def createMTFatm(D, m, k, wlum, zen, r0inmRef, model="vonK"):
"""
Generate the modulation transfer function (MTF) for atmosphere.
Arguments:
D {[float]} -- Side length of optical path difference (OPD) image in m.
m {[int]} -- Dimension of OPD image in pixel. The the number of pixel we want to have
to cover the length of D.
k {[int]} -- Use a k-times bigger array to pad the MTF. Use k=1 for the same size.
wlum {[float]} -- Wavelength in um.
zen {[float]} -- Telescope zenith angle in degree.
r0inmRef {[float]} -- Reference r0 in meter at the wavelength of 0.5 um.
Keyword Arguments:
model {str} -- Kolmogorov power spectrum ("Kolm") or van Karman power spectrum ("vonK").
(default: {"vonK"})
Returns:
[ndarray] -- MTF at specific atmosphere model.
"""
# Get the atmosphere phase structure function
sfa = atmSF(D, m, wlum, zen, r0inmRef, model)
# Get the modular transfer function for atmosphere
mtfa = np.exp(-0.5 * sfa)
# Add even number
N = int(m + np.rint((m * (k - 1) + 1e-5) / 2) * 2)
# Pad the matrix if necessary
mtfa = padArray(mtfa, N)
return mtfa
def createMTFatm(D, m, k, wlum, zen, r0inmRef):
"""
m is the number of pixel we want to have to cover the length of D.
If we want a k-times bigger array, we pad the mtf generated using k=1.
"""
sfa = atmSF('vonK', D, m, wlum, zen, r0inmRef)
mtfa = np.exp(-0.5 * sfa)
N = int(m + np.rint((m * (k - 1) + 1e-5) / 2) * 2) # add even number
mtfa = padArray(mtfa, N)
return mtfa
def createMTFatm(D, m, k, wlum, zen, r0inmRef):
"""
m is the number of pixel we want to have to cover the length of D/wl.
If we want a k-times bigger array, we pad the mtf generated using k=1.
"""
sfa = atmSF('vonK', D, m, wlum, zen, r0inmRef)
mtfa = np.exp(-0.5 * sfa)
N = np.rint(m * k + 1e-5)
mtfa = padArray(mtfa, N)
return mtfa
def calc_pssn(array,
wlum,
aType="opd",
D=8.36,
r0inmRef=0.1382,
zen=0,
pmask=0,
imagedelta=0,
fno=1.2335,
debugLevel=0):
"""
Calculate the normalized point source sensitivity (PSSN).
Arguments:
array {[ndarray]} -- Array that contains either opd or pdf. opd need to be in microns.
wlum {[float]} -- Wavelength in microns.
Keyword Arguments:
aType {str} -- What is used to calculate pssn - either opd or psf. (default: {"opd"})
D {float} -- Side length of OPD image in meter. (default: {8.36})
r0inmRef {float} -- Fidicial atmosphere r0 @ 500nm in meter, Konstantinos uses 0.20.
(default: {0.1382})
zen {float} -- Telescope zenith angle in degree. (default: {0})
pmask {float/ ndarray} -- Pupil mask. when opd is used, it can be generated using opd
image, we can put 0 or -1 or whatever here. When psf is used,
this needs to be provided separately with same size as array.
(default: {0})
imagedelta {float} -- Only needed when psf is used. use 0 for opd. (default: {0})
fno {float} -- Only needed when psf is used. use 0 for opd. (default: {1.2335})
debugLevel {int} -- Debug level. The higher value gives more information. (default: {0})
Returns:
[float] -- PSSN value.
"""
