/usr/lib/python2.7/dist-packages/dipy/align/imwarp.py is in python-dipy 0.10.1-1.
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from __future__ import print_function
import abc
from dipy.utils.six import with_metaclass
import numpy as np
import numpy.linalg as npl
import scipy as sp
import nibabel as nib
from . import vector_fields as vfu
from . import floating
from . import VerbosityLevels
from . import Bunch
from .scalespace import ScaleSpace
RegistrationStages = Bunch(INIT_START=0,
INIT_END=1,
OPT_START=2,
OPT_END=3,
SCALE_START=4,
SCALE_END=5,
ITER_START=6,
ITER_END=7)
r"""Registration Stages
This enum defines the different stages which the Volumetric Registration
may be in. The value of the stage is passed as a parameter to the call-back
function so that it can react accordingly.
INIT_START: optimizer initialization starts
INIT_END: optimizer initialization ends
OPT_START: optimization starts
OPT_END: optimization ends
SCALE_START: optimization at a new scale space resolution starts
SCALE_END: optimization at the current scale space resolution ends
ITER_START: a new iteration starts
ITER_END: the current iteration ends
"""
def mult_aff(A, B):
r"""Returns the matrix product A.dot(B) considering None as the identity
Parameters
----------
A : array, shape (n,k)
B : array, shape (k,m)
Returns
-------
The matrix product A.dot(B). If any of the input matrices is None, it is
treated as the identity matrix. If both matrices are None, None is returned
"""
if A is None:
return B
elif B is None:
return A
return A.dot(B)
def get_direction_and_spacings(affine, dim):
r"""Extracts the rotational and spacing components from a matrix
Extracts the rotational and spacing (voxel dimensions) components from a
matrix. An image gradient represents the local variation of the image's
gray values per voxel. Since we are iterating on the physical space, we
need to compute the gradients as variation per millimeter, so we need to
divide each gradient's component by the voxel size along the corresponding
axis, that's what the spacings are used for. Since the image's gradients
are oriented along the grid axes, we also need to re-orient the gradients
to be given in physical space coordinates.
Parameters
----------
affine : array, shape (k, k), k = 3, 4
the matrix transforming grid coordinates to physical space.
Returns
-------
direction : array, shape (k-1, k-1)
the rotational component of the input matrix
spacings : array, shape (k-1,)
the scaling component (voxel size) of the matrix
"""
if affine is None:
return np.eye(dim), np.ones(dim)
dim = affine.shape[1]-1
# Temporary hack: get the zooms by building a nifti image
affine4x4 = np.eye(4)
empty_volume = np.zeros((0, 0, 0))
affine4x4[:dim, :dim] = affine[:dim, :dim]
affine4x4[:dim, 3] = affine[:dim, dim-1]
nib_nifti = nib.Nifti1Image(empty_volume, affine4x4)
scalings = np.asarray(nib_nifti.get_header().get_zooms())
scalings = np.asarray(scalings[:dim], dtype=np.float64)
A = affine[:dim, :dim]
return A.dot(np.diag(1.0/scalings)), scalings
class DiffeomorphicMap(object):
def __init__(self,
dim,
disp_shape,
disp_grid2world=None,
domain_shape=None,
domain_grid2world=None,
codomain_shape=None,
codomain_grid2world=None,
prealign=None):
r""" DiffeomorphicMap
Implements a diffeomorphic transformation on the physical space. The
deformation fields encoding the direct and inverse transformations
share the same domain discretization (both the discretization grid
shape and voxel-to-space matrix). The input coordinates (physical
coordinates) are first aligned using prealign, and then displaced
using the corresponding vector field interpolated at the aligned
coordinates.
Parameters
----------
dim : int, 2 or 3
the transformation's dimension
disp_shape : array, shape (dim,)
the number of slices (if 3D), rows and columns of the deformation
field's discretization
disp_grid2world : the voxel-to-space transform between the def. fields
grid and space
domain_shape : array, shape (dim,)
the number of slices (if 3D), rows and columns of the default
discretizatio of this map's domain
domain_grid2world : array, shape (dim+1, dim+1)
the default voxel-to-space transformation between this map's
discretization and physical space
codomain_shape : array, shape (dim,)
the number of slices (if 3D), rows and columns of the images that
are 'normally' warped using this transformation in the forward
direction (this will provide default transformation parameters to
warp images under this transformation). By default, we assume that
the inverse transformation is 'normally' used to warp images with
the same discretization and voxel-to-space transformation as the
deformation field grid.
codomain_grid2world : array, shape (dim+1, dim+1)
the voxel-to-space transformation of images that are 'normally'
warped using this transformation (in the forward direction).
prealign : array, shape (dim+1, dim+1)
the linear transformation to be applied to align input images to
the reference space before warping under the deformation field.
"""
self.dim = dim
if(disp_shape is None):
raise ValueError("Invalid displacement field discretization")
self.disp_shape = np.asarray(disp_shape, dtype=np.int32)
# If the discretization affine is None, we assume it's the identity
self.disp_grid2world = disp_grid2world
if(self.disp_grid2world is None):
self.disp_world2grid = None
else:
self.disp_world2grid = npl.inv(self.disp_grid2world)
# If domain_shape isn't provided, we use the map's discretization shape
if(domain_shape is None):
self.domain_shape = self.disp_shape
else:
self.domain_shape = np.asarray(domain_shape, dtype=np.int32)
self.domain_grid2world = domain_grid2world
if(domain_grid2world is None):
self.domain_world2grid = None
else:
self.domain_world2grid = npl.inv(domain_grid2world)
# If codomain shape was not provided, we assume it is an endomorphism:
# use the same domain_shape and codomain_grid2world as the field domain
if codomain_shape is None:
self.codomain_shape = self.domain_shape
else:
self.codomain_shape = np.asarray(codomain_shape, dtype=np.int32)
self.codomain_grid2world = codomain_grid2world
if codomain_grid2world is None:
self.codomain_world2grid = None
else:
self.codomain_world2grid = npl.inv(codomain_grid2world)
self.prealign = prealign
if prealign is None:
self.prealign_inv = None
else:
self.prealign_inv = npl.inv(prealign)
self.is_inverse = False
self.forward = None
self.backward = None
def interpret_matrix(self, obj):
''' Try to interpret `obj` as a matrix
Some operations are performed faster if we know in advance if a matrix
is the identity (so we can skip the actual matrix-vector
multiplication). This function returns None if the given object
is None or the 'identity' string. It returns the same object if it is
a numpy array. It raises an exception otherwise.
