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"""watershed.py - watershed algorithm
This module implements a watershed algorithm that apportions pixels into
marked basins. The algorithm uses a priority queue to hold the pixels
with the metric for the priority queue being pixel value, then the time
of entry into the queue - this settles ties in favor of the closest marker.
Some ideas taken from
Soille, "Automated Basin Delineation from Digital Elevation Models Using
Mathematical Morphology", Signal Processing 20 (1990) 171-182.
The most important insight in the paper is that entry time onto the queue
solves two problems: a pixel should be assigned to the neighbor with the
largest gradient or, if there is no gradient, pixels on a plateau should
be split between markers on opposite sides.
Originally part of CellProfiler, code licensed under both GPL and BSD licenses.
Website: http://www.cellprofiler.org
Copyright (c) 2003-2009 Massachusetts Institute of Technology
Copyright (c) 2009-2011 Broad Institute
All rights reserved.
Original author: Lee Kamentsky
"""
import numpy as np
from scipy import ndimage as ndi
from . import _watershed
from ..util import crop, regular_seeds
def _validate_inputs(image, markers, mask):
"""Ensure that all inputs to watershed have matching shapes and types.
Parameters
----------
image : array
The input image.
markers : int or array of int
The marker image.
mask : array, or None
A boolean mask, True where we want to compute the watershed.
Returns
-------
image, markers, mask : arrays
The validated and formatted arrays. Image will have dtype float64,
markers int32, and mask int8. If ``None`` was given for the mask,
it is a volume of all 1s.
Raises
------
ValueError
If the shapes of the given arrays don't match.
"""
if not isinstance(markers, (np.ndarray, list, tuple)):
# not array-like, assume int
markers = regular_seeds(image.shape, markers)
elif markers.shape != image.shape:
raise ValueError("`markers` (shape {}) must have same shape "
"as `image` (shape {})".format(markers.shape, image.shape))
if mask is not None and mask.shape != image.shape:
raise ValueError("`mask` must have same shape as `image`")
if mask is None:
# Use a complete `True` mask if none is provided
mask = np.ones(image.shape, bool)
return (image.astype(np.float64),
markers.astype(np.int32),
mask.astype(np.int8))
def _validate_connectivity(image_dim, connectivity, offset):
"""Convert any valid connectivity to a structuring element and offset.
Parameters
----------
image_dim : int
The number of dimensions of the input image.
connectivity : int, array, or None
The neighborhood connectivity. An integer is interpreted as in
``scipy.ndimage.generate_binary_structure``, as the maximum number
of orthogonal steps to reach a neighbor. An array is directly
interpreted as a structuring element and its shape is validated against
the input image shape. ``None`` is interpreted as a connectivity of 1.
offset : tuple of int, or None
The coordinates of the center of the structuring element.
Returns
-------
c_connectivity : array of bool
The structuring element corresponding to the input `connectivity`.
offset : array of int
The offset corresponding to the center of the structuring element.
Raises
------
ValueError:
If the image dimension and the connectivity or offset dimensions don't
match.
"""
if connectivity is None:
connectivity = 1
if np.isscalar(connectivity):
c_connectivity = ndi.generate_binary_structure(image_dim, connectivity)
else:
c_connectivity = np.array(connectivity, bool)
if c_connectivity.ndim != image_dim:
raise ValueError("Connectivity dimension must be same as image")
if offset is None:
if any([x % 2 == 0 for x in c_connectivity.shape]):
raise ValueError("Connectivity array must have an unambiguous "
"center")
offset = np.array(c_connectivity.shape) // 2
return c_connectivity, offset
def _offsets_to_raveled_neighbors(image_shape, structure, center):
"""Compute offsets to a samples neighbors if the image would be raveled.
Parameters
----------
image_shape : tuple
The shape of the image for which the offsets are computed.
structure : ndarray
A structuring element determining the neighborhood expressed as an
n-D array of 1's and 0's.
center : sequence
Tuple of indices specifying the center of `selem`.
Returns
-------
offsets : ndarray
Linear offsets to a samples neighbors in the raveled image, sorted by
their Euclidean distance from the center.
Examples
--------
>>> _offsets_to_raveled_neighbors((4, 5), np.ones((4, 3)), (1, 1))
array([-5, -1, 1, 5, -6, -4, 4, 6, 10, 9, 11])
"""
structure = structure.copy() # Don't modify original input
structure[tuple(center)] = 0 # Ignore the center; it's not a neighbor
connection_indices = np.transpose(np.nonzero(structure))
offsets = (np.ravel_multi_index(connection_indices.T, image_shape) -
np.ravel_multi_index(center, image_shape))
squared_distances = np.sum((connection_indices - center) ** 2, axis=1)
return offsets[np.argsort(squared_distances)]
def watershed(image, markers, connectivity=1, offset=None, mask=None,
compactness=0, watershed_line=False):
"""Find watershed basins in `image` flooded from given `markers`.
