def derivatives(self,
x,
y,
theta_E,
e1,
e2,
s_scale,
center_x=0,
center_y=0):
"""
:param x: x-coordinate in image plane
:param y: y-coordinate in image plane
:param theta_E: Einstein radius
:param e1: eccentricity component
:param e2: eccentricity component
:param s_scale: smoothing scale
:param center_x: profile center
:param center_y: profile center
:return: alpha_x, alpha_y
"""
b, s, q, phi_G = self.param_conv(theta_E, e1, e2, s_scale)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f__x, f__y = self.nie_major_axis.derivatives(x__, y__, b, s, q)
# rotate back
f_x, f_y = util.rotate(f__x, f__y, -phi_G)
return f_x, f_y
def derivatives(self, x, y, a, s, e1, e2, center_x, center_y):
"""
:param x: coordinate in image plane (angle)
:param y: coordinate in image plane (angle)
:param a: lensing strength
:param s: core radius
:param e1: eccentricity
:param e2: eccentricity
:param center_x: center of profile
:param center_y: center of profile
:return: deflection in x- and y-direction
"""
phi_q, q = param_util.ellipticity2phi_q(e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_q)
f__x, f__y = self.major_axis_model.derivatives(x__, y__, a, s, q)
# rotate deflections back
f_x, f_y = util.rotate(f__x, f__y, -phi_q)
return f_x, f_y
def derivatives(self, x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0):
"""
returns df/dx and df/dy of the function, calculated as an elliptically distorted deflection angle of the
spherical NFW profile
:param x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param Rs: turn over point in the slope of the NFW profile in angular unit
:param alpha_Rs: deflection (angular units) at projected Rs
:param e1: eccentricity component in x-direction
:param e2: eccentricity component in y-direction
:param center_x: center of halo (in angular units)
:param center_y: center of halo (in angular units)
:return: deflection in x-direction, deflection in y-direction
"""
phi_q, q = param_util.ellipticity2phi_q(e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_q)
f__x, f__y = self.cse_major_axis_set.derivatives(
x__ / Rs, y__ / Rs, self._a_list, self._s_list, q)
# rotate deflections back
f_x, f_y = util.rotate(f__x, f__y, -phi_q)
const = self._normalization(alpha_Rs, Rs, q) / Rs
return const * f_x, const * f_y
def derivatives(self, x, y, theta_E, e1, e2, s_scale, center_x=0, center_y=0):
"""
:param x:
:param y:
:param theta_E:
:param e1:
:param e2:
:param s_scale:
:param center_x:
:param center_y:
:return:
"""
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
theta_E = self._theta_E_q_convert(theta_E, q)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f__x, f__y = self.nie_simple.derivatives(x__, y__, theta_E, s_scale, q)
# rotate back
f_x, f_y = util.rotate(f__x, f__y, -phi_G)
return f_x, f_y
def derivatives(self, x, y, theta_E, theta_c, e1, e2, center_x=0, center_y=0):
"""
:param x: x-coord (in angles)
:param y: y-coord (in angles)
:param theta_E: Einstein radius (in angles)
:param theta_c: core radius (in angles)
:param e1: eccentricity component, x direction(dimensionless)
:param e2: eccentricity component, y direction (dimensionless)
:return: deflection angle (in angles)
"""
theta_E_conv, theta_c_conv, eps, phi_G = self.param_conv(theta_E, theta_c, e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f__x, f__y = self.nie_potential_major_axis.derivatives(x__, y__, theta_E_conv, theta_c_conv, eps)
# rotate back
f_x, f_y = util.rotate(f__x, f__y, -phi_G)
return f_x, f_y
def hessian(self, x, y, theta_E, theta_c, e1, e2, center_x=0, center_y=0):
"""
:param x: x-coord (in angles)
:param y: y-coord (in angles)
:param theta_E: Einstein radius (in angles)
:param theta_c: core radius (in angles)
:param e1: eccentricity component, x direction(dimensionless)
:param e2: eccentricity component, y direction (dimensionless)
:return: hessian matrix (in angles)
"""
theta_E_conv, theta_c_conv, eps, phi_G = self.param_conv(
theta_E, theta_c, e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f__xx, f__yy, f__xy = self.nie_potential_major_axis.hessian(
x__, y__, theta_E_conv, theta_c_conv, eps)
# rotate back
kappa = 1. / 2 * (f__xx + f__yy)
gamma1__ = 1. / 2 * (f__xx - f__yy)
gamma2__ = f__xy
gamma1 = np.cos(2 * phi_G) * gamma1__ - np.sin(2 * phi_G) * gamma2__
gamma2 = +np.sin(2 * phi_G) * gamma1__ + np.cos(2 * phi_G) * gamma2__
