"""Metrics for Symmetric Diffeomorphic Registration"""
import abc
import numpy as np
from numpy import gradient
from scipy import ndimage
from dipy.align import (
crosscorr as cc,
expectmax as em,
floating,
sumsqdiff as ssd,
vector_fields as vfu,
)
from dipy.testing.decorators import warning_for_keywords
[docs]
class SimilarityMetric:
def __init__(self, dim):
r"""Similarity Metric abstract class
A similarity metric is in charge of keeping track of the numerical
value of the similarity (or distance) between the two given images. It
also computes the update field for the forward and inverse displacement
fields to be used in a gradient-based optimization algorithm. Note that
this metric does not depend on any transformation (affine or
non-linear) so it assumes the static and moving images are already
warped
Parameters
----------
dim : int (either 2 or 3)
the dimension of the image domain
"""
self.dim = dim
self.levels_above = None
self.levels_below = None
self.static_image = None
self.static_affine = None
self.static_spacing = None
self.static_direction = None
self.moving_image = None
self.moving_affine = None
self.moving_spacing = None
self.moving_direction = None
self.mask0 = False
[docs]
def set_levels_below(self, levels):
r"""Informs the metric how many pyramid levels are below the current one
Informs this metric the number of pyramid levels below the current one.
The metric may change its behavior (e.g. number of inner iterations)
accordingly
Parameters
----------
levels : int
the number of levels below the current Gaussian Pyramid level
"""
self.levels_below = levels
[docs]
def set_levels_above(self, levels):
r"""Informs the metric how many pyramid levels are above the current one
Informs this metric the number of pyramid levels above the current one.
The metric may change its behavior (e.g. number of inner iterations)
accordingly
Parameters
----------
levels : int
the number of levels above the current Gaussian Pyramid level
"""
self.levels_above = levels
[docs]
def set_static_image(
self, static_image, static_affine, static_spacing, static_direction
):
r"""Sets the static image being compared against the moving one.
Sets the static image. The default behavior (of this abstract class) is
simply to assign the reference to an attribute, but
generalizations of the metric may need to perform other operations
Parameters
----------
static_image : array, shape (R, C) or (S, R, C)
the static image
"""
self.static_image = static_image
self.static_affine = static_affine
self.static_spacing = static_spacing
self.static_direction = static_direction
[docs]
def use_static_image_dynamics(self, original_static_image, transformation):
r"""This is called by the optimizer just after setting the static image.
This method allows the metric to compute any useful
information from knowing how the current static image was generated
(as the transformation of an original static image). This method is
called by the optimizer just after it sets the static image.
Transformation will be an instance of DiffeomorficMap or None
if the original_static_image equals self.moving_image.
Parameters
----------
original_static_image : array, shape (R, C) or (S, R, C)
original image from which the current static image was generated
transformation : DiffeomorphicMap object
the transformation that was applied to original image to generate
the current static image
"""
pass
[docs]
def set_moving_image(
self, moving_image, moving_affine, moving_spacing, moving_direction
):
r"""Sets the moving image being compared against the static one.
Sets the moving image. The default behavior (of this abstract class) is
simply to assign the reference to an attribute, but
generalizations of the metric may need to perform other operations
Parameters
----------
moving_image : array, shape (R, C) or (S, R, C)
the moving image
"""
self.moving_image = moving_image
self.moving_affine = moving_affine
self.moving_spacing = moving_spacing
self.moving_direction = moving_direction
[docs]
def use_moving_image_dynamics(self, original_moving_image, transformation):
r"""This is called by the optimizer just after setting the moving image
This method allows the metric to compute any useful
information from knowing how the current static image was generated
(as the transformation of an original static image). This method is
called by the optimizer just after it sets the static image.
Transformation will be an instance of DiffeomorficMap or None if
the original_moving_image equals self.moving_image.
Parameters
----------
original_moving_image : array, shape (R, C) or (S, R, C)
original image from which the current moving image was generated
transformation : DiffeomorphicMap object
the transformation that was applied to the original image to generate
the current moving image
"""
pass
[docs]
@abc.abstractmethod
def initialize_iteration(self):
r"""Prepares the metric to compute one displacement field iteration.
This method will be called before any compute_forward or
compute_backward call, this allows the Metric to pre-compute any useful
information for speeding up the update computations. This
initialization was needed in ANTS because the updates are called once
per voxel. In Python this is unpractical, though.