Parameters
----------
image: ndarray (2-D, 3-D, ...) of integers
Data array where the lowest value points are labeled first.
markers: int, or ndarray of int, same shape as `image`
The desired number of markers, or an array marking the basins with the
values to be assigned in the label matrix. Zero means not a marker.
connectivity: ndarray, optional
An array with the same number of dimensions as `image` whose
non-zero elements indicate neighbors for connection.
Following the scipy convention, default is a one-connected array of
the dimension of the image.
offset: array_like of shape image.ndim, optional
offset of the connectivity (one offset per dimension)
mask: ndarray of bools or 0s and 1s, optional
Array of same shape as `image`. Only points at which mask == True
will be labeled.
compactness : float, optional
Use compact watershed [3]_ with given compactness parameter.
Higher values result in more regularly-shaped watershed basins.
watershed_line : bool, optional
If watershed_line is True, a one-pixel wide line separates the regions
obtained by the watershed algorithm. The line has the label 0.
Returns
-------
out: ndarray
A labeled matrix of the same type and shape as markers
See also
--------
skimage.segmentation.random_walker: random walker segmentation
A segmentation algorithm based on anisotropic diffusion, usually
slower than the watershed but with good results on noisy data and
boundaries with holes.
Notes
-----
This function implements a watershed algorithm [1]_ [2]_ that apportions
pixels into marked basins. The algorithm uses a priority queue to hold
the pixels with the metric for the priority queue being pixel value, then
the time of entry into the queue - this settles ties in favor of the
closest marker.
Some ideas taken from
Soille, "Automated Basin Delineation from Digital Elevation Models Using
Mathematical Morphology", Signal Processing 20 (1990) 171-182
The most important insight in the paper is that entry time onto the queue
solves two problems: a pixel should be assigned to the neighbor with the
largest gradient or, if there is no gradient, pixels on a plateau should
be split between markers on opposite sides.
This implementation converts all arguments to specific, lowest common
denominator types, then passes these to a C algorithm.
Markers can be determined manually, or automatically using for example
the local minima of the gradient of the image, or the local maxima of the
distance function to the background for separating overlapping objects
(see example).
References
----------
.. [1] https://en.wikipedia.org/wiki/Watershed_%28image_processing%29
.. [2] http://cmm.ensmp.fr/~beucher/wtshed.html
.. [3] Peer Neubert & Peter Protzel (2014). Compact Watershed and
Preemptive SLIC: On Improving Trade-offs of Superpixel Segmentation
Algorithms. ICPR 2014, pp 996-1001. :DOI:`10.1109/ICPR.2014.181`
https://www.tu-chemnitz.de/etit/proaut/forschung/rsrc/cws_pSLIC_ICPR.pdf
Examples
--------
The watershed algorithm is useful to separate overlapping objects.
We first generate an initial image with two overlapping circles:
>>> x, y = np.indices((80, 80))
>>> x1, y1, x2, y2 = 28, 28, 44, 52
>>> r1, r2 = 16, 20
>>> mask_circle1 = (x - x1)**2 + (y - y1)**2 < r1**2
>>> mask_circle2 = (x - x2)**2 + (y - y2)**2 < r2**2
>>> image = np.logical_or(mask_circle1, mask_circle2)
Next, we want to separate the two circles. We generate markers at the
maxima of the distance to the background:
>>> from scipy import ndimage as ndi
>>> distance = ndi.distance_transform_edt(image)
>>> from skimage.feature import peak_local_max
>>> local_maxi = peak_local_max(distance, labels=image,
... footprint=np.ones((3, 3)),
... indices=False)
>>> markers = ndi.label(local_maxi)[0]
Finally, we run the watershed on the image and markers:
>>> labels = watershed(-distance, markers, mask=image)
The algorithm works also for 3-D images, and can be used for example to
separate overlapping spheres.
"""
image, markers, mask = _validate_inputs(image, markers, mask)
connectivity, offset = _validate_connectivity(image.ndim, connectivity,
offset)
# pad the image, markers, and mask so that we can use the mask to
# keep from running off the edges
pad_width = [(p, p) for p in offset]
image = np.pad(image, pad_width, mode='constant')
mask = np.pad(mask, pad_width, mode='constant').ravel()
output = np.pad(markers, pad_width, mode='constant')
flat_neighborhood = _offsets_to_raveled_neighbors(
image.shape, connectivity, center=offset)
marker_locations = np.flatnonzero(output)
image_strides = np.array(image.strides, dtype=np.intp) // image.itemsize
_watershed.watershed_raveled(image.ravel(),
marker_locations, flat_neighborhood,
mask, image_strides, compactness,
output.ravel(),
watershed_line)
output = crop(output, pad_width, copy=True)
return output