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"""extrema.py - local minima and maxima
This module provides functions to find local maxima and minima of an image.
Here, local maxima (minima) are defined as connected sets of pixels with equal
gray level which is strictly greater (smaller) than the gray level of all
pixels in direct neighborhood of the connected set. In addition, the module
provides the related functions h-maxima and h-minima.
Soille, P. (2003). Morphological Image Analysis: Principles and Applications
(2nd ed.), Chapter 6. Springer-Verlag New York, Inc.
"""
from __future__ import absolute_import
import numpy as np
from scipy import ndimage as ndi
from ..util import dtype_limits, invert, crop
from .._shared.utils import warn
from . import greyreconstruct
from .watershed import _offsets_to_raveled_neighbors
from ._extrema_cy import _local_maxima
def _add_constant_clip(image, const_value):
"""Add constant to the image while handling overflow issues gracefully.
"""
min_dtype, max_dtype = dtype_limits(image, clip_negative=False)
if const_value > (max_dtype - min_dtype):
raise ValueError("The added constant is not compatible"
"with the image data type.")
result = image + const_value
result[image > max_dtype-const_value] = max_dtype
return(result)
def _subtract_constant_clip(image, const_value):
"""Subtract constant from image while handling underflow issues.
"""
min_dtype, max_dtype = dtype_limits(image, clip_negative=False)
if const_value > (max_dtype-min_dtype):
raise ValueError("The subtracted constant is not compatible"
"with the image data type.")
result = image - const_value
result[image < (const_value + min_dtype)] = min_dtype
return(result)
def h_maxima(image, h, selem=None):
"""Determine all maxima of the image with height >= h.
The local maxima are defined as connected sets of pixels with equal
grey level strictly greater than the grey level of all pixels in direct
neighborhood of the set.
A local maximum M of height h is a local maximum for which
there is at least one path joining M with a higher maximum on which the
minimal value is f(M) - h (i.e. the values along the path are not
decreasing by more than h with respect to the maximum's value) and no
path for which the minimal value is greater.
Parameters
----------
image : ndarray
The input image for which the maxima are to be calculated.
h : unsigned integer
The minimal height of all extracted maxima.
selem : ndarray, optional
The neighborhood expressed as an n-D array of 1's and 0's.
Default is the ball of radius 1 according to the maximum norm
(i.e. a 3x3 square for 2D images, a 3x3x3 cube for 3D images, etc.)
Returns
-------
h_max : ndarray
The maxima of height >= h. The result image is a binary image, where
pixels belonging to the selected maxima take value 1, the others
take value 0.
See also
--------
skimage.morphology.extrema.h_minima
skimage.morphology.extrema.local_maxima
skimage.morphology.extrema.local_minima
References
----------
.. [1] Soille, P., "Morphological Image Analysis: Principles and
Applications" (Chapter 6), 2nd edition (2003), ISBN 3540429883.
Examples
--------
>>> import numpy as np
>>> from skimage.morphology import extrema
We create an image (quadratic function with a maximum in the center and
4 additional constant maxima.
The heights of the maxima are: 1, 21, 41, 61, 81, 101
>>> 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(np.int)
We can calculate all maxima with a height of at least 40:
>>> maxima = extrema.h_maxima(f, 40)
The resulting image will contain 4 local maxima.
"""
if np.issubdtype(image.dtype, np.floating):
resolution = 2 * np.finfo(image.dtype).resolution
if h < resolution:
h = resolution
h_corrected = h - resolution / 2.0
shifted_img = image - h
else:
shifted_img = _subtract_constant_clip(image, h)
h_corrected = h
rec_img = greyreconstruct.reconstruction(shifted_img, image,
method='dilation', selem=selem)
residue_img = image - rec_img
h_max = np.zeros(image.shape, dtype=np.uint8)
h_max[residue_img >= h_corrected] = 1
return h_max
def h_minima(image, h, selem=None):
"""Determine all minima of the image with depth >= h.
The local minima are defined as connected sets of pixels with equal
grey level strictly smaller than the grey levels of all pixels in direct
neighborhood of the set.
A local minimum M of depth h is a local minimum for which
there is at least one path joining M with a deeper minimum on which the
maximal value is f(M) + h (i.e. the values along the path are not
increasing by more than h with respect to the minimum's value) and no
path for which the maximal value is smaller.
Parameters
----------
image : ndarray
The input image for which the minima are to be calculated.
h : unsigned integer
The minimal depth of all extracted minima.
selem : ndarray, optional
The neighborhood expressed as an n-D array of 1's and 0's.
