"""
Implements a range of cross-correlation coefficients.
Copyright (c) 2023-2024 European Molecular Biology Laboratory
Author: Valentin Maurer <valentin.maurer@embl-hamburg.de>
"""
import warnings
from typing import Callable, Tuple, Dict
import numpy as np
from .backends import backend as be
from .types import CallbackClass, BackendArray, shm_type
from .matching_utils import (
conditional_execute,
identity,
standardize,
to_padded,
)
[docs]
def cc_setup(
matching_data: type,
fast_shape: Tuple[int],
fast_ft_shape: Tuple[int],
shm_handler: type,
**kwargs,
) -> Dict:
"""
Setup function for computing the unnormalized cross-correlation between
``target`` (f) and ``template`` (g)
.. math::
\\mathcal{F}^{-1}(\\mathcal{F}(f) \\cdot \\mathcal{F}(g)^*).
Notes
-----
To be used with :py:meth:`corr_scoring`.
"""
target_pad = be.topleft_pad(
matching_data.target,
matching_data._batch_shape(fast_shape, matching_data._template_batch),
)
axes = matching_data._batch_axis(matching_data._batch_mask)
return {
"template": be.to_sharedarr(matching_data.template, shm_handler),
"ft_target": be.to_sharedarr(be.rfftn(target_pad, axes=axes), shm_handler),
"inv_denominator": be.to_sharedarr(
be.zeros(1, be._float_dtype) + 1, shm_handler
),
"numerator": be.to_sharedarr(be.zeros(1, be._float_dtype), shm_handler),
}
[docs]
def lcc_setup(matching_data, **kwargs) -> Dict:
"""
Setup function for computing the laplace cross-correlation between
``target`` (f) and ``template`` (g)
.. math::
\\mathcal{F}^{-1}(\\mathcal{F}(\\nabla^{2}f) \\cdot \\mathcal{F}(\\nabla^{2} g)^*)
Notes
-----
To be used with :py:meth:`corr_scoring`.
"""
matching_data.target = matching_data.transform_target("laplace")
matching_data.template = matching_data.transform_template("laplace")
return cc_setup(matching_data=matching_data, **kwargs)
[docs]
def corr_setup(
matching_data,
template_filter,
fast_shape: Tuple[int],
fast_ft_shape: Tuple[int],
shm_handler: type,
**kwargs,
) -> Dict:
"""
Setup for computing a normalized cross-correlation between a
``target`` (f), a ``template`` (g) given ``template_mask`` (m)
.. math::
\\frac{CC(f,g) - \\overline{g} \\cdot CC(f, m)}
{(CC(f^2, m) - \\frac{CC(f, m)^2}{N_g}) \\cdot \\sigma_{g}},
where
.. math::
CC(f,g) = \\mathcal{F}^{-1}(\\mathcal{F}(f) \\cdot \\mathcal{F}(g)^*).
Notes
-----
To be used with :py:meth:`corr_scoring`.
References
----------
.. [1] Lewis P. J. Fast Normalized Cross-Correlation, Industrial Light and Magic.
"""
template, template_mask = matching_data.template, matching_data.template_mask
target_pad = be.topleft_pad(
matching_data.target,
matching_data._batch_shape(fast_shape, matching_data._template_batch),
)
data_axes = matching_data._batch_axis(matching_data._batch_mask)
data_shape = tuple(fast_shape[i] for i in data_axes)
ft_window = be.rfftn(be.topleft_pad(template_mask, fast_shape), axes=data_axes)
ft_target = be.rfftn(be.square(target_pad), axes=data_axes)
ft_target = be.multiply(ft_target, ft_window)
denominator = be.irfftn(ft_target, s=data_shape, axes=data_axes)
ft_target = be.rfftn(target_pad, axes=data_axes)
ft_window = be.multiply(ft_target, ft_window)
window_sum = be.irfftn(ft_window, s=data_shape, axes=data_axes)
target_pad, ft_window = None, None
# TODO: Factor in template_filter here
if be.size(template_filter) != 1:
warnings.warn(
"CORR scores obtained with template_filter are not correctly scaled. "
"Please use a different score or consider only relative peak heights."
)
axis = matching_data._batch_axis(matching_data._template_batch)
n_obs = be.sum(
be.astype(template_mask, be._overflow_safe_dtype), axis=axis, keepdims=True
)
template_mean = be.multiply(template, template_mask)
template_mean = be.sum(template_mean, axis=axis, keepdims=True)
template_mean = be.divide(template_mean, n_obs)
template_ssd = be.square(template - template_mean) * template_mask
template_ssd = be.sum(template_ssd, axis=axis, keepdims=True)
template_volume = np.prod(
tuple(
int(x)
for i, x in enumerate(template.shape)
if matching_data._template_batch[i] == 0
)
)
template = be.multiply(template, template_mask, out=template)
numerator = be.multiply(window_sum, template_mean)
window_sum = be.square(window_sum, out=window_sum)
window_sum = be.divide(window_sum, template_volume, out=window_sum)
denominator = be.subtract(denominator, window_sum, out=denominator)
denominator = be.multiply(denominator, template_ssd, out=denominator)
denominator = be.maximum(denominator, 0, out=denominator)
denominator = be.sqrt(denominator, out=denominator)
mask = denominator > be.eps(be._float_dtype)
denominator = be.multiply(denominator, mask, out=denominator)
denominator = be.add(denominator, ~mask, out=denominator)
denominator = be.divide(1, denominator, out=denominator)
denominator = be.multiply(denominator, mask, out=denominator)
return {
"template": be.to_sharedarr(template, shm_handler),
"ft_target": be.to_sharedarr(ft_target, shm_handler),
"inv_denominator": be.to_sharedarr(denominator, shm_handler),
"numerator": be.to_sharedarr(numerator, shm_handler),
}
[docs]
def cam_setup(matching_data, **kwargs) -> Dict:
"""
Like :py:meth:`corr_setup` but with standardized ``target`` and ``template``
.. math::
f' = \\frac{f - \\overline{f}}{\\sigma_f}.