"""
[docs]
@abc.abstractmethod
def free_iteration(self):
r"""Releases the resources no longer needed by the metric
This method is called by the RegistrationOptimizer after the required
iterations have been computed (forward and / or backward) so that the
SimilarityMetric can safely delete any data it computed as part of the
initialization
"""
[docs]
@abc.abstractmethod
def compute_forward(self):
r"""Computes one step bringing the reference image towards the static.
Computes the forward update field to register the moving image towards
the static image in a gradient-based optimization algorithm
"""
[docs]
@abc.abstractmethod
def compute_backward(self):
r"""Computes one step bringing the static image towards the moving.
Computes the backward update field to register the static image towards
the moving image in a gradient-based optimization algorithm
"""
[docs]
@abc.abstractmethod
def get_energy(self):
r"""Numerical value assigned by this metric to the current image pair
Must return the numeric value of the similarity between the given
static and moving images
"""
[docs]
class CCMetric(SimilarityMetric):
@warning_for_keywords()
def __init__(self, dim, *, sigma_diff=2.0, radius=4):
r"""Normalized Cross-Correlation Similarity metric.
Parameters
----------
dim : int (either 2 or 3)
the dimension of the image domain
sigma_diff : the standard deviation of the Gaussian smoothing kernel to
be applied to the update field at each iteration
radius : int
the radius of the squared (cubic) neighborhood at each voxel to be
considered to compute the cross correlation
"""
super(CCMetric, self).__init__(dim)
self.sigma_diff = sigma_diff
self.radius = radius
self._connect_functions()
def _connect_functions(self):
r"""Assign the methods to be called according to the image dimension
Assigns the appropriate functions to be called for precomputing the
cross-correlation factors according to the dimension of the input
images
"""
if self.dim == 2:
self.precompute_factors = cc.precompute_cc_factors_2d
self.compute_forward_step = cc.compute_cc_forward_step_2d
self.compute_backward_step = cc.compute_cc_backward_step_2d
self.reorient_vector_field = vfu.reorient_vector_field_2d
elif self.dim == 3:
self.precompute_factors = cc.precompute_cc_factors_3d
self.compute_forward_step = cc.compute_cc_forward_step_3d
self.compute_backward_step = cc.compute_cc_backward_step_3d
self.reorient_vector_field = vfu.reorient_vector_field_3d
else:
raise ValueError(f"CC Metric not defined for dim. {self.dim}")
[docs]
def initialize_iteration(self):
r"""Prepares the metric to compute one displacement field iteration.
Pre-computes the cross-correlation factors for efficient computation
of the gradient of the Cross Correlation w.r.t. the displacement field.
It also pre-computes the image gradients in the physical space by
re-orienting the gradients in the voxel space using the corresponding
affine transformations.
"""
min_size = self.radius * 2 + 1
def invalid_image_size(image):
return any(size < min_size for size in image.shape)
msg = (
"Each image dimension should be superior to 2 * radius + 1 "
f"({min_size}). Decrease CCMetric radius ({self.radius}) or "
"increase your image size (shape=%(shape)s)."
)
if invalid_image_size(self.static_image):
raise ValueError(
"Static image size is too small. "
+ msg % {"shape": self.static_image.shape}
)
if invalid_image_size(self.moving_image):
raise ValueError(
"Moving image size is too small. "
+ msg % {"shape": self.moving_image.shape}
)
self.factors = self.precompute_factors(
self.static_image, self.moving_image, self.radius
)
self.factors = np.array(self.factors)
self.gradient_moving = np.empty(
shape=self.moving_image.shape + (self.dim,), dtype=floating
)
for i, grad in enumerate(gradient(self.moving_image)):
self.gradient_moving[..., i] = grad
# Convert moving image's gradient field from voxel to physical space
if self.moving_spacing is not None:
self.gradient_moving /= self.moving_spacing
if self.moving_direction is not None:
self.reorient_vector_field(self.gradient_moving, self.moving_direction)
self.gradient_static = np.empty(
shape=self.static_image.shape + (self.dim,), dtype=floating
)
for i, grad in enumerate(gradient(self.static_image)):
self.gradient_static[..., i] = grad
# Convert moving image's gradient field from voxel to physical space
if self.static_spacing is not None:
self.gradient_static /= self.static_spacing
if self.static_direction is not None:
self.reorient_vector_field(self.gradient_static, self.static_direction)
[docs]
def free_iteration(self):
r"""Frees the resources allocated during initialization"""
del self.factors
del self.gradient_moving
del self.gradient_static
[docs]
def compute_forward(self):
r"""Computes one step bringing the moving image towards the static.
