def morphological_geodesic_active_contour(
gimage,
iterations,
init_level_set="circle",
smoothing=1,
threshold="auto",
balloon=0,
iter_callback=lambda x: None,
):
"""Morphological Geodesic Active Contours (MorphGAC).
Geodesic active contours implemented with morphological operators. It can
be used to segment objects with visible but noisy, cluttered, broken
borders.
Parameters
----------
gimage : (M, N) or (L, M, N) array
Preprocessed image or volume to be segmented. This is very rarely the
original image. Instead, this is usually a preprocessed version of the
original image that enhances and highlights the borders (or other
structures) of the object to segment.
`morphological_geodesic_active_contour` will try to stop the contour
evolution in areas where `gimage` is small. See
`morphsnakes.inverse_gaussian_gradient` as an example function to
perform this preprocessing. Note that the quality of
`morphological_geodesic_active_contour` might greatly depend on this
preprocessing.
iterations : uint
Number of iterations to run.
init_level_set : str, (M, N) array, or (L, M, N) array
Initial level set. If an array is given, it will be binarized and used
as the initial level set. If a string is given, it defines the method
to generate a reasonable initial level set with the shape of the
`image`. Accepted values are 'checkerboard' and 'circle'. See the
documentation of `checkerboard_level_set` and `circle_level_set`
respectively for details about how these level sets are created.
smoothing : uint, optional
Number of times the smoothing operator is applied per iteration.
Reasonable values are around 1-4. Larger values lead to smoother
segmentations.
threshold : float, optional
Areas of the image with a value smaller than this threshold will be
considered borders. The evolution of the contour will stop in this
areas.
balloon : float, optional
Balloon force to guide the contour in non-informative areas of the
image, i.e., areas where the gradient of the image is too small to push
the contour towards a border. A negative value will shrink the contour,
while a positive value will expand the contour in these areas. Setting
this to zero will disable the balloon force.
iter_callback : function, optional
If given, this function is called once per iteration with the current
level set as the only argument. This is useful for debugging or for
plotting intermediate results during the evolution.
Returns
-------
out : (M, N) or (L, M, N) array
Final segmentation (i.e., the final level set)
See Also
--------
inverse_gaussian_gradient, circle_level_set, checkerboard_level_set
Notes
-----
This is a version of the Geodesic Active Contours (GAC) algorithm that uses
morphological operators instead of solving partial differential equations
(PDEs) for the evolution of the contour. The set of morphological operators
used in this algorithm are proved to be infinitesimally equivalent to the
GAC PDEs (see [1]_). However, morphological operators are do not suffer
from the numerical stability issues typically found in PDEs (e.g., it is
not necessary to find the right time step for the evolution), and are
computationally faster.
The algorithm and its theoretical derivation are described in [1]_.
References
----------
.. [1] A Morphological Approach to Curvature-based Evolution of Curves and
Surfaces, Pablo Márquez-Neila, Luis Baumela, Luis Álvarez. In IEEE
Transactions on Pattern Analysis and Machine Intelligence (PAMI),
2014, :DOI:`10.1109/TPAMI.2013.106`
"""
image = gimage
init_level_set = _init_level_set(init_level_set, image.shape)
_check_input(image, init_level_set)
if threshold == "auto":
threshold = cp.percentile(image, 40)
structure = cp.ones((3, ) * len(image.shape), dtype=cp.int8)
dimage = cnp.gradient(image)
# threshold_mask = image > threshold
if balloon != 0:
threshold_mask_balloon = image > threshold / cp.abs(balloon)
u = (init_level_set > 0).astype(cp.int8)
iter_callback(u)
for _ in range(iterations):
# Balloon
if balloon > 0:
aux = ndi.binary_dilation(u, structure)
elif balloon < 0:
aux = ndi.binary_erosion(u, structure)
if balloon != 0:
u[threshold_mask_balloon] = aux[threshold_mask_balloon]
# Image attachment
aux = cp.zeros_like(image)
du = cnp.gradient(u)
for el1, el2 in zip(dimage, du):
aux += el1 * el2
u[aux > 0] = 1
u[aux < 0] = 0
# Smoothing
for _ in range(smoothing):
u = _curvop(u)
iter_callback(u)
return u
def morphological_chan_vese(
image,
iterations,
init_level_set="checkerboard",
smoothing=1,
lambda1=1,
lambda2=1,
iter_callback=lambda x: None,
):
"""Morphological Active Contours without Edges (MorphACWE)
Active contours without edges implemented with morphological operators. It
can be used to segment objects in images and volumes without well defined
borders. It is required that the inside of the object looks different on
average than the outside (i.e., the inner area of the object should be
darker or lighter than the outer area on average).