Notes
-----
To be used with :py:meth:`corr_scoring`.
"""
matching_data.target = matching_data.transform_target("standardize")
matching_data.template = matching_data.transform_template("standardize")
return flcSphericalMask_setup(matching_data=matching_data, **kwargs)
def ncc_setup(matching_data, **kwargs) -> Dict:
matching_data.target = matching_data.transform_target("standardize")
return cc_setup(matching_data=matching_data, **kwargs)
[docs]
def flc_setup(
matching_data,
fast_shape: Tuple[int],
fast_ft_shape: Tuple[int],
shm_handler: type,
**kwargs,
) -> Dict:
"""
Setup function for :py:meth:`flc_scoring`.
"""
target_pad = be.topleft_pad(
matching_data.target,
matching_data._batch_shape(fast_shape, matching_data._template_batch),
)
data_axes = matching_data._batch_axis(matching_data._batch_mask)
ft_target = be.rfftn(target_pad, axes=data_axes)
target_pad = be.square(target_pad, out=target_pad)
ft_target2 = be.rfftn(target_pad, axes=data_axes)
return {
"template": be.to_sharedarr(matching_data.template, shm_handler),
"template_mask": be.to_sharedarr(matching_data.template_mask, shm_handler),
"ft_target": be.to_sharedarr(ft_target, shm_handler),
"ft_target2": be.to_sharedarr(ft_target2, shm_handler),
}
[docs]
def flcSphericalMask_setup(
matching_data,
fast_shape: Tuple[int],
fast_ft_shape: Tuple[int],
shm_handler: type,
**kwargs,
) -> Dict:
"""
Like :py:meth:`flc_setup` for rotation invariant masks
Notes
-----
To be used with :py:meth:`corr_scoring`.
"""
template_mask = matching_data.template_mask
axis = matching_data._batch_axis(matching_data._template_batch)
n_obs = be.sum(
be.astype(template_mask, be._overflow_safe_dtype), axis=axis, keepdims=True
)
target_pad = be.topleft_pad(
matching_data.target,
matching_data._batch_shape(fast_shape, matching_data._template_batch),
)
# Enable mask broadcasting
_out_shape = tuple(
y if i in axis else x
for i, (x, y) in enumerate(zip(template_mask.shape, fast_shape))
)
template_mask_pad = be.topleft_pad(
template_mask,
matching_data._batch_shape(_out_shape, matching_data._target_batch),
)
data_axes = matching_data._batch_axis(matching_data._batch_mask)
data_shape = tuple(fast_shape[i] for i in data_axes)
ft_temp = be.zeros(fast_ft_shape, be._complex_dtype)
ft_template_mask = be.rfftn(template_mask_pad, s=data_shape, axes=data_axes)
ft_target = be.rfftn(be.square(target_pad), axes=data_axes)
ft_temp = be.multiply(ft_target, ft_template_mask, out=ft_temp)
temp2 = be.irfftn(ft_temp, s=data_shape, axes=data_axes)
ft_target = be.rfftn(target_pad, axes=data_axes)
ft_temp = be.multiply(ft_target, ft_template_mask, out=ft_temp)
temp = be.irfftn(ft_temp, s=data_shape, axes=data_axes)
temp2 = be.norm_scores(1, temp2, temp, n_obs, be.eps(be._float_dtype), temp2)
return {
"template": be.to_sharedarr(matching_data.template, shm_handler),
"template_mask": be.to_sharedarr(template_mask, shm_handler),
"ft_target": be.to_sharedarr(ft_target, shm_handler),
"inv_denominator": be.to_sharedarr(temp2, shm_handler),
"numerator": be.to_sharedarr(be.zeros(1, be._float_dtype), shm_handler),
}
[docs]
def mcc_setup(
matching_data,
fast_shape: Tuple[int],
fast_ft_shape: Tuple[int],
shm_handler: Callable,
**kwargs,
) -> Dict:
"""
Setup function for :py:meth:`mcc_scoring`.
"""
target, target_mask = matching_data.target, matching_data.target_mask
target = be.multiply(target, target_mask, out=target)
ax = matching_data._batch_axis(matching_data._batch_mask)
shape = matching_data._batch_shape(fast_shape, matching_data._template_batch)
target = be.topleft_pad(target, shape)
target_mask = be.topleft_pad(target_mask, shape)
return {
"template": be.to_sharedarr(matching_data.template, shm_handler),
"template_mask": be.to_sharedarr(matching_data.template_mask, shm_handler),
"ft_target": be.to_sharedarr(be.rfftn(target, axes=ax), shm_handler),
"ft_target2": be.to_sharedarr(
be.rfftn(be.square(target), axes=ax), shm_handler
),
"ft_target_mask": be.to_sharedarr(be.rfftn(target_mask, axes=ax), shm_handler),
}
[docs]
def corr_scoring(
template: shm_type,
template_filter: shm_type,
ft_target: shm_type,
inv_denominator: shm_type,
numerator: shm_type,
fast_shape: Tuple[int],
fast_ft_shape: Tuple[int],
rotations: BackendArray,
callback: CallbackClass,
interpolation_order: int,
template_mask: shm_type = None,
score_mask: shm_type = None,
template_background: shm_type = None,
) -> CallbackClass:
"""
Calculates a normalized cross-correlation between a target f and a template g.