Computes the update displacement field to be used for registration of
the moving image towards the static image
"""
displacement, self.energy = self.compute_forward_step(
self.gradient_static, self.factors, self.radius
)
displacement = np.array(displacement)
for i in range(self.dim):
displacement[..., i] = ndimage.gaussian_filter(
displacement[..., i], self.sigma_diff
)
return displacement
[docs]
def compute_backward(self):
r"""Computes one step bringing the static image towards the moving.
Computes the update displacement field to be used for registration of
the static image towards the moving image
"""
displacement, energy = self.compute_backward_step(
self.gradient_moving, self.factors, self.radius
)
displacement = np.array(displacement)
for i in range(self.dim):
displacement[..., i] = ndimage.gaussian_filter(
displacement[..., i], self.sigma_diff
)
return displacement
[docs]
def get_energy(self):
r"""Numerical value assigned by this metric to the current image pair
Returns the Cross Correlation (data term) energy computed at the
largest iteration
"""
return self.energy
[docs]
class EMMetric(SimilarityMetric):
@warning_for_keywords()
def __init__(
self,
dim,
*,
smooth=1.0,
inner_iter=5,
q_levels=256,
double_gradient=True,
step_type="gauss_newton",
):
r"""Expectation-Maximization Metric
Similarity metric based on the Expectation-Maximization algorithm to
handle multi-modal images. The transfer function is modeled as a set of
hidden random variables that are estimated at each iteration of the
algorithm.
Parameters
----------
dim : int (either 2 or 3)
the dimension of the image domain
smooth : float
smoothness parameter, the larger the value the smoother the
deformation field
inner_iter : int
number of iterations to be performed at each level of the multi-
resolution Gauss-Seidel optimization algorithm (this is not the
number of steps per Gaussian Pyramid level, that parameter must
be set for the optimizer, not the metric)
q_levels : number of quantization levels (equal to the number of hidden
variables in the EM algorithm)
double_gradient : boolean
if True, the gradient of the expected static image under the moving
modality will be added to the gradient of the moving image,
similarly, the gradient of the expected moving image under the
static modality will be added to the gradient of the static image.
step_type : string ('gauss_newton', 'demons')
the optimization schedule to be used in the multi-resolution
Gauss-Seidel optimization algorithm (not used if Demons Step is
selected)
"""
super(EMMetric, self).__init__(dim)
self.smooth = smooth
self.inner_iter = inner_iter
self.q_levels = q_levels
self.use_double_gradient = double_gradient
self.step_type = step_type
self.static_image_mask = None
self.moving_image_mask = None
self.staticq_means_field = None
self.movingq_means_field = None
self.movingq_levels = None
self.staticq_levels = None
self._connect_functions()
def _connect_functions(self):
r"""Assign the methods to be called according to the image dimension
Assigns the appropriate functions to be called for image quantization,
statistics computation and multi-resolution iterations according to the
dimension of the input images
"""
if self.dim == 2:
self.quantize = em.quantize_positive_2d
self.compute_stats = em.compute_masked_class_stats_2d
self.reorient_vector_field = vfu.reorient_vector_field_2d
elif self.dim == 3:
self.quantize = em.quantize_positive_3d
self.compute_stats = em.compute_masked_class_stats_3d
self.reorient_vector_field = vfu.reorient_vector_field_3d
else:
raise ValueError(f"EM Metric not defined for dim. {self.dim}")
if self.step_type == "demons":
self.compute_step = self.compute_demons_step
elif self.step_type == "gauss_newton":
self.compute_step = self.compute_gauss_newton_step
else:
raise ValueError(f"Opt. step {self.step_type} not defined")
[docs]
def initialize_iteration(self):
r"""Prepares the metric to compute one displacement field iteration.