Default is the ball of radius 1 according to the maximum norm
(i.e. a 3x3 square for 2D images, a 3x3x3 cube for 3D images, etc.)
Returns
-------
h_min : ndarray
The minima of depth >= h. The result image is a binary image, where
pixels belonging to the selected minima take value 1, the other pixels
take value 0.
See also
--------
skimage.morphology.extrema.h_maxima
skimage.morphology.extrema.local_maxima
skimage.morphology.extrema.local_minima
References
----------
.. [1] Soille, P., "Morphological Image Analysis: Principles and
Applications" (Chapter 6), 2nd edition (2003), ISBN 3540429883.
Examples
--------
>>> import numpy as np
>>> from skimage.morphology import extrema
We create an image (quadratic function with a minimum in the center and
4 additional constant maxima.
The depth of the minima are: 1, 21, 41, 61, 81, 101
>>> w = 10
>>> x, y = np.mgrid[0:w,0:w]
>>> f = 180 + 0.2*((x - w/2)**2 + (y-w/2)**2)
>>> f[2:4,2:4] = 160; f[2:4,7:9] = 140; f[7:9,2:4] = 120; f[7:9,7:9] = 100
>>> f = f.astype(np.int)
We can calculate all minima with a depth of at least 40:
>>> minima = extrema.h_minima(f, 40)
The resulting image will contain 4 local minima.
"""
if np.issubdtype(image.dtype, np.floating):
resolution = 2 * np.finfo(image.dtype).resolution
if h < resolution:
h = resolution
h_corrected = h - resolution / 2.0
shifted_img = image + h
else:
shifted_img = _add_constant_clip(image, h)
h_corrected = h
rec_img = greyreconstruct.reconstruction(shifted_img, image,
method='erosion', selem=selem)
residue_img = rec_img - image
h_min = np.zeros(image.shape, dtype=np.uint8)
h_min[residue_img >= h_corrected] = 1
return h_min
def _set_edge_values_inplace(image, value):
"""Set edge values along all axes to a constant value.
Parameters
----------
image : ndarray
The array to modify inplace.
value : scalar
The value to use. Should be compatible with `image`'s dtype.
Examples
--------
>>> image = np.zeros((4, 5), dtype=int)
>>> _set_edge_values_inplace(image, 1)
>>> image
array([[1, 1, 1, 1, 1],
[1, 0, 0, 0, 1],
[1, 0, 0, 0, 1],
[1, 1, 1, 1, 1]])
"""
for axis in range(image.ndim):
sl = [slice(None)] * image.ndim
# Set edge in front
sl[axis] = 0
image[tuple(sl)] = value
# Set edge to the end
sl[axis] = -1
image[tuple(sl)] = value
def _fast_pad(image, value):
"""Pad an array on all axes with one constant value.
Parameters
----------
image : ndarray
Image to pad.
value : scalar
The value to use. Should be compatible with `image`'s dtype.
Returns
-------
padded_image : ndarray
The new image.
Notes
-----
The output of this function is equivalent to::
np.pad(image, mode="constant", constant_values=value)
However this method needs to only allocate and copy once which can result
in significant speed gains if `image` is large.
Examples
--------
>>> _fast_pad(np.zeros((2, 3), dtype=int), 4)
array([[4, 4, 4, 4, 4],
[4, 0, 0, 0, 4],
[4, 0, 0, 0, 4],
[4, 4, 4, 4, 4]])
"""
# Allocate padded image
new_shape = np.array(image.shape) + 2
new_image = np.empty(new_shape, dtype=image.dtype, order="C")
# Copy old image into new space
original_slice = tuple(slice(1, -1) for _ in range(image.ndim))
new_image[original_slice] = image
# and set the edge values
_set_edge_values_inplace(new_image, value)
return new_image
def _resolve_neighborhood(selem, connectivity, ndim):
"""Validate or create structuring element for use in `local_maxima`.
Depending on the values of `connectivity` and `selem` this function
either creates a new structuring element (`selem` is None) using
`connectivity` or validates the given structuring element (`selem` is not
None).
Parameters
----------
selem : array-like or None
The structuring element to validate. See same argument in
`local_maxima`.
connectivity : int or None
A number used to determine the neighborhood of each evaluated pixel.
See same argument in `local_maxima`.
ndim : int
Number of dimensions `selem` ought to have.
Returns
-------
selem : ndarray
Validated or new structuring element specifying the neighborhood.
