def test_sersic(self):
from lenstronomy.LensModel.Profiles.sersic import Sersic
from lenstronomy.LightModel.Profiles.sersic import Sersic as SersicLight
sersic_lens = Sersic()
sersic_light = SersicLight()
kwargs_light = {
'n_sersic': 2,
'R_sersic': 0.5,
'I0_sersic': 1,
'center_x': 0,
'center_y': 0
}
kwargs_lens = {
'n_sersic': 2,
'R_sersic': 0.5,
'k_eff': 1,
'center_x': 0,
'center_y': 0
}
deltaPix = 0.01
numPix = 1000
x_grid, y_grid = util.make_grid(numPix=numPix, deltapix=deltaPix)
x_grid2d = util.array2image(x_grid)
y_grid2d = util.array2image(y_grid)
f_xx, f_yy, _ = sersic_lens.hessian(x_grid, y_grid, **kwargs_lens)
f_x, f_y = sersic_lens.derivatives(x_grid, y_grid, **kwargs_lens)
f_x = util.array2image(f_x)
kappa = util.array2image((f_xx + f_yy) / 2.)
f_x_num, f_y_num = convergence_integrals.deflection_from_kappa_grid(
kappa, deltaPix)
x1, y1 = 500, 550
x0, y0 = int(numPix / 2.), int(numPix / 2.)
npt.assert_almost_equal(f_x[x1, y1], f_x_num[x1, y1], decimal=2)
f_num = convergence_integrals.potential_from_kappa_grid(
kappa, deltaPix)
f_ = sersic_lens.function(x_grid2d[x1, y1], y_grid2d[x1, y1],
**kwargs_lens)
f_00 = sersic_lens.function(x_grid2d[x0, y0], y_grid2d[x0, y0],
**kwargs_lens)
npt.assert_almost_equal(f_ - f_00,
f_num[x1, y1] - f_num[x0, y0],
decimal=2)
class TestSersic(object):
"""
tests the Gaussian methods
"""
def setup(self):
self.sersic_2 = SersicEllipseKappa()
self.sersic = Sersic()
self.sersic_light = Sersic_light()
def test_function(self):
x = 1
y = 2
n_sersic = 2.
R_sersic = 1.
k_eff = 0.2
values = self.sersic.function(x, y, n_sersic, R_sersic, k_eff)
npt.assert_almost_equal(values, 1.0272982586319199, decimal=10)
x = np.array([0])
y = np.array([0])
values = self.sersic.function(x, y, n_sersic, R_sersic, k_eff)
npt.assert_almost_equal(values[0], 0., decimal=9)
x = np.array([2, 3, 4])
y = np.array([1, 1, 1])
values = self.sersic.function(x, y, n_sersic, R_sersic, k_eff)
npt.assert_almost_equal(values[0], 1.0272982586319199, decimal=10)
npt.assert_almost_equal(values[1], 1.3318743892966658, decimal=10)
npt.assert_almost_equal(values[2], 1.584299393114988, decimal=10)
def test_derivatives(self):
x = np.array([1])
y = np.array([2])
n_sersic = 2.
R_sersic = 1.
k_eff = 0.2
f_x, f_y = self.sersic.derivatives(x, y, n_sersic, R_sersic, k_eff)
f_x2, f_y2 = self.sersic_2.derivatives(x, y, n_sersic, R_sersic, k_eff,
0, 0.00000001)
assert f_x[0] == 0.16556078301997193
assert f_y[0] == 0.33112156603994386
npt.assert_almost_equal(f_x2[0], f_x[0])
npt.assert_almost_equal(f_y2[0], f_y[0])
x = np.array([0])
y = np.array([0])
f_x, f_y = self.sersic.derivatives(x, y, n_sersic, R_sersic, k_eff)
f_x2, f_y2 = self.sersic_2.derivatives(x, y, n_sersic, R_sersic, k_eff,
0, 0.00000001)
assert f_x[0] == 0
assert f_y[0] == 0
npt.assert_almost_equal(f_x2[0], f_x[0])
npt.assert_almost_equal(f_y2[0], f_y[0])
x = np.array([1, 3, 4])
y = np.array([2, 1, 1])
values = self.sersic.derivatives(x, y, n_sersic, R_sersic, k_eff)
values2 = self.sersic_2.derivatives(x, y, n_sersic, R_sersic, k_eff, 0,
0.00000001)
assert values[0][0] == 0.16556078301997193
assert values[1][0] == 0.33112156603994386
assert values[0][1] == 0.2772992378623737
assert values[1][1] == 0.092433079287457892
npt.assert_almost_equal(values2[0][0], values[0][0])
npt.assert_almost_equal(values2[1][0], values[1][0])
npt.assert_almost_equal(values2[0][1], values[0][1])
npt.assert_almost_equal(values2[1][1], values[1][1])
values2 = self.sersic_2.derivatives(0.3, -0.2, n_sersic, R_sersic,
k_eff, 0, 0.00000001)
values = self.sersic.derivatives(0.3, -0.2, n_sersic, R_sersic, k_eff,
0, 0.00000001)
npt.assert_almost_equal(values2[0], values[0])
npt.assert_almost_equal(values2[1], values[1])
def test_differentails(self):
x_, y_ = 1., 1
n_sersic = 2.