f_xx = kappa + gamma1
f_yy = kappa - gamma1
f_xy = gamma2
return f_xx, f_yy, f_xy
def function(self, x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0):
"""
returns elliptically distorted NFW lensing potential
:param x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param Rs: turn over point in the slope of the NFW profile in angular unit
:param alpha_Rs: deflection (angular units) at projected Rs
:param e1: eccentricity component in x-direction
:param e2: eccentricity component in y-direction
:param center_x: center of halo (in angular units)
:param center_y: center of halo (in angular units)
:return: lensing potential
"""
phi_q, q = param_util.ellipticity2phi_q(e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_q)
# potential calculation
f_ = self.cse_major_axis_set.function(x__ / Rs, y__ / Rs, self._a_list,
self._s_list, q)
const = self._normalization(alpha_Rs, Rs, q)
return const * f_
def hessian(self, x, y, theta_E, e1, e2, s_scale, center_x=0, center_y=0):
"""
:param x: x-coordinate in image plane
:param y: y-coordinate in image plane
:param theta_E: Einstein radius
:param e1: eccentricity component
:param e2: eccentricity component
:param s_scale: smoothing scale
:param center_x: profile center
:param center_y: profile center
:return: f_xx, f_xy, f_yx, f_yy
"""
b, s, q, phi_G = self.param_conv(theta_E, e1, e2, s_scale)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f__xx, f__xy, _, f__yy = self.nie_major_axis.hessian(x__, y__, b, s, q)
# rotate back
kappa = 1. / 2 * (f__xx + f__yy)
gamma1__ = 1. / 2 * (f__xx - f__yy)
gamma2__ = f__xy
gamma1 = np.cos(2 * phi_G) * gamma1__ - np.sin(2 * phi_G) * gamma2__
gamma2 = +np.sin(2 * phi_G) * gamma1__ + np.cos(2 * phi_G) * gamma2__
f_xx = kappa + gamma1
f_yy = kappa - gamma1
f_xy = gamma2
return f_xx, f_xy, f_xy, f_yy
def hessian(self, x, y, theta_E, e1, e2, s_scale, center_x=0, center_y=0):
"""
:param x:
:param y:
:param theta_E:
:param e1:
:param e2:
:param s_scale:
:param center_x:
:param center_y:
:return:
"""
theta_E, phi_G, q = self.param_conv(theta_E, e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f__xx, f__yy, f__xy = self.nie_simple.hessian(x__, y__, theta_E, s_scale, q)
# rotate back
kappa = 1./2 * (f__xx + f__yy)
gamma1__ = 1./2 * (f__xx - f__yy)
gamma2__ = f__xy
gamma1 = np.cos(2 * phi_G) * gamma1__ - np.sin(2 * phi_G) * gamma2__
gamma2 = +np.sin(2 * phi_G) * gamma1__ + np.cos(2 * phi_G) * gamma2__
f_xx = kappa + gamma1
f_yy = kappa - gamma1
f_xy = gamma2
return f_xx, f_yy, f_xy
def hessian(self, x, y, sigma0, Rs, e1, e2, 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
"""
phi_q, q = param_util.ellipticity2phi_q(e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_q)
f__xx, f__xy, __, f__yy = self.cse_major_axis_set.hessian(
x__ / Rs, y__ / Rs, self._a_list, self._s_list, q)
# rotate back
kappa = 1. / 2 * (f__xx + f__yy)
gamma1__ = 1. / 2 * (f__xx - f__yy)
gamma2__ = f__xy
gamma1 = np.cos(2 * phi_q) * gamma1__ - np.sin(2 * phi_q) * gamma2__
gamma2 = +np.sin(2 * phi_q) * gamma1__ + np.cos(2 * phi_q) * gamma2__
f_xx = kappa + gamma1
f_yy = kappa - gamma1
f_xy = gamma2
const = self._normalization(sigma0, Rs, q) / Rs**2
return const * f_xx, const * f_xy, const * f_xy, const * f_yy
def hessian(self, x, y, a, s, e1, e2, center_x, center_y):
"""
:param x: coordinate in image plane (angle)
:param y: coordinate in image plane (angle)
:param a: lensing strength
:param s: core radius
:param e1: eccentricity
:param e2: eccentricity
:param center_x: center of profile
:param center_y: center of profile
:return: hessian elements f_xx, f_xy, f_yx, f_yy
"""
phi_q, q = param_util.ellipticity2phi_q(e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_q)
f__xx, f__xy, __, f__yy = self.major_axis_model.hessian(x__, y__, a, s, q)
# rotate back
kappa = 1. / 2 * (f__xx + f__yy)
gamma1__ = 1. / 2 * (f__xx - f__yy)
gamma2__ = f__xy
gamma1 = np.cos(2 * phi_q) * gamma1__ - np.sin(2 * phi_q) * gamma2__
gamma2 = +np.sin(2 * phi_q) * gamma1__ + np.cos(2 * phi_q) * gamma2__
f_xx = kappa + gamma1
f_yy = kappa - gamma1
f_xy = gamma2
return f_xx, f_xy, f_xy, f_yy
def derivatives(self, x, y, sigma0, Rs, e1, e2, center_x=0, center_y=0):
"""
returns df/dx and df/dy of the function (integral of NFW)
"""