"""
if selem is None:
if connectivity is None:
connectivity = ndim
selem = ndi.generate_binary_structure(ndim, connectivity)
else:
# Validate custom structured element
selem = np.asarray(selem, dtype=bool)
# Must specify neighbors for all dimensions
if selem.ndim != ndim:
raise ValueError(
"structuring element and image must have the same number of "
"dimensions"
)
# Must only specify direct neighbors
if any(s != 3 for s in selem.shape):
raise ValueError("dimension size in structuring element is not 3")
return selem
def local_maxima(image, selem=None, connectivity=None, indices=False,
allow_borders=True):
"""Find local maxima of n-dimensional array.
The local maxima are defined as connected sets of pixels with equal gray
level (plateaus) strictly greater than the gray levels of all pixels in the
neighborhood.
Parameters
----------
image : ndarray
An n-dimensional array.
selem : ndarray, optional
A structuring element used to determine the neighborhood of each
evaluated pixel. It must contain only 1's and 0's, have the same number
of dimensions as `image`. If not given, all adjacent pixels are
considered as part of the neighborhood.
connectivity : int, optional
A number used to determine the neighborhood of each evaluated pixel.
Adjacent pixels whose squared distance from the center is larger or
equal to `connectivity` are considered neighbors. Ignored if
`selem` is not None.
indices : bool, optional
If True, the output will be a tuple of one-dimensional arrays
representing the indices of local maxima in each dimension. If False,
the output will be an array of 0's and 1's with the same shape as
`image`.
allow_borders : bool, optional
If true, plateaus that touch the image border are valid maxima.
Returns
-------
maxima : ndarray or tuple[ndarray]
If `indices` is false, an array with the same shape as `image` is
returned with 1's indicating the position of local maxima
(0 otherwise). If `indices` is true, a tuple of one-dimensional arrays
containing the coordinates (indices) of all found maxima.
Warns
-----
UserWarning
If `allow_borders` is false and any dimension of the given `image` is
shorter than 3 samples, maxima can't exist and a warning is shown.
See Also
--------
skimage.morphology.local_minima
skimage.morphology.h_maxima
skimage.morphology.h_minima
Notes
-----
This function operates on the following ideas:
1. Make a first pass over the image's last dimension and flag candidates
for local maxima by comparing pixels in only one direction.
If the pixels aren't connected in the last dimension all pixels are
flagged as candidates instead.
For each candidate:
2. Perform a flood-fill to find all connected pixels that have the same
gray value and are part of the plateau.
3. Consider the connected neighborhood of a plateau: if no bordering sample
has a higher gray level, mark the plateau as a definite local maximum.
Examples
--------
>>> from skimage.morphology import local_maxima
>>> image = np.zeros((4, 7), dtype=int)
>>> image[1:3, 1:3] = 1
>>> image[3, 0] = 1
>>> image[1:3, 4:6] = 2
>>> image[3, 6] = 3
>>> image
array([[0, 0, 0, 0, 0, 0, 0],
[0, 1, 1, 0, 2, 2, 0],
[0, 1, 1, 0, 2, 2, 0],
[1, 0, 0, 0, 0, 0, 3]])
Find local maxima by comparing to all neighboring pixels (maximal
connectivity):
>>> local_maxima(image)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 1, 1, 0, 0, 0, 0],
[0, 1, 1, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0, 1]], dtype=uint8)
>>> local_maxima(image, indices=True)
(array([1, 1, 2, 2, 3, 3]), array([1, 2, 1, 2, 0, 6]))
Find local maxima without comparing to diagonal pixels (connectivity 1):
>>> local_maxima(image, connectivity=1)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 1, 1, 0, 1, 1, 0],
[0, 1, 1, 0, 1, 1, 0],
[1, 0, 0, 0, 0, 0, 1]], dtype=uint8)
and exclude maxima that border the image edge:
>>> local_maxima(image, connectivity=1, allow_borders=False)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 1, 1, 0, 1, 1, 0],
[0, 1, 1, 0, 1, 1, 0],
[0, 0, 0, 0, 0, 0, 0]], dtype=uint8)
"""
image = np.asarray(image, order="C")
if image.size == 0:
# Return early for empty input
if indices:
# Make sure that output is a tuple of 1 empty array per dimension
return np.nonzero(image)
else:
return np.zeros(image.shape, dtype=np.uint8)
if allow_borders:
# Ensure that local maxima are always at least one smaller sample away
# from the image border
image = _fast_pad(image, image.min())
