def plot(method, strength=None):
# test distribution
source = SampleImage(n=2 * rmax - 1).image / scale
Isrc, _ = Ibeta(source)
Inorm = (Isrc**2).sum()
# simulated projection fith Poissonian noise
proj, _ = rbasex_transform(source, direction='forward')
proj[proj < 0] = 0
proj = np.random.RandomState(0).poisson(proj) # (reproducible, see NEP 19)
# reconstructed image and intensity
if strength is None:
reg = method
else:
reg = (method, strength)
im, distr = rbasex_transform(proj, reg=reg)
I, _ = distr.Ibeta()
# plot...
fig = plt.figure(figsize=(7, 3.5), frameon=False)
# image
plt.subplot(121)
fig = plt.imshow(rescaleI(im), vmin=-vlim, vmax=vlim, cmap='bwr')
plt.axis('off')
plt.text(0, 2 * rmax, method, va='top')
# intensity
ax = plt.subplot2grid((3, 2), (0, 1), rowspan=2)
ax.plot(Isrc, c='r', lw=1)
ax.plot(I, c='k', lw=1)
ax.set_xlim((0, rmax))
ax.set_ylim(Ilim)
# error
plt.subplot(326)
plt.axhline(c='r', lw=1)
plt.plot(I - Isrc, c='b', lw=1)
plt.xlim((0, rmax))
plt.ylim(dIlim)
# finish
plt.subplots_adjust(left=0,
right=0.97,
wspace=0.1,
bottom=0.08,
top=0.98,
hspace=0.5)
def test_rbasex_zeros():
rmax = 11
n = 2 * rmax - 1
im = np.zeros((n, n), dtype='float')
fwd_im, fwd_distr = rbasex_transform(im, direction='forward')
assert fwd_im.shape == (n, n)
assert fwd_distr.r.shape == (rmax,)
inv_im, inv_distr = rbasex_transform(im)
assert inv_im.shape == (n, n)
assert inv_distr.r.shape == (rmax,)
def rmse(method, strengths=None):
if strengths is None:
_, distr = rbasex_transform(proj, reg=method, out=None)
I, _ = distr.Ibeta()
return ((I - Isrc)**2).sum() / Inorm
err = np.empty_like(strengths, dtype=float)
for n, strength in enumerate(strengths):
_, distr = rbasex_transform(proj, reg=(method, strength), out=None)
I, _ = distr.Ibeta()
err[n] = ((I - Isrc)**2).sum() / Inorm
return err
def test_rbasex_gaussian():
"""Check an isotropic gaussian solution for rBasex"""
rmax = 100
sigma = 30
n = 2 * rmax - 1
ref = GaussianAnalytical(n, rmax, symmetric=True, sigma=sigma)
# images as direct products
src = ref.func * ref.func[:, None]
proj = ref.abel * ref.func[:, None] # (vertical is not Abel-transformed)
fwd_im, fwd_distr = rbasex_transform(src, direction='forward')
# whole image
assert_allclose(fwd_im, proj, rtol=0.02, atol=0.001)
# radial intensity profile (without r = 0)
assert_allclose(fwd_distr.harmonics()[0, 1:], ref.abel[rmax:],
rtol=0.02, atol=5e-4)
inv_im, inv_distr = rbasex_transform(proj)
# whole image
assert_allclose(inv_im, src, rtol=0.02, atol=0.02)
# radial intensity profile (without r = 0)
assert_allclose(inv_distr.harmonics()[0, 1:], ref.func[rmax:],
rtol=0.02, atol=1e-4)
def run_orders(odd=False):
"""
Test angular orders using Gaussian peaks by comparison with hansenlaw.
"""
maxorder = 6
sigma = 5.0 # Gaussian sigma
step = 6 * sigma # distance between peak centers
for order in range(maxorder + 1):
rmax = int((order + 2) * step)
if odd:
if order == 0:
continue # 0th order cannot be odd
height = 2 * rmax + 1
else: # even only
if order % 2:
continue # skip odd
height = rmax + 1
# coordinates (Q0 or right half):
x = np.arange(float(rmax + 1))
y = rmax - np.arange(float(height))[:, None]
# radius
r = np.sqrt(x**2 + y**2)
# cos, sin
r[rmax, 0] = np.inf
c = y / r
s = x / r
r[rmax, 0] = 0
# Gaussian peak with one cossin angular term
def peak(i):
m = i # cos power
k = (order - m) & ~1 # sin power (round down to even)
return c ** m * s ** k * \
np.exp(-(r - (i + 1) * step) ** 2 / (2 * sigma**2))
# create source distribution
src = peak(0)
for i in range(1, order + 1):
if not odd and i % 2:
continue # skip odd
src += peak(i)
# reference forward transform
abel = hansenlaw_transform(src, direction='forward', hold_order=1)
param = ', order = {}, odd = {}, '.format(order, odd)
# test forward transform
for mode in ['clean', 'cached']:
if mode == 'clean':
cache_cleanup()
proj, _ = rbasex_transform(src, origin=(rmax, 0),
order=order, odd=odd,
direction='forward', out='fold')
assert_allclose(proj, abel, rtol=0.003, atol=0.4,
err_msg='-> forward' + param + mode)
# test inverse transforms
for reg in [None, ('L2', 1), ('diff', 1), ('SVD', 1 / rmax)]:
for mode in ['clean', 'cached']:
if mode == 'clean':
cache_cleanup()
recon, _ = rbasex_transform(abel, origin=(rmax, 0),
order=order, odd=odd,
reg=reg, out='fold')
recon[rmax-2:rmax+3, :2] = 0 # exclude pixels near center
assert_allclose(recon, src, atol=0.03,
err_msg='-> reg = ' + str(reg) + param + mode)
def run_out(odd=False):
"""
Test some output shapes.