# Only needed for psf: pmask, imagedelta, fno
# THE INTERNAL RESOLUTION THAT FFTS OPERATE ON IS VERY IMPORTANT
# TO THE ACCUARCY OF PSSN.
# WHEN TYPE='OPD', NRESO=SIZE(ARRAY,1)
# WHEN TYPE='PSF', NRESO=SIZE(PMASK,1)
# for the psf option, we can not first convert psf back to opd then
# start over,
# because psf=|exp(-2*OPD)|^2. information has been lost in the | |^2.
# we need to go forward with psf->mtf,
# and take care of the coordinates properly.
# PSSN = (n_eff)_atm / (n_eff)_atm+sys
# (n_eff))_atm = 1 / (int (PSF^2)_atm dOmega)
# (n_eff))_atm+sys = 1 / (int (PSF^2)_atm+sys dOmega)
# Check the type is "OPD" or "PSF"
if aType not in ("opd", "psf"):
raise ValueError("The type of %s is not allowed." % aType)
# Squeeze the array if necessary
if (array.ndim == 3):
array2D = array[0, :, :].squeeze()
# Get the k value (magnification ratio used in creating MTF)
if (aType == "opd"):
try:
m = max(array2D.shape)
except NameError:
m = max(array.shape)
k = 1
elif (aType == "psf"):
m = max(pmask.shape)
# Pupil needs to be padded k times larger to get imagedelta
# Do not know where to find this formular. Check with Bo.
k = fno * wlum / imagedelta
# Get the modulation transfer function with the van Karman power spectrum
mtfa = createMTFatm(D, m, k, wlum, zen, r0inmRef, model="vonK")
# Get the pupil function
if (aType == "opd"):
try:
iad = (array2D != 0)
except NameError:
iad = (array != 0)
elif (aType == "psf"):
# Add even number
mk = int(m + np.rint((m * (k - 1) + 1e-5) / 2) * 2)
# padArray(pmask, m)
iad = pmask
# OPD --> PSF --> OTF --> OTF' (OTF + atmosphere) --> PSF'
# Check with Bo that we could get OTF' or PSF' from PhoSim or not directly.
# The above question might not be a concern in the simulation.
# However, for the real image, it loooks like this is hard to do
# What should be the standard way to judge the PSSN in the real telescope?
# OPD is zero for perfect telescope
opdt = np.zeros((m, m))
# OPD to PSF
psft = opd2psf(opdt,
iad,
wlum,
imagedelta=imagedelta,
sensorFactor=1,
fno=fno,
debugLevel=debugLevel)
# PSF to optical transfer function (OTF)
otft = psf2otf(psft)
# Add atmosphere to perfect telescope
otfa = otft * mtfa
# OTF to PSF
psfa = otf2psf(otfa)
# Atmospheric PSS (point spread sensitivity) = 1/neff_atm
pssa = np.sum(psfa**2)
# Calculate PSF with error (atmosphere + system)
if (aType == "opd"):
if (array.ndim == 2):
ninst = 1
else:
ninst = array.shape[0]
for ii in range(ninst):
if (array.ndim == 2):
array2D = array
else:
array2D = array[ii, :, :].squeeze()
psfei = opd2psf(array2D, iad, wlum, debugLevel=debugLevel)
if (ii == 0):
psfe = psfei
else:
psfe += psfei
# Do the normalization based on the number of instrument
psfe = psfe / ninst
elif (aType == "psf"):
if (array.shape[0] == mk):
psfe = array
elif (array.shape[0] > mk):
psfe = extractArray(array, mk)
else:
print("calc_pssn: image provided too small, %d < %d x %6.4f." %
(array.shape[0], m, k))
print("IQ is over-estimated !!!")
psfe = padArray(array, mk)
# Do the normalization of PSF
psfe = psfe / np.sum(psfe) * np.sum(psft)
# OTF with system error
otfe = psf2otf(psfe)
# Add the atmosphere error
# OTF with system and atmosphere errors
otftot = otfe * mtfa
# PSF with system and atmosphere errors
psftot = otf2psf(otftot)
# atmospheric + error PSS
pss = np.sum(psftot**2)
# normalized PSS
pssn = pss / pssa
if (debugLevel >= 3):
print("pssn = %10.8e/%10.8e = %6.4f." % (pss, pssa, pssn))
return pssn
def opd2psf(opd,
pupil,
wavelength,
imagedelta=0,
sensorFactor=1,
fno=1.2335,
debugLevel=0):
"""
Optical path difference (OPD) to point spread function (PSF).
Arguments:
opd {[ndarray]} -- Optical path difference.
pupil {[ndarray/ float/ int]} -- Pupil function. If pupil is a number, not an array, we will
get pupil geometry from OPD.
wavelength {[float]} -- Wavelength in um.
Keyword Arguments:
imagedelta {float} -- Pixel size in um. Use 0 if pixel size is not specified. (default: {0})
sensorFactor {float} -- Factor of sensor (check with Bo for this). Only need this if
imagedelta != 0. (default: {1})
fno {float} -- ? Check with Bo. Only need this if imagedelta=0. (default: {1.2335})
debugLevel {int} -- Debug level. The higher value gives more information. (default: {0})
Returns:
[ndarray] -- Normalized PSF.
