def _test_cross_zernikes(testj=4, nterms=10, npix=500):
"""Verify the functions are orthogonal, by taking the
integrals of a given Zernike times N other ones.
Parameters :
--------------
testj : int
Index of the Zernike polynomial to test against the others
nterms : int
Test that polynomial against those from 1 to this N
npix : int
Size of array to use for this test
"""
zj = zernike.zernike1(testj, npix=npix)
assert np.sum(
np.isfinite(zj)) > 0, "Zernike calculation failure; all NaNs."
zbasis = zernike.zernike_basis(nterms=nterms, npix=npix)
for idx, z in enumerate(zbasis):
j = idx + 1
if j == testj or j == 1:
continue # discard piston term and self
prod = z * zj
wg = np.where(np.isfinite(prod))
cross_sum = np.abs(prod[wg].sum())
assert cross_sum < 1e-9, (
"orthogonality failure, Sum[Zernike(j={}) * Zernike(j={})] = {} (> 1e-9)"
.format(j, testj, cross_sum))
def test_cached_zernike1(nterms=10):
radius = 1.1
osys = poppy_core.OpticalSystem()
osys.addPupil(optics.CircularAperture(radius=radius))
wave = osys.inputWavefront()
y, x = wave.coordinates()
rho = np.sqrt(y ** 2 + x ** 2) / radius
theta = np.arctan2(y, x)
cached_results = []
for j in range(1, nterms + 1):
cached_output = zernike.cached_zernike1(j, wave.shape, wave.pixelscale, radius, outside=0.0)
cached_results.append(cached_output)
uncached_output = zernike.zernike1(j, rho=rho, theta=theta, outside=0.0)
assert np.allclose(cached_output, uncached_output)
try:
cached_output[0, 0] = np.nan
assert False, "Shouldn't be able to assign to a cached output array!"
except ValueError:
pass
# Check that we're getting cached copies
for j, array_ref in enumerate(cached_results, start=1):
cached_array_ref = zernike.cached_zernike1(j, wave.shape, wave.pixelscale, radius, outside=0.0)
assert id(array_ref) == id(cached_array_ref), "cached_zernike1 returned a new object for the same arguments"
def _test_cross_zernikes(testj=4, nterms=10, npix=500):
"""Verify the functions are orthogonal, by taking the
integrals of a given Zernike times N other ones.
Parameters :
--------------
testj : int
Index of the Zernike polynomial to test against the others
nterms : int
Test that polynomial against those from 1 to this N
npix : int
Size of array to use for this test
"""
Zj = zernike.zernike1(testj, npix=npix)
Zbasis = zernike.zernike_basis(nterms=nterms, npix=npix)
for idx, Z in enumerate(Zbasis):
j = idx + 1
if j == testj or j == 1:
continue # discard piston term and self
prod = Z * Zj
wg = np.where(np.isfinite(prod))
cross_sum = np.abs(prod[wg].sum())
assert cross_sum < 1e-9, (
"orthogonality failure, Sum[Zernike(j={}) * Zernike(j={})] = {} (> 1e-9)".format(
j, testj, cross_sum)
)
def test_cached_zernike1(nterms=10):
radius = 1.1
osys = poppy_core.OpticalSystem()
osys.add_pupil(optics.CircularAperture(radius=radius))
wave = osys.input_wavefront()
y, x = wave.coordinates()
rho = np.sqrt(y ** 2 + x ** 2) / radius
theta = np.arctan2(y, x)
cached_results = []
for j in range(1, nterms + 1):
cached_output = zernike.cached_zernike1(j, wave.shape, wave.pixelscale, radius, outside=0.0)
cached_results.append(cached_output)
uncached_output = zernike.zernike1(j, rho=rho, theta=theta, outside=0.0)
assert np.allclose(cached_output, uncached_output)
try:
cached_output[0, 0] = np.nan
assert False, "Shouldn't be able to assign to a cached output array!"
except ValueError:
pass
# Check that we're getting cached copies
for j, array_ref in enumerate(cached_results, start=1):
cached_array_ref = zernike.cached_zernike1(j, wave.shape, wave.pixelscale, radius, outside=0.0)
assert id(array_ref) == id(cached_array_ref), "cached_zernike1 returned a new object for the same arguments"
def test_zernikes_rms(nterms=10, size=500):
"""Verify RMS(Zernike[n,m]) == 1."""
assert np.nanstd(zernike.zernike1(1)) == 0.0, "Zernike(j=0) has nonzero RMS"
for j in range(2, nterms):
n, m = zernike.noll_indices(j)
z = zernike.zernike(n, m, npix=size)
rms = np.nanstd(z) # exclude masked pixels
assert abs(1.0 - rms) < 0.001, "Zernike(j={}) has RMS value of {}".format(j, rms)
def test_zernikes_rms(nterms=10, size=500):
"""Verify RMS(Zernike[n,m]) == 1."""