Pre-computes the transfer functions (hidden random variables) and
variances of the estimators. Also pre-computes the gradient of both
input images. Note that once the images are transformed to the opposite
modality, the gradient of the transformed images can be used with the
gradient of the corresponding modality in the same fashion as
diff-demons does for mono-modality images. If the flag
self.use_double_gradient is True these gradients are averaged.
"""
sampling_mask = self.static_image_mask * self.moving_image_mask
self.sampling_mask = sampling_mask
staticq, self.staticq_levels, hist = self.quantize(
self.static_image, self.q_levels
)
staticq = np.array(staticq, dtype=np.int32)
self.staticq_levels = np.array(self.staticq_levels)
staticq_means, staticq_vars = self.compute_stats(
sampling_mask, self.moving_image, self.q_levels, staticq
)
staticq_means[0] = 0
self.staticq_means = np.array(staticq_means)
self.staticq_variances = np.array(staticq_vars)
self.staticq_sigma_sq_field = self.staticq_variances[staticq]
self.staticq_means_field = self.staticq_means[staticq]
self.gradient_moving = np.empty(
shape=self.moving_image.shape + (self.dim,), dtype=floating
)
for i, grad in enumerate(gradient(self.moving_image)):
self.gradient_moving[..., i] = grad
# Convert moving image's gradient field from voxel to physical space
if self.moving_spacing is not None:
self.gradient_moving /= self.moving_spacing
if self.moving_direction is not None:
self.reorient_vector_field(self.gradient_moving, self.moving_direction)
self.gradient_static = np.empty(
shape=self.static_image.shape + (self.dim,), dtype=floating
)
for i, grad in enumerate(gradient(self.static_image)):
self.gradient_static[..., i] = grad
# Convert moving image's gradient field from voxel to physical space
if self.static_spacing is not None:
self.gradient_static /= self.static_spacing
if self.static_direction is not None:
self.reorient_vector_field(self.gradient_static, self.static_direction)
movingq, self.movingq_levels, hist = self.quantize(
self.moving_image, self.q_levels
)
movingq = np.array(movingq, dtype=np.int32)
self.movingq_levels = np.array(self.movingq_levels)
movingq_means, movingq_variances = self.compute_stats(
sampling_mask, self.static_image, self.q_levels, movingq
)
movingq_means[0] = 0
self.movingq_means = np.array(movingq_means)
self.movingq_variances = np.array(movingq_variances)
self.movingq_sigma_sq_field = self.movingq_variances[movingq]
self.movingq_means_field = self.movingq_means[movingq]
if self.use_double_gradient:
for i, grad in enumerate(gradient(self.staticq_means_field)):
self.gradient_moving[..., i] += grad
for i, grad in enumerate(gradient(self.movingq_means_field)):
self.gradient_static[..., i] += grad
[docs]
def free_iteration(self):
r"""
Frees the resources allocated during initialization
"""
del self.sampling_mask
del self.staticq_levels
del self.movingq_levels
del self.staticq_sigma_sq_field
del self.staticq_means_field
del self.movingq_sigma_sq_field
del self.movingq_means_field
del self.gradient_moving
del self.gradient_static
[docs]
def compute_forward(self):
"""Computes one step bringing the reference image towards the static.
Computes the forward update field to register the moving image towards
the static image in a gradient-based optimization algorithm
"""
return self.compute_step(forward_step=True)
[docs]
def compute_backward(self):
r"""Computes one step bringing the static image towards the moving.
Computes the update displacement field to be used for registration of
the static image towards the moving image
"""
return self.compute_step(forward_step=False)
[docs]
@warning_for_keywords()
def compute_gauss_newton_step(self, *, forward_step=True):
r"""Computes the Gauss-Newton energy minimization step
Computes the Newton step to minimize this energy, i.e., minimizes the
linearized energy function with respect to the
regularized displacement field (this step does not require
post-smoothing, as opposed to the demons step, which does not include
regularization). To accelerate convergence we use the multi-grid
Gauss-Seidel algorithm proposed by :footcite:t:`Bruhn2005`.