Parameters
----------
obj : object
any object
Returns
----------
obj : object
the same object given as argument if `obj` is None or a numpy
array. None if `obj` is the 'identity' string.
'''
if (obj is None) or isinstance(obj, np.ndarray):
return obj
if isinstance(obj, str) and (obj == 'identity'):
return None
raise ValueError('Invalid matrix')
def get_forward_field(self):
r"""Deformation field to transform an image in the forward direction
Returns the deformation field that must be used to warp an image under
this transformation in the forward direction (note the 'is_inverse'
flag).
"""
if self.is_inverse:
return self.backward
else:
return self.forward
def get_backward_field(self):
r"""Deformation field to transform an image in the backward direction
Returns the deformation field that must be used to warp an image under
this transformation in the backward direction (note the 'is_inverse'
flag).
"""
if self.is_inverse:
return self.forward
else:
return self.backward
def allocate(self):
r"""Creates a zero displacement field
Creates a zero displacement field (the identity transformation).
"""
self.forward = np.zeros(tuple(self.disp_shape) + (self.dim,),
dtype=floating)
self.backward = np.zeros(tuple(self.disp_shape) + (self.dim,),
dtype=floating)
def _get_warping_function(self, interpolation):
r"""Appropriate warping function for the given interpolation type
Returns the right warping function from vector_fields that must be
called for the specified data dimension and interpolation type
"""
if self.dim == 2:
if interpolation == 'linear':
return vfu.warp_2d
else:
return vfu.warp_2d_nn
else:
if interpolation == 'linear':
return vfu.warp_3d
else:
return vfu.warp_3d_nn
def _warp_forward(self, image, interpolation='linear',
image_world2grid=None, out_shape=None,
out_grid2world=None):
r"""Warps an image in the forward direction
Deforms the input image under this diffeomorphic map in the forward
direction. Since the mapping is defined in the physical space, the user
must specify the sampling grid shape and its space-to-voxel mapping.
By default, the transformation will use the discretization information
given at initialization.
Parameters
----------
image : array, shape (s, r, c) if dim = 3 or (r, c) if dim = 2
the image to be warped under this transformation in the forward
direction
interpolation : string, either 'linear' or 'nearest'
the type of interpolation to be used for warping, either 'linear'
(for k-linear interpolation) or 'nearest' for nearest neighbor
image_world2grid : array, shape (dim+1, dim+1)
the transformation bringing world (space) coordinates to voxel
coordinates of the image given as input
out_shape : array, shape (dim,)
the number of slices, rows and columns of the desired warped image
out_grid2world : the transformation bringing voxel coordinates of the
warped image to physical space
Returns
-------
warped : array, shape = out_shape or self.codomain_shape if None
the warped image under this transformation in the forward direction
Notes
-----
A diffeomorphic map must be thought as a mapping between points
in space. Warping an image J towards an image I means transforming
each voxel with (discrete) coordinates i in I to (floating-point) voxel
coordinates j in J. The transformation we consider 'forward' is
precisely mapping coordinates i from the input image to coordinates j
from reference image, which has the effect of warping an image with
reference discretization (typically, the "static image") "towards" an
image with input discretization (typically, the "moving image"). More
precisely, the warped image is produced by the following interpolation:
warped[i] = image[W * forward[Dinv * P * S * i] + W * P * S * i )]
where i denotes the coordinates of a voxel in the input grid, W is
the world-to-grid transformation of the image given as input, Dinv
is the world-to-grid transformation of the deformation field
discretization, P is the pre-aligning matrix (transforming input
points to reference points), S is the voxel-to-space transformation of
the sampling grid (see comment below) and forward is the forward
deformation field.
If we want to warp an image, we also must specify on what grid we
want to sample the resulting warped image (the images are considered as
points in space and its representation on a grid depends on its
grid-to-space transform telling us for each grid voxel what point in
space we need to bring via interpolation). So, S is the matrix that
converts the sampling grid (whose shape is given as parameter
'out_shape' ) to space coordinates.
"""
# if no world-to-image transform is provided, we use the codomain info
if image_world2grid is None:
image_world2grid = self.codomain_world2grid
# if no sampling info is provided, we use the domain info
if out_shape is None:
if self.domain_shape is None:
raise ValueError('Unable to infer sampling info. '
'Provide a valid out_shape.')
out_shape = self.domain_shape
else:
out_shape = np.asarray(out_shape, dtype=np.int32)
if out_grid2world is None:
out_grid2world = self.domain_grid2world
W = self.interpret_matrix(image_world2grid)
Dinv = self.disp_world2grid
P = self.prealign
S = self.interpret_matrix(out_grid2world)
# this is the matrix which we need to multiply the voxel coordinates
# to interpolate on the forward displacement field ("in"side the
# 'forward' brackets in the expression above)
affine_idx_in = mult_aff(Dinv, mult_aff(P, S))
# this is the matrix which we need to multiply the voxel coordinates
# to add to the displacement ("out"side the 'forward' brackets in the
# expression above)
affine_idx_out = mult_aff(W, mult_aff(P, S))
# this is the matrix which we need to multiply the displacement vector
# prior to adding to the transformed input point
affine_disp = W
# Convert the data to required types to use the cythonized functions
if interpolation == 'nearest':
if image.dtype is np.dtype('float64') and floating is np.float32:
image = image.astype(floating)
elif image.dtype is np.dtype('int64'):
image = image.astype(np.int32)
else:
image = np.asarray(image, dtype=floating)
warp_f = self._get_warping_function(interpolation)
warped = warp_f(image, self.forward, affine_idx_in, affine_idx_out,
affine_disp, out_shape)
return warped
def _warp_backward(self, image, interpolation='linear',
image_world2grid=None, out_shape=None,
out_grid2world=None):
r"""Warps an image in the backward direction
Deforms the input image under this diffeomorphic map in the backward
direction. Since the mapping is defined in the physical space, the user
must specify the sampling grid shape and its space-to-voxel mapping.
