671 lines
27 KiB
Python
671 lines
27 KiB
Python
"""max_tree.py - max_tree representation of images.
|
|
|
|
This module provides operators based on the max-tree representation of images.
|
|
A grayscale image can be seen as a pile of nested sets, each of which is the
|
|
result of a threshold operation. These sets can be efficiently represented by
|
|
max-trees, where the inclusion relation between connected components at
|
|
different levels are represented by parent-child relationships.
|
|
|
|
These representations allow efficient implementations of many algorithms, such
|
|
as attribute operators. Unlike morphological openings and closings, these
|
|
operators do not require a fixed structuring element, but rather act with a
|
|
flexible structuring element that meets a certain criterion.
|
|
|
|
This implementation provides functions for:
|
|
1. max-tree generation
|
|
2. area openings / closings
|
|
3. diameter openings / closings
|
|
4. local maxima
|
|
|
|
References:
|
|
.. [1] Salembier, P., Oliveras, A., & Garrido, L. (1998). Antiextensive
|
|
Connected Operators for Image and Sequence Processing.
|
|
IEEE Transactions on Image Processing, 7(4), 555-570.
|
|
:DOI:10.1109/83.663500
|
|
.. [2] Berger, C., Geraud, T., Levillain, R., Widynski, N., Baillard, A.,
|
|
Bertin, E. (2007). Effective Component Tree Computation with
|
|
Application to Pattern Recognition in Astronomical Imaging.
|
|
In International Conference on Image Processing (ICIP) (pp. 41-44).
|
|
:DOI:10.1109/ICIP.2007.4379949
|
|
.. [3] Najman, L., & Couprie, M. (2006). Building the component tree in
|
|
quasi-linear time. IEEE Transactions on Image Processing, 15(11),
|
|
3531-3539.
|
|
:DOI:10.1109/TIP.2006.877518
|
|
.. [4] Carlinet, E., & Geraud, T. (2014). A Comparative Review of
|
|
Component Tree Computation Algorithms. IEEE Transactions on Image
|
|
Processing, 23(9), 3885-3895.
|
|
:DOI:10.1109/TIP.2014.2336551
|
|
"""
|
|
|
|
import numpy as np
|
|
|
|
from ._util import _validate_connectivity, _offsets_to_raveled_neighbors
|
|
from ..util import invert
|
|
|
|
from . import _max_tree
|
|
|
|
unsigned_int_types = [np.uint8, np.uint16, np.uint32, np.uint64]
|
|
signed_int_types = [np.int8, np.int16, np.int32, np.int64]
|
|
signed_float_types = [np.float16, np.float32, np.float64]
|
|
|
|
|
|
# building the max tree.
|
|
def max_tree(image, connectivity=1):
|
|
"""Build the max tree from an image.
|
|
|
|
Component trees represent the hierarchical structure of the connected
|
|
components resulting from sequential thresholding operations applied to an
|
|
image. A connected component at one level is parent of a component at a
|
|
higher level if the latter is included in the first. A max-tree is an
|
|
efficient representation of a component tree. A connected component at
|
|
one level is represented by one reference pixel at this level, which is
|
|
parent to all other pixels at that level and to the reference pixel at the
|
|
level above. The max-tree is the basis for many morphological operators,
|
|
namely connected operators.
|
|
|
|
Parameters
|
|
----------
|
|
image : ndarray
|
|
The input image for which the max-tree is to be calculated.
|
|
This image can be of any type.
|
|
connectivity : unsigned int, optional
|
|
The neighborhood connectivity. The integer represents the maximum
|
|
number of orthogonal steps to reach a neighbor. In 2D, it is 1 for
|
|
a 4-neighborhood and 2 for a 8-neighborhood. Default value is 1.
|
|
|
|
Returns
|
|
-------
|
|
parent : ndarray, int64
|
|
Array of same shape as image. The value of each pixel is the index of
|
|
its parent in the ravelled array.