Raises:
ValueError -- Shapes of OPD and pupil are different.
ValueError -- OPD shape is not square.
ValueError -- Padding value is less than 1.
"""
# Make sure all NaN in OPD to be 0
opd[np.isnan(opd)] = 0
# Get the pupil function from OPD if necessary
if (not isinstance(pupil, np.ndarray)):
pupil = (opd != 0)
# Check the dimension of pupil and OPD should be the same
if (opd.shape != pupil.shape):
raise ValueError("Shapes of OPD and pupil are different.")
# For the PSF
if (imagedelta != 0):
# Check the dimension of OPD
if (opd.shape[0] != opd.shape[1]):
raise ValueError("Error (opd2psf): OPD image size = (%d, %d)." %
(opd.shape[0], opd.shape[1]))
# Get the k value and the padding
k = fno * wavelength / imagedelta
padding = k / sensorFactor
# Check the padding
if (padding < 1):
errorMes = "opd2psf: Sampling too low, data inaccurate.\n"
errorMes += "Imagedelta needs to be smaller than fno * wlum = %4.2f um.\n" % (
fno * wavelength)
errorMes += "So that the padding factor > 1.\n"
errorMes += "Otherwise we have to cut pupil to be < D."
raise ValueError(errorMes)
# Size of sensor
sensorSamples = opd.shape[0]
# Add even number for padding
N = int(sensorSamples +
np.rint(((padding - 1) * sensorSamples + 1e-5) / 2) * 2)
pupil = padArray(pupil, N)
opd = padArray(opd, N)
# Show the padding information or not
if (debugLevel >= 3):
print("padding = %8.6f." % padding)
# If imagedelta = 0, we don't do any padding, and go with below
z = pupil * np.exp(-2j * np.pi * opd / wavelength)
z = np.fft.fftshift(np.fft.fft2(np.fft.fftshift(z), s=z.shape))
z = np.absolute(z**2)
# Normalize the PSF
z = z / np.sum(z)
# Show the information of PSF from OPD
if (debugLevel >= 3):
print("opd2psf(): imagedelta = %8.6f." % imagedelta, end="")
if (imagedelta == 0):
print("0 means using OPD with padding as provided.")
print("Verify psf has been normalized: %4.1f." % np.sum(z))
return z
def psf2FWHMring(array,
wlum,
type='opd',
D=8.36,
r0inmRef=0.1382,
zen=0,
pmask=0,
imagedelta=0,
fno=1.2335,
fwhm_thresh=0.01,
power=2,
debugLevel=0):
'''
wavefront OPD in micron
'''
wl = wlum * 1.e-6
if array.ndim == 3:
array2D = array[0, :, :].squeeze()
if type == 'opd':
try:
m = max(array2D.shape)
except NameError:
m = max(array.shape)
k = 1
imagedelta = fno * wlum
else:
m = max(pmask.shape)
k = fno * wlum / imagedelta
m = int(np.round(m * k))
D = D * k
mtfa = createMTFatm(D, m, k, wlum, zen, r0inmRef)
if type == 'opd':
try:
iad = (array2D != 0)
except NameError:
iad = (array != 0)
elif type == 'psf':
mk = int(m + np.rint((m * (k - 1) + 1e-5) / 2) * 2) # add even number
iad = pmask # padArray(pmask, m)
# coordinates of the PSF in mas
conv = 206265000. #=3600*180/pi*1000; const. for converting radian to mas
da = conv * wl / D #in arcsec; if type==psf, D includes the padding, so this is still valid
ha = da * (m - 1) / 2
ha1d = np.linspace(-ha, ha, m)
xxr, yyr = np.meshgrid(ha1d, ha1d)
# Perfect telescope
opdt = np.zeros((m, m))
psft = opd2psf(opdt, iad, wlum, imagedelta, 1, fno, debugLevel)
otft = psf2otf(psft) # OTF of perfect telescope
otfa = otft * mtfa # add atmosphere to perfect telescope
psfa = otf2psf(otfa)
dm = np.max(psfa)
idxmax = (psfa == dm)
idx = np.abs(psfa - 0.5 * dm) < fwhm_thresh * dm
r = np.sqrt((xxr[idx] - xxr[idxmax])**2 + (yyr[idx] - yyr[idxmax])**2)
fwhmatm = 2 * np.mean(r)
# Error;
if type == 'opd':
if array.ndim == 2:
ninst = 1
else:
ninst = array.shape[0]
for i in range(ninst):
if array.ndim == 2:
array2D = array
else:
array2D = array[i, :, :].squeeze()
psfei = opd2psf(array2D, iad, wlum, 0, 0, 0, debugLevel)
if i == 0:
psfe = psfei
else:
psfe += psfei
psfe = psfe / ninst
else:
if array.shape[0] == mk:
psfe = array
elif array.shape[0] > mk:
psfe = extractArray(array, mk)
else:
print('calc_pssn: image provided too small, %d < %d x %6.4f' %
(array.shape[0], m, k))
print('IQ is over-estimated !!!')