# Array of flags used to store the state of each pixel during evaluation.
# See _extrema_cy.pyx for their meaning
flags = np.zeros(image.shape, dtype=np.uint8)
_set_edge_values_inplace(flags, value=3)
if any(s < 3 for s in image.shape):
# Warn and skip if any dimension is smaller than 3
# -> no maxima can exist & structuring element can't be applied
warn(
"maxima can't exist for an image with any dimension smaller 3 "
"if borders aren't allowed",
stacklevel=3
)
else:
selem = _resolve_neighborhood(selem, connectivity, image.ndim)
neighbor_offsets = _offsets_to_raveled_neighbors(
image.shape, selem, center=((1,) * image.ndim)
)
try:
_local_maxima(image.ravel(), flags.ravel(), neighbor_offsets)
except TypeError:
if image.dtype == np.float16:
# Provide the user with clearer error message
raise TypeError("dtype of `image` is float16 which is not "
"supported, try upcasting to float32")
else:
raise # Otherwise raise original message
if allow_borders:
# Revert padding performed at the beginning of the function
flags = crop(flags, 1)
else:
# No padding was performed but set edge values back to 0
_set_edge_values_inplace(flags, value=0)
if indices:
return np.nonzero(flags)
else:
return flags
def local_minima(image, selem=None, connectivity=None, indices=False,
allow_borders=True):
"""Find local minima of n-dimensional array.
The local minima are defined as connected sets of pixels with equal gray
level (plateaus) strictly smaller than the gray levels of all pixels in the
neighborhood.
Parameters
----------
image : ndarray
An n-dimensional array.
selem : ndarray, optional
A structuring element used to determine the neighborhood of each
evaluated pixel. It must contain only 1's and 0's, have the same number
of dimensions as `image`. If not given, all adjacent pixels are
considered as part of the neighborhood.
connectivity : int, optional
A number used to determine the neighborhood of each evaluated pixel.
Adjacent pixels whose squared distance from the center is larger or
equal to `connectivity` are considered neighbors. Ignored if
`selem` is not None.
indices : bool, optional
If True, the output will be a tuple of one-dimensional arrays
representing the indices of local minima in each dimension. If False,
the output will be an array of 0's and 1's with the same shape as
`image`.
allow_borders : bool, optional
If true, plateaus that touch the image border are valid minima.
Returns
-------
minima : ndarray or tuple[ndarray]
If `indices` is false, an array with the same shape as `image` is
returned with 1's indicating the position of local minima
(0 otherwise). If `indices` is true, a tuple of one-dimensional arrays
containing the coordinates (indices) of all found minima.
See Also
--------
skimage.morphology.local_maxima
skimage.morphology.h_maxima
skimage.morphology.h_minima
Notes
-----
This function operates on the following ideas:
1. Make a first pass over the image's last dimension and flag candidates
for local minima by comparing pixels in only one direction.
If the pixels aren't connected in the last dimension all pixels are
flagged as candidates instead.
For each candidate:
2. Perform a flood-fill to find all connected pixels that have the same
gray value and are part of the plateau.
3. Consider the connected neighborhood of a plateau: if no bordering sample
has a smaller gray level, mark the plateau as a definite local minimum.
Examples
--------
>>> from skimage.morphology import local_minima
>>> image = np.zeros((4, 7), dtype=int)
>>> image[1:3, 1:3] = -1
>>> image[3, 0] = -1
>>> image[1:3, 4:6] = -2
>>> image[3, 6] = -3
>>> image
array([[ 0, 0, 0, 0, 0, 0, 0],
[ 0, -1, -1, 0, -2, -2, 0],
[ 0, -1, -1, 0, -2, -2, 0],
[-1, 0, 0, 0, 0, 0, -3]])
Find local minima by comparing to all neighboring pixels (maximal
connectivity):
>>> local_minima(image)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 1, 1, 0, 0, 0, 0],
[0, 1, 1, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0, 1]], dtype=uint8)
>>> local_minima(image, indices=True)
(array([1, 1, 2, 2, 3, 3]), array([1, 2, 1, 2, 0, 6]))
Find local minima without comparing to diagonal pixels (connectivity 1):
>>> local_minima(image, connectivity=1)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 1, 1, 0, 1, 1, 0],
[0, 1, 1, 0, 1, 1, 0],
[1, 0, 0, 0, 0, 0, 1]], dtype=uint8)
and exclude minima that border the image edge:
>>> local_minima(image, connectivity=1, allow_borders=False)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 1, 1, 0, 1, 1, 0],
[0, 1, 1, 0, 1, 1, 0],
[0, 0, 0, 0, 0, 0, 0]], dtype=uint8)
"""
return local_maxima(
image=invert(image),
selem=selem,
connectivity=connectivity,
indices=indices,
allow_borders=allow_borders
)