13) == np.sign(prod34)) & (np.sign(prod34) == np.sign(prod41)))] = 2
# In the following, the choices 'image == 1' and 'image == 2' stand for
# upper and lower triangles (or viceversa).
# first kind of images (first triangle)
images1 = np.argwhere(image == 1)
xi_images_ = images1[:, 1]
yi_images_ = images1[:, 0]
xi_images = xi_images_[(xi_images_ > 0) & (yi_images_ > 0)]
yi_images = yi_images_[(xi_images_ > 0) & (yi_images_ > 0)]
# compute the weights
w = np.array([1. / np.sqrt(x1[xi_images, yi_images] ** 2 + y1[xi_images, yi_images] ** 2),
1. / np.sqrt(x2[xi_images, yi_images] ** 2 + y2[xi_images, yi_images] ** 2),
1. / np.sqrt(x3[xi_images, yi_images] ** 2 + y3[xi_images, yi_images] ** 2)])
xif1, yif1 = self.refineImagePositions(xi_images, yi_images, w, 1)
# second kind of images
images1 = np.argwhere(image == 2)
xi_images_ = images1[:, 1]
yi_images_ = images1[:, 0]
xi_images = xi_images_[(xi_images_ > 0) & (yi_images_ > 0)]
yi_images = yi_images_[(xi_images_ > 0) & (yi_images_ > 0)]
# compute the weights
w = np.array([1. / np.sqrt(x1[xi_images, yi_images] ** 2 + y1[xi_images, yi_images] ** 2),
1. / np.sqrt(x3[xi_images, yi_images] ** 2 + y3[xi_images, yi_images] ** 2),
1. / np.sqrt(x4[xi_images, yi_images] ** 2 + y4[xi_images, yi_images] ** 2)])
xif2, yif2 = self.refineImagePositions(xi_images, yi_images, w, 2)
xi = np.concatenate([xif1, xif2])
yi = np.concatenate([yif1, yif2])
mui=self.mu_image(xi,yi)
poti=self.pot_image(xi,yi)
xi = (xi - 1 - len(self.df.thetax) / 2.0) * self.df.pixel_scale
yi = (yi - 1 - len(self.df.thetay) / 2.0) * self.df.pixel_scale
tdi=(0.5*((self.ys1-xi)**2+(self.ys2-yi)**2)-poti)*self.df.conv_fact_time.value
isel = mui > 1e-2
tdi[isel] = tdi[isel]-tdi[isel].min()
return (xi[isel], yi[isel], mui[isel], tdi[isel])
def refineImagePositions(self, x, y, w, typ):
"""Image positions are computed as weighted means of the positions
of the triangle vertices. The weights are the distances between the
vertices mapped onto the source plane, and the source position."""
if (typ == 2):
xp = np.array([x, x + 1, x + 1])
yp = np.array([y, y, y + 1])
else:
xp = np.array([x, x + 1, x])
yp = np.array([y, y + 1, y + 1])
xi = np.zeros(x.size)
yi = np.zeros(y.size)
for i in range(x.size):
xi[i] = (xp[:, i] / w[:, i]).sum() / (1. / w[:, i]).sum()
yi[i] = (yp[:, i] / w[:, i]).sum() / (1. / w[:, i]).sum()
return (xi, yi)
def mu_image(self,xi1,xi2):
mu = map_coordinates((1.0-self.df.ka)**2-self.df.g1**2-self.df.g2**2,
[xi2-1, xi1-1], order=1, prefilter=True)
return(np.abs(1./mu))
def pot_image(self,xi1,xi2):
pot = map_coordinates(self.df.pot,
[xi2-1, xi1-1], order=1, prefilter=True)
return pot
#### Functions to be used with SIE models
def x_ima(self,phi):
x=self.ys1*np.cos(phi)+self.ys2*np.sin(phi)+(self.psi_tilde(phi+self.pa))
return x
def psi_tilde(self,phi):
if (self.df.q < 1.0):
fp=np.sqrt(1.0-self.df.q**2)
return np.sqrt(self.df.q)/fp*(np.sin(phi-self.df.pa)*np.arcsin(fp*np.sin(phi-self.df.pa))+
np.cos(phi-self.df.pa)*np.arcsinh(fp/self.df.q*np.cos(phi-self.df.pa)))
else:
return(1.0)
def psi(self,x,phi):
psi=x*self.psi_tilde(phi)
return psi
def x_ima(self,y1,y2,phi):
x=y1*np.cos(phi)+y2*np.sin(phi)+(self.psi_tilde(phi+self.df.pa))
return x
def alpha(self,phi):
fp=np.sqrt(1.0-self.df.q**2)
a1=np.sqrt(self.df.q)/fp*np.arcsinh(fp/self.df.q*np.cos(phi))
a2=np.sqrt(self.df.q)/fp*np.arcsin(fp*np.sin(phi))
return a1,a2
def phi_ima(self,checkplot=True,verbose=True):
y1_ = self.ys1 * np.cos(self.df.pa) + self.ys2 * np.sin(self.df.pa)
y2_ = - self.ys1 * np.sin(self.df.pa) + self.ys2 * np.cos(self.df.pa)
y1_= y1_/self.df.bsie()/np.sqrt(self.df.q)
y2_= y2_/self.df.bsie()/np.sqrt(self.df.q)
def phi_func(phi):
a1,a2=self.alpha(phi)
func=(y1_+a1)*np.sin(phi)-(y2_+a2)*np.cos(phi)
