def test_em_demons_step_2d():
r"""
Compares the output of the demons step in 2d against an analytical
step. The fixed image is given by $F(x) = \frac{1}{2}||x - c_f||^2$, the
moving image is given by $G(x) = \frac{1}{2}||x - c_g||^2$,
$x, c_f, c_g \in R^{2}$
References
----------
[Vercauteren09] Vercauteren, T., Pennec, X., Perchant, A., & Ayache, N.
(2009). Diffeomorphic demons: efficient non-parametric
image registration. NeuroImage, 45(1 Suppl), S61-72.
doi:10.1016/j.neuroimage.2008.10.040
"""
#Select arbitrary images' shape (same shape for both images)
sh = (20, 10)
#Select arbitrary centers
c_f = np.asarray(sh)/2
c_g = c_f + 0.5
#Compute the identity vector field I(x) = x in R^2
x_0 = np.asarray(range(sh[0]))
x_1 = np.asarray(range(sh[1]))
X = np.ndarray(sh + (2,), dtype = np.float64)
O = np.ones(sh)
X[...,0]= x_0[:, None] * O
X[...,1]= x_1[None, :] * O
#Compute the gradient fields of F and G
grad_F = X - c_f
grad_G = X - c_g
#The squared norm of grad_G to be used later
sq_norm_grad_F = np.sum(grad_F**2,-1)
sq_norm_grad_G = np.sum(grad_G**2,-1)
#Compute F and G
F = 0.5 * sq_norm_grad_F
G = 0.5 * sq_norm_grad_G
#Create an instance of EMMetric
metric = EMMetric(2)
metric.static_spacing = np.array([1.2, 1.2])
#The $\sigma_x$ (eq. 4 in [Vercauteren09]) parameter is computed in ANTS
#based on the image's spacing
sigma_x_sq = np.sum(metric.static_spacing**2)/metric.dim
#Set arbitrary values for $\sigma_i$ (eq. 4 in [Vercauteren09])
#The original Demons algorithm used simply |F(x) - G(x)| as an
#estimator, so let's use it as well
sigma_i_sq = (F - G)**2
#Set the properties relevant to the demons methods
metric.smooth = 3.0
metric.gradient_static = np.array(grad_F, dtype = floating)
metric.gradient_moving = np.array(grad_G, dtype = floating)
metric.static_image = np.array(F, dtype = floating)
metric.moving_image = np.array(G, dtype = floating)
metric.staticq_means_field = np.array(F, dtype = floating)
metric.staticq_sigma_sq_field = np.array(sigma_i_sq, dtype = floating)
metric.movingq_means_field = np.array(G, dtype = floating)
metric.movingq_sigma_sq_field = np.array(sigma_i_sq, dtype = floating)
#compute the step using the implementation under test
actual_forward = metric.compute_demons_step(True)
actual_backward = metric.compute_demons_step(False)
#Now directly compute the demons steps according to eq 4 in [Vercauteren09]
num_fwd = sigma_x_sq * (G - F)
den_fwd = sigma_x_sq * sq_norm_grad_F + sigma_i_sq
expected_fwd = -1 * np.array(grad_F) #This is $J^{P}$ in eq. 4 [Vercauteren09]
expected_fwd[..., 0] *= num_fwd / den_fwd
expected_fwd[..., 1] *= num_fwd / den_fwd
#apply Gaussian smoothing
expected_fwd[..., 0] = ndimage.filters.gaussian_filter(expected_fwd[..., 0], 3.0)
expected_fwd[..., 1] = ndimage.filters.gaussian_filter(expected_fwd[..., 1], 3.0)
num_bwd = sigma_x_sq * (F - G)
den_bwd = sigma_x_sq * sq_norm_grad_G + sigma_i_sq
expected_bwd = -1 * np.array(grad_G) #This is $J^{P}$ in eq. 4 [Vercauteren09]
expected_bwd[..., 0] *= num_bwd / den_bwd
expected_bwd[..., 1] *= num_bwd / den_bwd
#apply Gaussian smoothing
expected_bwd[..., 0] = ndimage.filters.gaussian_filter(expected_bwd[..., 0], 3.0)
expected_bwd[..., 1] = ndimage.filters.gaussian_filter(expected_bwd[..., 1], 3.0)
assert_array_almost_equal(actual_forward, expected_fwd)
assert_array_almost_equal(actual_backward, expected_bwd)
def test_em_demons_step_2d():
r"""
Compares the output of the demons step in 2d against an analytical
step. The fixed image is given by $F(x) = \frac{1}{2}||x - c_f||^2$, the
moving image is given by $G(x) = \frac{1}{2}||x - c_g||^2$,
$x, c_f, c_g \in R^{2}$
References
----------
[Vercauteren09] Vercauteren, T., Pennec, X., Perchant, A., & Ayache, N.
