Module: morphology¶

Generates a ball-shaped structuring element.

This is the 3D equivalent of a disk. A pixel is within the neighborhood if the euclidean distance between it and the origin is no greater than radius.

Return fast binary morphological closing of an image.

This function returns the same result as greyscale closing but performs faster for binary images.

The morphological closing on an image is defined as a dilation followed by an erosion. Closing can remove small dark spots (i.e. “pepper”) and connect small bright cracks. This tends to “close” up (dark) gaps between (bright) features.

Return fast binary morphological dilation of an image.

This function returns the same result as greyscale dilation but performs faster for binary images.

Morphological dilation sets a pixel at (i,j) to the maximum over all pixels in the neighborhood centered at (i,j). Dilation enlarges bright regions and shrinks dark regions.

Return fast binary morphological erosion of an image.

This function returns the same result as greyscale erosion but performs faster for binary images.

Morphological erosion sets a pixel at (i,j) to the minimum over all pixels in the neighborhood centered at (i,j). Erosion shrinks bright regions and enlarges dark regions.

Return fast binary morphological opening of an image.

This function returns the same result as greyscale opening but performs faster for binary images.

The morphological opening on an image is defined as an erosion followed by a dilation. Opening can remove small bright spots (i.e. “salt”) and connect small dark cracks. This tends to “open” up (dark) gaps between (bright) features.

Return black top hat of an image.

The black top hat of an image is defined as its morphological closing minus the original image. This operation returns the dark spots of the image that are smaller than the structuring element. Note that dark spots in the original image are bright spots after the black top hat.

Examples

>>> # Change dark peak to bright peak and subtract background
>>> import numpy as np
>>> from skimage.morphology import square
>>> dark_on_grey = np.array([[7, 6, 6, 6, 7],
...                          [6, 5, 4, 5, 6],
...                          [6, 4, 0, 4, 6],
...                          [6, 5, 4, 5, 6],
...                          [7, 6, 6, 6, 7]], dtype=np.uint8)
>>> black_tophat(dark_on_grey, square(3))
array([[0, 0, 0, 0, 0],
       [0, 0, 1, 0, 0],
       [0, 1, 5, 1, 0],
       [0, 0, 1, 0, 0],
       [0, 0, 0, 0, 0]], dtype=uint8)

Return greyscale morphological closing of an image.

The morphological closing on an image is defined as a dilation followed by an erosion. Closing can remove small dark spots (i.e. “pepper”) and connect small bright cracks. This tends to “close” up (dark) gaps between (bright) features.

Examples

>>> # Close a gap between two bright lines
>>> import numpy as np
>>> from skimage.morphology import square
>>> broken_line = np.array([[0, 0, 0, 0, 0],
...                         [0, 0, 0, 0, 0],
...                         [1, 1, 0, 1, 1],
...                         [0, 0, 0, 0, 0],
...                         [0, 0, 0, 0, 0]], dtype=np.uint8)
>>> closing(broken_line, square(3))
array([[0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0],
       [1, 1, 1, 1, 1],
       [0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0]], dtype=uint8)

Compute the convex hull image of a binary image.

The convex hull is the set of pixels included in the smallest convex polygon that surround all white pixels in the input image.

References

Compute the convex hull image of individual objects in a binary image.

The convex hull is the set of pixels included in the smallest convex polygon that surround all white pixels in the input image.

Notes

This function uses skimage.morphology.label to define unique objects, finds the convex hull of each using convex_hull_image, and combines these regions with logical OR. Be aware the convex hulls of unconnected objects may overlap in the result. If this is suspected, consider using convex_hull_image separately on each object.

Generates a cube-shaped structuring element.

This is the 3D equivalent of a square. Every pixel along the perimeter has a chessboard distance no greater than radius (radius=floor(width/2)) pixels.

Generates a flat, diamond-shaped structuring element.

A pixel is part of the neighborhood (i.e. labeled 1) if the city block/Manhattan distance between it and the center of the neighborhood is no greater than radius.

Return greyscale morphological dilation of an image.

Morphological dilation sets a pixel at (i,j) to the maximum over all pixels in the neighborhood centered at (i,j). Dilation enlarges bright regions and shrinks dark regions.

Notes

For uint8 (and uint16 up to a certain bit-depth) data, the lower algorithm complexity makes the skimage.filters.rank.maximum function more efficient for larger images and structuring elements.