Parameters
----------
image : (M, N) or (L, M, N) array
Grayscale image or volume to be segmented.
iterations : uint
Number of iterations to run
init_level_set : str, (M, N) array, or (L, M, N) array
Initial level set. If an array is given, it will be binarized and used
as the initial level set. If a string is given, it defines the method
to generate a reasonable initial level set with the shape of the
`image`. Accepted values are 'checkerboard' and 'circle'. See the
documentation of `checkerboard_level_set` and `circle_level_set`
respectively for details about how these level sets are created.
smoothing : uint, optional
Number of times the smoothing operator is applied per iteration.
Reasonable values are around 1-4. Larger values lead to smoother
segmentations.
lambda1 : float, optional
Weight parameter for the outer region. If `lambda1` is larger than
`lambda2`, the outer region will contain a larger range of values than
the inner region.
lambda2 : float, optional
Weight parameter for the inner region. If `lambda2` is larger than
`lambda1`, the inner region will contain a larger range of values than
the outer region.
iter_callback : function, optional
If given, this function is called once per iteration with the current
level set as the only argument. This is useful for debugging or for
plotting intermediate results during the evolution.
Returns
-------
out : (M, N) or (L, M, N) array
Final segmentation (i.e., the final level set)
See Also
--------
circle_level_set, checkerboard_level_set
Notes
-----
This is a version of the Chan-Vese algorithm that uses morphological
operators instead of solving a partial differential equation (PDE) for the
evolution of the contour. The set of morphological operators used in this
algorithm are proved to be infinitesimally equivalent to the Chan-Vese PDE
(see [1]_). However, morphological operators are do not suffer from the
numerical stability issues typically found in PDEs (it is not necessary to
find the right time step for the evolution), and are computationally
faster.
The algorithm and its theoretical derivation are described in [1]_.
References
----------
.. [1] A Morphological Approach to Curvature-based Evolution of Curves and
Surfaces, Pablo Márquez-Neila, Luis Baumela, Luis Álvarez. In IEEE
Transactions on Pattern Analysis and Machine Intelligence (PAMI),
2014, :DOI:`10.1109/TPAMI.2013.106`
"""
init_level_set = _init_level_set(init_level_set, image.shape)
_check_input(image, init_level_set)
u = (init_level_set > 0).astype(cp.int8)
iter_callback(u)
for _ in range(iterations):
# inside = u > 0
# outside = u <= 0
c0 = (image * (1 - u)).sum() / float((1 - u).sum() + 1e-8)
c1 = (image * u).sum() / float(u.sum() + 1e-8)
# Image attachment
du = cnp.gradient(u)
abs_du = cp.abs(cp.stack(du, axis=0)).sum(0)
aux = abs_du * (lambda1 * (image - c1)**2 - lambda2 * (image - c0)**2)
u[aux < 0] = 1
u[aux > 0] = 0
# Smoothing
for _ in range(smoothing):
u = _curvop(u)
iter_callback(u)
return u
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.
"""
def invalid_image_size(image):
min_size = self.radius * 2 + 1
return any([size < min_size for size in image.shape])
msg = ("Each image dimension should be superior to 2 * radius + 1."