psfe = padArray(array, mk)
psfe = psfe / np.sum(psfe) * np.sum(psft)
otfe = psf2otf(psfe) # OTF of error
otftot = otfe * mtfa # add atmosphere to error
psftot = otf2psf(otftot)
dm = np.max(psftot)
idxmax = (psftot == dm)
idx = np.abs(psftot - 0.5 * dm) < fwhm_thresh * dm
r = np.sqrt((xxr[idx] - xxr[idxmax])**2 + (yyr[idx] - yyr[idxmax])**2)
fwhmtot = 2 * np.mean(r)
fwhm_mas = np.max(
(0,
(fwhmtot**power - fwhmatm**power)**(1 / power))) #cannot be negative
return fwhm_mas
def opd2psf(opd, pupil, wavelength, imagedelta, sensorFactor, fno, debugLevel):
"""
wavefront OPD in micron
imagedelta in micron, use 0 if pixel size is not specified
wavelength in micron
if pupil is a number, not an array, we will get pupil geometry from opd
The following are not needed if imagedelta=0,
sensorFactor, fno
"""
opd[np.isnan(opd)] = 0
try:
if (pupil.shape == opd.shape):
pass
else:
raise AttributeError
except AttributeError:
pupil = (opd != 0)
if imagedelta != 0:
try:
if opd.shape[0] != opd.shape[1]:
raise (nonSquareImageError)
except nonSquareImageError:
print('Error (opd2psf): Only square images are accepted.')
print('image size = (%d, %d)' % (opd.shape[0], opd.shape[1]))
sys.exit()
k = fno * wavelength / imagedelta
padding = k / sensorFactor
try:
if padding < 1:
raise (psfSamplingTooLowError)
except psfSamplingTooLowError:
print('opd2psf: sampling too low, data inaccurate')
print('imagedelta needs to be smaller than fno*wlum=%4.2f um' %
(fno * wavelength))
print(' so that the padding factor > 1')
print(' otherwise we have to cut pupil to be < D')
sys.exit()
sensorSamples = opd.shape[0]
# add even number for padding
N = int(sensorSamples + \
np.rint(((padding - 1) * sensorSamples + 1e-5) / 2) * 2)
pupil = padArray(pupil, N)
opd = padArray(opd, N)
if debugLevel >= 3:
print('padding=%8.6f' % padding)
# if imagedelta = 0, we don't do any padding, and go with below
z = pupil * np.exp(-2j * np.pi * opd / wavelength)
z = np.fft.fftshift(np.fft.fft2(np.fft.fftshift(z),
s=z.shape)) # /sqrt(miad2/m^2)
z = np.absolute(z**2)
z = z / np.sum(z)
if debugLevel >= 3:
print('opd2psf(): imagedelta=%8.6f' % imagedelta, end='')
if imagedelta == 0:
print('0 means using OPD with padding as provided')
else:
print('')
print('verify psf has been normalized: %4.1f' % np.sum(z))
return z
def calc_pssn(array,
wlum,
type='opd',
D=8.36,
r0inmRef=0.1382,
zen=0,
pmask=0,
imagedelta=0,
fno=1.2335,
debugLevel=0):
"""
array: the array that contains either opd or pdf
opd need to be in microns
wlum: wavelength in microns
type: what is used to calculate pssn - either opd or psf
psf doesn't matter, will be normalized anyway
D: side length of OPD image in meter
r0inmRef: fidicial atmosphere r0@500nm in meter, Konstantinos uses 0.20
Now that we use vonK atmosphere, r0in=0.1382 -> fwhm=0.6"
earlier, we used Kolm atmosphere, r0in=0.1679 -> fwhm=0.6"
zen: telescope zenith angle
The following are only needed when the input array is psf -
pmask: pupil mask. when opd is used, it can be generated using opd image,
we can put 0 or -1 or whatever here.
when psf is used, this needs to be provided separately with same
size as array.
imagedelta and fno are only needed when psf is used. use 0,0 for opd
THE INTERNAL RESOLUTION THAT FFTS OPERATE ON IS VERY IMPORTANT
TO THE ACCUARCY OF PSSN.