phi_q, q = param_util.ellipticity2phi_q(e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_q)
f__x, f__y = self.cse_major_axis_set.derivatives(
x__ / Rs, y__ / Rs, self._a_list, self._s_list, q)
# rotate deflections back
f_x, f_y = util.rotate(f__x, f__y, -phi_q)
const = self._normalization(sigma0, Rs, q) / Rs
return const * f_x, const * f_y
def derivatives(self, x, y, tangential_stretch, curvature, direction,
center_x, center_y):
"""
:param x:
:param y:
:param tangential_stretch: float, stretch of intrinsic source in tangential direction
:param curvature: 1/curvature radius
:param direction: float, angle in radian
:param center_x: center of source in image plane
:param center_y: center of source in image plane
:return:
"""
r = 1 / curvature
# deflection angle to allow for tangential stretch
# (ratio of source position around zero point relative to radius is tangential stretch)
alpha = r * (1 / tangential_stretch + 1)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, direction)
# evaluate
# move x-coordinate to circle intercept with x-axis
if isinstance(x, int) or isinstance(x, float):
if abs(y__) > r:
f__x, f__y = 0, 0
else:
f__x, f__y = self._deflection(y__, r, tangential_stretch)
else:
f__x, f__y = np.zeros_like(x__), np.zeros_like(y__)
_y__ = y__[y__ <= r]
_f__x, _f__y = self._deflection(_y__, r, tangential_stretch)
f__x[y__ <= r] = _f__x
f__y[y__ <= r] = _f__y
# rotate back
f_x, f_y = util.rotate(f__x, f__y, -direction)
return f_x, f_y
def magnification_finite(self,
x_pos,
y_pos,
kwargs_lens,
source_sigma=0.003,
window_size=0.1,
grid_number=100,
polar_grid=False,
aspect_ratio=0.5):
"""
returns the magnification of an extended source with Gaussian light profile
:param x_pos: x-axis positons of point sources
:param y_pos: y-axis position of point sources
:param kwargs_lens: lens model kwargs
:param source_sigma: Gaussian sigma in arc sec in source
:param window_size: size of window to compute the finite flux
:param grid_number: number of grid cells per axis in the window to numerically compute the flux
:return: numerically computed brightness of the sources
"""
mag_finite = np.zeros_like(x_pos)
deltaPix = float(window_size) / grid_number
from lenstronomy.LightModel.Profiles.gaussian import Gaussian
quasar = Gaussian()
x_grid, y_grid = util.make_grid(numPix=grid_number,
deltapix=deltaPix,
subgrid_res=1)
if polar_grid is True:
a = window_size * 0.5
b = window_size * 0.5 * aspect_ratio
ellipse_inds = (x_grid * a**-1)**2 + (y_grid * b**-1)**2 <= 1
x_grid, y_grid = x_grid[ellipse_inds], y_grid[ellipse_inds]
for i in range(len(x_pos)):
ra, dec = x_pos[i], y_pos[i]
center_x, center_y = self._lensModel.ray_shooting(
ra, dec, kwargs_lens)
if polar_grid is True:
theta = np.arctan2(dec, ra)
xcoord, ycoord = util.rotate(x_grid, y_grid, theta)
else:
xcoord, ycoord = x_grid, y_grid
betax, betay = self._lensModel.ray_shooting(
xcoord + ra, ycoord + dec, kwargs_lens)
I_image = quasar.function(betax, betay, 1., source_sigma, center_x,
center_y)
mag_finite[i] = np.sum(I_image) * deltaPix**2
return mag_finite
def hessian(self, x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0):
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
gamma, q = self._param_bounds(gamma, q)
phi_E_new = theta_E * q
#x_shift = x - center_x
#y_shift = y - center_y
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
E = phi_E_new / (((3 - gamma) / 2.)**(1. / (1 - gamma)) * np.sqrt(q))
if E <= 0:
return np.zeros_like(x), np.zeros_like(x), np.zeros_like(
x), np.zeros_like(x)
# E = phi_E
eta = float(-gamma + 3)
#xt1 = np.cos(phi_G)*x_shift+np.sin(phi_G)*y_shift
#xt2 = -np.sin(phi_G)*x_shift+np.cos(phi_G)*y_shift
xt1, xt2 = x__, y__
P2 = xt1**2 + xt2**2 / q**2
if isinstance(P2, int) or isinstance(P2, float):
a = max(0.000001, P2)
else:
a = np.empty_like(P2)
p2 = P2[P2 > 0] #in the SIS regime
a[P2 == 0] = 0.000001
a[P2 > 0] = p2
s2 = 0. # softening
kappa = 1. / eta * (a / E**2)**(eta / 2 - 1) * (
(eta - 2) * (xt1**2 + xt2**2 / q**4) / a + (1 + 1 / q**2))
gamma1_value = 1. / eta * (a / E**2)**(
eta / 2 - 1) * (1 - 1 / q**2 + (eta / 2 - 1) *
(2 * xt1**2 - 2 * xt2**2 / q**4) / a)
gamma2_value = 4 * xt1 * xt2 / q**2 * (1. / 2 - 1 / eta) * (
a / E**2)**(eta / 2 - 2) / E**2
gamma1 = np.cos(2 * phi_G) * gamma1_value - np.sin(
2 * phi_G) * gamma2_value
gamma2 = +np.sin(2 * phi_G) * gamma1_value + np.cos(