R_sersic = 1.
k_eff = 0.2
r = np.sqrt(x_**2 + y_**2)
d_alpha_dr = self.sersic.d_alpha_dr(x_, y_, n_sersic, R_sersic, k_eff)
alpha = self.sersic.alpha_abs(x_, y_, n_sersic, R_sersic, k_eff)
f_xx_ = d_alpha_dr * calc_util.d_r_dx(
x_, y_) * x_ / r + alpha * calc_util.d_x_diffr_dx(x_, y_)
f_yy_ = d_alpha_dr * calc_util.d_r_dy(
x_, y_) * y_ / r + alpha * calc_util.d_y_diffr_dy(x_, y_)
f_xy_ = d_alpha_dr * calc_util.d_r_dy(
x_, y_) * x_ / r + alpha * calc_util.d_x_diffr_dy(x_, y_)
f_xx = (d_alpha_dr / r - alpha / r**2) * y_**2 / r + alpha / r
f_yy = (d_alpha_dr / r - alpha / r**2) * x_**2 / r + alpha / r
f_xy = (d_alpha_dr / r - alpha / r**2) * x_ * y_ / r
npt.assert_almost_equal(f_xx, f_xx_, decimal=10)
npt.assert_almost_equal(f_yy, f_yy_, decimal=10)
npt.assert_almost_equal(f_xy, f_xy_, decimal=10)
def test_hessian(self):
x = np.array([1])
y = np.array([2])
n_sersic = 2.
R_sersic = 1.
k_eff = 0.2
f_xx, f_xy, f_yx, f_yy = self.sersic.hessian(x, y, n_sersic, R_sersic,
k_eff)
assert f_xx[0] == 0.1123170666045793
npt.assert_almost_equal(f_yy[0], -0.047414082641598576, decimal=10)
npt.assert_almost_equal(f_xy[0], -0.10648743283078525, decimal=10)
npt.assert_almost_equal(f_xy, f_yx, decimal=5)
x = np.array([1, 3, 4])
y = np.array([2, 1, 1])
values = self.sersic.hessian(x, y, n_sersic, R_sersic, k_eff)
assert values[0][0] == 0.1123170666045793
npt.assert_almost_equal(values[3][0],
-0.047414082641598576,
decimal=10)
npt.assert_almost_equal(values[1][0], -0.10648743283078525, decimal=10)
npt.assert_almost_equal(values[0][1],
-0.053273787681591328,
decimal=10)
npt.assert_almost_equal(values[3][1], 0.076243427402007985, decimal=10)
npt.assert_almost_equal(values[1][1],
-0.048568955656349749,
decimal=10)
f_xx2, f_xy2, f_yx2, f_yy2 = self.sersic_2.hessian(
x, y, n_sersic, R_sersic, k_eff, 0.0000001, 0)
npt.assert_almost_equal(f_xx2, values[0])
npt.assert_almost_equal(f_yy2, values[3], decimal=6)
npt.assert_almost_equal(f_xy2, values[1], decimal=6)
npt.assert_almost_equal(f_yx2, values[2], decimal=6)
def test_alpha_abs(self):
x = 1.
dr = 0.0000001
n_sersic = 2.5
R_sersic = .5
k_eff = 0.2
alpha_abs = self.sersic.alpha_abs(x, 0, n_sersic, R_sersic, k_eff)
f_dr = self.sersic.function(x + dr, 0, n_sersic, R_sersic, k_eff)
f_ = self.sersic.function(x, 0, n_sersic, R_sersic, k_eff)
alpha_abs_num = -(f_dr - f_) / dr
npt.assert_almost_equal(alpha_abs_num, alpha_abs, decimal=3)
def test_dalpha_dr(self):
x = 1.
dr = 0.0000001
n_sersic = 1.