.. math::
(CC(f,g) - \\text{numerator}) \\cdot \\text{inv_denominator},
where
.. math::
CC(f,g) = \\mathcal{F}^{-1}(\\mathcal{F}(f) \\cdot \\mathcal{F}(g)^*).
Parameters
----------
template : Union[Tuple[type, tuple of ints, type], BackendArray]
Template data buffer, its shape and datatype.
template_filter : Union[Tuple[type, tuple of ints, type], BackendArray]
Template filter data buffer, its shape and datatype.
ft_target : Union[Tuple[type, tuple of ints, type], BackendArray]
Fourier transformed target data buffer, its shape and datatype.
inv_denominator : Union[Tuple[type, tuple of ints, type], BackendArray]
Inverse denominator data buffer, its shape and datatype.
numerator : Union[Tuple[type, tuple of ints, type], BackendArray]
Numerator data buffer, its shape, and its datatype.
fast_shape: tuple of ints
Data shape for the forward Fourier transform.
fast_ft_shape: tuple of ints
Data shape for the inverse Fourier transform.
rotations : BackendArray
Rotation matrices to be sampled (n, d, d).
callback : CallbackClass
A callable for processing the result of each rotation.
interpolation_order : int
Spline order for template rotations.
template_mask : Union[Tuple[type, tuple of ints, type], BackendArray], optional
Template mask data buffer, its shape and datatype, None by default.
score_mask : Union[Tuple[type, tuple of ints, type], BackendArray], optional
Score mask data buffer, its shape and datatype, None by default.
Returns
-------
CallbackClass
"""
template = be.from_sharedarr(template)
ft_target = be.from_sharedarr(ft_target)
inv_denominator = be.from_sharedarr(inv_denominator)
numerator = be.from_sharedarr(numerator)
template_filter = be.from_sharedarr(template_filter)
score_mask = be.from_sharedarr(score_mask)
n_obs = None
if template_mask is not None:
template_mask = be.from_sharedarr(template_mask)
n_obs = be.sum(template_mask) if template_mask is not None else None
norm_template = conditional_execute(standardize, n_obs is not None)
norm_sub = conditional_execute(be.subtract, numerator.shape != (1,))
norm_mul = conditional_execute(be.multiply, inv_denominator.shape != (1))
norm_mask = conditional_execute(be.multiply, score_mask.shape != (1,))
arr = be.zeros(fast_shape, be._float_dtype)
ft_temp = be.zeros(fast_ft_shape, be._complex_dtype)
template_rot = be.zeros(template.shape, be._float_dtype)
tmpl_filter_func = _create_filter_func(template.shape, template_filter)
center = be.divide(be.to_backend_array(template.shape) - 1, 2)
unpadded_slice = tuple(slice(0, stop) for stop in template.shape)
background_correction = template_background is not None
if background_correction:
scores_alt, compute_norm = _setup_background_correction(
fast_shape=fast_shape,
template_background=template_background,
rotation_buffer=template_rot,
unpadded_slice=unpadded_slice,
interpolation_order=interpolation_order,
tmpl_filter_func=tmpl_filter_func,
norm_template=norm_template,
)
for index in range(rotations.shape[0]):
rotation = rotations[index]
matrix = be._build_transform_matrix(
rotation_matrix=rotation, center=center, shape=template.shape
)
_ = be.rigid_transform(
arr=template,
rotation_matrix=matrix,
out=template_rot,
order=interpolation_order,
cache=True,
use_geometric_center=True,
)
template_rot = tmpl_filter_func(template_rot, ft_temp)
template_rot = norm_template(template_rot, template_mask, n_obs)
arr = to_padded(arr, template_rot, unpadded_slice)
ft_temp = be.rfftn(arr, s=fast_shape, out=ft_temp)
arr = _correlate_fts(ft_target, ft_temp, ft_temp, arr, fast_shape)
arr = norm_sub(arr, numerator, out=arr)
arr = norm_mul(arr, inv_denominator, out=arr)
arr = norm_mask(arr, score_mask, out=arr)
callback(arr, rotation_matrix=rotation)
if background_correction:
arr = compute_norm(arr, ft_target, ft_temp, matrix, template_mask, n_obs)
arr = norm_sub(arr, numerator, out=arr)
arr = norm_mul(arr, inv_denominator, out=arr)
arr = norm_mask(arr, score_mask, out=arr)
scores_alt = be.maximum(arr, scores_alt, out=scores_alt)
if background_correction:
scores_alt = norm_mask(scores_alt, score_mask, out=scores_alt)
scores_alt = be.subtract(scores_alt, be.mean(scores_alt), out=scores_alt)
callback.correct_background(scores_alt)
return callback
[docs]
def flc_scoring(
template: shm_type,
template_mask: shm_type,
ft_target: shm_type,
ft_target2: shm_type,
template_filter: shm_type,
fast_shape: Tuple[int],
fast_ft_shape: Tuple[int],
rotations: BackendArray,
callback: CallbackClass,
interpolation_order: int,
score_mask: shm_type = None,
template_background: shm_type = None,
) -> CallbackClass:
"""
Computes a normalized cross-correlation between ``target`` (f),
``template`` (g), and ``template_mask`` (m)
.. math::
\\frac{CC(f, \\frac{g*m - \\overline{g*m}}{\\sigma_{g*m}})}
{N_m * \\sqrt{
\\frac{CC(f^2, m)}{N_m} - (\\frac{CC(f, m)}{N_m})^2}
},
where
.. math::
CC(f,g) = \\mathcal{F}^{-1}(\\mathcal{F}(f) \\cdot \\mathcal{F}(g)^*)
and Nm is the sum of g.