|
|
|
|
tree_traverser : 1D array, int64
|
|
The ordered pixel indices (referring to the ravelled array). The pixels
|
|
are ordered such that every pixel is preceded by its parent (except for
|
|
the root which has no parent).
|
|
|
|
References
|
|
----------
|
|
.. [1] Salembier, P., Oliveras, A., & Garrido, L. (1998). Antiextensive
|
|
Connected Operators for Image and Sequence Processing.
|
|
IEEE Transactions on Image Processing, 7(4), 555-570.
|
|
:DOI:`10.1109/83.663500`
|
|
.. [2] Berger, C., Geraud, T., Levillain, R., Widynski, N., Baillard, A.,
|
|
Bertin, E. (2007). Effective Component Tree Computation with
|
|
Application to Pattern Recognition in Astronomical Imaging.
|
|
In International Conference on Image Processing (ICIP) (pp. 41-44).
|
|
:DOI:`10.1109/ICIP.2007.4379949`
|
|
.. [3] Najman, L., & Couprie, M. (2006). Building the component tree in
|
|
quasi-linear time. IEEE Transactions on Image Processing, 15(11),
|
|
3531-3539.
|
|
:DOI:`10.1109/TIP.2006.877518`
|
|
.. [4] Carlinet, E., & Geraud, T. (2014). A Comparative Review of
|
|
Component Tree Computation Algorithms. IEEE Transactions on Image
|
|
Processing, 23(9), 3885-3895.
|
|
:DOI:`10.1109/TIP.2014.2336551`
|
|
|
|
Examples
|
|
--------
|
|
We create a small sample image (Figure 1 from [4]) and build the max-tree.
|
|
|
|
>>> image = np.array([[15, 13, 16], [12, 12, 10], [16, 12, 14]])
|
|
>>> P, S = max_tree(image, connectivity=2)
|
|
"""
|
|
# User defined masks are not allowed, as there might be more than one
|
|
# connected component in the mask (and therefore not a single tree that
|
|
# represents the image). Mask here is an image that is 0 on the border
|
|
# and 1 everywhere else.
|
|
mask = np.ones(image.shape)
|
|
for k in range(len(image.shape)):
|
|
np.moveaxis(mask, k, 0)[0] = 0
|
|
np.moveaxis(mask, k, 0)[-1] = 0
|
|
|
|
neighbors, offset = _validate_connectivity(image.ndim, connectivity,
|
|
offset=None)
|
|
|
|
# initialization of the parent image
|
|
parent = np.zeros(image.shape, dtype=np.int64)
|
|
|
|
# flat_neighborhood contains a list of offsets allowing one to find the
|
|
# neighbors in the ravelled image.
|
|
flat_neighborhood = _offsets_to_raveled_neighbors(image.shape, neighbors,
|
|
offset).astype(np.int32)
|
|
|
|
# pixels need to be sorted according to their gray level.
|
|
tree_traverser = np.argsort(image.ravel()).astype(np.int64)
|
|
|
|
# call of cython function.
|
|
_max_tree._max_tree(image.ravel(), mask.ravel().astype(np.uint8),
|
|
flat_neighborhood, offset.astype(np.int32),
|
|
np.array(image.shape, dtype=np.int32),
|
|
parent.ravel(), tree_traverser)
|
|
|
|
return parent, tree_traverser
|
|
|
|
|
|
def area_opening(image, area_threshold=64, connectivity=1,
|
|
parent=None, tree_traverser=None):
|
|
"""Perform an area opening of the image.
|
|
|
|
Area opening removes all bright structures of an image with
|
|
a surface smaller than area_threshold.
|
|
The output image is thus the largest image smaller than the input
|
|
for which all local maxima have at least a surface of
|
|
area_threshold pixels.