return func
U=np.linspace(0.,2.0*np.pi+0.001,100)
c = phi_func(U)
s = np.sign(c)
phi=[]
xphi=[]
for i in range(len(U)-1):
if s[i] + s[i+1] == 0: # opposite signs
u = brentq(phi_func, U[i], U[i+1])
z = phi_func(u)
if np.isnan(z) or abs(z) > 1e-3:
continue
x=self.x_ima(y1_,y2_,u)
if (x>0):
phi.append(u)
xphi.append(x)
if (verbose):
print('found zero at {}'.format(u))
if (x<0):
print ('discarded because x is negative ({})'.format(x))
else:
print ('accepted because x is positive ({})'.format(x))
xphi=np.array(xphi)
phi=np.array(phi)
if (checkplot):
phi_=np.linspace(0.,2.0*np.pi,100)
ax[0].plot(phi_,phi_func(phi_),label=r'$F(\varphi)$')
ax[0].plot(phi_,self.x_ima(y1,y2,phi_),label=r'$x(\varphi)$')
#ax[0].plot(phi_,psi_tilde(phi_,f)-1)
ax[0].plot(phi,phi_func(phi),'o',markersize=8)
ax[0].set_xlabel(r'$\varphi$',fontsize=20)
ax[0].set_ylabel(r'$F(\varphi),x(\varphi)$',fontsize=20)
ax[0].legend(fontsize=16)
return(xphi*self.df.bsie()*np.sqrt(self.df.q),phi+self.df.pa)
class sersic(gensrc):
def __init__(self, size=100.0, Npix=100, gl=None, sizex=None, sizey=None, pcx=None, pcy=None, mask=None, save_unlensed=False, rmaxf=100, **kwargs):
if ('n' in kwargs):
self.n = kwargs['n']
else:
self.n = 4
if ('re' in kwargs):
self.re = kwargs['re']
else:
self.re = 5.0
if ('q' in kwargs):
self.q = kwargs['q']
else:
self.q = 1.0
if ('pa' in kwargs):
self.pa = kwargs['pa']
else:
self.pa = 0.0
if ('ys1' in kwargs):
self.ys1 = kwargs['ys1']
else:
self.ys1 = 0.0
if ('ys2' in kwargs):
self.ys2 = kwargs['ys2']
else:
self.ys2 = 0.0
if ('c' in kwargs):
self.c = kwargs['c']
else:
self.c = 0.0
if ('Ie' in kwargs):
self.Ie = kwargs['Ie']
else:
self.Ie = 100.0
if ('zs' in kwargs):
self.zs = kwargs['zs']
else:
self.zs = 1.0
if gl != None:
if self.zs != gl.zs:
if self.zs > gl.zl:
ds = gl.co.angular_diameter_distance(self.zs).value
dls = gl.co.angular_diameter_distance_z1z2(gl.zl,self.zs).value
self.rescf=dls/ds*gl.ds/gl.dls
else:
self.rescf = 0.0
else:
self.rescf=1.0
else:
self.rescf=1.0
self.df = gl
self.mask = mask
self.save_unlensed = save_unlensed
# define the pixel coordinates
if pcx is not None or pcy is not None:
self.Nx = len(pcx)
self.Ny = len(pcx)
self.sizex = np.max(pcx)-np.min(pcx)
self.sizey = np.max(pcy)-np.min(pcy)
if np.round((pcx[1]-pcx[0]),6) != np.round((pcy[1]-pcy[0]),6):
raise Exception('Pixels of pcx and pcy must have the same dimension')
elif sizex is not None or sizey is not None:
self.N = Npix
pcx = np.linspace(sizex[0], sizex[1], self.N)
pcy = np.linspace(sizey[0], sizey[1], self.N)
self.Nx = len(pcx)
self.Ny = len(pcx)
self.sizex = np.max(pcx)-np.min(pcx)
self.sizey = np.max(pcy)-np.min(pcy)
else:
self.size = size
self.N = Npix
pcx = np.linspace(-self.size / 2.0, self.size / 2.0, self.N)
pcy = np.linspace(-self.size / 2.0, self.size / 2.0, self.N)
self.Nx = Npix
self.Ny = Npix
self.sizex = float(size)
self.sizey = float(size)
self.x1, self.x2 = np.meshgrid(pcx, pcy)
if mask is not None:
image_mask = np.full(self.x1.shape,np.nan)
self.x1, self.x2 = self.x1[self.mask], self.x2[self.mask]
if self.df != None:
y1, y2 = self.ray_trace()
else:
y1, y2 = self.x1, self.x2
self.y1 = y1
self.y2 = y2
self.image = self.brightness(y1, y2, rmaxf)
if (save_unlensed):
self.image_unlensed = self.brightness(self.x1, self.x2, rmaxf=rmaxf)
if self.mask is not None:
image_mask[self.mask] = self.image
self.image = image_mask
def brightness(self, y1, y2, rmaxf):
px = self.sizex / (self.Nx - 1)
brightness = np.zeros_like(y1)
isel = (y1 < rmaxf*self.re + self.ys1) & (y2 < rmaxf*self.re + self.ys2)
s = Sersic2D(amplitude=self.Ie, r_eff=self.re, n=self.n, x_0=self.ys1, y_0=self.ys2,
ellip=np.sqrt(1-self.q**2), theta=self.pa + np.pi/2)
brightness[isel] = s(y1[isel], y2[isel])*px*px
return brightness