By default, the transformation will use the discretization information
given at initialization.
Parameters
----------
image : array, shape (s, r, c) if dim = 3 or (r, c) if dim = 2
the image to be warped under this transformation in the backward
direction
interpolation : string, either 'linear' or 'nearest'
the type of interpolation to be used for warping, either 'linear'
(for k-linear interpolation) or 'nearest' for nearest neighbor
image_world2grid : array, shape (dim+1, dim+1)
the transformation bringing world (space) coordinates to voxel
coordinates of the image given as input
out_shape : array, shape (dim,)
the number of slices, rows and columns of the desired warped image
out_grid2world : the transformation bringing voxel coordinates of the
warped image to physical space
Returns
-------
warped : array, shape = out_shape or self.domain_shape if None
the warped image under this transformation in the backward
direction
Notes
-----
A diffeomorphic map must be thought as a mapping between points
in space. Warping an image J towards an image I means transforming
each voxel with (discrete) coordinates i in I to (floating-point) voxel
coordinates j in J. The transformation we consider 'backward' is
precisely mapping coordinates i from the reference grid to coordinates
j from the input image (that's why it's "backward"), which has the
effect of warping the input image (moving) "towards" the reference.
More precisely, the warped image is produced by the following
interpolation:
warped[i]=image[W * Pinv * backward[Dinv * S * i] + W * Pinv * S * i )]
where i denotes the coordinates of a voxel in the input grid, W is
the world-to-grid transformation of the image given as input, Dinv
is the world-to-grid transformation of the deformation field
discretization, Pinv is the pre-aligning matrix's inverse (transforming
reference points to input points), S is the grid-to-space
transformation of the sampling grid (see comment below) and backward is
the backward deformation field.
If we want to warp an image, we also must specify on what grid we
want to sample the resulting warped image (the images are considered as
points in space and its representation on a grid depends on its
grid-to-space transform telling us for each grid voxel what point in
space we need to bring via interpolation). So, S is the matrix that
converts the sampling grid (whose shape is given as parameter
'out_shape' ) to space coordinates.
"""
# if no world-to-image transform is provided, we use the domain info
if image_world2grid is None:
image_world2grid = self.domain_world2grid
# if no sampling info is provided, we use the codomain info
if out_shape is None:
if self.codomain_shape is None:
msg = 'Unknown sampling info. Provide a valid out_shape.'
raise ValueError(msg)
out_shape = self.codomain_shape
if out_grid2world is None:
out_grid2world = self.codomain_grid2world
W = self.interpret_matrix(image_world2grid)
Dinv = self.disp_world2grid
Pinv = self.prealign_inv
S = self.interpret_matrix(out_grid2world)
# this is the matrix which we need to multiply the voxel coordinates
# to interpolate on the backward displacement field ("in"side the
# 'backward' brackets in the expression above)
affine_idx_in = mult_aff(Dinv, S)
# this is the matrix which we need to multiply the voxel coordinates
# to add to the displacement ("out"side the 'backward' brackets in the
# expression above)
affine_idx_out = mult_aff(W, mult_aff(Pinv, S))
# this is the matrix which we need to multiply the displacement vector
# prior to adding to the transformed input point
affine_disp = mult_aff(W, Pinv)
if interpolation == 'nearest':
if image.dtype is np.dtype('float64') and floating is np.float32:
image = image.astype(floating)
elif image.dtype is np.dtype('int64'):
image = image.astype(np.int32)
else:
image = np.asarray(image, dtype=floating)
warp_f = self._get_warping_function(interpolation)
warped = warp_f(image, self.backward, affine_idx_in, affine_idx_out,
affine_disp, out_shape)
return warped
def transform(self, image, interpolation='linear', image_world2grid=None,
out_shape=None, out_grid2world=None):
r"""Warps an image in the forward direction
Transforms the input image under this transformation in the forward
direction. It uses the "is_inverse" flag to switch between "forward"
and "backward" (if is_inverse is False, then transform(...) warps the
image forwards, else it warps the image backwards).
Parameters
----------
image : array, shape (s, r, c) if dim = 3 or (r, c) if dim = 2
the image to be warped under this transformation in the forward
direction
interpolation : string, either 'linear' or 'nearest'
the type of interpolation to be used for warping, either 'linear'
(for k-linear interpolation) or 'nearest' for nearest neighbor
image_world2grid : array, shape (dim+1, dim+1)
the transformation bringing world (space) coordinates to voxel
coordinates of the image given as input
out_shape : array, shape (dim,)
the number of slices, rows and columns of the desired warped image
out_grid2world : the transformation bringing voxel coordinates of the
warped image to physical space
Returns
-------
warped : array, shape = out_shape or self.codomain_shape if None
the warped image under this transformation in the forward direction
Notes
-----
See _warp_forward and _warp_backward documentation for further
information.