Parameters
----------
template : Union[Tuple[type, tuple of ints, type], BackendArray]
Template data buffer, its shape and datatype.
template_mask : Union[Tuple[type, tuple of ints, type], BackendArray]
Template mask data buffer, its shape and datatype.
template_filter : Union[Tuple[type, tuple of ints, type], BackendArray]
Template filter data buffer, its shape and datatype.
ft_target : Union[Tuple[type, tuple of ints, type], BackendArray]
Fourier transformed target data buffer, its shape and datatype.
ft_target2 : Union[Tuple[type, tuple of ints, type], BackendArray]
Fourier transformed squared target data buffer, its shape and datatype.
fast_shape : tuple of ints
Data shape for the forward Fourier transform.
fast_ft_shape : tuple of ints
Data shape for the inverse Fourier transform.
rotations : BackendArray
Rotation matrices to be sampled (n, d, d).
callback : CallbackClass
A callable for processing the result of each rotation.
interpolation_order : int
Spline order for template rotations.
Returns
-------
CallbackClass
References
----------
.. [1] Hrabe T. et al, J. Struct. Biol. 178, 177 (2012).
"""
template = be.from_sharedarr(template)
template_mask = be.from_sharedarr(template_mask)
ft_target = be.from_sharedarr(ft_target)
ft_target2 = be.from_sharedarr(ft_target2)
template_filter = be.from_sharedarr(template_filter)
score_mask = be.from_sharedarr(score_mask)
arr = be.zeros(fast_shape, be._float_dtype)
temp = be.zeros(fast_shape, be._float_dtype)
temp2 = be.zeros(fast_shape, be._float_dtype)
ft_temp = be.zeros(fast_ft_shape, be._complex_dtype)
ft_denom = be.zeros(fast_ft_shape, be._complex_dtype)
template_rot = be.zeros(template.shape, be._float_dtype)
template_mask_rot = be.zeros(template.shape, be._float_dtype)
tmpl_filter_func = _create_filter_func(template.shape, template_filter)
norm_mask = conditional_execute(be.multiply, score_mask.shape != (1,))
eps = be.eps(be._float_dtype)
center = be.divide(be.to_backend_array(template.shape) - 1, 2)
unpadded_slice = tuple(slice(0, stop) for stop in template.shape)
background_correction = template_background is not None
if background_correction:
scores_alt, compute_norm = _setup_background_correction(
fast_shape=fast_shape,
template_background=template_background,
rotation_buffer=template_rot,
unpadded_slice=unpadded_slice,
interpolation_order=interpolation_order,
tmpl_filter_func=tmpl_filter_func,
norm_template=standardize,
)
for index in range(rotations.shape[0]):
rotation = rotations[index]
matrix = be._build_transform_matrix(
rotation_matrix=rotation, center=center, shape=template.shape
)
_ = be.rigid_transform(
arr=template,
arr_mask=template_mask,
rotation_matrix=matrix,
out=template_rot,
out_mask=template_mask_rot,
order=interpolation_order,
cache=True,
use_geometric_center=True,
)
n_obs = be.sum(template_mask_rot)
template_rot = tmpl_filter_func(template_rot, ft_temp)
template_rot = standardize(template_rot, template_mask_rot, n_obs)
arr = to_padded(arr, template_rot, unpadded_slice)
temp = to_padded(temp, template_mask_rot, unpadded_slice)
ft_temp = be.rfftn(temp, out=ft_temp, s=fast_shape)
temp = _correlate_fts(ft_target, ft_temp, ft_denom, temp, fast_shape)
temp2 = _correlate_fts(ft_target2, ft_temp, ft_denom, temp, fast_shape)
ft_temp = be.rfftn(arr, out=ft_temp, s=fast_shape)
arr = _correlate_fts(ft_target, ft_temp, ft_temp, arr, fast_shape)
inv_sdev = be.norm_scores(1, temp2, temp, n_obs, eps, temp2)
arr = be.multiply(arr, inv_sdev, out=arr)
arr = norm_mask(arr, score_mask, out=arr)
callback(arr, rotation_matrix=rotation)
if background_correction:
arr = compute_norm(arr, ft_target, ft_temp, matrix, template_mask, n_obs)
arr = be.multiply(arr, inv_sdev, out=arr)
scores_alt = be.maximum(arr, scores_alt, out=scores_alt)
if background_correction:
scores_alt = norm_mask(scores_alt, score_mask, out=scores_alt)
scores_alt = be.subtract(scores_alt, be.mean(scores_alt), out=scores_alt)
callback.correct_background(scores_alt)
return callback
def ncc_scoring(
template: shm_type,
ft_target: shm_type,
fast_shape: Tuple[int],
fast_ft_shape: Tuple[int],
rotations: BackendArray,
callback: CallbackClass,
interpolation_order: int,
template_filter: shm_type = None,
score_mask: shm_type = None,
template_background: shm_type = None,
**kwargs,
) -> CallbackClass:
template = be.from_sharedarr(template)
ft_target = be.from_sharedarr(ft_target)
score_mask = be.from_sharedarr(score_mask)