"Decrease CCMetric radius or increase your image size")
if invalid_image_size(self.static_image):
raise ValueError("Static image size is too small. " + msg)
if invalid_image_size(self.moving_image):
raise ValueError("Moving image size is too small. " + msg)
self.factors = self.precompute_factors(self.static_image,
self.moving_image, self.radius)
if self.coord_axis == -1:
self.gradient_moving = cp.empty(shape=(self.moving_image.shape) +
(self.dim, ),
dtype=floating)
for i, grad in enumerate(gradient(self.moving_image)):
self.gradient_moving[..., i] = grad
else:
self.gradient_moving = cp.empty(shape=(self.dim, ) +
(self.moving_image.shape),
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:
if self.coord_axis == -1:
self.gradient_moving /= self.moving_spacing
else:
temp = self.moving_spacing.reshape((-1, ) + (1, ) * self.dim)
self.gradient_moving /= temp
if self.moving_direction is not None:
self.reorient_vector_field(self.gradient_moving,
self.moving_direction,
coord_axis=self.coord_axis)
if self.coord_axis == -1:
self.gradient_static = cp.empty(shape=(self.static_image.shape) +
(self.dim, ),
dtype=floating)
for i, grad in enumerate(gradient(self.static_image)):
self.gradient_static[..., i] = grad
else:
self.gradient_static = cp.empty(shape=(self.dim, ) +
(self.static_image.shape),
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:
if self.coord_axis == -1:
self.gradient_static /= self.static_spacing
else:
temp = self.moving_spacing.reshape((-1, ) + (1, ) * self.dim)
self.gradient_static /= temp
if self.static_direction is not None:
self.reorient_vector_field(self.gradient_static,
self.static_direction,
coord_axis=self.coord_axis)
def _tvl1(
reference_image,
moving_image,
flow0,
attachment,
tightness,
num_warp,
num_iter,
tol,
prefilter,
):
"""TV-L1 solver for optical flow estimation.
Parameters
----------
reference_image : ndarray, shape (M, N[, P[, ...]])
The first gray scale image of the sequence.
moving_image : ndarray, shape (M, N[, P[, ...]])
The second gray scale image of the sequence.
flow0 : ndarray, shape (image0.ndim, M, N[, P[, ...]])
Initialization for the vector field.
attachment : float
Attachment parameter. The smaller this parameter is,
the smoother is the solutions.
tightness : float
Tightness parameter. It should have a small value in order to
maintain attachement and regularization parts in
correspondence.
num_warp : int
Number of times image1 is warped.
num_iter : int
Number of fixed point iteration.
tol : float
Tolerance used as stopping criterion based on the L² distance
between two consecutive values of (u, v).
prefilter : bool
Whether to prefilter the estimated optical flow before each
image warp.
Returns
-------
flow : ndarray, shape ((image0.ndim, M, N[, P[, ...]])
The estimated optical flow components for each axis.