"""
if out_shape is not None:
out_shape = np.asarray(out_shape, dtype=np.int32)
if self.is_inverse:
warped = self._warp_backward(image, interpolation,
image_world2grid, out_shape,
out_grid2world)
else:
warped = self._warp_forward(image, interpolation, image_world2grid,
out_shape, out_grid2world)
return np.asarray(warped)
def transform_inverse(self, image, interpolation='linear',
image_world2grid=None, out_shape=None,
out_grid2world=None):
r"""Warps an image in the backward direction
Transforms the input image under this transformation in the backward
direction. It uses the "is_inverse" flag to switch between "forward"
and "backward" (if is_inverse is False, then transform_inverse(...)
warps the image backwards, else it warps the image forwards)
Parameters
----------
image : array, shape (s, r, c) if dim = 3 or (r, c) if dim = 2
the image to be warped under this transformation in the forward
direction
interpolation : string, either 'linear' or 'nearest'
the type of interpolation to be used for warping, either 'linear'
(for k-linear interpolation) or 'nearest' for nearest neighbor
image_world2grid : array, shape (dim+1, dim+1)
the transformation bringing world (space) coordinates to voxel
coordinates of the image given as input
out_shape : array, shape (dim,)
the number of slices, rows and columns of the desired warped image
out_grid2world : the transformation bringing voxel coordinates of the
warped image to physical space
Returns
-------
warped : array, shape = out_shape or self.codomain_shape if None
warped image under this transformation in the backward direction
Notes
-----
See _warp_forward and _warp_backward documentation for further
information.
"""
if self.is_inverse:
warped = self._warp_forward(image, interpolation, image_world2grid,
out_shape, out_grid2world)
else:
warped = self._warp_backward(image, interpolation,
image_world2grid, out_shape,
out_grid2world)
return np.asarray(warped)
def inverse(self):
r"""Inverse of this DiffeomorphicMap instance
Returns a diffeomorphic map object representing the inverse of this
transformation. The internal arrays are not copied but just referenced.
Returns
-------
inv : DiffeomorphicMap object
the inverse of this diffeomorphic map.
"""
inv = DiffeomorphicMap(self.dim,
self.disp_shape,
self.disp_grid2world,
self.domain_shape,
self.domain_grid2world,
self.codomain_shape,
self.codomain_grid2world,
self.prealign)
inv.forward = self.forward
inv.backward = self.backward
inv.is_inverse = True
return inv
def expand_fields(self, expand_factors, new_shape):
r"""Expands the displacement fields from current shape to new_shape
Up-samples the discretization of the displacement fields to be of
new_shape shape.
Parameters
----------
expand_factors : array, shape (dim,)
the factors scaling current spacings (voxel sizes) to spacings in
the expanded discretization.
new_shape : array, shape (dim,)
the shape of the arrays holding the up-sampled discretization
"""
if self.dim == 2:
expand_f = vfu.resample_displacement_field_2d
else:
expand_f = vfu.resample_displacement_field_3d
expanded_forward = expand_f(self.forward, expand_factors, new_shape)
expanded_backward = expand_f(self.backward, expand_factors, new_shape)
expand_factors = np.append(expand_factors, [1])
expanded_grid2world = mult_aff(self.disp_grid2world,
np.diag(expand_factors))
expanded_world2grid = npl.inv(expanded_grid2world)
self.forward = expanded_forward
self.backward = expanded_backward
self.disp_shape = new_shape
self.disp_grid2world = expanded_grid2world
self.disp_world2grid = expanded_world2grid
def compute_inversion_error(self):
r"""Inversion error of the displacement fields
Estimates the inversion error of the displacement fields by computing
statistics of the residual vectors obtained after composing the forward
and backward displacement fields.
Returns
-------
residual : array, shape (R, C) or (S, R, C)
the displacement field resulting from composing the forward and
backward displacement fields of this transformation (the residual
should be zero for a perfect diffeomorphism)
stats : array, shape (3,)
statistics from the norms of the vectors of the residual
displacement field: maximum, mean and standard deviation
Notes
-----
Since the forward and backward displacement fields have the same
discretization, the final composition is given by
comp[i] = forward[ i + Dinv * backward[i]]
where Dinv is the space-to-grid transformation of the displacement
fields
"""
Dinv = self.disp_world2grid
if self.dim == 2:
compose_f = vfu.compose_vector_fields_2d
else:
compose_f = vfu.compose_vector_fields_3d
residual, stats = compose_f(self.backward, self.forward,
None, Dinv, 1.0, None)
return np.asarray(residual), np.asarray(stats)
def shallow_copy(self):
r"""Shallow copy of this DiffeomorphicMap instance
Creates a shallow copy of this diffeomorphic map (the arrays are not
copied but just referenced)
Returns
-------
new_map : DiffeomorphicMap object
the shallow copy of this diffeomorphic map
"""
new_map = DiffeomorphicMap(self.dim,
self.disp_shape,
self.disp_grid2world,
self.domain_shape,
self.domain_grid2world,
self.codomain_shape,
self.codomain_grid2world,
self.prealign)
new_map.forward = self.forward
new_map.backward = self.backward
new_map.is_inverse = self.is_inverse
return new_map
def warp_endomorphism(self, phi):
r"""Composition of this DiffeomorphicMap with a given endomorphism
Creates a new DiffeomorphicMap C with the same properties as self and
composes its displacement fields with phi's corresponding fields.
The resulting diffeomorphism is of the form C(x) = phi(self(x)) with
inverse C^{-1}(y) = self^{-1}(phi^{-1}(y)). We assume that phi is an
endomorphism with the same discretization and domain affine as self
to ensure that the composition inherits self's properties (we also
assume that the pre-aligning matrix of phi is None or identity).
Parameters
----------
phi : DiffeomorphicMap object
the endomorphism to be warped by this diffeomorphic map
Returns
-------
composition : the composition of this diffeomorphic map with the
endomorphism given as input
Notes
-----
The problem with our current representation of a DiffeomorphicMap is
that the set of Diffeomorphism that can be represented this way (a
pre-aligning matrix followed by a non-linear endomorphism given as a
displacement field) is not closed under the composition operation.
Supporting a general DiffeomorphicMap class, closed under composition,
may be extremely costly computationally, and the kind of
transformations we actually need for Avants' mid-point algorithm (SyN)
are much simpler.