WHEN TYPE='OPD', NRESO=SIZE(ARRAY,1)
WHEN TYPE='PSF', NRESO=SIZE(PMASK,1)
for the psf option, we can not first convert psf back to opd then
start over,
because psf=|exp(-2*OPD)|^2. information has been lost in the | |^2.
we need to go forward with psf->mtf,
and take care of the coordinates properly.
"""
if array.ndim == 3:
array2D = array[0, :, :].squeeze()
if type == 'opd':
try:
m = max(array2D.shape)
except NameError:
m = max(array.shape)
k = 1
imagedelta = fno * wlum
else:
m = max(pmask.shape)
# pupil needs to be padded k times larger to get imagedelta
k = fno * wlum / imagedelta
mtfa = createMTFatm(D, m, k, wlum, zen, r0inmRef)
if type == 'opd':
try:
iad = (array2D != 0)
except NameError:
iad = (array != 0)
elif type == 'psf':
mk = int(m + np.rint((m * (k - 1) + 1e-5) / 2) * 2) # add even number
iad = pmask # padArray(pmask, m)
# number of non-zero elements, used for normalization later
# miad2 = np.count_nonzero(iad)
# Perfect telescope
opdt = np.zeros((m, m))
psft = opd2psf(opdt, iad, wlum, imagedelta, 1, fno, debugLevel)
otft = psf2otf(psft) # OTF of perfect telescope
otfa = otft * mtfa # add atmosphere to perfect telescope
psfa = otf2psf(otfa)
pssa = np.sum(psfa**2) # atmospheric PSS = 1/neff_atm
# Error;
if type == 'opd':
if array.ndim == 2:
ninst = 1
else:
ninst = array.shape[0]
for i in range(ninst):
if array.ndim == 2:
array2D = array
else:
array2D = array[i, :, :].squeeze()
psfei = opd2psf(array2D, iad, wlum, 0, 0, 0, debugLevel)
if i == 0:
psfe = psfei
else:
psfe += psfei
psfe = psfe / ninst
else:
if array.shape[0] == mk:
psfe = array
elif array.shape[0] > mk:
psfe = extractArray(array, mk)
else:
print('calc_pssn: image provided too small, %d < %d x %6.4f' %
(array.shape[0], m, k))
print('IQ is over-estimated !!!')
psfe = padArray(array, mk)
psfe = psfe / np.sum(psfe) * np.sum(psft)
pixmas = imagedelta * 20
aa = psfe / np.sum(psfe)
neff = 1 / np.sum(aa**2)
fwhmeff = 0.664 * pixmas * np.sqrt(neff)
otfe = psf2otf(psfe) # OTF of error
otftot = otfe * mtfa # add atmosphere to error
psftot = otf2psf(otftot)
pss = np.sum(psftot**2) # atmospheric + error PSS
pssn = pss / pssa # normalized PSS
if debugLevel >= 3:
print('pssn = %10.8e/%10.8e = %6.4f' % (pss, pssa, pssn))
return pssn, fwhmeff
def opd2psf(opd, pupil, wavelength, imagedelta, sensorFactor, fno, debugLevel):
"""
wavefront OPD in micron
imagedelta in micron, use 0 if pixel size is not specified
wavelength in micron
if pupil is a number, not an array, we will get pupil geometry from opd
The following are not needed if imagedelta=0,
sensorFactor, fno
"""
opd[np.isnan(opd)] = 0
try:
if (pupil.shape == opd.shape):
pass
else:
raise AttributeError
except AttributeError:
pupil = (opd != 0)
if imagedelta != 0:
try:
if opd.shape[0] != opd.shape[1]:
raise(nonSquareImageError)
except nonSquareImageError:
print('Error (opd2psf): Only square images are accepted.')