|
|
|
|
Area openings are similar to morphological openings, but
|
|
they do not use a fixed structuring element, but rather a deformable
|
|
one, with surface = area_threshold. Consequently, the area_opening
|
|
with area_threshold=1 is the identity.
|
|
|
|
In the binary case, area openings are equivalent to
|
|
remove_small_objects; this operator is thus extended to gray-level images.
|
|
|
|
Technically, this operator is based on the max-tree representation of
|
|
the image.
|
|
|
|
Parameters
|
|
----------
|
|
image : ndarray
|
|
The input image for which the area_opening is to be calculated.
|
|
This image can be of any type.
|
|
area_threshold : unsigned int
|
|
The size parameter (number of pixels). The default value is arbitrarily
|
|
chosen to be 64.
|
|
connectivity : unsigned int, optional
|
|
The neighborhood connectivity. The integer represents the maximum
|
|
number of orthogonal steps to reach a neighbor. In 2D, it is 1 for
|
|
a 4-neighborhood and 2 for a 8-neighborhood. Default value is 1.
|
|
parent : ndarray, int64, optional
|
|
Parent image representing the max tree of the image. The
|
|
value of each pixel is the index of its parent in the ravelled array.
|
|
tree_traverser : 1D array, int64, optional
|
|
The ordered pixel indices (referring to the ravelled array). The pixels
|
|
are ordered such that every pixel is preceded by its parent (except for
|
|
the root which has no parent).
|
|
|
|
Returns
|
|
-------
|
|
output : ndarray
|
|
Output image of the same shape and type as the input image.
|
|
|
|
See also
|
|
--------
|
|
skimage.morphology.area_closing
|
|
skimage.morphology.diameter_opening
|
|
skimage.morphology.diameter_closing
|
|
skimage.morphology.max_tree
|
|
skimage.morphology.remove_small_objects
|
|
skimage.morphology.remove_small_holes
|
|
|
|
|
|
References
|
|
----------
|
|
.. [1] Vincent L., Proc. "Grayscale area openings and closings,
|
|
their efficient implementation and applications",
|
|
EURASIP Workshop on Mathematical Morphology and its
|
|
Applications to Signal Processing, Barcelona, Spain, pp.22-27,
|
|
May 1993.
|
|
.. [2] Soille, P., "Morphological Image Analysis: Principles and
|
|
Applications" (Chapter 6), 2nd edition (2003), ISBN 3540429883.
|
|
:DOI:10.1007/978-3-662-05088-0
|
|
.. [3] Salembier, P., Oliveras, A., & Garrido, L. (1998). Antiextensive
|
|
Connected Operators for Image and Sequence Processing.
|
|
IEEE Transactions on Image Processing, 7(4), 555-570.
|
|
:DOI:10.1109/83.663500
|
|
.. [4] Najman, L., & Couprie, M. (2006). Building the component tree in
|
|
quasi-linear time. IEEE Transactions on Image Processing, 15(11),
|
|
3531-3539.
|
|
:DOI:10.1109/TIP.2006.877518
|
|
.. [5] Carlinet, E., & Geraud, T. (2014). A Comparative Review of
|
|
Component Tree Computation Algorithms. IEEE Transactions on Image
|
|
Processing, 23(9), 3885-3895.
|
|
:DOI:10.1109/TIP.2014.2336551
|
|
|
|
Examples
|
|
--------
|
|
|
|
We create an image (quadratic function with a maximum in the center and
|
|
4 additional local maxima.
|
|
|
|
>>> w = 12
|
|
>>> x, y = np.mgrid[0:w,0:w]
|
|
>>> f = 20 - 0.2*((x - w/2)**2 + (y-w/2)**2)
|
|
>>> f[2:3,1:5] = 40; f[2:4,9:11] = 60; f[9:11,2:4] = 80
|
|
>>> f[9:10,9:11] = 100; f[10,10] = 100
|
|
>>> f = f.astype(int)
|
|
|
|
We can calculate the area opening:
|
|
|
|
>>> open = area_opening(f, 8, connectivity=1)
|
|
|
|
The peaks with a surface smaller than 8 are removed.