"""
# Compose the forward deformation fields
d1 = self.get_forward_field()
d2 = phi.get_forward_field()
d1_inv = self.get_backward_field()
d2_inv = phi.get_backward_field()
premult_disp = self.disp_world2grid
if self.dim == 2:
compose_f = vfu.compose_vector_fields_2d
else:
compose_f = vfu.compose_vector_fields_3d
forward, stats = compose_f(d1, d2, None, premult_disp, 1.0, None)
backward, stats, = compose_f(d2_inv, d1_inv, None, premult_disp, 1.0,
None)
composition = self.shallow_copy()
composition.forward = forward
composition.backward = backward
return composition
def get_simplified_transform(self):
r""" Constructs a simplified version of this Diffeomorhic Map
The simplified version incorporates the pre-align transform, as well as
the domain and codomain affine transforms into the displacement field.
The resulting transformation may be regarded as operating on the
image spaces given by the domain and codomain discretization. As a
result, self.prealign, self.disp_grid2world, self.domain_grid2world and
self.codomain affine will be None (denoting Identity) in the resulting
diffeomorphic map.
"""
if self.dim == 2:
simplify_f = vfu.simplify_warp_function_2d
else:
simplify_f = vfu.simplify_warp_function_3d
# Simplify the forward transform
D = self.domain_grid2world
P = self.prealign
Rinv = self.disp_world2grid
Cinv = self.codomain_world2grid
# this is the matrix which we need to multiply the voxel coordinates
# to interpolate on the forward displacement field ("in"side the
# 'forward' brackets in the expression above)
affine_idx_in = mult_aff(Rinv, mult_aff(P, D))
# this is the matrix which we need to multiply the voxel coordinates
# to add to the displacement ("out"side the 'forward' brackets in the
# expression above)
affine_idx_out = mult_aff(Cinv, mult_aff(P, D))
# this is the matrix which we need to multiply the displacement vector
# prior to adding to the transformed input point
affine_disp = Cinv
new_forward = simplify_f(self.forward, affine_idx_in,
affine_idx_out, affine_disp,
self.domain_shape)
# Simplify the backward transform
C = self.codomain_world2grid
Pinv = self.prealign_inv
Dinv = self.domain_world2grid
affine_idx_in = mult_aff(Rinv, C)
affine_idx_out = mult_aff(Dinv, mult_aff(Pinv, C))
affine_disp = mult_aff(Dinv, Pinv)
new_backward = simplify_f(self.backward, affine_idx_in,
affine_idx_out, affine_disp,
self.codomain_shape)
simplified = DiffeomorphicMap(self.dim,
self.disp_shape,
None,
self.domain_shape,
None,
self.codomain_shape,
None,
None)
simplified.forward = new_forward
simplified.backward = new_backward
return simplified
class DiffeomorphicRegistration(with_metaclass(abc.ABCMeta, object)):
def __init__(self, metric=None):
r""" Diffeomorphic Registration
This abstract class defines the interface to be implemented by any
optimization algorithm for diffeomorphic registration.
Parameters
----------
metric : SimilarityMetric object
the object measuring the similarity of the two images. The
registration algorithm will minimize (or maximize) the provided
similarity.
"""
if metric is None:
raise ValueError('The metric cannot be None')
self.metric = metric
self.dim = metric.dim
def set_level_iters(self, level_iters):
r"""Sets the number of iterations at each pyramid level
Establishes the maximum number of iterations to be performed at each
level of the Gaussian pyramid, similar to ANTS.
Parameters
----------
level_iters : list
the number of iterations at each level of the Gaussian pyramid.
level_iters[0] corresponds to the finest level, level_iters[n-1]
the coarsest, where n is the length of the list
"""
self.levels = len(level_iters) if level_iters else 0
self.level_iters = level_iters
@abc.abstractmethod
def optimize(self):
r"""Starts the metric optimization
This is the main function each specialized class derived from this must
implement. Upon completion, the deformation field must be available
from the forward transformation model.
"""
@abc.abstractmethod
def get_map(self):
r"""
Returns the resulting diffeomorphic map after optimization
"""
class SymmetricDiffeomorphicRegistration(DiffeomorphicRegistration):
def __init__(self,
metric,
level_iters=None,
step_length=0.25,
ss_sigma_factor=0.2,
opt_tol=1e-5,
inv_iter=20,
inv_tol=1e-3,
callback=None):
r""" Symmetric Diffeomorphic Registration (SyN) Algorithm
Performs the multi-resolution optimization algorithm for non-linear
registration using a given similarity metric.
Parameters
----------
metric : SimilarityMetric object
the metric to be optimized
level_iters : list of int
the number of iterations at each level of the Gaussian Pyramid (the
length of the list defines the number of pyramid levels to be
used)
opt_tol : float
the optimization will stop when the estimated derivative of the
energy profile w.r.t. time falls below this threshold
inv_iter : int
the number of iterations to be performed by the displacement field
inversion algorithm
step_length : float
the length of the maximum displacement vector of the update
displacement field at each iteration
ss_sigma_factor : float
parameter of the scale-space smoothing kernel. For example, the
std. dev. of the kernel will be factor*(2^i) in the isotropic case
where i = 0, 1, ..., n_scales is the scale
inv_tol : float
the displacement field inversion algorithm will stop iterating
when the inversion error falls below this threshold
callback : function(SymmetricDiffeomorphicRegistration)
a function receiving a SymmetricDiffeomorphicRegistration object
to be called after each iteration (this optimizer will call this
function passing self as parameter)
"""
super(SymmetricDiffeomorphicRegistration, self).__init__(metric)
if level_iters is None:
level_iters = [100, 100, 25]
if len(level_iters) == 0:
raise ValueError('The iterations list cannot be empty')
self.set_level_iters(level_iters)
self.step_length = step_length
self.ss_sigma_factor = ss_sigma_factor
self.opt_tol = opt_tol
self.inv_tol = inv_tol
self.inv_iter = inv_iter
self.energy_window = 12
self.energy_list = []
self.full_energy_profile = []
self.verbosity = VerbosityLevels.STATUS
self.callback = callback
self.moving_ss = None
self.static_ss = None
self.static_direction = None
self.moving_direction = None
self.mask0 = metric.mask0
def update(self, current_displacement, new_displacement,
disp_world2grid, time_scaling):
r"""Composition of the current displacement field with the given field
Interpolates new displacement at the locations defined by
current_displacement. Equivalently, computes the composition C of the
given displacement fields as C(x) = B(A(x)), where A is
current_displacement and B is new_displacement. This function is
intended to be used with deformation fields of the same sampling
(e.g. to be called by a registration algorithm).