Parameters
----------
forward_step : boolean
if True, computes the Newton step in the forward direction
(warping the moving towards the static image). If False,
computes the backward step (warping the static image to the
moving image)
Returns
-------
displacement : array, shape (R, C, 2) or (S, R, C, 3)
the Newton step
References
----------
.. footbibliography::
"""
reference_shape = self.static_image.shape
if forward_step:
gradient = self.gradient_static
delta = self.staticq_means_field - self.moving_image
sigma_sq_field = self.staticq_sigma_sq_field
else:
gradient = self.gradient_moving
delta = self.movingq_means_field - self.static_image
sigma_sq_field = self.movingq_sigma_sq_field
displacement = np.zeros(shape=reference_shape + (self.dim,), dtype=floating)
if self.dim == 2:
self.energy = v_cycle_2d(
self.levels_below,
self.inner_iter,
delta,
sigma_sq_field,
gradient,
None,
self.smooth,
displacement,
)
else:
self.energy = v_cycle_3d(
self.levels_below,
self.inner_iter,
delta,
sigma_sq_field,
gradient,
None,
self.smooth,
displacement,
)
return displacement
[docs]
@warning_for_keywords()
def compute_demons_step(self, *, forward_step=True):
r"""Demons step for EM metric
Parameters
----------
forward_step : boolean
if True, computes the Demons step in the forward direction
(warping the moving towards the static image). If False,
computes the backward step (warping the static image to the
moving image)
Returns
-------
displacement : array, shape (R, C, 2) or (S, R, C, 3)
the Demons step
"""
sigma_reg_2 = np.sum(self.static_spacing**2) / self.dim
if forward_step:
gradient = self.gradient_static
delta_field = self.static_image - self.movingq_means_field
sigma_sq_field = self.movingq_sigma_sq_field
else:
gradient = self.gradient_moving
delta_field = self.moving_image - self.staticq_means_field
sigma_sq_field = self.staticq_sigma_sq_field
if self.dim == 2:
step, self.energy = em.compute_em_demons_step_2d(
delta_field, sigma_sq_field, gradient, sigma_reg_2, None
)
else:
step, self.energy = em.compute_em_demons_step_3d(
delta_field, sigma_sq_field, gradient, sigma_reg_2, None
)
for i in range(self.dim):
step[..., i] = ndimage.gaussian_filter(step[..., i], self.smooth)
return step
[docs]
def get_energy(self):
r"""The numerical value assigned by this metric to the current image pair
Returns the EM (data term) energy computed at the largest
iteration
"""
return self.energy
[docs]
def use_static_image_dynamics(self, original_static_image, transformation):
r"""This is called by the optimizer just after setting the static image.
EMMetric takes advantage of the image dynamics by computing the
current static image mask from the originalstaticImage mask (warped
by nearest neighbor interpolation)
Parameters
----------
original_static_image : array, shape (R, C) or (S, R, C)
the original static image from which the current static image was
generated, the current static image is the one that was provided
via 'set_static_image(...)', which may not be the same as the
original static image but a warped version of it (even the static
image changes during Symmetric Normalization, not only the moving
one).
transformation : DiffeomorphicMap object
the transformation that was applied to the original_static_image
to generate the current static image
"""
self.static_image_mask = (original_static_image > 0).astype(np.int32)
if transformation is None:
return
shape = np.array(self.static_image.shape, dtype=np.int32)
affine = self.static_affine
self.static_image_mask = transformation.transform(
self.static_image_mask,
interpolation="nearest",
image_world2grid=None,
out_shape=shape,
out_grid2world=affine,
)
[docs]
def use_moving_image_dynamics(self, original_moving_image, transformation):
r"""This is called by the optimizer just after setting the moving image.
EMMetric takes advantage of the image dynamics by computing the
current moving image mask from the original_moving_image mask (warped
by nearest neighbor interpolation)
Parameters
----------
original_moving_image : array, shape (R, C) or (S, R, C)
the original moving image from which the current moving image was
generated, the current moving image is the one that was provided
via 'set_moving_image(...)', which may not be the same as the
original moving image but a warped version of it.