2 * phi_G) * gamma2_value
f_xx = kappa + gamma1
f_yy = kappa - gamma1
f_xy = gamma2
return f_xx, f_xy, f_xy, f_yy
def function(self, x, y, sigma0, Rs, e1, e2, center_x=0, center_y=0):
"""
returns double integral of NFW profile
"""
phi_q, q = param_util.ellipticity2phi_q(e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_q)
# potential calculation
f_ = self.cse_major_axis_set.function(x__ / Rs, y__ / Rs, self._a_list,
self._s_list, q)
const = self._normalization(sigma0, Rs, q)
return const * f_
def coord2image_pixel(ra, dec, center_x, center_y, phi_G, scale):
"""
:param ra: angular coordinate
:param dec: angular coordinate
:param center_x: center of image in angular coordinates
:param center_y: center of image in angular coordinates
:param phi_G: rotation angle
:param scale: pixel scale of image
:return: pixel coordinates
"""
ra_ = ra - center_x
dec_ = dec - center_y
x_ = ra_ / scale
y_ = dec_ / scale
x, y = util.rotate(x_, y_, phi_G)
return x, y
def mask_ellipse(x, y, center_x, center_y, a, b, angle):
"""
:param x: x-coordinates of pixels
:param y: y-coordinates of pixels
:param center_x: center of mask
:param center_y: center of mask
:param a: major axis
:param b: minor axis
:param angle: angle of major axis
:return: mask (list of zeros and ones)
"""
x_shift = x - center_x
y_shift = y - center_y
x_rot, y_rot = util.rotate(x_shift, y_shift, angle)
r_ab = x_rot**2 / a**2 + y_rot**2 / b**2
mask = np.empty_like(r_ab)
mask[r_ab > 1] = 0
mask[r_ab <= 1] = 1
return mask
def function(self, x, y, amp, e1, e2, s_scale, center_x=0, center_y=0):
"""
:param x: x-coordinate
:param y: y-coordinate
:param amp: surface brightness normalization
:param e1: eccentricity component
:param e2: eccentricity component
:param s_scale: smoothing scale (square averaged of minor and major axis)
:param center_x: center of profile
:param center_y: center of profile
:return: surface brightness of NIE profile
"""
x_ = x - center_x
y_ = y - center_y
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
s = s_scale * np.sqrt((1 + q ** 2) / (2 * q ** 2))
f_ = amp/2. * (q**2 * (s**2 + x__**2) + y__**2)**(-1./2)
return f_
def function(self, x, y, a, s, e1, e2, center_x, center_y):
"""
:param x: coordinate in image plane (angle)
:param y: coordinate in image plane (angle)
:param a: lensing strength
:param s: core radius
:param e1: eccentricity
:param e2: eccentricity
:param center_x: center of profile
:param center_y: center of profile
:return: lensing potential
"""
phi_q, q = param_util.ellipticity2phi_q(e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_q)
# potential calculation
f_ = self.major_axis_model.function(x__, y__, a, s, q)
return f_
def function(self, x, y, amp, e1, e2, s_scale, center_x=0, center_y=0):
"""
:param x:
:param y:
:param theta_E:
:param e1:
:param e2:
:param s_scale:
:param center_x:
:param center_y:
:return:
"""
phi_G, q = param_util.ellipticity2phi_q(e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f_ = self._nie_simple_function(x__, y__, amp, s_scale, q)
# rotate back
return f_
def function(self, x, y, theta_E, e1, e2, s_scale, center_x=0, center_y=0):
"""
:param x:
:param y:
:param theta_E:
:param e1:
:param e2:
:param s_scale:
:param center_x:
:param center_y:
:return:
"""
theta_E, phi_G, q = self.param_conv(theta_E, e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f_ = self.nie_simple.function(x__, y__, theta_E, s_scale, q)
# rotate back
return f_
def function(self, x, y, theta_E, e1, e2, s_scale, center_x=0, center_y=0):
"""
:param x: x-coordinate in image plane
:param y: y-coordinate in image plane
:param theta_E: Einstein radius
:param e1: eccentricity component
:param e2: eccentricity component
:param s_scale: smoothing scale
:param center_x: profile center
:param center_y: profile center
:return: lensing potential
"""
b, s, q, phi_G = self.param_conv(theta_E, e1, e2, s_scale)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_G)
# evaluate
f_ = self.nie_major_axis.function(x__, y__, b, s, q)
# rotate back
return f_
def hessian(self, x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0):
"""
returns Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy
the calculation is performed as a numerical differential from the deflection field.
Analytical relations are possible.
