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This morphological reconstruction routine was adapted from CellProfiler, code
licensed under both GPL and BSD licenses.
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 ..filters._rank_order import rank_order
def reconstruction(seed, mask, method='dilation', selem=None, offset=None):
"""Perform a morphological reconstruction of an image.
Morphological reconstruction by dilation is similar to basic morphological
dilation: high-intensity values will replace nearby low-intensity values.
The basic dilation operator, however, uses a structuring element to
determine how far a value in the input image can spread. In contrast,
reconstruction uses two images: a "seed" image, which specifies the values
that spread, and a "mask" image, which gives the maximum allowed value at
each pixel. The mask image, like the structuring element, limits the spread
of high-intensity values. Reconstruction by erosion is simply the inverse:
low-intensity values spread from the seed image and are limited by the mask
image, which represents the minimum allowed value.
Alternatively, you can think of reconstruction as a way to isolate the
connected regions of an image. For dilation, reconstruction connects
regions marked by local maxima in the seed image: neighboring pixels
less-than-or-equal-to those seeds are connected to the seeded region.
Local maxima with values larger than the seed image will get truncated to
the seed value.
seed : ndarray
The seed image (a.k.a. marker image), which specifies the values that
are dilated or eroded.
mask : ndarray
The maximum (dilation) / minimum (erosion) allowed value at each pixel.
method : {'dilation'|'erosion'}
Perform reconstruction by dilation or erosion. In dilation (or
erosion), the seed image is dilated (or eroded) until limited by the
mask image. For dilation, each seed value must be less than or equal
to the corresponding mask value; for erosion, the reverse is true.
selem : ndarray
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.)
reconstructed : ndarray
The result of morphological reconstruction.
>>> import numpy as np
>>> from skimage.morphology import reconstruction
First, we create a sinusoidal mask image with peaks at middle and ends.
>>> x = np.linspace(0, 4 * np.pi)
>>> y_mask = np.cos(x)
Then, we create a seed image initialized to the minimum mask value (for
reconstruction by dilation, min-intensity values don't spread) and add
"seeds" to the left and right peak, but at a fraction of peak value (1).
>>> y_seed = y_mask.min() * np.ones_like(x)
>>> y_seed[0] = 0.5
>>> y_seed[-1] = 0
>>> y_rec = reconstruction(y_seed, y_mask)
The reconstructed image (or curve, in this case) is exactly the same as the
mask image, except that the peaks are truncated to 0.5 and 0. The middle
peak disappears completely: Since there were no seed values in this peak
region, its reconstructed value is truncated to the surrounding value (-1).
As a more practical example, we try to extract the bright features of an
image by subtracting a background image created by reconstruction.
>>> y, x = np.mgrid[:20:0.5, :20:0.5]
>>> bumps = np.sin(x) + np.sin(y)
To create the background image, set the mask image to the original image,
and the seed image to the original image with an intensity offset, `h`.
>>> h = 0.3
>>> seed = bumps - h
>>> background = reconstruction(seed, bumps)
The resulting reconstructed image looks exactly like the original image,
but with the peaks of the bumps cut off. Subtracting this reconstructed
image from the original image leaves just the peaks of the bumps
>>> hdome = bumps - background
This operation is known as the h-dome of the image and leaves features
of height `h` in the subtracted image.
The algorithm is taken from [1]_. Applications for greyscale reconstruction
are discussed in [2]_ and [3]_.
.. [1] Robinson, "Efficient morphological reconstruction: a downhill
filter", Pattern Recognition Letters 25 (2004) 1759-1767.
.. [2] Vincent, L., "Morphological Grayscale Reconstruction in Image
Analysis: Applications and Efficient Algorithms", IEEE Transactions
on Image Processing (1993)
.. [3] Soille, P., "Morphological Image Analysis: Principles and
Applications", Chapter 6, 2nd edition (2003), ISBN 3540429883.
assert tuple(seed.shape) == tuple(mask.shape)
if method == 'dilation' and np.any(seed > mask):
raise ValueError("Intensity of seed image must be less than that "
"of the mask image for reconstruction by dilation.")
elif method == 'erosion' and np.any(seed < mask):
raise ValueError("Intensity of seed image must be greater than that "
"of the mask image for reconstruction by erosion.")
from ._greyreconstruct import reconstruction_loop
except ImportError:
raise ImportError("_greyreconstruct extension not available.")
if selem is None:
selem = np.ones([3] * seed.ndim, dtype=bool)
selem = selem.astype(bool)
if offset is None:
if not all([d % 2 == 1 for d in selem.shape]):
raise ValueError("Footprint dimensions must all be odd")
offset = np.array([d // 2 for d in selem.shape])
# Cross out the center of the selem
selem[tuple(slice(d, d + 1) for d in offset)] = False
# Make padding for edges of reconstructed image so we can ignore boundaries
padding = (np.array(selem.shape) / 2).astype(int)
dims = np.zeros(seed.ndim + 1, dtype=int)
dims[1:] = np.array(seed.shape) + 2 * padding
dims[0] = 2
inside_slices = tuple(slice(p, -p) for p in padding)
# Set padded region to minimum image intensity and mask along first axis so
# we can interleave image and mask pixels when sorting.
if method == 'dilation':
pad_value = np.min(seed)
elif method == 'erosion':
pad_value = np.max(seed)
raise ValueError("Reconstruction method can be one of 'erosion' "
"or 'dilation'. Got '%s'." % method)
images = np.full(dims, pad_value, dtype='float64')
images[(0, *inside_slices)] = seed
images[(1, *inside_slices)] = mask
# Create a list of strides across the array to get the neighbors within
# a flattened array
value_stride = np.array(images.strides[1:]) // images.dtype.itemsize
image_stride = images.strides[0] // images.dtype.itemsize
selem_mgrid = np.mgrid[[slice(-o, d - o)
for d, o in zip(selem.shape, offset)]]
selem_offsets = selem_mgrid[:, selem].transpose()
nb_strides = np.array([np.sum(value_stride * selem_offset)
for selem_offset in selem_offsets], np.int32)
images = images.flatten()
# Erosion goes smallest to largest; dilation goes largest to smallest.
index_sorted = np.argsort(images).astype(np.int32)
if method == 'dilation':
index_sorted = index_sorted[::-1]
# Make a linked list of pixels sorted by value. -1 is the list terminator.
prev = np.full(len(images), -1, np.int32)
next = np.full(len(images), -1, np.int32)
prev[index_sorted[1:]] = index_sorted[:-1]
next[index_sorted[:-1]] = index_sorted[1:]
# Cython inner-loop compares the rank of pixel values.
if method == 'dilation':
value_rank, value_map = rank_order(images)
elif method == 'erosion':
value_rank, value_map = rank_order(-images)
value_map = -value_map
start = index_sorted[0]
reconstruction_loop(value_rank, prev, next, nb_strides, start,
# Reshape reconstructed image to original image shape and remove padding.
rec_img = value_map[value_rank[:image_stride]]
rec_img.shape = np.array(seed.shape) + 2 * padding
return rec_img[inside_slices]