transformation : DiffeomorphicMap object
the transformation that was applied to the original_moving_image
to generate the current moving image
"""
self.moving_image_mask = (original_moving_image > 0).astype(np.int32)
if transformation is None:
return
shape = np.array(self.moving_image.shape, dtype=np.int32)
affine = self.moving_affine
self.moving_image_mask = transformation.transform(
self.moving_image_mask,
interpolation="nearest",
image_world2grid=None,
out_shape=shape,
out_grid2world=affine,
)
[docs]
class SSDMetric(SimilarityMetric):
@warning_for_keywords()
def __init__(self, dim, *, smooth=4, inner_iter=10, step_type="demons"):
r"""Sum of Squared Differences (SSD) Metric
Similarity metric for (mono-modal) nonlinear image registration defined
by the sum of squared differences (SSD)
Parameters
----------
dim : int (either 2 or 3)
the dimension of the image domain
smooth : float
smoothness parameter, the larger the value the smoother the
deformation field
inner_iter : int
number of iterations to be performed at each level of the multi-
resolution Gauss-Seidel optimization algorithm (this is not the
number of steps per Gaussian Pyramid level, that parameter must
be set for the optimizer, not the metric)
step_type : string
the displacement field step to be computed when 'compute_forward'
and 'compute_backward' are called. Either 'demons' or
'gauss_newton'
"""
super(SSDMetric, self).__init__(dim)
self.smooth = smooth
self.inner_iter = inner_iter
self.step_type = step_type
self.levels_below = 0
self._connect_functions()
def _connect_functions(self):
r"""Assign the methods to be called according to the image dimension
Assigns the appropriate functions to be called for vector field
reorientation and displacement field steps according to the
dimension of the input images and the select type of step (either
Demons or Gauss Newton)
"""
if self.dim == 2:
self.reorient_vector_field = vfu.reorient_vector_field_2d
elif self.dim == 3:
self.reorient_vector_field = vfu.reorient_vector_field_3d
else:
raise ValueError(f"SSD Metric not defined for dim. {self.dim}")
if self.step_type == "gauss_newton":
self.compute_step = self.compute_gauss_newton_step
elif self.step_type == "demons":
self.compute_step = self.compute_demons_step
else:
raise ValueError(f"Opt. step {self.step_type} not defined")
[docs]
def initialize_iteration(self):
r"""Prepares the metric to compute one displacement field iteration.
Pre-computes the gradient of the input images to be used in the
computation of the forward and backward steps.
"""
self.gradient_moving = np.empty(
shape=self.moving_image.shape + (self.dim,), dtype=floating
)
for i, grad in enumerate(gradient(self.moving_image)):
self.gradient_moving[..., i] = grad
# Convert static image's gradient field from voxel to physical space
if self.moving_spacing is not None:
self.gradient_moving /= self.moving_spacing
if self.moving_direction is not None:
self.reorient_vector_field(self.gradient_moving, self.moving_direction)
self.gradient_static = np.empty(
shape=self.static_image.shape + (self.dim,), dtype=floating
)
for i, grad in enumerate(gradient(self.static_image)):
self.gradient_static[..., i] = grad
# Convert static image's gradient field from voxel to physical space
if self.static_spacing is not None:
self.gradient_static /= self.static_spacing
if self.static_direction is not None:
self.reorient_vector_field(self.gradient_static, self.static_direction)
[docs]
def compute_forward(self):
r"""Computes one step bringing the reference image towards the static.
Computes the update displacement field to be used for registration of
the moving image towards the static image
"""
return self.compute_step(forward_step=True)
[docs]
def compute_backward(self):
r"""Computes one step bringing the static image towards the moving.
Computes the updated displacement field to be used for registration of
the static image towards the moving image
"""
return self.compute_step(forward_step=False)
[docs]
@warning_for_keywords()
def compute_gauss_newton_step(self, *, forward_step=True):
r"""Computes the Gauss-Newton energy minimization step
Minimizes the linearized energy function (Newton step) defined by the
sum of squared differences of corresponding pixels of the input images
with respect to the displacement field.
Parameters
----------
forward_step : boolean
if True, computes the Newton step in the forward direction
(warping the moving towards the static image). If False,
computes the backward step (warping the static image to the
moving image)
Returns
-------
displacement : array, shape = static_image.shape + (3,)
if forward_step==True, the forward SSD Gauss-Newton step,
else, the backward step
"""
reference_shape = self.static_image.shape
if forward_step:
gradient = self.gradient_static
delta_field = self.static_image - self.moving_image
else:
gradient = self.gradient_moving
delta_field = self.moving_image - self.static_image
displacement = np.zeros(shape=reference_shape + (self.dim,), dtype=floating)
if self.dim == 2:
self.energy = v_cycle_2d(
self.levels_below,
self.inner_iter,
delta_field,
None,
gradient,
None,
self.smooth,
displacement,
)
else:
self.energy = v_cycle_3d(
self.levels_below,
self.inner_iter,
delta_field,
None,
gradient,
None,
self.smooth,
displacement,
)
return displacement
[docs]
@warning_for_keywords()
def compute_demons_step(self, *, forward_step=True):
r"""Demons step for SSD metric
Computes the demons step proposed by :footcite:t:`Vercauteren2009` for
the SSD metric.