|
|
"""
|
|
output = image.copy()
|
|
|
|
if parent is None or tree_traverser is None:
|
|
parent, tree_traverser = max_tree(image, connectivity)
|
|
|
|
area = _max_tree._compute_area(image.ravel(),
|
|
parent.ravel(), tree_traverser)
|
|
|
|
_max_tree._direct_filter(image.ravel(), output.ravel(), parent.ravel(),
|
|
tree_traverser, area, area_threshold)
|
|
return output
|
|
|
|
|
|
def diameter_opening(image, diameter_threshold=8, connectivity=1,
|
|
parent=None, tree_traverser=None):
|
|
"""Perform a diameter opening of the image.
|
|
|
|
Diameter opening removes all bright structures of an image with
|
|
maximal extension smaller than diameter_threshold. The maximal
|
|
extension is defined as the maximal extension of the bounding box.
|
|
The operator is also called Bounding Box Opening. In practice,
|
|
the result is similar to a morphological opening, but long and thin
|
|
structures are not removed.
|
|
|
|
Technically, this operator is based on the max-tree representation of
|
|
the image.
|
|
|
|
Parameters
|
|
----------
|
|
image : ndarray
|
|
The input image for which the area_opening is to be calculated.
|
|
This image can be of any type.
|
|
diameter_threshold : unsigned int
|
|
The maximal extension parameter (number of pixels). The default value
|
|
is 8.
|
|
connectivity : unsigned int, optional
|
|
The neighborhood connectivity. The integer represents the maximum
|
|
number of orthogonal steps to reach a neighbor. In 2D, it is 1 for
|
|
a 4-neighborhood and 2 for a 8-neighborhood. Default value is 1.
|
|
parent : ndarray, int64, optional
|
|
Parent image representing the max tree of the image. The
|
|
value of each pixel is the index of its parent in the ravelled array.
|
|
tree_traverser : 1D array, int64, optional
|
|
The ordered pixel indices (referring to the ravelled array). The pixels
|
|
are ordered such that every pixel is preceded by its parent (except for
|
|
the root which has no parent).
|
|
|
|
Returns
|
|
-------
|
|
output : ndarray
|
|
Output image of the same shape and type as the input image.
|
|
|
|
See also
|
|
--------
|
|
skimage.morphology.area_opening
|
|
skimage.morphology.area_closing
|
|
skimage.morphology.diameter_closing
|
|
skimage.morphology.max_tree
|
|
|
|
References
|
|
----------
|
|
.. [1] Walter, T., & Klein, J.-C. (2002). Automatic Detection of
|
|
Microaneurysms in Color Fundus Images of the Human Retina by Means
|
|
of the Bounding Box Closing. In A. Colosimo, P. Sirabella,
|
|
A. Giuliani (Eds.), Medical Data Analysis. Lecture Notes in Computer
|
|
Science, vol 2526, pp. 210-220. Springer Berlin Heidelberg.
|
|
:DOI:`10.1007/3-540-36104-9_23`
|
|
.. [2] Carlinet, E., & Geraud, T. (2014). A Comparative Review of
|
|
Component Tree Computation Algorithms. IEEE Transactions on Image
|
|
Processing, 23(9), 3885-3895.
|
|
:DOI:`10.1109/TIP.2014.2336551`
|
|
|
|
Examples
|
|
--------
|
|
We create an image (quadratic function with a maximum in the center and
|
|
4 additional local maxima.
|
|
|
|
>>> w = 12
|
|
>>> x, y = np.mgrid[0:w,0:w]
|
|
>>> f = 20 - 0.2*((x - w/2)**2 + (y-w/2)**2)
|
|
>>> f[2:3,1:5] = 40; f[2:4,9:11] = 60; f[9:11,2:4] = 80
|
|
>>> f[9:10,9:11] = 100; f[10,10] = 100
|
|
>>> f = f.astype(int)
|
|
|
|
We can calculate the diameter opening:
|
|
|
|
>>> open = diameter_opening(f, 3, connectivity=1)
|
|
|
|
The peaks with a maximal extension of 2 or less are removed.