:param x: angular position (normally in units of arc seconds)
:param y: angular position (normally in units of arc seconds)
:param Rs: turn over point in the slope of the NFW profile in angular unit
:param alpha_Rs: deflection (angular units) at projected Rs
:param e1: eccentricity component in x-direction
:param e2: eccentricity component in y-direction
:param center_x: center of halo (in angular units)
:param center_y: center of halo (in angular units)
:return: d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2
"""
phi_q, q = param_util.ellipticity2phi_q(e1, e2)
# shift
x_ = x - center_x
y_ = y - center_y
# rotate
x__, y__ = util.rotate(x_, y_, phi_q)
f__xx, f__xy, __, f__yy = self.cse_major_axis_set.hessian(
x__ / Rs, y__ / Rs, self._a_list, self._s_list, q)
# rotate back
kappa = 1. / 2 * (f__xx + f__yy)
gamma1__ = 1. / 2 * (f__xx - f__yy)
gamma2__ = f__xy
gamma1 = np.cos(2 * phi_q) * gamma1__ - np.sin(2 * phi_q) * gamma2__
gamma2 = +np.sin(2 * phi_q) * gamma1__ + np.cos(2 * phi_q) * gamma2__
f_xx = kappa + gamma1
f_yy = kappa - gamma1
f_xy = gamma2
const = self._normalization(alpha_Rs, Rs, q) / Rs**2
return const * f_xx, const * f_xy, const * f_xy, const * f_yy
def test_derivatives(self):
x = np.array([1])
y = np.array([2])
mu_r = 1.
mu_t = 10.
# positive parity
parity = 1
############
# rotation 1
############
phi_G = np.pi
f_x, f_y = self.const_mag.derivatives(x, y, mu_r, mu_t, parity, phi_G)
# rotate
x__, y__ = util.rotate(x, y, phi_G)
# evaluate
f__x, f__y = self.const_mag.derivatives(x__, y__, mu_r, mu_t, parity,
0.0)
# rotate back
f_x_rot, f_y_rot = util.rotate(f__x, f__y, -phi_G)
# compare
npt.assert_almost_equal(f_x, f_x_rot, decimal=4)
npt.assert_almost_equal(f_y, f_y_rot, decimal=4)
############
# rotation 2
############
phi_G = np.pi / 3.
f_x, f_y = self.const_mag.derivatives(x, y, mu_r, mu_t, parity, phi_G)
# rotate
x__, y__ = util.rotate(x, y, phi_G)
# evaluate
f__x, f__y = self.const_mag.derivatives(x__, y__, mu_r, mu_t, parity,
0.0)
# rotate back
f_x_rot, f_y_rot = util.rotate(f__x, f__y, -phi_G)
# compare
npt.assert_almost_equal(f_x, f_x_rot, decimal=4)
npt.assert_almost_equal(f_y, f_y_rot, decimal=4)
#===========================================================
# negative parity
parity = -1
############
# rotation 1
############
phi_G = np.pi
f_x, f_y = self.const_mag.derivatives(x, y, mu_r, mu_t, parity, phi_G)
# rotate
x__, y__ = util.rotate(x, y, phi_G)
# evaluate
f__x, f__y = self.const_mag.derivatives(x__, y__, mu_r, mu_t, parity,
0.0)
# rotate back
f_x_rot, f_y_rot = util.rotate(f__x, f__y, -phi_G)
# compare
npt.assert_almost_equal(f_x, f_x_rot, decimal=4)
npt.assert_almost_equal(f_y, f_y_rot, decimal=4)
############
# rotation 2
############
phi_G = np.pi / 3.
f_x, f_y = self.const_mag.derivatives(x, y, mu_r, mu_t, parity, phi_G)
# rotate
x__, y__ = util.rotate(x, y, phi_G)
# evaluate
f__x, f__y = self.const_mag.derivatives(x__, y__, mu_r, mu_t, parity,
0.0)
# rotate back
f_x_rot, f_y_rot = util.rotate(f__x, f__y, -phi_G)
# compare
npt.assert_almost_equal(f_x, f_x_rot, decimal=4)
npt.assert_almost_equal(f_y, f_y_rot, decimal=4)
def magnification_finite_adaptive(self, x_image, y_image, source_x, source_y, kwargs_lens,
source_fwhm_parsec, z_source,
cosmo=None, grid_resolution=None,
grid_radius_arcsec=None, axis_ratio=0.5,
tol=0.001, step_size=0.05,
use_largest_eigenvalue=True,
source_light_model='SINGLE_GAUSSIAN',
dx=None, dy=None, size_scale=None, amp_scale=None,
fixed_aperture_size=False):
"""
This method computes image magnifications with a finite-size background source assuming a Gaussian or a
double Gaussian source light profile. It can be much faster that magnification_finite for lens models with many
deflectors and a compact source. This is because most pixels in a rectangular window around a lensed
image of a compact source do not map onto the source, and therefore don't contribute to the integrated flux in
the image plane.
Rather than ray tracing through a rectangular grid, this routine accelerates the computation of image
magnifications with finite-size sources by ray tracing through an elliptical region oriented such that
tracks the surface brightness of the lensed image. The aperture size is initially quite small,
and increases in size until the flux inside of it (and hence the magnification) converges. The orientation of
the elliptical aperture is computed from the magnification tensor evaluated at the image coordinate.
If for whatever reason you prefer a circular aperture to the elliptical approximation using the hessian
eigenvectors, you can just set axis_ratio = 1.