print('image size = (%d, %d)' % (
opd.shape[0], opd.shape[1]))
sys.exit()
k = fno * wavelength / imagedelta
padding = k / sensorFactor
try:
if padding < 1:
raise(psfSamplingTooLowError)
except psfSamplingTooLowError:
print('opd2psf: sampling too low, data inaccurate')
print('imagedelta needs to be smaller than fno*wlum=%4.2f um' % (
fno * wavelength))
print(' so that the padding factor > 1')
print(' otherwise we have to cut pupil to be < D')
sys.exit()
sensorSamples = opd.shape[0]
N = np.rint(padding * sensorSamples)
pupil = padArray(pupil, N)
opd = padArray(opd, N)
if debugLevel >= 3:
print('padding=%8.6f' % padding)
z = pupil * np.exp(-2j * np.pi * opd / wavelength)
z = np.fft.fftshift(np.fft.fft2(np.fft.fftshift(z),
s=z.shape)) # /sqrt(miad2/m^2)
z = np.absolute(z**2)
z = z / np.sum(z)
if debugLevel >= 3:
print('opd2psf(): imagedelta=%8.6f' % imagedelta)
print('verify psf has been normalized: %4.1f' % np.sum(z))
return z
def calc_pssn(array, wlum, type='opd', D=8.36, r0inmRef=0.1382, zen=0,
pmask=0, imagedelta=0.2, fno=1.2335, debugLevel=0):
"""
array: the array that contains eitehr opd or pdf
opd need to be in microns
wlum: wavelength in microns
type: what is used to calculate pssn - either opd or psf
psf doesn't matter, will be normalized anyway
D: side length of OPD image in meter
r0inmRef: fidicial atmosphere r0@500nm in meter, Konstantinos uses 0.20
Now that we use vonK atmosphere, r0in=0.1382 -> fwhm=0.6"
earlier, we used Kolm atmosphere, r0in=0.1679 -> fwhm=0.6"
zen: telescope zenith angle
The following are only needed when the input array is psf -
pmask: pupil mask. when opd is used, it can be generated using opd image,
we can put 0 or -1 or whatever here.
when psf is used, this needs to be provided separately with same
size as array
imagedelta and fno are only needed when psf is used. use 0,0 for opd
THE INTERNAL RESOLUTION THAT FFTS OPERATE ON IS VERY IMPORTANT
TO THE ACCUARCY OF PSSN.
WHEN TYPE='OPD', NRESO=SIZE(ARRAY,1)
WHEN TYPE='PSF', NRESO=SIZE(PMASK,1)
for the psf option, we can not first convert psf back to opd then
start over,
because psf=|exp(-2*OPD)|^2. information has been lost in the | |^2.
we need to go forward with psf->mtf,
and take care of the coordinates properly.
"""
if array.ndim == 3:
array2D = array[0, :, :].squeeze()
if type == 'opd':
try:
m = max(array2D.shape)
except NameError:
m = max(array.shape)
k = 1
else:
m = max(pmask.shape)
# pupil needs to be padded k times larger to get imagedelta
k = fno * wlum / imagedelta
mtfa = createMTFatm(D, m, k, wlum, zen, r0inmRef)
if type == 'opd':
try:
iad = (array2D != 0)
except NameError:
iad = (array != 0)
elif type == 'psf':
iad = padArray(pmask, m)
# number of non-zero elements, used for normalization later
# miad2 = np.count_nonzero(iad)
# Perfect telescope
opdt = np.zeros((m, m))
psft = opd2psf(opdt, iad, wlum, 0, 0, 0, debugLevel)
otft = psf2otf(psft) # OTF of perfect telescope
otfa = otft * mtfa # add atmosphere to perfect telescope
psfa = otf2psf(otfa)
pssa = np.sum(psfa**2) # atmospheric PSS = 1/neff_atm
# Error;
if type == 'opd':
if array.ndim == 2:
ninst = 1
else:
ninst = array.shape[0]
for i in range(ninst):
if array.ndim == 2:
array2D = array
else:
array2D = array[i, :, :].squeeze()
psfei = opd2psf(array2D, iad, wlum, 0, 0, 0, debugLevel)
if i == 0:
psfe = psfei
else:
psfe += psfei
psfe = psfe / ninst
else:
psfe = padArray(array, m)
psfe = psfe / np.sum(psfe) * np.sum(psft)
otfe = psf2otf(psfe) # OTF of error
otftot = otfe * mtfa # add atmosphere to error
psftot = otf2psf(otftot)
pss = np.sum(psftot**2) # atmospheric + error PSS
pssn = pss / pssa # normalized PSS
return pssn