template_filter = be.from_sharedarr(template_filter)
arr = be.zeros(fast_shape, be._float_dtype)
ft_temp = be.zeros(fast_ft_shape, be._complex_dtype)
template_rot = be.zeros(template.shape, be._float_dtype)
# Welford arrays for global statistics
pixel_mean = be.zeros(fast_shape, be._float_dtype)
pixel_M2 = be.zeros(fast_shape, be._float_dtype)
tmpl_filter_func = _create_filter_func(template.shape, template_filter)
norm_mask = conditional_execute(be.multiply, score_mask.shape != (1,))
size = be.size(template)
center = be.divide(be.to_backend_array(template.shape) - 1, 2)
unpadded_slice = tuple(slice(0, stop) for stop in template.shape)
n_angles = rotations.shape[0]
# Scale forward transform by 1/n i.e. norm 'forward'
ft_target = be.multiply(ft_target, 1 / be.size(arr))
background_correction = template_background is not None
for index in range(n_angles):
arr = be.fill(arr, 0)
rotation = rotations[index]
matrix = be._build_transform_matrix(
rotation_matrix=rotation, center=center, shape=template.shape
)
be.rigid_transform(
template,
rotation_matrix=matrix,
out=template_rot,
order=interpolation_order,
cache=True,
use_geometric_center=True,
)
template_rot = tmpl_filter_func(template_rot, ft_temp)
template_rot = standardize(template_rot, 1, size)
arr = to_padded(arr, template_rot, unpadded_slice)
ft_temp = be.rfftn(arr, s=fast_shape, norm="forward")
ft_temp = be.multiply(ft_temp, ft_target, out=ft_temp)
arr = be.irfftn(ft_temp, s=fast_shape, norm="forward")
arr = norm_mask(arr, score_mask, out=arr)
callback(arr, rotation_matrix=rotation)
delta = be.subtract(arr, pixel_mean)
pixel_mean = be.add(pixel_mean, be.divide(delta, index + 1), out=pixel_mean)
delta2 = be.subtract(arr, pixel_mean)
delta = be.multiply(delta, delta2, out=delta)
pixel_M2 = be.add(pixel_M2, delta, out=pixel_M2)
global_mean = be.mean(pixel_mean)
pixel_variance = be.divide(pixel_M2, n_angles - 1)
global_std = be.sqrt(be.mean(pixel_variance))
callback.correct_background(global_mean, global_std)
if background_correction:
# Adapt units for local normalization
pixel_mean = be.subtract(pixel_mean, global_mean, out=pixel_mean)
pixel_mean = be.divide(pixel_mean, global_std, out=pixel_mean)
pixel_std = be.sqrt(pixel_variance, out=pixel_variance)
pixel_std = be.divide(pixel_std, global_std, out=pixel_variance)
pixel_std = be.where(pixel_std > 1e-4, 1 / pixel_std, 0.0)
callback.correct_background(pixel_mean, pixel_std)
return callback
[docs]
def mcc_scoring(
template: shm_type,
template_mask: shm_type,
template_filter: shm_type,
ft_target: shm_type,
ft_target2: shm_type,
ft_target_mask: shm_type,
fast_shape: Tuple[int],
fast_ft_shape: Tuple[int],
rotations: BackendArray,
callback: CallbackClass,
interpolation_order: int,
overlap_ratio: float = 0.3,
**kwargs,
) -> CallbackClass:
"""
Computes a normalized cross-correlation score between ``target`` (f),
``template`` (g), ``template_mask`` (m) and ``target_mask`` (t)
.. math::
\\frac{
CC(f, g) - \\frac{CC(f, m) \\cdot CC(t, g)}{CC(t, m)}
}{
\\sqrt{
(CC(f ^ 2, m) - \\frac{CC(f, m) ^ 2}{CC(t, m)}) \\cdot
(CC(t, g^2) - \\frac{CC(t, g) ^ 2}{CC(t, m)})
}
},
where
.. math::
CC(f,g) = \\mathcal{F}^{-1}(\\mathcal{F}(f) \\cdot \\mathcal{F}(g)^*).
Parameters
----------
template : Union[Tuple[type, tuple of ints, type], BackendArray]
Template data buffer, its shape and datatype.
template_mask : Union[Tuple[type, tuple of ints, type], BackendArray]
Template mask data buffer, its shape and datatype.
template_filter : Union[Tuple[type, tuple of ints, type], BackendArray]
Template filter data buffer, its shape and datatype.
ft_target : Union[Tuple[type, tuple of ints, type], BackendArray]
Fourier transformed target data buffer, its shape and datatype.
ft_target2 : Union[Tuple[type, tuple of ints, type], BackendArray]
Fourier transformed squared target data buffer, its shape and datatype.
ft_target_mask : Union[Tuple[type, tuple of ints, type], BackendArray]
Fourier transformed target mask data buffer, its shape and datatype.
fast_shape: tuple of ints
Data shape for the forward Fourier transform.
fast_ft_shape: tuple of ints
Data shape for the inverse Fourier transform.
rotations : BackendArray
Rotation matrices to be sampled (n, d, d).
callback : CallbackClass
A callable for processing the result of each rotation.
interpolation_order : int
Spline order for template rotations.
overlap_ratio : float, optional
Required fractional mask overlap, 0.3 by default.