R_sersic = .5
k_eff = 0.2
d_alpha_dr = self.sersic.d_alpha_dr(x, 0, n_sersic, R_sersic, k_eff)
alpha_dr = self.sersic.alpha_abs(x + dr, 0, n_sersic, R_sersic, k_eff)
alpha = self.sersic.alpha_abs(x, 0, n_sersic, R_sersic, k_eff)
d_alpha_dr_num = (alpha_dr - alpha) / dr
npt.assert_almost_equal(d_alpha_dr, d_alpha_dr_num, decimal=3)
def test_mag_sym(self):
"""
:return:
"""
r = 2.
angle1 = 0.
angle2 = 1.5
x1 = r * np.cos(angle1)
y1 = r * np.sin(angle1)
x2 = r * np.cos(angle2)
y2 = r * np.sin(angle2)
n_sersic = 4.5
R_sersic = 2.5
k_eff = 0.8
f_xx1, f_xy1, f_yx1, f_yy1 = self.sersic.hessian(
x1, y1, n_sersic, R_sersic, k_eff)
f_xx2, f_xy2, f_yx2, f_yy2 = self.sersic.hessian(
x2, y2, n_sersic, R_sersic, k_eff)
kappa_1 = (f_xx1 + f_yy1) / 2
kappa_2 = (f_xx2 + f_yy2) / 2
npt.assert_almost_equal(kappa_1, kappa_2, decimal=10)
A_1 = (1 - f_xx1) * (1 - f_yy1) - f_xy1 * f_yx1
A_2 = (1 - f_xx2) * (1 - f_yy2) - f_xy2 * f_yx2
npt.assert_almost_equal(A_1, A_2, decimal=10)
def test_convergernce(self):
"""
test the convergence and compares it with the original Sersic profile
:return:
"""
x = np.array([0, 0, 0, 0, 0])
y = np.array([0.5, 1, 1.5, 2, 2.5])
n_sersic = 4.5
R_sersic = 2.5
k_eff = 0.2
f_xx, f_xy, f_yx, f_yy = self.sersic.hessian(x, y, n_sersic, R_sersic,
k_eff)
kappa = (f_xx + f_yy) / 2.
assert kappa[0] > 0
flux = self.sersic_light.function(x,
y,
amp=1.,
R_sersic=R_sersic,
n_sersic=n_sersic)
flux /= flux[0]
kappa /= kappa[0]
npt.assert_almost_equal(flux[1], kappa[1], decimal=5)
xvalues = np.linspace(0.5, 3., 100)
e1, e2 = 0.4, 0.
q = ellipticity2phi_q(e1, e2)[1]
kappa_ellipse = self.sersic_2.projected_mass(xvalues, 0, q, n_sersic,
R_sersic, k_eff)
fxx, _, _, fyy = self.sersic_2.hessian(xvalues, 0, n_sersic, R_sersic,
k_eff, e1, e2)
npt.assert_almost_equal(kappa_ellipse, 0.5 * (fxx + fyy), decimal=5)
def test_sersic_util(self):
n = 1.
Re = 2.
k, bn = self.sersic.k_bn(n, Re)
Re_new = self.sersic.k_Re(n, k)
assert Re == Re_new
class TestSersicEllipseGaussDec(object):
"""
This class tests the methods for Gauss-decomposed elliptic Sersic
convergence.
"""
def setup(self):
self.sersic_gauss = SersicEllipseGaussDec()
self.sersic_light = SersicElliptic()
self.sersic_sphere = Sersic()
def test_function(self):
"""
Test the potential function of Gauss-decomposed elliptical Sersic by
asserting that the numerical derivative of the computed potential
matches with the analytical derivative values.
:return:
:rtype:
"""
k_eff = 1.
R_sersic = 1.
n_sersic = 1.
e1 = 0.2
e2 = 0.2
center_x = 0.
center_y = 0.
diff = 1.e-6
n = 5
xs = np.linspace(0.5 * R_sersic, 2. * R_sersic, n)
ys = np.linspace(0.5 * R_sersic, 2. * R_sersic, n)
for x, y in zip(xs, ys):
func = self.sersic_gauss.function(x, y, e1=e1, e2=e2,
center_x=center_x,
center_y=center_y,
n_sersic=n_sersic,
R_sersic=R_sersic,
k_eff=k_eff
)
func_dx = self.sersic_gauss.function(x+diff, y, e1=e1, e2=e2,
center_x=center_x,
center_y=center_y,
n_sersic=n_sersic,
R_sersic=R_sersic,
k_eff=k_eff
)
func_dy = self.sersic_gauss.function(x, y+diff, e1=e1, e2=e2,
center_x=center_x,
center_y=center_y,
n_sersic=n_sersic,
R_sersic=R_sersic,
k_eff=k_eff
)
f_x_num = (func_dx - func) / diff
f_y_num = (func_dy - func) / diff
f_x, f_y = self.sersic_gauss.derivatives(x, y, e1=e1, e2=e2,
center_x=center_x,
center_y=center_y,
n_sersic=n_sersic,
R_sersic=R_sersic,
k_eff=k_eff
)
npt.assert_almost_equal(f_x_num, f_x, decimal=4)
npt.assert_almost_equal(f_y_num, f_y, decimal=4)
def test_derivatives(self):
"""
Test the derivative function of Gauss-decomposed elliptical Sersic by
matching with the spherical case.