Parameters
----------
forward_step : boolean
if True, computes the Demons step in the forward direction
(warping the moving towards the static image). If False,
computes the backward step (warping the static image to the
moving image)
Returns
-------
displacement : array, shape (R, C, 2) or (S, R, C, 3)
the Demons step
References
----------
.. footbibliography::
"""
sigma_reg_2 = np.sum(self.static_spacing**2) / self.dim
if forward_step:
gradient = self.gradient_static
delta_field = self.static_image - self.moving_image
else:
gradient = self.gradient_moving
delta_field = self.moving_image - self.static_image
if self.dim == 2:
step, self.energy = ssd.compute_ssd_demons_step_2d(
delta_field, gradient, sigma_reg_2, None
)
else:
step, self.energy = ssd.compute_ssd_demons_step_3d(
delta_field, gradient, sigma_reg_2, None
)
for i in range(self.dim):
step[..., i] = ndimage.gaussian_filter(step[..., i], self.smooth)
return step
[docs]
def get_energy(self):
r"""The numerical value assigned by this metric to the current image pair
Returns the Sum of Squared Differences (data term) energy computed at
the largest iteration
"""
return self.energy
[docs]
def free_iteration(self):
r"""
Nothing to free for the SSD metric
"""
pass
[docs]
@warning_for_keywords()
def v_cycle_2d(
n,
k,
delta_field,
sigma_sq_field,
gradient_field,
target,
lambda_param,
displacement,
*,
depth=0,
):
r"""Multi-resolution Gauss-Seidel solver using V-type cycles
Multi-resolution Gauss-Seidel solver: solves the Gauss-Newton linear system
by first filtering (GS-iterate) the current level, then solves for the
residual at a coarser resolution and finally refines the solution at the
current resolution. This scheme corresponds to the V-cycle proposed by
:footcite:t:`Bruhn2005`.
Parameters
----------
n : int
number of levels of the multi-resolution algorithm (it will be called
recursively until level n == 0)
k : int
the number of iterations at each multi-resolution level
delta_field : array, shape (R, C)
the difference between the static and moving image (the 'derivative
w.r.t. time' in the optical flow model)
sigma_sq_field : array, shape (R, C)
the variance of the gray level value at each voxel, according to the
EM model (for SSD, it is 1 for all voxels). Inf and 0 values
are processed specially to support infinite and zero variance.
gradient_field : array, shape (R, C, 2)
the gradient of the moving image
target : array, shape (R, C, 2)
right-hand side of the linear system to be solved in the Weickert's
multi-resolution algorithm
lambda_param : float
smoothness parameter, the larger its value the smoother the
displacement field
displacement : array, shape (R, C, 2)
the displacement field to start the optimization from
Returns
-------
energy : the energy of the EM (or SSD if sigmafield[...]==1) metric at this
iteration
References
----------
.. footbibliography::
"""
# pre-smoothing
for _ in range(k):
ssd.iterate_residual_displacement_field_ssd_2d(
delta_field,
sigma_sq_field,
gradient_field,
target,
lambda_param,
displacement,
)
if n == 0:
energy = ssd.compute_energy_ssd_2d(delta_field)
return energy
# solve at coarser grid
residual = None
residual = ssd.compute_residual_displacement_field_ssd_2d(
delta_field,
sigma_sq_field,
gradient_field,
target,
lambda_param,
displacement,
residual,
)
sub_residual = np.array(vfu.downsample_displacement_field_2d(residual))
del residual
subsigma_sq_field = None
if sigma_sq_field is not None:
subsigma_sq_field = vfu.downsample_scalar_field_2d(sigma_sq_field)
subdelta_field = vfu.downsample_scalar_field_2d(delta_field)
subgradient_field = np.array(vfu.downsample_displacement_field_2d(gradient_field))
shape = np.array(displacement.shape).astype(np.int32)
half_shape = ((shape[0] + 1) // 2, (shape[1] + 1) // 2, 2)
sub_displacement = np.zeros(shape=half_shape, dtype=floating)