Returns
-------
CallbackClass
References
----------
.. [1] Masked FFT registration, Dirk Padfield, CVPR 2010 conference
.. [2] https://scikit-image.org/docs/stable/api/skimage.registration.html
"""
float_dtype, complex_dtype = be._float_dtype, be._complex_dtype
template = be.from_sharedarr(template)
target_ft = be.from_sharedarr(ft_target)
target_ft2 = be.from_sharedarr(ft_target2)
template_mask = be.from_sharedarr(template_mask)
target_mask_ft = be.from_sharedarr(ft_target_mask)
template_filter = be.from_sharedarr(template_filter)
axes = tuple(range(template.ndim))
eps = be.eps(float_dtype)
# Allocate score and process specific arrays
template_rot = be.zeros(fast_shape, float_dtype)
mask_overlap = be.zeros(fast_shape, float_dtype)
numerator = be.zeros(fast_shape, float_dtype)
temp = be.zeros(fast_shape, float_dtype)
temp2 = be.zeros(fast_shape, float_dtype)
temp3 = be.zeros(fast_shape, float_dtype)
temp_ft = be.zeros(fast_ft_shape, complex_dtype)
tmpl_filter_func = _create_filter_func(
arr_shape=template.shape,
template_filter=template_filter,
arr_padded=True,
)
for index in range(rotations.shape[0]):
rotation = rotations[index]
template_rot = be.fill(template_rot, 0)
temp = be.fill(temp, 0)
be.rigid_transform(
arr=template,
arr_mask=template_mask,
rotation_matrix=rotation,
out=template_rot,
out_mask=temp,
use_geometric_center=True,
order=interpolation_order,
cache=True,
)
template_rot = tmpl_filter_func(template_rot, temp_ft)
template_rot = standardize(template_rot, temp, be.sum(temp))
temp_ft = be.rfftn(template_rot, out=temp_ft, s=fast_shape)
temp2 = be.irfftn(target_mask_ft * temp_ft, out=temp2, s=fast_shape)
numerator = be.irfftn(target_ft * temp_ft, out=numerator, s=fast_shape)
# temp template_mask_rot | temp_ft template_mask_rot_ft
# Calculate overlap of masks at every point in the convolution.
# Locations with high overlap should not be taken into account.
temp_ft = be.rfftn(temp, out=temp_ft, s=fast_shape)
mask_overlap = be.irfftn(
temp_ft * target_mask_ft, out=mask_overlap, s=fast_shape
)
be.maximum(mask_overlap, eps, out=mask_overlap)
temp = be.irfftn(temp_ft * target_ft, out=temp, s=fast_shape)
be.subtract(
numerator,
be.divide(be.multiply(temp, temp2), mask_overlap),
out=numerator,
)
# temp_3 = fixed_denom
be.multiply(temp_ft, target_ft2, out=temp_ft)
temp3 = be.irfftn(temp_ft, out=temp3, s=fast_shape)
be.subtract(temp3, be.divide(be.square(temp), mask_overlap), out=temp3)
be.maximum(temp3, 0.0, out=temp3)
# temp = moving_denom
temp_ft = be.rfftn(be.square(template_rot), out=temp_ft, s=fast_shape)
be.multiply(target_mask_ft, temp_ft, out=temp_ft)
temp = be.irfftn(temp_ft, out=temp, s=fast_shape)
be.subtract(temp, be.divide(be.square(temp2), mask_overlap), out=temp)
be.maximum(temp, 0.0, out=temp)
# temp_2 = denom
be.multiply(temp3, temp, out=temp)
be.sqrt(temp, out=temp2)