|
|
The remaining peaks have all a maximal extension of at least 3.
|
|
"""
|
|
output = image.copy()
|
|
|
|
if parent is None or tree_traverser is None:
|
|
parent, tree_traverser = max_tree(image, connectivity)
|
|
|
|
diam = _max_tree._compute_extension(image.ravel(),
|
|
np.array(image.shape, dtype=np.int32),
|
|
parent.ravel(), tree_traverser)
|
|
|
|
_max_tree._direct_filter(image.ravel(), output.ravel(), parent.ravel(),
|
|
tree_traverser, diam, diameter_threshold)
|
|
return output
|
|
|
|
|
|
def area_closing(image, area_threshold=64, connectivity=1,
|
|
parent=None, tree_traverser=None):
|
|
"""Perform an area closing of the image.
|
|
|
|
Area closing removes all dark structures of an image with
|
|
a surface smaller than area_threshold.
|
|
The output image is larger than or equal to the input image
|
|
for every pixel and all local minima have at least a surface of
|
|
area_threshold pixels.
|
|
|
|
Area closings are similar to morphological closings, but
|
|
they do not use a fixed structuring element, but rather a deformable
|
|
one, with surface = area_threshold.
|
|
|
|
In the binary case, area closings are equivalent to
|
|
remove_small_holes; this operator is thus extended to gray-level images.
|
|
|
|
Technically, this operator is based on the max-tree representation of
|
|
the image.
|
|
|
|
Parameters
|
|
----------
|
|
image : ndarray
|
|
The input image for which the area_closing is to be calculated.
|
|
This image can be of any type.
|
|
area_threshold : unsigned int
|
|
The size parameter (number of pixels). The default value is arbitrarily
|
|
chosen to be 64.
|
|
connectivity : unsigned int, optional
|
|
The neighborhood connectivity. The integer represents the maximum
|
|
number of orthogonal steps to reach a neighbor. In 2D, it is 1 for
|
|
a 4-neighborhood and 2 for a 8-neighborhood. Default value is 1.
|
|
parent : ndarray, int64, optional
|
|
Parent image representing the max tree of the inverted image. The
|
|
value of each pixel is the index of its parent in the ravelled array.
|
|
See Note for further details.
|
|
tree_traverser : 1D array, int64, optional
|
|
The ordered pixel indices (referring to the ravelled array). The pixels
|
|
are ordered such that every pixel is preceded by its parent (except for
|
|
the root which has no parent).
|
|
|
|
Returns
|
|
-------
|
|
output : ndarray
|
|
Output image of the same shape and type as input image.
|
|
|
|
See also
|
|
--------
|
|
skimage.morphology.area_opening
|
|
skimage.morphology.diameter_opening
|
|
skimage.morphology.diameter_closing
|
|
skimage.morphology.max_tree
|
|
skimage.morphology.remove_small_objects
|
|
skimage.morphology.remove_small_holes
|
|
|
|
References
|
|
----------
|
|
.. [1] Vincent L., Proc. "Grayscale area openings and closings,
|
|
their efficient implementation and applications",
|
|
EURASIP Workshop on Mathematical Morphology and its
|
|
Applications to Signal Processing, Barcelona, Spain, pp.22-27,
|
|
May 1993.
|
|
.. [2] Soille, P., "Morphological Image Analysis: Principles and
|
|
Applications" (Chapter 6), 2nd edition (2003), ISBN 3540429883.
|
|
:DOI:`10.1007/978-3-662-05088-0`
|
|
.. [3] Salembier, P., Oliveras, A., & Garrido, L. (1998). Antiextensive
|
|
Connected Operators for Image and Sequence Processing.
