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lensutil.py
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lensutil.py
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#
# lensdemo_funcs.py
#
# Function module for strong lensing demos
#
# Intended for use with lensdemo_script.py
#
# Copyright 2009 by Adam S. Bolton
# Creative Commons Attribution-Noncommercial-ShareAlike 3.0 license applies:
# http://creativecommons.org/licenses/by-nc-sa/3.0/
# All redistributions, modified or otherwise, must include this
# original copyright notice, licensing statement, and disclaimer.
# DISCLAIMER: ABSOLUTELY NO WARRANTY EXPRESS OR IMPLIED.
# AUTHOR ASSUMES NO LIABILITY IN CONNECTION WITH THIS COMPUTER CODE.
#
import numpy as N
def xy_rotate(x, y, xcen, ycen, phi):
"""
NAME: xy_rotate
PURPOSE: Transform input (x, y) coordiantes into the frame of a new
(x, y) coordinate system that has its origin at the point
(xcen, ycen) in the old system, and whose x-axis is rotated
c.c.w. by phi degrees with respect to the original x axis.
USAGE: (xnew,ynew) = xy_rotate(x, y, xcen, ycen, phi)
ARGUMENTS:
x, y: numpy ndarrays with (hopefully) matching sizes
giving coordinates in the old system
xcen: old-system x coordinate of the new origin
ycen: old-system y coordinate of the new origin
phi: angle c.c.w. in degrees from old x to new x axis
RETURNS: 2-item tuple containing new x and y coordinate arrays
WRITTEN: Adam S. Bolton, U. of Utah, 2009
"""
phirad = N.deg2rad(phi)
xnew = (x - xcen) * N.cos(phirad) + (y - ycen) * N.sin(phirad)
ynew = (y - ycen) * N.cos(phirad) - (x - xcen) * N.sin(phirad)
return (xnew,ynew)
def delta_2d(x, y, par):
"""
NAME: delta_2d
PURPOSE: Implement 2D Delta function
USAGE: z = delta_2d(x, y, par)
ARGUMENTS:
x, y: vecors or images of coordinates;
should be matching numpy ndarrays
par: vector of parameters, defined as follows:
par[0]: x-center
par[1]: y-center
par[2]: radius of a single pixel
par[3-5]: unused
RETURNS: 2D Delta function evaluated at x-y coords
WRITTEN: R. S. Bussmann, 2013 November, Cornell University
"""
square = x.copy()
square[:] = 0.
offx = x - par[0]
offy = y - par[1]
offset = N.sqrt(offx ** 2 + offy ** 2)
index = offset < par[2]
#xspot = x[:, 0] == par[0]
#yspot = y[0, :] == par[1]
square[index] = 1.0
#import pdb; pdb.set_trace()
#import matplotlib.pyplot as plt
#plt.imshow(square, origin='lower')
#plt.show()
return square
def ellipse_2d(x, y, par):
"""
NAME: ellipse_2d
PURPOSE: Implement 2D Elliptical function
USAGE: z = ellipse_2d(x, y, par)
ARGUMENTS:
x, y: vecors or images of coordinates;
should be matching numpy ndarrays
par: vector of parameters, defined as follows:
par[0]: amplitude
par[1]: intermediate-axis sigma
par[2]: x-center
par[3]: y-center
par[4]: axis ratio
par[5]: c.c.w. major-axis rotation w.r.t. x-axis
RETURNS: 2D Gaussian evaluated at x-y coords
NOTE: amplitude = 1 is peak flux, not normalized total flux
WRITTEN: Adam S. Bolton, U. of Utah, 2009
"""
(xnew,ynew) = xy_rotate(x, y, -par[2], par[3], par[5])
r_ell_sq = ((xnew**2)*par[4] + (ynew**2)/par[4]) / N.abs(par[1])**2
ellipse = r_ell_sq.copy()
ellipse[:] = 0.