To use the eigenvalues of the hessian matrix to estimate the optimum axis ratio, set axis_ratio = 0.
The default settings for the grid resolution and ray tracing window size work well for sources with fwhm between
0.5 - 100 pc.
:param x_image: a list or array of x coordinates [units arcsec]
:param y_image: a list or array of y coordinates [units arcsec]
:param source_x: float, source position
:param source_y: float, source position
:param kwargs_lens: keyword arguments for the lens model
:param source_fwhm_parsec: the size of the background source [units parsec]
:param z_source: the source redshift
:param cosmo: (optional) an instance of astropy.cosmology; if not specified, a default cosmology will be used
:param grid_resolution: the grid resolution in units arcsec/pixel; if not specified, an appropriate value will
be estimated from the source size
:param grid_radius_arcsec: (optional) the size of the ray tracing region in arcsec; if not specified, an appropriate value
will be estimated from the source size
:param axis_ratio: the axis ratio of the ellipse used for ray tracing; if axis_ratio = 0, then the eigenvalues
the hessian matrix will be used to estimate an appropriate axis ratio. Be warned: if the image is highly
magnified it will tend to curve out of the resulting ellipse
:param tol: tolerance for convergence in the magnification
:param step_size: sets the increment for the successively larger ray tracing windows
:param use_largest_eigenvalue: bool; if True, then the major axis of the ray tracing ellipse region
will be aligned with the eigenvector corresponding to the largest eigenvalue of the hessian matrix
:param source_light_model: the model for backgourn source light; currently implemented are 'SINGLE_GAUSSIAN' and
'DOUBLE_GAUSSIAN'.
:param dx: used with source model 'DOUBLE_GAUSSIAN', the offset of the second source light profile from the first
[arcsec]
:param dy: used with source model 'DOUBLE_GAUSSIAN', the offset of the second source light profile from the first
[arcsec]
:param size_scale: used with source model 'DOUBLE_GAUSSIAN', the size of the second source light profile relative
to the first
:param amp_scale: used with source model 'DOUBLE_GAUSSIAN', the peak brightness of the second source light profile
relative to the first
:param fixed_aperture_size: bool, if True the flux is computed inside a fixed aperture size with radius
grid_radius_arcsec
:return: an array of image magnifications
"""
grid_x_0, grid_y_0, source_model, kwargs_source, grid_resolution, grid_radius_arcsec = setup_mag_finite(cosmo,
self._lensModel,
grid_radius_arcsec,
grid_resolution,
source_fwhm_parsec,
source_light_model,
z_source,
source_x,
source_y,
dx, dy,
amp_scale,
size_scale)
grid_x_0, grid_y_0 = grid_x_0.ravel(), grid_y_0.ravel()
minimum_magnification = 1e-5
magnifications = []
for xi, yi in zip(x_image, y_image):
if axis_ratio == 1:
grid_r = np.hypot(grid_x_0, grid_y_0)
else:
w1, w2, v11, v12, v21, v22 = self.hessian_eigenvectors(xi, yi, kwargs_lens)
_v = [np.array([v11, v12]), np.array([v21, v22])]
_w = [abs(w1), abs(w2)]
if use_largest_eigenvalue:
idx = int(np.argmax(_w))
else:
idx = int(np.argmin(_w))
v = _v[idx]
rotation_angle = np.arctan(v[1] / v[0]) - np.pi / 2
grid_x, grid_y = util.rotate(grid_x_0, grid_y_0, rotation_angle)
if axis_ratio == 0:
sort = np.argsort(_w)
q = _w[sort[0]] / _w[sort[1]]
grid_r = np.hypot(grid_x, grid_y / q).ravel()
else:
grid_r = np.hypot(grid_x, grid_y / axis_ratio).ravel()
flux_array = np.zeros_like(grid_x_0)
step = step_size * grid_radius_arcsec
r_min = 0
if fixed_aperture_size:
r_max = grid_radius_arcsec
else:
r_max = step
magnification_current = 0.
while True:
flux_array = self._magnification_adaptive_iteration(flux_array, xi, yi, grid_x_0, grid_y_0, grid_r,
r_min, r_max, self._lensModel, kwargs_lens,
source_model, kwargs_source)
new_magnification = np.sum(flux_array) * grid_resolution ** 2
diff = abs(new_magnification - magnification_current) / new_magnification
if r_max >= grid_radius_arcsec:
break
elif diff < tol and new_magnification > minimum_magnification:
break
else:
r_min += step
r_max += step
magnification_current = new_magnification
magnifications.append(new_magnification)
return np.array(magnifications)
def test_function(self):
y = np.array([1., 2])
x = np.array([0., 0.])
mu_r = 1.
mu_t = 10.