"""
sigma = 5.0 # Gaussian sigma
step = int(6 * sigma) # distance between peak centers
rmax = (2 + 2) * step
size = 2 * rmax + 1
# coordinates (full image):
x = np.arange(float(size)) - rmax
y = rmax - np.arange(float(size))[:, None]
# radius
r = np.sqrt(x**2 + y**2)
# cos
r[rmax, rmax] = np.inf
c = y / r
r[rmax, rmax] = 0
# Gaussian peak with one cos^n angular term
def peak(i):
n = i if odd else 2 * i
return c ** n * \
np.exp(-(r - (i + 1) * step) ** 2 / (2 * sigma**2))
# create source distribution
src = peak(0) + peak(1)
# reference forward transform
abel = Transform(src, direction='forward', method='hansenlaw',
transform_options={'hold_order': 1}).transform
param = '-> odd = {}, '.format(odd)
# Test forward transform:
# rmax < MIN => larger 'same'
proj, _ = rbasex_transform(src, rmax=rmax - step, odd=odd,
direction='forward', out='same')
assert_allclose(proj, abel, rtol=0.005, atol=0.2,
err_msg=param + 'rmax < MIN, out = same')
# cropped, rmax = HOR < VER -> same
crop = (slice(3 * step, -step // 2),
slice(3 * step, -step))
proj, _ = rbasex_transform(src[crop], origin=(step, step),
odd=odd, direction='forward', out='same')
assert_allclose(proj, abel[crop], rtol=0.005, atol=0.2,
err_msg=param + 'rmax = HOR < VER, out = same')
# cropped, rmax = VER < HOR -> same
crop = (slice(step, -3 * step),
slice(3 * step, -step // 2))
proj, _ = rbasex_transform(src[crop], origin=(3 * step, step),
odd=odd, direction='forward', out='same')
assert_allclose(proj, abel[crop], rtol=0.0003, atol=0.3,
err_msg=param + 'rmax = VER < HOR, out = same')
# cropped, rmax = VER > HOR -> same
crop = (slice(3 * step, -step // 2),
slice(3 * step, -step))
proj, _ = rbasex_transform(src[crop], origin=(step, step), rmax='MAX',
odd=odd, direction='forward', out='same')
assert_allclose(proj, abel[crop], rtol=0.005, atol=0.2,
err_msg=param + 'rmax = VER > HOR, out = same')
# cropped, rmax = HOR > VER -> same
crop = (slice(step, -3 * step),
slice(3 * step, -step // 2))
proj, _ = rbasex_transform(src[crop], origin=(3 * step, step), rmax='MAX',
odd=odd, direction='forward', out='same')
assert_allclose(proj, abel[crop], rtol=0.0003, atol=0.3,
err_msg=param + 'rmax = HOR > VER, out = same')
# cropped, rmax = rmax -> full
crop = (slice(3 * step, -step),
slice(3 * step, -step))
# (in multiples of step: VER^2 + HOR^2 = 2 * 3^2 > rmax^2 = 4^2)
proj, _ = rbasex_transform(src[crop], origin=(step, step), rmax=rmax,
odd=odd, direction='forward', out='full')
assert_allclose(proj, abel, rtol=0.002, atol=0.3,
err_msg=param + 'rmax = rmax, out = full')
def test_rbasex_pos():
"""
Test positive regularization as in run_orders().