Parameters
----------
current_displacement : array, shape (R', C', 2) or (S', R', C', 3)
the displacement field defining where to interpolate
new_displacement
new_displacement : array, shape (R, C, 2) or (S, R, C, 3)
the displacement field to be warped by current_displacement
disp_world2grid : array, shape (dim+1, dim+1)
the space-to-grid transform associated with the displacements'
grid (we assume that both displacements are discretized over the
same grid)
time_scaling : float
scaling factor applied to d2. The effect may be interpreted as
moving d1 displacements along a factor (`time_scaling`) of d2.
Returns
-------
updated : array, shape (the same as new_displacement)
the warped displacement field
mean_norm : the mean norm of all vectors in current_displacement
"""
sq_field = np.sum((np.array(current_displacement) ** 2), -1)
mean_norm = np.sqrt(sq_field).mean()
# We assume that both displacement fields have the same
# grid2world transform, which implies premult_index=Identity
# and premult_disp is the world2grid transform associated with
# the displacements' grid
self.compose(current_displacement, new_displacement, None,
disp_world2grid, time_scaling, current_displacement)
return np.array(current_displacement), np.array(mean_norm)
def get_map(self):
r"""Returns the resulting diffeomorphic map
Returns the DiffeomorphicMap registering the moving image towards
the static image.
"""
return self.static_to_ref
def _connect_functions(self):
r"""Assign the methods to be called according to the image dimension
Assigns the appropriate functions to be called for displacement field
inversion, Gaussian pyramid, and affine / dense deformation composition
according to the dimension of the input images e.g. 2D or 3D.
"""
if self.dim == 2:
self.invert_vector_field = vfu.invert_vector_field_fixed_point_2d
self.compose = vfu.compose_vector_fields_2d
else:
self.invert_vector_field = vfu.invert_vector_field_fixed_point_3d
self.compose = vfu.compose_vector_fields_3d
def _init_optimizer(self, static, moving,
static_grid2world, moving_grid2world, prealign):
r"""Initializes the registration optimizer
Initializes the optimizer by computing the scale space of the input
images and allocating the required memory for the transformation models
at the coarsest scale.
Parameters
----------
static : array, shape (S, R, C) or (R, C)
the image to be used as reference during optimization. The
displacement fields will have the same discretization as the static
image.
moving : array, shape (S, R, C) or (R, C)
the image to be used as "moving" during optimization. Since the
deformation fields' discretization is the same as the static image,
it is necessary to pre-align the moving image to ensure its domain
lies inside the domain of the deformation fields. This is assumed
to be accomplished by "pre-aligning" the moving image towards the
static using an affine transformation given by the 'prealign'
matrix
static_grid2world : array, shape (dim+1, dim+1)
the voxel-to-space transformation associated to the static image
moving_grid2world : array, shape (dim+1, dim+1)
the voxel-to-space transformation associated to the moving image
prealign : array, shape (dim+1, dim+1)
the affine transformation (operating on the physical space)
pre-aligning the moving image towards the static
"""
self._connect_functions()
# Extract information from affine matrices to create the scale space
static_direction, static_spacing = \
get_direction_and_spacings(static_grid2world, self.dim)
moving_direction, moving_spacing = \
get_direction_and_spacings(moving_grid2world, self.dim)
# the images' directions don't change with scale
self.static_direction = np.eye(self.dim + 1)
self.moving_direction = np.eye(self.dim + 1)
self.static_direction[:self.dim, :self.dim] = static_direction
self.moving_direction[:self.dim, :self.dim] = moving_direction
# Build the scale space of the input images
if self.verbosity >= VerbosityLevels.DIAGNOSE:
print('Applying zero mask: ' + str(self.mask0))
if self.verbosity >= VerbosityLevels.STATUS:
print('Creating scale space from the moving image. Levels: %d. '
'Sigma factor: %f.' % (self.levels, self.ss_sigma_factor))
self.moving_ss = ScaleSpace(moving, self.levels, moving_grid2world,
moving_spacing, self.ss_sigma_factor,
self.mask0)
if self.verbosity >= VerbosityLevels.STATUS:
print('Creating scale space from the static image. Levels: %d. '
'Sigma factor: %f.' % (self.levels, self.ss_sigma_factor))
self.static_ss = ScaleSpace(static, self.levels, static_grid2world,
static_spacing, self.ss_sigma_factor,
self.mask0)
if self.verbosity >= VerbosityLevels.DEBUG:
print('Moving scale space:')
for level in range(self.levels):
self.moving_ss.print_level(level)
print('Static scale space:')
for level in range(self.levels):
self.static_ss.print_level(level)
# Get the properties of the coarsest level from the static image. These
# properties will be taken as the reference discretization.
disp_shape = self.static_ss.get_domain_shape(self.levels-1)
disp_grid2world = self.static_ss.get_affine(self.levels-1)