(2009). Diffeomorphic demons: efficient non-parametric
image registration. NeuroImage, 45(1 Suppl), S61-72.
doi:10.1016/j.neuroimage.2008.10.040
"""
# Select arbitrary images' shape (same shape for both images)
sh = (20, 10)
# Select arbitrary centers
c_f = np.asarray(sh) / 2
c_g = c_f + 0.5
# Compute the identity vector field I(x) = x in R^2
x_0 = np.asarray(range(sh[0]))
x_1 = np.asarray(range(sh[1]))
X = np.ndarray(sh + (2, ), dtype=np.float64)
O = np.ones(sh)
X[..., 0] = x_0[:, None] * O
X[..., 1] = x_1[None, :] * O
# Compute the gradient fields of F and G
grad_F = X - c_f
grad_G = X - c_g
# The squared norm of grad_G to be used later
sq_norm_grad_F = np.sum(grad_F**2, -1)
sq_norm_grad_G = np.sum(grad_G**2, -1)
# Compute F and G
F = 0.5 * sq_norm_grad_F
G = 0.5 * sq_norm_grad_G
# Create an instance of EMMetric
metric = EMMetric(2)
metric.static_spacing = np.array([1.2, 1.2])
# The $\sigma_x$ (eq. 4 in [Vercauteren09]) parameter is computed in ANTS
# based on the image's spacing
sigma_x_sq = np.sum(metric.static_spacing**2) / metric.dim
# Set arbitrary values for $\sigma_i$ (eq. 4 in [Vercauteren09])
# The original Demons algorithm used simply |F(x) - G(x)| as an
# estimator, so let's use it as well
sigma_i_sq = (F - G)**2
# Set the properties relevant to the demons methods
metric.smooth = 3.0
metric.gradient_static = np.array(grad_F, dtype=floating)
metric.gradient_moving = np.array(grad_G, dtype=floating)
metric.static_image = np.array(F, dtype=floating)
metric.moving_image = np.array(G, dtype=floating)
metric.staticq_means_field = np.array(F, dtype=floating)
metric.staticq_sigma_sq_field = np.array(sigma_i_sq, dtype=floating)
metric.movingq_means_field = np.array(G, dtype=floating)
metric.movingq_sigma_sq_field = np.array(sigma_i_sq, dtype=floating)
# compute the step using the implementation under test
actual_forward = metric.compute_demons_step(True)
actual_backward = metric.compute_demons_step(False)
# Now directly compute the demons steps according to eq 4 in
# [Vercauteren09]
num_fwd = sigma_x_sq * (G - F)
den_fwd = sigma_x_sq * sq_norm_grad_F + sigma_i_sq
# This is $J^{P}$ in eq. 4 [Vercauteren09]
expected_fwd = -1 * np.array(grad_F)
expected_fwd[..., 0] *= num_fwd / den_fwd
expected_fwd[..., 1] *= num_fwd / den_fwd
# apply Gaussian smoothing
expected_fwd[...,
0] = ndimage.filters.gaussian_filter(expected_fwd[..., 0],
3.0)
expected_fwd[...,
1] = ndimage.filters.gaussian_filter(expected_fwd[..., 1],
3.0)
num_bwd = sigma_x_sq * (F - G)
den_bwd = sigma_x_sq * sq_norm_grad_G + sigma_i_sq
# This is $J^{P}$ in eq. 4 [Vercauteren09]
expected_bwd = -1 * np.array(grad_G)
expected_bwd[..., 0] *= num_bwd / den_bwd
expected_bwd[..., 1] *= num_bwd / den_bwd
# apply Gaussian smoothing
expected_bwd[...,
0] = ndimage.filters.gaussian_filter(expected_bwd[..., 0],
3.0)
expected_bwd[...,
1] = ndimage.filters.gaussian_filter(expected_bwd[..., 1],
3.0)
assert_array_almost_equal(actual_forward, expected_fwd)
assert_array_almost_equal(actual_backward, expected_bwd)