assert np.nanstd(zernike.zernike1(1)) == 0.0, "Zernike(j=0) has nonzero RMS"
for j in range(2, nterms):
n, m = zernike.noll_indices(j)
Z = zernike.zernike(n, m, npix=size)
rms = np.nanstd(Z) # exclude masked pixels
assert 1.0 - rms < 0.001, "Zernike(j={}) has RMS value of {}".format(j, rms)
def test_ones_zernikes(nterms=10):
"""Verify the radial scaling function is correctly normalized"""
rho = np.ones(3)
theta = np.array([0, 1, 2])
for j in np.arange(nterms)+1:
n, m = zernike.noll_indices(j)
Z = zernike.zernike1(j, rho=rho, theta=theta)
Rs = zernike.R(n, m, rho)
print "j=%d\tZ_(%d,%d) [1] = \t %s" % (j, n, m, str(Rs))
assert Rs[0] == Rs[1] == Rs[2], "Radial polynomial is not radially symmetric"
def test_ones_zernikes(nterms=10):
"""Verify the radial scaling function is correctly normalized"""
rho = np.ones(3)
theta = np.array([0, 1, 2])
for j in np.arange(nterms)+1:
n, m = zernike.noll_indices(j)
Z = zernike.zernike1(j, rho=rho, theta=theta)
Rs = zernike.R(n, m, rho)
print "j=%d\tZ_(%d,%d) [1] = \t %s" % (j, n, m, str(Rs))
assert Rs[0] == Rs[1] == Rs[2], "Radial polynomial is not radially symmetric"
def zern_seg(self, segment, vmax=150, unit='nm'):
""" Show the Zernike terms applied to a given segment"""
# the actual wavefront is always in units of microns.
if unit == 'nm':
scalefact = 1000.
elif unit =='micron':
scalefact = 1.0
else:
raise ValueError("unknown unit keyword")
nzerns = 11
title = "Zernike components for "+segment
zerns = np.zeros((nzerns))
if segment+"-decenter" in self.state.keys():
zerns += self._move(segment, type='decenter', vector=self.state[segment+"-decenter"], return_zernikes=True)
title += ", displacement=[%.2e, %.2e, %.2e] um" % tuple(self.state[segment+"-decenter"])
if segment+"-tilt" in self.state.keys():
zerns += self._move(segment, type='tilt', vector=self.state[segment+"-tilt"], return_zernikes=True)
title += ", tilts =[%.5f, %.5f, %.5f] urad" % tuple(self.state[segment+"-tilt"])
_log.info("Zerns: "+str(zerns))
fig = plt.gcf()
fig.clf()
npix=200
hexap = zernike.hex_aperture(npix)
hexap[np.where(hexap == 0)] = np.nan
cmap = matplotlib.cm.jet
cmap.set_bad('0.5', alpha=0.0)
for j in np.arange(nzerns)+1:
ax = fig.add_subplot(3, 4, j, frameon=False, xticks=[], yticks=[])
# n, m = zernike.noll_indices(j)
Z = zernike.zernike1(j, npix=npix)
ax.imshow(Z * zerns[j-1] * hexap*scalefact, vmin=-1*vmax, vmax=vmax, cmap=cmap)
ax.text(npix*0.95, npix*0.8, "$Z%d$" % (j), fontsize=20, horizontalalignment='right')
ax.text(npix*0.95, npix*0.1, "%.2e" % (zerns[j-1]), fontsize=15, horizontalalignment='right')
fig.text(0.5, 0.95, title, horizontalalignment='center', fontsize=15)
fig.text(0.95, 0.15, 'Segment RMS WFE = %.2f %s ' % (self.rms(segment)*(scalefact/1000), unit), fontsize=10, horizontalalignment='right')
#fig.text(0.95, 0.05, 'OPDs scaled from %.2e - %.2e um' % (-vmax, vmax), horizontalalignment='right', fontsize=10)
fig.text(0.95, 0.05, 'OPDs scaled from %.2f - %.2f %s' % (-vmax, vmax, unit), horizontalalignment='right', fontsize=10)
plt.draw()
def _apply_zernikes_to_seg(self, segment, zernike_coeffs):
""" Apply Zernike perturbations to a given segment """
assert(segment in self.segnames)
iseg = np.where(self.segnames == segment)[0][0]+1 # segment index from 1 - 18
wseg = np.where(self._segment_masks == iseg)
# determine the X and Y zernike coefficients for each segment
# determine the center of each segment, as the mean of the min and max X and Y values
Y, X = np.indices(self._opdHDU.data.shape)
cx = np.mean([X[wseg].min(), X[wseg].max()])
cy = np.mean([Y[wseg].min(), Y[wseg].max()])
seg_radius = (X[wseg].max()-X[wseg].min())/2.0
_log.debug("Segment %s is centered at pixel loc (%.1f, %.1f) with radius %.1f pix" % (segment, cx, cy, seg_radius))
Xw = X[wseg].astype(np.float64)
Yw = Y[wseg].astype(np.float64)
Yw -= cy
Xw -= cx
ang = self._rotations[segment]*np.pi/180 # This definitely has to be a positive sign,
# to get the right X, Y for locally-defined zernikes
Xr = Xw * np.cos(ang) + Yw * np.sin(ang)
Yr = Xw *-np.sin(ang) + Yw * np.cos(ang)
theta = np.arctan2(Yr, Xr)
Rw = np.sqrt(Xr**2 + Yr**2)/seg_radius
#zernike_coeffs *= 0
#zernike_coeffs[2]=1
if self.remove_piston_tip_tilt:
zernike_coeffs[0:3] = 0
for i in range(len(zernike_coeffs)):
zern = zernike.zernike1(i+1, rho=Rw, theta=theta, outside=0.0) * zernike_coeffs[i]
#print "Z%d = %f" % (i+1, zernike_coeffs[i])
self._opdHDU.data[wseg] += zern
#self._opdHDU.data[wseg] = Yw
outtxt="Zs=["+", ".join(['%.1e'%z for z in zernike_coeffs])+"]"
#print outtxt
_log.debug(" "+outtxt)