# The codomain discretization of both diffeomorphic maps is
# precisely the discretization of the static image
codomain_shape = static.shape
codomain_grid2world = static_grid2world
# The forward model transforms points from the static image
# to points on the reference (which is the static as well). So the
# domain properties are taken from the static image. Since its the same
# as the reference, we don't need to pre-align.
domain_shape = static.shape
domain_grid2world = static_grid2world
self.static_to_ref = DiffeomorphicMap(self.dim,
disp_shape,
disp_grid2world,
domain_shape,
domain_grid2world,
codomain_shape,
codomain_grid2world,
None)
self.static_to_ref.allocate()
# The backward model transforms points from the moving image
# to points on the reference (which is the static). So the input
# properties are taken from the moving image, and we need to pre-align
# points on the moving physical space to the reference physical space
# by applying the inverse of pre-align. This is done this way to make
# it clear for the user: the pre-align matrix is usually obtained by
# doing affine registration of the moving image towards the static
# image, which results in a matrix transforming points in the static
# physical space to points in the moving physical space
prealign_inv = None if prealign is None else npl.inv(prealign)
domain_shape = moving.shape
domain_grid2world = moving_grid2world
self.moving_to_ref = DiffeomorphicMap(self.dim,
disp_shape,
disp_grid2world,
domain_shape,
domain_grid2world,
codomain_shape,
codomain_grid2world,
prealign_inv)
self.moving_to_ref.allocate()
def _end_optimizer(self):
r"""Frees the resources allocated during initialization
"""
del self.moving_ss
del self.static_ss
def _iterate(self):
r"""Performs one symmetric iteration
Performs one iteration of the SyN algorithm:
1.Compute forward
2.Compute backward
3.Update forward
4.Update backward
5.Compute inverses
6.Invert the inverses
Returns
-------
der : float
the derivative of the energy profile, computed by fitting a
quadratic function to the energy values at the latest T iterations,
where T = self.energy_window. If the current iteration is less than
T then np.inf is returned instead.
"""
# Acquire current resolution information from scale spaces
current_moving = self.moving_ss.get_image(self.current_level)
current_static = self.static_ss.get_image(self.current_level)
current_disp_shape = \
self.static_ss.get_domain_shape(self.current_level)
current_disp_grid2world = \
self.static_ss.get_affine(self.current_level)
current_disp_world2grid = \
self.static_ss.get_affine_inv(self.current_level)
current_disp_spacing = \
self.static_ss.get_spacing(self.current_level)
# Warp the input images (smoothed to the current scale) to the common
# (reference) space at the current resolution
wstatic = self.static_to_ref.transform_inverse(current_static,
'linear',
None,
current_disp_shape,
current_disp_grid2world)
wmoving = self.moving_to_ref.transform_inverse(current_moving,
'linear',
None,
current_disp_shape,
current_disp_grid2world)
# Pass both images to the metric. Now both images are sampled on the
# reference grid (equal to the static image's grid) and the direction
# doesn't change across scales
self.metric.set_moving_image(wmoving, current_disp_grid2world,
current_disp_spacing,
self.static_direction)
self.metric.use_moving_image_dynamics(
current_moving, self.moving_to_ref.inverse())
self.metric.set_static_image(wstatic, current_disp_grid2world,
current_disp_spacing,
self.static_direction)
self.metric.use_static_image_dynamics(
current_static, self.static_to_ref.inverse())
# Initialize the metric for a new iteration
self.metric.initialize_iteration()
if self.callback is not None:
self.callback(self, RegistrationStages.ITER_START)
# Compute the forward step (to be used to update the forward transform)
fw_step = np.array(self.metric.compute_forward())
# set zero displacements at the boundary
fw_step[0, ...] = 0
fw_step[:, 0, ...] = 0
fw_step[-1, ...] = 0
fw_step[:, -1, ...] = 0
if(self.dim == 3):
fw_step[:, :, 0, ...] = 0
fw_step[:, :, -1, ...] = 0
# Normalize the forward step
nrm = np.sqrt(np.sum((fw_step/current_disp_spacing)**2, -1)).max()
if nrm > 0:
fw_step /= nrm
# Add to current total field
self.static_to_ref.forward, md_forward = self.update(
self.static_to_ref.forward, fw_step,
current_disp_world2grid, self.step_length)
del fw_step
# Keep track of the forward energy
fw_energy = self.metric.get_energy()
# Compose backward step (to be used to update the backward transform)
bw_step = np.array(self.metric.compute_backward())
# set zero displacements at the boundary
bw_step[0, ...] = 0
bw_step[:, 0, ...] = 0
if(self.dim == 3):
bw_step[:, :, 0, ...] = 0
# Normalize the backward step
nrm = np.sqrt(np.sum((bw_step/current_disp_spacing) ** 2, -1)).max()
if nrm > 0:
bw_step /= nrm
# Add to current total field
self.moving_to_ref.forward, md_backward = self.update(
self.moving_to_ref.forward, bw_step,
current_disp_world2grid, self.step_length)
del bw_step
# Keep track of the energy
bw_energy = self.metric.get_energy()
der = np.inf
n_iter = len(self.energy_list)
if len(self.energy_list) >= self.energy_window:
der = self._get_energy_derivative()
if self.verbosity >= VerbosityLevels.DIAGNOSE:
ch = '-' if np.isnan(der) else der
print('%d:\t%0.6f\t%0.6f\t%0.6f\t%s' %
(n_iter, fw_energy, bw_energy, fw_energy + bw_energy, ch))
self.energy_list.append(fw_energy + bw_energy)
# Invert the forward model's forward field
self.static_to_ref.backward = np.array(
self.invert_vector_field(
self.static_to_ref.forward,
current_disp_world2grid,
current_disp_spacing,
self.inv_iter, self.inv_tol, self.static_to_ref.backward))
# Invert the backward model's forward field
self.moving_to_ref.backward = np.array(
self.invert_vector_field(
self.moving_to_ref.forward,
current_disp_world2grid,
current_disp_spacing,
self.inv_iter, self.inv_tol, self.moving_to_ref.backward))
# Invert the forward model's backward field
self.static_to_ref.forward = np.array(
self.invert_vector_field(
self.static_to_ref.backward,
current_disp_world2grid,
current_disp_spacing,
self.inv_iter, self.inv_tol, self.static_to_ref.forward))
# Invert the backward model's backward field
self.moving_to_ref.forward = np.array(
self.invert_vector_field(
self.moving_to_ref.backward,
current_disp_world2grid,
current_disp_spacing,
self.inv_iter, self.inv_tol, self.moving_to_ref.forward))
# Free resources no longer needed to compute the forward and backward
# steps
if self.callback is not None:
self.callback(self, RegistrationStages.ITER_END)
self.metric.free_iteration()
return der
def _approximate_derivative_direct(self, x, y):
r"""Derivative of the degree-2 polynomial fit of the given x, y pairs
Directly computes the derivative of the least-squares-fit quadratic
function estimated from (x[...],y[...]) pairs.