|
|
IEEE Transactions on Image Processing, 7(4), 555-570.
|
|
:DOI:`10.1109/83.663500`
|
|
.. [4] Najman, L., & Couprie, M. (2006). Building the component tree in
|
|
quasi-linear time. IEEE Transactions on Image Processing, 15(11),
|
|
3531-3539.
|
|
:DOI:`10.1109/TIP.2006.877518`
|
|
.. [5] Carlinet, E., & Geraud, T. (2014). A Comparative Review of
|
|
Component Tree Computation Algorithms. IEEE Transactions on Image
|
|
Processing, 23(9), 3885-3895.
|
|
:DOI:`10.1109/TIP.2014.2336551`
|
|
|
|
|
|
Examples
|
|
--------
|
|
We create an image (quadratic function with a minimum in the center and
|
|
4 additional local minima.
|
|
|
|
>>> w = 12
|
|
>>> x, y = np.mgrid[0:w,0:w]
|
|
>>> f = 180 + 0.2*((x - w/2)**2 + (y-w/2)**2)
|
|
>>> f[2:3,1:5] = 160; f[2:4,9:11] = 140; f[9:11,2:4] = 120
|
|
>>> f[9:10,9:11] = 100; f[10,10] = 100
|
|
>>> f = f.astype(int)
|
|
|
|
We can calculate the area closing:
|
|
|
|
>>> closed = area_closing(f, 8, connectivity=1)
|
|
|
|
All small minima are removed, and the remaining minima have at least
|
|
a size of 8.
|
|
|
|
|
|
Notes
|
|
-----
|
|
If a max-tree representation (parent and tree_traverser) are given to the
|
|
function, they must be calculated from the inverted image for this
|
|
function, i.e.:
|
|
>>> P, S = max_tree(invert(f))
|
|
>>> closed = diameter_closing(f, 3, parent=P, tree_traverser=S)
|
|
"""
|
|
# inversion of the input image
|
|
image_inv = invert(image)
|
|
output = image_inv.copy()
|
|
|
|
if parent is None or tree_traverser is None:
|
|
parent, tree_traverser = max_tree(image_inv, connectivity)
|
|
|
|
area = _max_tree._compute_area(image_inv.ravel(),
|
|
parent.ravel(), tree_traverser)
|
|
|
|
_max_tree._direct_filter(image_inv.ravel(), output.ravel(), parent.ravel(),
|
|
tree_traverser, area, area_threshold)
|
|
|
|
# inversion of the output image
|
|
output = invert(output)
|
|
|
|
return output
|
|
|
|
|
|
def diameter_closing(image, diameter_threshold=8, connectivity=1,
|
|
parent=None, tree_traverser=None):
|
|
"""Perform a diameter closing of the image.
|
|
|
|
Diameter closing removes all dark structures of an image with
|
|
maximal extension smaller than diameter_threshold. The maximal
|
|
extension is defined as the maximal extension of the bounding box.
|
|
The operator is also called Bounding Box Closing. In practice,
|
|
the result is similar to a morphological closing, but long and thin
|
|
structures are not removed.
|
|
|
|
Technically, this operator is based on the max-tree representation of
|
|
the image.
|
|
|
|
Parameters
|
|
----------
|
|
image : ndarray
|
|
The input image for which the diameter_closing is to be calculated.
|
|
This image can be of any type.
|
|
diameter_threshold : unsigned int
|
|
The maximal extension parameter (number of pixels). The default value
|
|
is 8.
|
|
connectivity : unsigned int, optional
|
|
The neighborhood connectivity. The integer represents the maximum
|
|
number of orthogonal steps to reach a neighbor. In 2D, it is 1 for
|
|
a 4-neighborhood and 2 for a 8-neighborhood. Default value is 1.