# positive parity
parity = 1
############
# rotation 1
############
phi_G = np.pi
values = self.const_mag.function(x, y, mu_r, mu_t, parity, phi_G)
delta_pot = values[1] - values[0]
# rotate
x__, y__ = util.rotate(x, y, phi_G)
# evaluate
f_ = self.const_mag.function(x__, y__, mu_r, mu_t, parity, 0.0)
# rotate back
delta_pot_rot = f_[1] - f_[0]
# compare
npt.assert_almost_equal(delta_pot, delta_pot_rot, decimal=4)
############
# rotation 2
############
phi_G = np.pi / 3.
values = self.const_mag.function(x, y, mu_r, mu_t, parity, phi_G)
delta_pot = values[1] - values[0]
# rotate
x__, y__ = util.rotate(x, y, phi_G)
# evaluate
f_ = self.const_mag.function(x__, y__, mu_r, mu_t, parity, 0.0)
# rotate back
delta_pot_rot = f_[1] - f_[0]
# compare
npt.assert_almost_equal(delta_pot, delta_pot_rot, decimal=4)
#===========================================================
# negative parity
parity = -1
############
# rotation 1
############
phi_G = np.pi
values = self.const_mag.function(x, y, mu_r, mu_t, parity, phi_G)
delta_pot = values[1] - values[0]
# rotate
x__, y__ = util.rotate(x, y, phi_G)
# evaluate
f_ = self.const_mag.function(x__, y__, mu_r, mu_t, parity, 0.0)
# rotate back
delta_pot_rot = f_[1] - f_[0]
# compare
npt.assert_almost_equal(delta_pot, delta_pot_rot, decimal=4)
############
# rotation 2
############
phi_G = np.pi / 3.
values = self.const_mag.function(x, y, mu_r, mu_t, parity, phi_G)
delta_pot = values[1] - values[0]
# rotate
x__, y__ = util.rotate(x, y, phi_G)
# evaluate
f_ = self.const_mag.function(x__, y__, mu_r, mu_t, parity, 0.0)
# rotate back
delta_pot_rot = f_[1] - f_[0]
# compare
npt.assert_almost_equal(delta_pot, delta_pot_rot, decimal=4)
def test_hessian(self):
x = np.array([1])
y = np.array([2])
mu_r = 1.
mu_t = 10.
# positive parity
parity = 1
############
# rotation 1
############
phi_G = np.pi
f_xx, f_xy, f_yx, f_yy = self.const_mag.hessian(
x, y, mu_r, mu_t, parity, phi_G)
# rotate
x__, y__ = util.rotate(x, y, phi_G)
# evaluate
f__xx, f__xy, f__yx, f__yy = self.const_mag.hessian(
x__, y__, mu_r, mu_t, parity, 0.0)
# rotate back
kappa = 1. / 2 * (f__xx + f__yy)
gamma1__ = 1. / 2 * (f__xx - f__yy)
gamma2__ = f__xy
gamma1 = np.cos(2 * phi_G) * gamma1__ - np.sin(2 * phi_G) * gamma2__
gamma2 = +np.sin(2 * phi_G) * gamma1__ + np.cos(2 * phi_G) * gamma2__
f_xx_rot = kappa + gamma1
f_yy_rot = kappa - gamma1
f_xy_rot = gamma2
# compare
npt.assert_almost_equal(f_xx, f_xx_rot, decimal=4)
npt.assert_almost_equal(f_yy, f_yy_rot, decimal=4)
npt.assert_almost_equal(f_xy, f_xy_rot, decimal=4)
npt.assert_almost_equal(f_xy, f_yx, decimal=8)
############
# rotation 2
############
phi_G = np.pi / 3.
f_xx, f_xy, f_yx, f_yy = self.const_mag.hessian(
x, y, mu_r, mu_t, parity, phi_G)
# rotate
x__, y__ = util.rotate(x, y, phi_G)
# evaluate
f__xx, f__xy, f__yx, f__yy = self.const_mag.hessian(
x__, y__, mu_r, mu_t, parity, 0.0)
# rotate back
kappa = 1. / 2 * (f__xx + f__yy)
gamma1__ = 1. / 2 * (f__xx - f__yy)
gamma2__ = f__xy
gamma1 = np.cos(2 * phi_G) * gamma1__ - np.sin(2 * phi_G) * gamma2__
gamma2 = +np.sin(2 * phi_G) * gamma1__ + np.cos(2 * phi_G) * gamma2__
f_xx_rot = kappa + gamma1
f_yy_rot = kappa - gamma1
f_xy_rot = gamma2
# compare
npt.assert_almost_equal(f_xx, f_xx_rot, decimal=4)
npt.assert_almost_equal(f_yy, f_yy_rot, decimal=4)
npt.assert_almost_equal(f_xy, f_xy_rot, decimal=4)
npt.assert_almost_equal(f_yx, f_xy_rot, decimal=4)
def magnification_finite_adaptive(self,
x_image,
y_image,
source_x,
source_y,
kwargs_lens,
source_fwhm_parsec,
z_source,
cosmo=None,
grid_resolution=None,
grid_radius_arcsec=None,
axis_ratio=0.5,
tol=0.001,
step_size=0.05,
use_largest_eigenvalue=True):
"""
This method computes image magnifications with a finite-size background source assuming a Gaussian
source light profile. It can be much faster that magnification_finite for lens models with many
deflectors and a relatively compact source. This is because most pixels in a rectangular window around a lensed
image of a compact source will contain zero flux, and therefore don't contribute to the image brightness.