inside = r_ell_sq < 1
ellipse[inside] = par[0]
#import matplotlib.pyplot as plt
#plt.imshow(r_ell_sq, origin='lower', vmax=10*par[1])
#plt.colorbar()
#plt.contour(ellipse)
#plt.show()
return ellipse
def gauss_2d(x, y, par):
"""
NAME: gauss_2d
PURPOSE: Implement 2D Gaussian function
USAGE: z = gauss_2d(x, y, par)
ARGUMENTS:
x, y: vecors or images of coordinates;
should be matching numpy ndarrays
par: vector of parameters, defined as follows:
par[0]: total flux # in ASB original code, this was amplitude
par[1]: intermediate-axis sigma
par[2]: x-center
par[3]: y-center
par[4]: axis ratio
par[5]: c.c.w. major-axis rotation w.r.t. y-axis
RETURNS: 2D Gaussian evaluated at x-y coords
NOTE: amplitude = 1 is peak flux, not normalized total flux
WRITTEN: Adam S. Bolton, U. of Utah, 2009
"""
(xnew,ynew) = xy_rotate(x, y, -par[2], par[3], par[5] + 90)
r_ell_sq = ((xnew**2)*par[4] + (ynew**2)/par[4]) / N.abs(par[1])**2
expgauss = N.exp(-0.5*r_ell_sq)
#import pdb; pdb.set_trace()
return par[0] * expgauss
def sie_grad(x, y, par):
"""
NAME: sie_grad
PURPOSE: compute the deflection of an SIE potential
USAGE: (xg, yg) = sie_grad(x, y, par)
ARGUMENTS:
x, y: vectors or images of coordinates;
should be matching numpy ndarrays
par: vector of parameters with 1 to 5 elements, defined as follows:
par[0]: lens strength, or 'Einstein radius'
par[1]: (optional) x-center (default = 0.0)
par[2]: (optional) y-center (default = 0.0)
par[3]: (optional) axis ratio (default=1.0)
par[4]: (optional) major axis Position Angle
in degrees c.c.w. of y axis. (default = 0.0)
RETURNS: tuple (xg, yg) of gradients at the positions (x, y)
NOTES: This routine implements an 'intermediate-axis' convention.
Analytic forms for the SIE potential can be found in:
Kassiola & Kovner 1993, ApJ, 417, 450
Kormann et al. 1994, A&A, 284, 285
Keeton & Kochanek 1998, ApJ, 495, 157
The parameter-order convention in this routine differs from that
of a previous IDL routine of the same name by ASB.
WRITTEN: Adam S. Bolton, U of Utah, 2009
"""
# Set parameters:
b = N.abs(par[0]) # can't be negative!!!
xzero = 0. if (len(par) < 2) else -par[1]
yzero = 0. if (len(par) < 3) else par[2]
q = 1. if (len(par) < 4) else N.abs(par[3])
phiq = 0. if (len(par) < 5) else par[4]
eps = 0.001 # for sqrt(1/q - q) < eps, a limit expression is used.
# Handle q > 1 gracefully:
if (q > 1.):
q = 1.0 / q
phiq = phiq + 90.0
# Go into shifted coordinats of the potential:
phirad = N.deg2rad(phiq + 90)
xsie = (x-xzero) * N.cos(phirad) + (y-yzero) * N.sin(phirad)
ysie = (y-yzero) * N.cos(phirad) - (x-xzero) * N.sin(phirad)
# Compute potential gradient in the transformed system:
r_ell = N.sqrt(q * xsie**2 + ysie**2 / q)
qfact = N.sqrt(1./q - q)
# (r_ell == 0) terms prevent divide-by-zero problems
if (qfact >= eps):
xtg = (b/qfact) * N.arctan(qfact * xsie / (r_ell + (r_ell == 0)))
ytg = (b/qfact) * N.arctanh(qfact * ysie / (r_ell + (r_ell == 0)))
# force r_ell to be greater than 0.1
thresh = 1e-3
toolow = (r_ell - b < thresh) & (r_ell - b > 0)
r_ell[toolow] = r_ell[toolow].mean()#b + thresh
toolow = (r_ell - b > -thresh) & (r_ell - b < 0)
r_ell[toolow] = r_ell[toolow].mean()#b - thresh
M = 1 - b / (r_ell + (r_ell == 0))
mu = N.abs(1. / M)
#plt.imshow(mu, origin='lower')
#plt.colorbar()
#plt.show()
else:
xtg = b * xsie / (r_ell + (r_ell == 0))
ytg = b * ysie / (r_ell + (r_ell == 0))