# Pixels where `denom` is very small will introduce large
# numbers after division. To get around this problem,
# we zero-out problematic pixels.
tol = 1e3 * eps * be.max(be.abs(temp2), axis=axes, keepdims=True)
temp2[temp2 < tol] = 1
temp = be.divide(numerator, temp2, out=temp)
temp = be.clip(temp, a_min=-1, a_max=1, out=temp)
# Apply overlap ratio threshold
number_px_threshold = overlap_ratio * be.max(
mask_overlap, axis=axes, keepdims=True
)
temp[mask_overlap < number_px_threshold] = 0.0
callback(temp, rotation_matrix=rotation)
return callback
def flc_scoring2(
template: shm_type,
template_mask: shm_type,
ft_target: shm_type,
ft_target2: shm_type,
template_filter: shm_type,
fast_shape: Tuple[int],
fast_ft_shape: Tuple[int],
rotations: BackendArray,
callback: CallbackClass,
interpolation_order: int,
score_mask: shm_type = None,
template_background: shm_type = None,
) -> CallbackClass:
template = be.from_sharedarr(template)
template_mask = be.from_sharedarr(template_mask)
ft_target = be.from_sharedarr(ft_target)
ft_target2 = be.from_sharedarr(ft_target2)
template_filter = be.from_sharedarr(template_filter)
score_mask = be.from_sharedarr(score_mask)
tar_batch, tmpl_batch = _get_batch_dim(ft_target, template)
nd = len(fast_shape)
sqz_slice = tuple(slice(0, 1) if i in tar_batch else slice(None) for i in range(nd))
tmpl_subset = tuple(0 if i in tar_batch else slice(None) for i in range(nd))
axes, shape, batched = None, fast_shape, len(tmpl_batch) > 0
if len(tar_batch) or len(tmpl_batch):
axes = tuple(i for i in range(nd) if i not in (*tar_batch, *tmpl_batch))
shape = tuple(fast_shape[i] for i in axes)
arr = be.zeros(fast_shape, be._float_dtype)
temp = be.zeros(fast_shape, be._float_dtype)
temp2 = be.zeros(fast_shape, be._float_dtype)
ft_denom = be.zeros(fast_ft_shape, be._complex_dtype)
tmp_sqz, arr_sqz, ft_temp = temp[sqz_slice], arr[sqz_slice], ft_denom[sqz_slice]
tmpl_filter_func = _create_filter_func(
arr_shape=template.shape,
template_filter=template_filter,
arr_padded=True,
)
norm_mask = conditional_execute(be.multiply, score_mask.shape != (1,))
background_correction = template_background is not None
if background_correction:
scores_alt, compute_norm = _setup_background_correction(
fast_shape=fast_shape,
template_background=template_background,
rotation_buffer=arr_sqz[tmpl_subset],
unpadded_slice=tmpl_subset,
interpolation_order=interpolation_order,
tmpl_filter_func=tmpl_filter_func,
norm_template=standardize,
)
eps = be.eps(be._float_dtype)
for index in range(rotations.shape[0]):
rotation = rotations[index]
be.fill(arr, 0)
be.fill(temp, 0)
_, _ = be.rigid_transform(
arr=template[tmpl_subset],
arr_mask=template_mask[tmpl_subset],
rotation_matrix=rotation,
out=arr_sqz[tmpl_subset],
out_mask=tmp_sqz[tmpl_subset],
use_geometric_center=True,
order=interpolation_order,
cache=False,
batched=batched,
)
n_obs = be.sum(tmp_sqz, axis=axes, keepdims=True)
arr_norm = tmpl_filter_func(arr_sqz, ft_temp)
arr_norm = standardize(arr_norm, tmp_sqz, n_obs, axis=axes)
ft_temp = be.rfftn(tmp_sqz, out=ft_temp, axes=axes, s=shape)
temp = _correlate_fts(ft_target, ft_temp, ft_denom, temp, shape, axes)
temp2 = _correlate_fts(ft_target2, ft_temp, ft_denom, temp2, shape, axes)
ft_temp = be.rfftn(arr_norm, out=ft_temp, axes=axes, s=shape)
arr = _correlate_fts(ft_target, ft_temp, ft_denom, arr, shape, axes)
inv_sdev = be.norm_scores(1, temp2, temp, n_obs, eps, temp2)
arr = be.multiply(arr, inv_sdev, out=arr)
arr = norm_mask(arr, score_mask, out=arr)
callback(arr, rotation_matrix=rotation)
if background_correction:
arr = compute_norm(arr, ft_target, ft_temp, rotation, template_mask, n_obs)
arr = be.multiply(arr, inv_sdev, out=arr)
scores_alt = be.maximum(arr, scores_alt, out=scores_alt)
if background_correction:
scores_alt = norm_mask(scores_alt, score_mask, out=scores_alt)
scores_alt = be.subtract(scores_alt, be.mean(scores_alt), out=scores_alt)
callback.correct_background(scores_alt)
return callback
def corr_scoring2(
template: shm_type,
template_filter: shm_type,
ft_target: shm_type,
inv_denominator: shm_type,
numerator: shm_type,
fast_shape: Tuple[int],
fast_ft_shape: Tuple[int],
rotations: BackendArray,
callback: CallbackClass,
interpolation_order: int,
target_filter: shm_type = None,
template_mask: shm_type = None,
score_mask: shm_type = None,
template_background: shm_type = None,
) -> CallbackClass:
template = be.from_sharedarr(template)
ft_target = be.from_sharedarr(ft_target)
inv_denominator = be.from_sharedarr(inv_denominator)
numerator = be.from_sharedarr(numerator)
template_filter = be.from_sharedarr(template_filter)
score_mask = be.from_sharedarr(score_mask)
tar_batch, tmpl_batch = _get_batch_dim(ft_target, template)
nd = len(fast_shape)
sqz_slice = tuple(slice(0, 1) if i in tar_batch else slice(None) for i in range(nd))
tmpl_subset = tuple(0 if i in tar_batch else slice(None) for i in range(nd))
axes, shape, batched = None, fast_shape, len(tmpl_batch) > 0
if len(tar_batch) or len(tmpl_batch):
axes = tuple(i for i in range(nd) if i not in (*tar_batch, *tmpl_batch))
shape = tuple(fast_shape[i] for i in axes)
unpadded_slice = tuple(
slice(None) if i in (*tar_batch, *tmpl_batch) else slice(0, x)
for i, x in enumerate(template.shape)
)
arr = be.zeros(fast_shape, be._float_dtype)