Rather than ray tracing through a rectangular grid, this routine accelerates the computation of image
magnifications with finite-size sources by ray tracing through an elliptical aperture oriented such that
it resembles the surface brightness of the lensed image itself. The aperture size is initially quite small,
and increases in size until the flux inside of it (and hence the magnification) converges. The orientation of
the elliptical aperture is computed from the magnification tensor at the image coordinate.
If for whatever reason you prefer a circular aperture to the elliptical approximation using the hessian eigenvectors,
you can just set axis_ratio = 1.
To use the eigenvalues of the hessian matrix to estimate the optimum axis ratio, set axis_ratio = 0.
The default settings for the grid resolution and ray tracing window size work well for sources with fwhm between
0.5 - 100 pc.
:param x_image: a list or array of x coordinates [units arcsec]
:param y_image: a list or array of y coordinates [units arcsec]
:param kwargs_lens: keyword arguments for the lens model
:param source_fwhm_parsec: the size of the background source [units parsec]
:param z_source: the source redshift
:param cosmo: (optional) an instance of astropy.cosmology; if not specified, a default cosmology will be used
:param grid_resolution: the grid resolution in units arcsec/pixel; if not specified, an appropriate value will
be estimated from the source size
:param grid_radius_arcsec: (optional) the size of the ray tracing region in arcsec; if not specified, an appropriate value
will be estimated from the source size
:param axis_ratio: the axis ratio of the ellipse used for ray tracing; if axis_ratio = 0, then the eigenvalues
the hessian matrix will be used to estimate an appropriate axis ratio. Be warned: if the image is highly
magnified it will tend to curve out of the resulting ellipse
:param tol: tolerance for convergence in the magnification
:param step_size: sets the increment for the successively larger ray tracing windows
:param use_largest_eigenvalue: bool; if True, then the major axis of the ray tracing ellipse region
will be aligned with the eigenvector corresponding to the largest eigenvalue of the hessian matrix
:return: an array of image magnifications
"""
if cosmo is None:
cosmo = self._lensModel.cosmo
if grid_radius_arcsec is None:
grid_radius_arcsec = auto_raytracing_grid_size(source_fwhm_parsec)
if grid_resolution is None:
grid_resolution = auto_raytracing_grid_resolution(
source_fwhm_parsec)
pc_per_arcsec = 1000 / cosmo.arcsec_per_kpc_proper(z_source).value
source_fwhm_arcsec = source_fwhm_parsec / pc_per_arcsec
source_sigma_arcsec = fwhm2sigma(source_fwhm_arcsec)
kwargs_source = [{
'amp': 1.,
'center_x': source_x,
'center_y': source_y,
'sigma': source_sigma_arcsec
}]
source_model = LightModel(['GAUSSIAN'])
npix = int(2 * grid_radius_arcsec / grid_resolution)
_grid_x = np.linspace(-grid_radius_arcsec, grid_radius_arcsec, npix)
_grid_y = np.linspace(-grid_radius_arcsec, grid_radius_arcsec, npix)
magnifications = []
minimum_magnification = 1e-4
grid_x_0, grid_y_0 = np.meshgrid(_grid_x, _grid_y)
grid_x_0, grid_y_0 = grid_x_0.ravel(), grid_y_0.ravel()
for xi, yi in zip(x_image, y_image):
w1, w2, v11, v12, v21, v22 = self.hessian_eigenvectors(
xi, yi, kwargs_lens)
_v = [np.array([v11, v12]), np.array([v21, v22])]
_w = [abs(w1), abs(w2)]
if use_largest_eigenvalue:
idx = int(np.argmax(_w))
else:
idx = int(np.argmin(_w))
v = _v[idx]
rotation_angle = np.arctan(v[1] / v[0]) - np.pi / 2
grid_x, grid_y = util.rotate(grid_x_0, grid_y_0, rotation_angle)
if axis_ratio == 0:
sort = np.argsort(_w)
q = _w[sort[0]] / _w[sort[1]]
grid_r = np.hypot(grid_x, grid_y / q).ravel()
else:
grid_r = np.hypot(grid_x, grid_y / axis_ratio).ravel()
flux_array = np.zeros_like(grid_x_0)
step = step_size * grid_radius_arcsec
r_min = 0
r_max = step
magnification_current = 0.
while True:
flux_array = self._magnification_adaptive_iteration(
flux_array, xi, yi, grid_x_0, grid_y_0, grid_r, r_min,
r_max, self._lensModel, kwargs_lens, source_model,
kwargs_source)
new_magnification = np.sum(flux_array) * grid_resolution**2
diff = abs(new_magnification -
magnification_current) / new_magnification
# the sqrt(2) will allow this algorithm to fill up the entire square window
if r_max > np.sqrt(2) * grid_radius_arcsec:
break
elif diff < tol and new_magnification > minimum_magnification:
break
else:
r_min += step
r_max += step
magnification_current = new_magnification
magnifications.append(new_magnification)
return np.array(magnifications)