|
|
parent : ndarray, int64, optional
|
|
Precomputed parent image representing the max tree of the inverted
|
|
image. This function is fast, if precomputed parent and tree_traverser
|
|
are provided. See Note for further details.
|
|
tree_traverser : 1D array, int64, optional
|
|
Precomputed traverser, where the pixels are ordered such that every
|
|
pixel is preceded by its parent (except for the root which has no
|
|
parent). This function is fast, if precomputed parent and
|
|
tree_traverser are provided. See Note for further details.
|
|
|
|
Returns
|
|
-------
|
|
output : ndarray
|
|
Output image of the same shape and type as input image.
|
|
|
|
See also
|
|
--------
|
|
skimage.morphology.area_opening
|
|
skimage.morphology.area_closing
|
|
skimage.morphology.diameter_opening
|
|
skimage.morphology.max_tree
|
|
|
|
References
|
|
----------
|
|
.. [1] Walter, T., & Klein, J.-C. (2002). Automatic Detection of
|
|
Microaneurysms in Color Fundus Images of the Human Retina by Means
|
|
of the Bounding Box Closing. In A. Colosimo, P. Sirabella,
|
|
A. Giuliani (Eds.), Medical Data Analysis. Lecture Notes in Computer
|
|
Science, vol 2526, pp. 210-220. Springer Berlin Heidelberg.
|
|
:DOI:`10.1007/3-540-36104-9_23`
|
|
.. [2] Carlinet, E., & Geraud, T. (2014). A Comparative Review of
|
|
Component Tree Computation Algorithms. IEEE Transactions on Image
|
|
Processing, 23(9), 3885-3895.
|
|
:DOI:`10.1109/TIP.2014.2336551`
|
|
|
|
Examples
|
|
--------
|
|
We create an image (quadratic function with a minimum in the center and
|
|
4 additional local minima.
|
|
|
|
>>> w = 12
|
|
>>> x, y = np.mgrid[0:w,0:w]
|
|
>>> f = 180 + 0.2*((x - w/2)**2 + (y-w/2)**2)
|
|
>>> f[2:3,1:5] = 160; f[2:4,9:11] = 140; f[9:11,2:4] = 120
|
|
>>> f[9:10,9:11] = 100; f[10,10] = 100
|
|
>>> f = f.astype(int)
|
|
|
|
We can calculate the diameter closing:
|
|
|
|
>>> closed = diameter_closing(f, 3, connectivity=1)
|
|
|
|
All small minima with a maximal extension of 2 or less are removed.
|
|
The remaining minima have all a maximal extension of at least 3.
|
|
|
|
|
|
Notes
|
|
-----
|
|
If a max-tree representation (parent and tree_traverser) are given to the
|
|
function, they must be calculated from the inverted image for this
|
|
function, i.e.:
|
|
>>> P, S = max_tree(invert(f))
|
|
>>> closed = diameter_closing(f, 3, parent=P, tree_traverser=S)
|
|
"""
|
|
# inversion of the input image
|
|
image_inv = invert(image)
|
|
output = image_inv.copy()
|
|
|
|
if parent is None or tree_traverser is None:
|
|
parent, tree_traverser = max_tree(image_inv, connectivity)
|
|
|
|
diam = _max_tree._compute_extension(image_inv.ravel(),
|
|
np.array(image_inv.shape,
|
|
dtype=np.int32),
|
|
parent.ravel(), tree_traverser)
|
|
|
|
_max_tree._direct_filter(image_inv.ravel(), output.ravel(), parent.ravel(),
|
|
tree_traverser, diam, diameter_threshold)
|
|
output = invert(output)
|
|
return output
|
|
|
|
|
|
def max_tree_local_maxima(image, connectivity=1,
|
|
parent=None, tree_traverser=None):
|
|
"""Determine all local maxima of the image.
|
|
|
|
The local maxima are defined as connected sets of pixels with equal
|
|
gray level strictly greater than the gray levels of all pixels in direct
|
|
neighborhood of the set. The function labels the local maxima.