"""
dtype = reference_image.dtype
grid = cp.stack(
cp.meshgrid(
*[cp.arange(n, dtype=dtype) for n in reference_image.shape],
indexing="ij",
),
axis=0,
)
dt = 0.5 / reference_image.ndim
reg_num_iter = 2
f0 = attachment * tightness
f1 = dt / tightness
tol *= reference_image.size
flow_current = flow_previous = flow0
g = cp.zeros((reference_image.ndim, ) + reference_image.shape, dtype=dtype)
proj = cp.zeros(
(reference_image.ndim, reference_image.ndim) + reference_image.shape,
dtype=dtype,
)
s_g = [slice(None)] * g.ndim
s_p = [slice(None)] * proj.ndim
s_d = [slice(None)] * (proj.ndim - 2)
for _ in range(num_warp):
if prefilter:
flow_current = ndi.median_filter(flow_current,
[1] + reference_image.ndim * [3])
image1_warp = warp(moving_image, grid + flow_current, mode="nearest")
grad = cp.stack(cnp.gradient(image1_warp))
NI = (grad * grad).sum(0)
NI[NI == 0] = 1
rho_0 = image1_warp - reference_image - (grad * flow_current).sum(0)
for _ in range(num_iter):
# Data term
rho = rho_0 + (grad * flow_current).sum(0)
idx = abs(rho) <= f0 * NI
flow_auxiliary = flow_current
flow_auxiliary[:, idx] -= rho[idx] * grad[:, idx] / NI[idx]
idx = ~idx
srho = f0 * cp.sign(rho[idx])
flow_auxiliary[:, idx] -= srho * grad[:, idx]
# Regularization term
flow_current = flow_auxiliary.copy()
for idx in range(reference_image.ndim):
s_p[0] = idx
for _ in range(reg_num_iter):
for ax in range(reference_image.ndim):
s_g[0] = ax
s_g[ax + 1] = slice(0, -1)
g[tuple(s_g)] = cp.diff(flow_current[idx], axis=ax)
s_g[ax + 1] = slice(None)
norm = cp.sqrt((g * g).sum(0, keepdims=True))
norm *= f1
norm += 1.0
proj[idx] -= dt * g
proj[idx] /= norm
# d will be the (negative) divergence of proj[idx]
d = -proj[idx].sum(0)
for ax in range(reference_image.ndim):
s_p[1] = ax
s_p[ax + 2] = slice(0, -1)
s_d[ax] = slice(1, None)
d[tuple(s_d)] += proj[tuple(s_p)]
s_p[ax + 2] = slice(None)
s_d[ax] = slice(None)
flow_current[idx] = flow_auxiliary[idx] + d
flow_previous -= flow_current # The difference as stopping criteria
if (flow_previous * flow_previous).sum() < tol:
break
flow_previous = flow_current
return flow_current
def hessian_matrix(image, sigma=1, mode="constant", cval=0, order="rc"):
"""Compute Hessian matrix.
The Hessian matrix is defined as::
H = [Hrr Hrc]
[Hrc Hcc]
which is computed by convolving the image with the second derivatives
of the Gaussian kernel in the respective r- and c-directions.
Parameters
----------
image : ndarray
Input image.
sigma : float
Standard deviation used for the Gaussian kernel, which is used as
weighting function for the auto-correlation matrix.
mode : {'constant', 'reflect', 'wrap', 'nearest', 'mirror'}, optional
How to handle values outside the image borders.
cval : float, optional
Used in conjunction with mode 'constant', the value outside
the image boundaries.
order : {'rc', 'xy'}, optional
This parameter allows for the use of reverse or forward order of
the image axes in gradient computation. 'rc' indicates the use of
the first axis initially (Hrr, Hrc, Hcc), whilst 'xy' indicates the
usage of the last axis initially (Hxx, Hxy, Hyy)
Returns
-------
Hrr : ndarray
Element of the Hessian matrix for each pixel in the input image.
Hrc : ndarray
Element of the Hessian matrix for each pixel in the input image.
Hcc : ndarray
Element of the Hessian matrix for each pixel in the input image.
Examples
--------
>>> import cupy as cp
>>> from cupyimg.skimage.feature import hessian_matrix
>>> square = cp.zeros((5, 5))
>>> square[2, 2] = 4
>>> Hrr, Hrc, Hcc = hessian_matrix(square, sigma=0.1, order='rc')
>>> Hrc
array([[ 0., 0., 0., 0., 0.],
[ 0., 1., 0., -1., 0.],
[ 0., 0., 0., 0., 0.],
[ 0., -1., 0., 1., 0.],
[ 0., 0., 0., 0., 0.]])
"""
image = img_as_float(image)
gaussian_filtered = ndi.gaussian_filter(image,
sigma=sigma,
mode=mode,
cval=cval)
gradients = cnp.gradient(gaussian_filtered)
axes = range(image.ndim)
if order == "rc":
axes = reversed(axes)
H_elems = [
cnp.gradient(gradients[ax0], axis=ax1)
for ax0, ax1 in combinations_with_replacement(axes, 2)
]
return H_elems