Parameters
----------
x : array, shape (n,)
increasing array representing the x-coordinates of the points to
be fit
y : array, shape (n,)
array representing the y-coordinates of the points to be fit
Returns
-------
y0 : float
the estimated derivative at x0 = 0.5*len(x)
"""
x = np.asarray(x)
y = np.asarray(y)
X = np.row_stack((x**2, x, np.ones_like(x)))
XX = (X).dot(X.T)
b = X.dot(y)
beta = npl.solve(XX, b)
x0 = 0.5 * len(x)
y0 = 2.0 * beta[0] * (x0) + beta[1]
return y0
def _get_energy_derivative(self):
r"""Approximate derivative of the energy profile
Returns the derivative of the estimated energy as a function of "time"
(iterations) at the last iteration
"""
n_iter = len(self.energy_list)
if n_iter < self.energy_window:
raise ValueError('Not enough data to fit the energy profile')
x = range(self.energy_window)
y = self.energy_list[(n_iter - self.energy_window):n_iter]
ss = sum(y)
if(ss > 0):
ss *= -1
y = [v / ss for v in y]
der = self._approximate_derivative_direct(x, y)
return der
def _optimize(self):
r"""Starts the optimization
The main multi-scale symmetric optimization algorithm
"""
self.full_energy_profile = []
if self.callback is not None:
self.callback(self, RegistrationStages.OPT_START)
for level in range(self.levels - 1, -1, -1):
if self.verbosity >= VerbosityLevels.STATUS:
print('Optimizing level %d' % level)
self.current_level = level
self.metric.set_levels_below(self.levels - level)
self.metric.set_levels_above(level)
if level < self.levels - 1:
expand_factors = \
self.static_ss.get_expand_factors(level+1, level)
new_shape = self.static_ss.get_domain_shape(level)
self.static_to_ref.expand_fields(expand_factors, new_shape)
self.moving_to_ref.expand_fields(expand_factors, new_shape)
self.niter = 0
self.energy_list = []
derivative = np.inf
if self.callback is not None:
self.callback(self, RegistrationStages.SCALE_START)
while ((self.niter < self.level_iters[self.levels - 1 - level]) and
(self.opt_tol < derivative)):
derivative = self._iterate()
self.niter += 1
self.full_energy_profile.extend(self.energy_list)
if self.callback is not None:
self.callback(self, RegistrationStages.SCALE_END)
# Reporting mean and std in stats[1] and stats[2]
residual, stats = self.static_to_ref.compute_inversion_error()
if self.verbosity >= VerbosityLevels.DIAGNOSE:
print('Static-Reference Residual error: %0.6f (%0.6f)'
% (stats[1], stats[2]))
residual, stats = self.moving_to_ref.compute_inversion_error()
if self.verbosity >= VerbosityLevels.DIAGNOSE:
print('Moving-Reference Residual error :%0.6f (%0.6f)'
% (stats[1], stats[2]))
# Compose the two partial transformations
self.static_to_ref = self.moving_to_ref.warp_endomorphism(
self.static_to_ref.inverse()).inverse()
# Report mean and std for the composed deformation field
residual, stats = self.static_to_ref.compute_inversion_error()
if self.verbosity >= VerbosityLevels.DIAGNOSE:
print('Final residual error: %0.6f (%0.6f)' % (stats[1], stats[2]))
if self.callback is not None:
self.callback(self, RegistrationStages.OPT_END)
def optimize(self, static, moving, static_grid2world=None,
moving_grid2world=None, prealign=None):
r"""
Starts the optimization
Parameters
----------
static : array, shape (S, R, C) or (R, C)
the image to be used as reference during optimization. The
displacement fields will have the same discretization as the static
image.
moving : array, shape (S, R, C) or (R, C)
the image to be used as "moving" during optimization. Since the
deformation fields' discretization is the same as the static image,
it is necessary to pre-align the moving image to ensure its domain
lies inside the domain of the deformation fields. This is assumed
to be accomplished by "pre-aligning" the moving image towards the
static using an affine transformation given by the 'prealign'
matrix
static_grid2world : array, shape (dim+1, dim+1)
the voxel-to-space transformation associated to the static image
moving_grid2world : array, shape (dim+1, dim+1)
the voxel-to-space transformation associated to the moving image
prealign : array, shape (dim+1, dim+1)
the affine transformation (operating on the physical space)
pre-aligning the moving image towards the static
Returns
-------
static_to_ref : DiffeomorphicMap object
the diffeomorphic map that brings the moving image towards the
static one in the forward direction (i.e. by calling
static_to_ref.transform) and the static image towards the
moving one in the backward direction (i.e. by calling
static_to_ref.transform_inverse).
"""
if self.verbosity >= VerbosityLevels.DEBUG:
print("Pre-align:", prealign)
self._init_optimizer(static.astype(floating), moving.astype(floating),
static_grid2world, moving_grid2world, prealign)
self._optimize()
self._end_optimizer()
self.static_to_ref.forward = np.array(self.static_to_ref.forward)
self.static_to_ref.backward = np.array(self.static_to_ref.backward)
return self.static_to_ref
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