ft_temp = be.zeros(fast_ft_shape, be._complex_dtype)
arr_sqz, ft_sqz = arr[sqz_slice], ft_temp[sqz_slice]
n_obs = None
if template_mask is not None:
template_mask = be.from_sharedarr(template_mask)
n_obs = be.sum(template_mask, axis=axes, keepdims=True)
norm_template = conditional_execute(standardize, n_obs is not None)
norm_sub = conditional_execute(be.subtract, numerator.shape != (1,))
norm_mul = conditional_execute(be.multiply, inv_denominator.shape != (1,))
norm_mask = conditional_execute(be.multiply, score_mask.shape != (1,))
template_filter_func = _create_filter_func(
arr_shape=template.shape,
template_filter=template_filter,
arr_padded=True,
)
for index in range(rotations.shape[0]):
be.fill(arr, 0)
rotation = rotations[index]
_, _ = be.rigid_transform(
arr=template[tmpl_subset],
rotation_matrix=rotation,
out=arr_sqz[tmpl_subset],
use_geometric_center=True,
order=interpolation_order,
cache=False,
batched=batched,
)
arr_norm = template_filter_func(arr_sqz, ft_sqz)
norm_template(arr_norm[unpadded_slice], template_mask, n_obs, axis=axes)
ft_sqz = be.rfftn(arr_norm, out=ft_sqz, axes=axes, s=shape)
arr = _correlate_fts(ft_target, ft_sqz, ft_temp, arr, shape, axes)
arr = norm_sub(arr, numerator, out=arr)
arr = norm_mul(arr, inv_denominator, out=arr)
arr = norm_mask(arr, score_mask, out=arr)
callback(arr, rotation_matrix=rotation)
return callback
def _get_batch_dim(target, template):
target_batch, template_batch = [], []
for i in range(len(target.shape)):
if target.shape[i] == 1 and template.shape[i] != 1:
template_batch.append(i)
if target.shape[i] != 1 and template.shape[i] == 1:
target_batch.append(i)
return target_batch, template_batch
def _correlate_fts(ft_tar, ft_tmpl, ft_buffer, real_buffer, fast_shape, axes=None):
ft_buffer = be.multiply(ft_tar, ft_tmpl, out=ft_buffer)
return be.irfftn(ft_buffer, out=real_buffer, s=fast_shape, axes=axes)
def _create_filter_func(
arr_shape: Tuple[int],
template_filter: BackendArray,
arr_padded: bool = False,
axes=None,
) -> Callable:
"""
Configure template filtering function for Fourier transforms.
Conceptually we distinguish between three cases. The base case
is that both template and the corresponding filter have the same
shape. Padding is used when the template filter is larger than
the template, for instance to better resolve Fourier filters. Finally
this function also handles the case when a filter is supposed to be
broadcasted over the template batch dimension.
Parameters
----------
arr_shape : tuple of ints
Shape of the array to be filtered.
template_filter : BackendArray
Precomputed filter to apply in the frequency domain.
arr_padded : bool, optional
Whether the input template is padded and will need to be cropped
to arr_shape prior to filter applications. Defaults to False.
axes : tuple of ints, optional
Axes to perform Fourier transform over.
Returns
-------
Callable
Filter function with parameters template, ft_temp and template_filter.
"""
filter_shape = template_filter.shape
if filter_shape == (1,):
return conditional_execute(identity, execute_operation=True)
# Default case, all shapes are correctly matched
def _apply_filter(template, ft_temp):
ft_temp = be.rfftn(template, out=ft_temp, s=template.shape)
ft_temp = be.multiply(ft_temp, template_filter, out=ft_temp)
return be.irfftn(ft_temp, out=template, s=template.shape)
if not arr_padded:
return _apply_filter
# Array is padded but filter is w.r.t to the original template
real_subset = tuple(slice(0, x) for x in arr_shape)
_template = be.zeros(arr_shape, be._float_dtype)
_ft_temp = be.zeros(filter_shape, be._complex_dtype)
def _apply_filter_subset(template, ft_temp):
_template[:] = template[real_subset]
template[real_subset] = _apply_filter(_template, _ft_temp)
return template
return _apply_filter_subset
def _setup_background_correction(
fast_shape: Tuple[int],
template_background: BackendArray,
rotation_buffer: BackendArray,
unpadded_slice: Tuple[slice],
interpolation_order: int = 3,
tmpl_filter_func: Callable = identity,
norm_template: Callable = identity,
axes=None,
shape=None,
):
scores_noise = be.zeros(fast_shape, be._float_dtype)
template_background = be.from_sharedarr(template_background)
fwd_shape = shape
if shape is not None:
fwd_shape = shape
def compute_norm(arr, ft_target, ft_temp, matrix, template_mask, n_obs):
_ = be.rigid_transform(
arr=template_background,
rotation_matrix=matrix,
out=rotation_buffer,
use_geometric_center=True,
order=interpolation_order,
cache=True,
)
template_rot = tmpl_filter_func(rotation_buffer, ft_temp)
template_rot = norm_template(template_rot, template_mask, n_obs, axis=axes)
arr = to_padded(arr, template_rot, unpadded_slice)
ft_temp = be.rfftn(arr, out=ft_temp, axes=axes, s=fwd_shape)
return _correlate_fts(ft_target, ft_temp, ft_temp, arr, fast_shape, axes)
return scores_noise, compute_norm
MATCHING_EXHAUSTIVE_REGISTER = {
"CC": (cc_setup, corr_scoring),
"LCC": (lcc_setup, corr_scoring),
"CORR": (corr_setup, corr_scoring),
"CAM": (cam_setup, corr_scoring),
# "NCC": (ncc_setup, ncc_scoring),
"FLCSphericalMask": (flcSphericalMask_setup, corr_scoring),
"FLC": (flc_setup, flc_scoring),
"MCC": (mcc_setup, mcc_scoring),
"batchFLCSphericalMask": (flcSphericalMask_setup, corr_scoring2),
"batchFLC": (flc_setup, flc_scoring2),
}