"""
sigma = 5.0 # Gaussian sigma
step = 6 * sigma # distance between peak centers
for order in [0, 1, 2, 4, 6]:
rmax = int((order + 2) * step)
if order == 1: # odd
rmax += int(step) # 3 peaks instead of 2
height = 2 * rmax + 1
else: # even only
height = rmax + 1
# coordinates (Q0 or right half):
x = np.arange(float(rmax + 1))
y = rmax - np.arange(float(height))[:, None]
# radius
r = np.sqrt(x**2 + y**2)
# cos, sin
r[rmax, 0] = np.inf
c = y / r
s = x / r
r[rmax, 0] = 0
# Gaussian peak with one cossin angular term
def peak(i, isotropic=False):
if isotropic:
return np.exp(-(r - (i + 1) * step) ** 2 / (2 * sigma**2))
m = i # cos power
k = (order - m) & ~1 # sin power (round down to even)
return c ** m * s ** k * \
np.exp(-(r - (i + 1) * step) ** 2 / (2 * sigma**2))
# create source distribution
src = peak(0)
if order == 1: # special case
src += (1 + c) * peak(1, True) # 1 + cos >= 0
src += (1 - c) * peak(2, True) # 1 - cos >= 0
else: # other even orders
for i in range(2, order + 1, 2):
src += peak(i)
# array for nonnegativity test (with some tolerance)
zero = np.full_like(src, -1e-15)
# reference forward transform
abel = hansenlaw_transform(src, direction='forward', hold_order=1)
# with some noise
abel += 0.05 * np.random.RandomState(0).rand(*abel.shape)
param = '-> order = {}, '.format(order)
# test inverse transform
for mode in ['clean', 'cached']:
if mode == 'clean':
cache_cleanup()
recon, _ = rbasex_transform(abel, origin=(rmax, 0),
order=order,
reg='pos', out='fold')
recon[rmax-3:rmax+4, :2] = 0 # exclude pixels near center
assert_allclose(recon, src, atol=0.05,
err_msg=param + mode)
# check nonnegativity
assert_array_less(zero, recon)
import numpy as np
import matplotlib.pyplot as plt
from abel.tools.analytical import SampleImage
from abel.tools.vmi import Ibeta
from abel.rbasex import rbasex_transform
# test distribution
rmax = 200
scale = 10000
source = SampleImage(n=2 * rmax - 1).image / scale
Isrc, _ = Ibeta(source)
Inorm = (Isrc**2).sum()
# simulated projection fith Poissonian noise
proj, _ = rbasex_transform(source, direction='forward')
proj[proj < 0] = 0
proj = np.random.RandomState(0).poisson(proj) # (reproducible, see NEP 19)
# calculate relative RMS error for given regularization parameters
def rmse(method, strengths=None):
if strengths is None:
_, distr = rbasex_transform(proj, reg=method, out=None)
I, _ = distr.Ibeta()
return ((I - Isrc)**2).sum() / Inorm
err = np.empty_like(strengths, dtype=float)
for n, strength in enumerate(strengths):
_, distr = rbasex_transform(proj, reg=(method, strength), out=None)
I, _ = distr.Ibeta()
dmask = 1 * (mask1 - binary_erosion(mask1, structure=brush)) + \
2 * (mask2 - binary_erosion(mask2, structure=brush)) + \
3 * (mask3 - binary_erosion(mask3, structure=brush))
dmask[dmask == 0] = np.nan
from matplotlib.colors import ListedColormap
rgb = ListedColormap(['#CC0000', '#00AA00', '#0055FF'])
plt.imshow(dmask, extent=(0, w, 0, h), cmap=rgb, interpolation='nearest')
# Analyze the left part and plot results
plt.subplot(222)
plt.title('Distributions: left image')
# the reconstructed image is not used in this example, so it is not created;
# also notice that order=0 is enough for this totally isotropic case
_, distr = rbasex_transform(im,
origin=origin1,
rmax=r1,
order=0,
weights=mask1,
out=None)
r, I = distr.rIbeta()
plt.plot(r, I, c=rgb(0), label='$I(r)$')
plt.legend()
plt.autoscale(enable=True, tight=True)
# Analyze the central part and plot results
plt.subplot(223)
plt.title('Distributions: central image')
# here the default order=2 is needed and used
_, distr = rbasex_transform(im,
origin=origin2,
rmax=r2,
weights=mask2,
R = 150 # image radius N = 2 * R + 1 # full image width and height block_r = 20 # beam-block disk radius block_w = 5 # beam-block holder width vlim = 3.6 # intensity limits for images ylim = (-1.3, 2.7) # limits for plots # create source distribution and its profiles for reference source = SampleImage(N).func / 100 r_src, P0_src, P2_src = rharmonics(source) # simulate experimental image: # projection im, _ = rbasex_transform(source, direction='forward') # Poissonian noise im[im < 0] = 0 im = np.random.RandomState(0).poisson(im) # image coordinates im_x = np.arange(float(N)) - R im_y = R - np.arange(float(N))[:, None] im_r = np.sqrt(im_x**2 + im_y**2) # simulate beam-block shadow im = im / (1 + np.exp(-(im_r - block_r))) im[:R] *= 1 / (1 + np.exp(-(np.abs(im_x) - block_w))) # create mask that fully covers beam-block shadow mask_r = block_r + 5 mask_w = block_w + 5 mask = np.ones_like(im)