sublambda_param = lambda_param * 0.25
v_cycle_2d(
n - 1,
k,
subdelta_field,
subsigma_sq_field,
subgradient_field,
sub_residual,
sublambda_param,
sub_displacement,
depth=depth + 1,
)
# displacement += np.array(
# vfu.upsample_displacement_field(sub_displacement, shape))
displacement += vfu.resample_displacement_field_2d(
sub_displacement, np.array([0.5, 0.5]), shape
)
# post-smoothing
for _ in range(k):
ssd.iterate_residual_displacement_field_ssd_2d(
delta_field,
sigma_sq_field,
gradient_field,
target,
lambda_param,
displacement,
)
energy = ssd.compute_energy_ssd_2d(delta_field)
return energy
[docs]
@warning_for_keywords()
def v_cycle_3d(
n,
k,
delta_field,
sigma_sq_field,
gradient_field,
target,
lambda_param,
displacement,
*,
depth=0,
):
r"""Multi-resolution Gauss-Seidel solver using V-type cycles
Multi-resolution Gauss-Seidel solver: solves the linear system by first
filtering (GS-iterate) the current level, then solves for the residual
at a coarser resolution and finally refines the solution at the current
resolution. This scheme corresponds to the V-cycle proposed by
:footcite:t:`Bruhn2005`.
Parameters
----------
n : int
number of levels of the multi-resolution algorithm (it will be called
recursively until level n == 0)
k : int
the number of iterations at each multi-resolution level
delta_field : array, shape (S, R, C)
the difference between the static and moving image (the 'derivative
w.r.t. time' in the optical flow model)
sigma_sq_field : array, shape (S, R, C)
the variance of the gray level value at each voxel, according to the
EM model (for SSD, it is 1 for all voxels). Inf and 0 values
are processed specially to support infinite and zero variance.
gradient_field : array, shape (S, R, C, 3)
the gradient of the moving image
target : array, shape (S, R, C, 3)
right-hand side of the linear system to be solved in the Weickert's
multi-resolution algorithm
lambda_param : float
smoothness parameter, the larger its value the smoother the
displacement field
displacement : array, shape (S, R, C, 3)
the displacement field to start the optimization from
Returns
-------
energy : the energy of the EM (or SSD if sigmafield[...]==1) metric at this
iteration
References
----------
.. footbibliography::
"""
# pre-smoothing
for _ in range(k):
ssd.iterate_residual_displacement_field_ssd_3d(
delta_field,
sigma_sq_field,
gradient_field,
target,
lambda_param,
displacement,
)
if n == 0:
energy = ssd.compute_energy_ssd_3d(delta_field)
return energy
# solve at coarser grid
residual = ssd.compute_residual_displacement_field_ssd_3d(
delta_field,
sigma_sq_field,
gradient_field,
target,
lambda_param,
displacement,
None,
)
sub_residual = np.array(vfu.downsample_displacement_field_3d(residual))
del residual
subsigma_sq_field = None
if sigma_sq_field is not None:
subsigma_sq_field = vfu.downsample_scalar_field_3d(sigma_sq_field)
subdelta_field = vfu.downsample_scalar_field_3d(delta_field)
subgradient_field = np.array(vfu.downsample_displacement_field_3d(gradient_field))
shape = np.array(displacement.shape).astype(np.int32)
sub_displacement = np.zeros(
shape=((shape[0] + 1) // 2, (shape[1] + 1) // 2, (shape[2] + 1) // 2, 3),
dtype=floating,
)
sublambda_param = lambda_param * 0.25
v_cycle_3d(
n - 1,
k,
subdelta_field,
subsigma_sq_field,
subgradient_field,
sub_residual,
sublambda_param,
sub_displacement,
depth=depth + 1,
)
del subdelta_field
del subsigma_sq_field
del subgradient_field
del sub_residual
displacement += vfu.resample_displacement_field_3d(
sub_displacement, 0.5 * np.ones(3), shape
)
del sub_displacement
# post-smoothing
for _ in range(k):
ssd.iterate_residual_displacement_field_ssd_3d(
delta_field,
sigma_sq_field,
gradient_field,
target,
lambda_param,
displacement,
)
energy = ssd.compute_energy_ssd_3d(delta_field)
return energy