M = 1 - b / (r_ell + (r_ell == 0))
mu = N.abs(1. / M)
# Transform back to un-rotated system:
xg = xtg * N.cos(phirad) - ytg * N.sin(phirad)
yg = ytg * N.cos(phirad) + xtg * N.sin(phirad)
#import matplotlib.pyplot as plt
#plt.imshow(M, origin='lower')
#plt.colorbar()
#plt.show()
# Return value:
return (xg, yg, mu)
def sbmap(x, y, nlens, nsource, parameters, model_types, computeamp=True):
# define the x, y, and magnification maps
dx = x.copy()
dy = y.copy()
dmu = N.zeros(x.shape)
nparlens = 5
# loop over each lens
for i in range(nlens):
# Set SIE lens-model parameters and pack them into an array:
i5 = i * nparlens
lpar = []
for ip in range(nparlens):
lpar.append(parameters[i5 + ip])
lpar = N.asarray(lpar)
# Compute the lensing potential gradients and magnification map:
(xg, yg, mu) = sie_grad(x, y, lpar)
# apply the gradients and magnifications
dx -= xg
dy -= yg
dmu += mu
# hack to get the right index from the pzero vector
interindx = nparlens * nlens
# loop over each source
g_lensimage = N.zeros(x.shape)
g_image = N.zeros(x.shape)
e_lensimage = N.zeros(x.shape)
e_image = N.zeros(x.shape)
amp1 = []
amp2 = []
for i in range(nsource):
i6 = i * 6
# Set Gaussian source parameters and pack them into an array:
nparsource = 6
gpar = []
for ip in range(nparsource):
gpar.append(parameters[i6 + interindx + ip])
gpar = N.asarray(gpar)
# compute the peak flux of the unlensed Gaussian
model_type = model_types[i]
if model_type == 'Delta':
g_image = delta_2d(x, y, gpar)
if model_type == 'Gaussian':
g_image = gauss_2d(x, y, gpar)
if model_type == 'cylinder':
g_image = ellipse_2d(x, y, gpar)
totalflux = g_image.sum()
if totalflux == 0:
totalflux = 1.
normflux = parameters[i6 + interindx] / totalflux
if model_types[i] != 'Delta':
gpar[0] *= normflux * 1e-3
# re-evaluate unlensed image with normalized flux
if computeamp:
if model_type == 'Gaussian':
g_image = gauss_2d(x, y, gpar)
if model_type == 'cylinder':
g_image = ellipse_2d(x, y, gpar)
if nlens > 0:
# Evaluate lensed Gaussian image:
if model_type == 'Delta':
tmplens = delta_2d(dx, dy, gpar)
if model_type == 'Gaussian':
tmplens = gauss_2d(dx, dy, gpar)
if model_type == 'cylinder':
tmplens = ellipse_2d(dx, dy, gpar)
g_lensimage += tmplens
else:
# Use the unlensed (but normalized) Gaussian image
if model_type == 'Delta':
tmplens = delta_2d(x, y, gpar)
if model_type == 'Gaussian':
tmplens = gauss_2d(x, y, gpar)
if model_type == 'cylinder':
tmplens = ellipse_2d(x, y, gpar)
g_lensimage += tmplens
if nlens > 0:
if computeamp:
# Set elliptical source parameters and pack them into an array:
epar = gpar.copy()
epar[0] = 1.0
epar[1] *= 2.5
# Evaluate lensed and unlensed elliptical masks:
lensellipse = ellipse_2d(dx, dy, epar)
e_lensimage += lensellipse
ellipse = ellipse_2d(x, y, epar)
e_image += ellipse
# Evaluate amplification for each source
lensmask = lensellipse == 1
mask = ellipse == 1
numer = tmplens[lensmask].sum()
denom = g_image[mask].sum()
if denom > 0:
amp_mask = numer / denom
else:
amp_mask = 1e2
numer = tmplens.sum()
denom = g_image.sum()
if denom > 0:
amp_tot = numer / denom
else:
amp_tot = 1e2
if amp_tot > 1e2:
amp_tot = 1e2
if amp_mask > 1e2:
amp_mask = 1e2
amp1.extend([amp_tot])
amp2.extend([amp_mask])
else:
amp1.extend([1.0])
amp2.extend([1.0])
e_image = 1
e_lensimage = 1
return g_image, g_lensimage, e_image, e_lensimage, amp1, amp2