:return:
:rtype:
"""
k_eff = 1.
R_sersic = 1.
n_sersic = 1.
e1 = 5.e-5
e2 = 0.
center_x = 0.
center_y = 0.
n = 10
x = np.linspace(0.5*R_sersic, 2.*R_sersic, n)
y = np.linspace(0.5*R_sersic, 2.*R_sersic, n)
X, Y = np.meshgrid(x, y)
f_x_s, f_y_s = self.sersic_sphere.derivatives(X, Y, center_x=center_x,
center_y=center_y,
n_sersic=n_sersic,
R_sersic=R_sersic,
k_eff=k_eff
)
f_x, f_y = self.sersic_gauss.derivatives(X, Y, e1=e1, e2=e2,
center_x=center_x,
center_y=center_y,
n_sersic=n_sersic,
R_sersic=R_sersic,
k_eff=k_eff
)
npt.assert_allclose(f_x, f_x_s, rtol=1e-3, atol=0.)
npt.assert_allclose(f_y, f_y_s, rtol=1e-3, atol=0.)
npt.assert_almost_equal(f_x, f_x_s, decimal=3)
npt.assert_almost_equal(f_y, f_y_s, decimal=3)
def test_hessian(self):
"""
Test the Hessian function of Gauss-decomposed elliptical Sersic by
matching with the spherical case.
:return:
:rtype:
"""
k_eff = 1.
R_sersic = 1.
n_sersic = 1.
e1 = 5e-5
e2 = 0.
center_x = 0.
center_y = 0.
n = 10
x = np.linspace(0.5 * R_sersic, 2. * R_sersic, n)
y = np.linspace(0.5 * R_sersic, 2. * R_sersic, n)
X, Y = np.meshgrid(x, y)
f_xx_s, f_yy_s, f_xy_s = self.sersic_sphere.hessian(X, Y,
center_x=center_x,
center_y=center_y,
n_sersic=n_sersic,
R_sersic=R_sersic,
k_eff=k_eff)
f_xx, f_yy, f_xy = self.sersic_gauss.hessian(X, Y, e1=e1, e2=e2,
center_x=center_x,
center_y=center_y,
n_sersic=n_sersic,
R_sersic=R_sersic,
k_eff=k_eff)
npt.assert_almost_equal(f_xx_s, f_xx, decimal=3)
npt.assert_almost_equal(f_yy_s, f_yy, decimal=3)
npt.assert_almost_equal(f_xy_s, f_xy, decimal=3)
def test_density_2d(self):
"""
Test the density function of Gauss-decomposed elliptical Sersic by
checking with the spherical case.
:return:
:rtype:
"""
k_eff = 1.
R_sersic = 1.
n_sersic = 1.
e1 = 0.2
e2 = 0.2
center_x = 0.
center_y = 0.
n = 100
x = np.logspace(-1., 1., n)
y = np.logspace(-1., 1., n)
X, Y = np.meshgrid(x, y)
sersic_analytic = self.sersic_light.function(X, Y, e1=e1, e2=e2,
center_x=center_x,
center_y=center_y,
n_sersic=n_sersic,
R_sersic=R_sersic,
amp=k_eff)
sersic_gauss = self.sersic_gauss.density_2d(X, Y, e1=e1, e2=e2,
center_x=center_x,
center_y=center_y,
n_sersic=n_sersic,
R_sersic=R_sersic,
k_eff=k_eff)
assert np.all(
np.abs(sersic_analytic - sersic_gauss) / np.sqrt(sersic_analytic)
* 100. < 1.)
def test_gauss_decompose_sersic(self):
"""
Test that `gauss_decompose_sersic()` decomposes the Sersic profile within 1%
Poission noise at R_sersic.
:return:
:rtype:
"""
y = np.logspace(-1., 1., 100)
k_eff = 1.
R_sersic = 1.
n_sersic = 1.
amps, sigmas = self.sersic_gauss.gauss_decompose(n_sersic=n_sersic,
R_sersic=R_sersic, k_eff=k_eff)
sersic = self.sersic_gauss.get_kappa_1d(y, n_sersic=n_sersic,
R_sersic=R_sersic, k_eff=k_eff)
back_sersic = np.zeros_like(y)
for a, s in zip(amps, sigmas):
back_sersic += a * np.exp(-y ** 2 / 2. / s ** 2)
assert np.all(np.abs(sersic-back_sersic)/np.sqrt(sersic)*100. < 1.)