|
|
|
|
Technically, the implementation is based on the max-tree representation
|
|
of an image. The function is very efficient if the max-tree representation
|
|
has already been computed. Otherwise, it is preferable to use
|
|
the function local_maxima.
|
|
|
|
Parameters
|
|
----------
|
|
image : ndarray
|
|
The input image for which the maxima are to be calculated.
|
|
connectivity: unsigned int, optional
|
|
The neighborhood connectivity. The integer represents the maximum
|
|
number of orthogonal steps to reach a neighbor. In 2D, it is 1 for
|
|
a 4-neighborhood and 2 for a 8-neighborhood. Default value is 1.
|
|
parent: ndarray, int64, optional
|
|
The value of each pixel is the index of its parent in the ravelled
|
|
array.
|
|
tree_traverser: 1D array, int64, optional
|
|
The ordered pixel indices (referring to the ravelled array). The pixels
|
|
are ordered such that every pixel is preceded by its parent (except for
|
|
the root which has no parent).
|
|
|
|
Returns
|
|
-------
|
|
local_max : ndarray, uint64
|
|
Labeled local maxima of the image.
|
|
|
|
See also
|
|
--------
|
|
skimage.morphology.local_maxima
|
|
skimage.morphology.max_tree
|
|
|
|
References
|
|
----------
|
|
.. [1] Vincent L., Proc. "Grayscale area openings and closings,
|
|
their efficient implementation and applications",
|
|
EURASIP Workshop on Mathematical Morphology and its
|
|
Applications to Signal Processing, Barcelona, Spain, pp.22-27,
|
|
May 1993.
|
|
.. [2] Soille, P., "Morphological Image Analysis: Principles and
|
|
Applications" (Chapter 6), 2nd edition (2003), ISBN 3540429883.
|
|
:DOI:`10.1007/978-3-662-05088-0`
|
|
.. [3] Salembier, P., Oliveras, A., & Garrido, L. (1998). Antiextensive
|
|
Connected Operators for Image and Sequence Processing.
|
|
IEEE Transactions on Image Processing, 7(4), 555-570.
|
|
:DOI:`10.1109/83.663500`
|
|
.. [4] Najman, L., & Couprie, M. (2006). Building the component tree in
|
|
quasi-linear time. IEEE Transactions on Image Processing, 15(11),
|
|
3531-3539.
|
|
:DOI:`10.1109/TIP.2006.877518`
|
|
.. [5] Carlinet, E., & Geraud, T. (2014). A Comparative Review of
|
|
Component Tree Computation Algorithms. IEEE Transactions on Image
|
|
Processing, 23(9), 3885-3895.
|
|
:DOI:`10.1109/TIP.2014.2336551`
|
|
|
|
Examples
|
|
--------
|
|
We create an image (quadratic function with a maximum in the center and
|
|
4 additional constant maxima.
|
|
|
|
>>> w = 10
|
|
>>> x, y = np.mgrid[0:w,0:w]
|
|
>>> f = 20 - 0.2*((x - w/2)**2 + (y-w/2)**2)
|
|
>>> f[2:4,2:4] = 40; f[2:4,7:9] = 60; f[7:9,2:4] = 80; f[7:9,7:9] = 100
|
|
>>> f = f.astype(int)
|
|
|
|
We can calculate all local maxima:
|
|
|
|
>>> maxima = max_tree_local_maxima(f)
|
|
|
|
The resulting image contains the labeled local maxima.
|
|
"""
|
|
|
|
output = np.ones(image.shape, dtype=np.uint64)
|
|
|
|
if parent is None or tree_traverser is None:
|
|
parent, tree_traverser = max_tree(image, connectivity)
|
|
|
|
_max_tree._max_tree_local_maxima(image.ravel(), output.ravel(),
|
|
parent.ravel(), tree_traverser)
|
|
|
|
return output
|