def test_normalisations_real(self):
log = logging.getLogger('TestZern.test_normalisations_real')
n_alpha = 6
L, K = 400, 357
# polar grid
pol = RZern(n_alpha)
fitAlpha = FitZern(pol, L, K)
t1 = time()
pol.make_pol_grid(fitAlpha.rho_j, fitAlpha.theta_i)
t2 = time()
log.debug('make pol grid {:.6f}'.format(t2 - t1))
# cartesian grid
cart = RZern(n_alpha)
dd = np.linspace(-1.0, 1.0, max(L, K))
xx, yy = np.meshgrid(dd, dd)
t1 = time()
cart.make_cart_grid(xx, yy)
t2 = time()
log.debug('make cart grid {:.6f}'.format(t2 - t1))
smap = np.isfinite(cart.eval_grid(np.zeros(cart.nk)))
scale = (1.0 / np.sum(smap))
log.debug('')
log.debug('{} modes, {} x {} grid'.format(n_alpha, L, K))
for i in range(pol.nk):
a = np.zeros(pol.nk)
a[i] = 1.0
Phi_a = cart.eval_grid(a)
for j in range(pol.nk):
b = np.zeros(pol.nk)
b[j] = 1.0
Phi_b = cart.eval_grid(b)
ip = scale * np.sum(Phi_a[smap] * Phi_b[smap])
if i == j:
eip = 1.0
else:
eip = 0.0
iperr = abs(ip - eip)
log.debug('<{:02},{:02}> = {:+e} {:+e}'.format(
i + 1, j + 1, ip, iperr))
self.assertTrue(iperr < self.max_ip_err)
def test_normalisations_real(self):
log = logging.getLogger('TestZern.test_normalisations_real')
n_alpha = 6
L, K = 400, 357
# polar grid
pol = RZern(n_alpha)
fitAlpha = FitZern(pol, L, K)
t1 = time()
pol.make_pol_grid(fitAlpha.rho_j, fitAlpha.theta_i)
t2 = time()
log.debug('make pol grid {:.6f}'.format(t2 - t1))
# cartesian grid
cart = RZern(n_alpha)
dd = np.linspace(-1.0, 1.0, max(L, K))
xx, yy = np.meshgrid(dd, dd)
t1 = time()
cart.make_cart_grid(xx, yy)
t2 = time()
log.debug('make cart grid {:.6f}'.format(t2 - t1))
smap = np.isfinite(cart.eval_grid(np.zeros(cart.nk)))
scale = (1.0/np.sum(smap))
log.debug('')
log.debug('{} modes, {} x {} grid'.format(n_alpha, L, K))
for i in range(pol.nk):
a = np.zeros(pol.nk)
a[i] = 1.0
Phi_a = cart.eval_grid(a)
for j in range(pol.nk):
b = np.zeros(pol.nk)
b[j] = 1.0
Phi_b = cart.eval_grid(b)
ip = scale*np.sum(Phi_a[smap]*Phi_b[smap])
if i == j:
eip = 1.0
else:
eip = 0.0
iperr = abs(ip - eip)
log.debug('<{:02},{:02}> = {:+e} {:+e}'.format(
i + 1, j + 1, ip, iperr))
self.assertTrue(iperr < self.max_ip_err)
try:
alpha[args.noll - 1] = args.rms
except:
print('Cannot set the required Noll index k.')
print('Index k must be between 1 and ' + str(alpha.size))
sys.exit(1)
# set the alpha coefficients randomly
if args.random:
alpha1 = normal(size=alpha.size-1)
alpha1 = (args.rms/norm(alpha1))*alpha1
alpha[1:] = alpha1
del alpha1
# evaluate the phase corresponding to alpha
Phi = phase_pol.eval_grid(alpha)
# evaluate the generalised pupil function P corresponding to alpha
P = np.exp(1j*Phi)
# estimate the beta coefficients from P
beta_hat = ip.fit(P)
# plot the results
phaseplot = PhasePlot(n=args.n_alpha) # to plot beta and the PSF
betaplot = BetaPlot(args) # to plot the phase
# plot alpha
p.figure(10)
p.subplot2grid((1, 3), (0, 0), colspan=2)
h1 = p.plot(range(1, phase_pol.nk + 1), alpha, marker='o')
try:
alpha[args.noll - 1] = args.rms
except BaseException:
print('Cannot set the required Noll index k.')
print('Index k must be between 1 and ' + str(alpha.size))
sys.exit(1)
# set the alpha coefficients randomly
if args.random:
alpha1 = normal(size=alpha.size-1)
alpha1 = (args.rms/norm(alpha1))*alpha1
alpha[1:] = alpha1
del alpha1
# evaluate the phase corresponding to alpha
Phi = phase_pol.eval_grid(alpha)
# evaluate the generalised pupil function P corresponding to alpha
P = np.exp(1j*Phi)
# estimate the beta coefficients from P
beta_hat = ip.fit(P)
# plot the results
phaseplot = PhasePlot(n=args.n_alpha) # to plot beta and the PSF
betaplot = BetaPlot(args) # to plot the phase
# plot alpha
p.figure(10)
p.subplot2grid((1, 3), (0, 0), colspan=2)
h1 = p.plot(range(1, phase_pol.nk + 1), alpha, marker='o')
# grid sizes
L, K = 95, 105
# real-valued Zernike polynomials up to the 6-th radial order
phase_pol = RZern(6)
# FitZern computes the approximate inner products, see Eq. (B2) in [A2015]
ip = FitZern(phase_pol, L, K)
phase_pol.make_pol_grid(ip.rho_j, ip.theta_i) # make a polar grid
# random vector of Zernike coefficients to be estimated
alpha_true = normal(size=phase_pol.nk)
# phase grid
Phi = phase_pol.eval_grid(alpha_true)
# estimate the random vector from the phase grid
alpha_hat = ip.fit(Phi)
# plot the results
p.figure(2)
# Zernike coefficients of alpha_true and alpha_hat
ax = p.subplot2grid((3, 3), (0, 0), colspan=3)
h1 = ax.plot(range(1, phase_pol.nk + 1), alpha_true, marker='o')
h2 = ax.plot(range(1, phase_pol.nk + 1), alpha_hat, marker='x')
p.legend(h1 + h2, [r'$\alpha$', r'$\hat{\alpha}$'])
p.ylabel('[rad]')
p.xlabel('$k$')
# grid sizes
L, K = 95, 105
# real-valued Zernike polynomials up to the 6-th radial order
phase_pol = RZern(6)
# FitZern computes the approximate inner products, see Eq. (B2) in [A2015]
ip = FitZern(phase_pol, L, K)
phase_pol.make_pol_grid(ip.rho_j, ip.theta_i) # make a polar grid
# random vector of Zernike coefficients to be estimated
alpha_true = normal(size=phase_pol.nk)
# phase grid
Phi = phase_pol.eval_grid(alpha_true)
# estimate the random vector from the phase grid
alpha_hat = ip.fit(Phi)
# plot the results
p.figure(2)
# Zernike coefficients of alpha_true and alpha_hat
ax = p.subplot2grid((3, 3), (0, 0), colspan=3)
h1 = ax.plot(range(1, phase_pol.nk + 1), alpha_true, marker='o')
h2 = ax.plot(range(1, phase_pol.nk + 1), alpha_hat, marker='x')
p.legend(h1 + h2, [r'$\alpha$', r'$\hat{\alpha}$'])
p.ylabel('[rad]')
p.xlabel('$k$')