Examples

>>> # Dilation enlarges bright regions
>>> import numpy as np
>>> from skimage.morphology import square
>>> bright_pixel = np.array([[0, 0, 0, 0, 0],
...                          [0, 0, 0, 0, 0],
...                          [0, 0, 1, 0, 0],
...                          [0, 0, 0, 0, 0],
...                          [0, 0, 0, 0, 0]], dtype=np.uint8)
>>> dilation(bright_pixel, square(3))
array([[0, 0, 0, 0, 0],
       [0, 1, 1, 1, 0],
       [0, 1, 1, 1, 0],
       [0, 1, 1, 1, 0],
       [0, 0, 0, 0, 0]], dtype=uint8)

Generates a flat, disk-shaped structuring element.

A pixel is within the neighborhood if the euclidean distance between it and the origin is no greater than radius.

Return greyscale morphological erosion of an image.

Morphological erosion sets a pixel at (i,j) to the minimum over all pixels in the neighborhood centered at (i,j). Erosion shrinks bright regions and enlarges dark regions.

Notes

For uint8 (and uint16 up to a certain bit-depth) data, the lower algorithm complexity makes the skimage.filters.rank.minimum function more efficient for larger images and structuring elements.

Examples

>>> # Erosion shrinks bright regions
>>> import numpy as np
>>> from skimage.morphology import square
>>> bright_square = np.array([[0, 0, 0, 0, 0],
...                           [0, 1, 1, 1, 0],
...                           [0, 1, 1, 1, 0],
...                           [0, 1, 1, 1, 0],
...                           [0, 0, 0, 0, 0]], dtype=np.uint8)
>>> erosion(bright_square, square(3))
array([[0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0],
       [0, 0, 1, 0, 0],
       [0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0]], dtype=uint8)

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.

References

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.

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.

References

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.

Label connected regions of an integer array.

Two pixels are connected when they are neighbors and have the same value. In 2D, they can be neighbors either in a 1- or 2-connected sense. The value refers to the maximum number of orthogonal hops to consider a pixel/voxel a neighbor:

1-connectivity      2-connectivity     diagonal connection close-up

     [ ]           [ ]  [ ]  [ ]         [ ]
      |               \  |  /             |  <- hop 2
[ ]--[x]--[ ]      [ ]--[x]--[ ]    [x]--[ ]
      |               /  |  \         hop 1
     [ ]           [ ]  [ ]  [ ]

References

Examples

>>> import numpy as np
>>> x = np.eye(3).astype(int)
>>> print(x)
[[1 0 0]
 [0 1 0]
 [0 0 1]]
>>> print(label(x, connectivity=1))
[[1 0 0]
 [0 2 0]
 [0 0 3]]
>>> print(label(x, connectivity=2))
[[1 0 0]
 [0 1 0]
 [0 0 1]]
>>> print(label(x, background=-1))
[[1 2 2]
 [2 1 2]
 [2 2 1]]
>>> x = np.array([[1, 0, 0],
...               [1, 1, 5],
...               [0, 0, 0]])
>>> print(label(x))
[[1 0 0]
 [1 1 2]
 [0 0 0]]

Determine all local maxima of the image.

The local maxima are defined as connected sets of pixels with equal grey level strictly greater than the grey levels of all pixels in direct neighborhood of the set.

For integer typed images, this corresponds to the h-maxima with h=1. For float typed images, h is determined as the smallest difference between grey levels.

References

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 local maxima:

>>> maxima = extrema.local_maxima(f)

The resulting image will contain all 6 local maxima.

Determine all local minima of the image.

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.

For integer typed images, this corresponds to the h-minima with h=1. For float typed images, h is determined as the smallest difference between grey levels.

References

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 local minima:

>>> minima = extrema.local_minima(f)

The resulting image will contain all 6 local minima.

Compute the medial axis transform of a binary image

Notes

This algorithm computes the medial axis transform of an image as the ridges of its distance transform.

The different steps of the algorithm are as follows

• A lookup table is used, that assigns 0 or 1 to each configuration of the 3x3 binary square, whether the central pixel should be removed or kept. We want a point to be removed if it has more than one neighbor and if removing it does not change the number of connected components.
• The distance transform to the background is computed, as well as the cornerness of the pixel.
• The foreground (value of 1) points are ordered by the distance transform, then the cornerness.
• A cython function is called to reduce the image to its skeleton. It processes pixels in the order determined at the previous step, and removes or maintains a pixel according to the lookup table. Because of the ordering, it is possible to process all pixels in only one pass.

Examples

>>> square = np.zeros((7, 7), dtype=np.uint8)
>>> square[1:-1, 2:-2] = 1
>>> square
array([[0, 0, 0, 0, 0, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 0, 0, 0, 0, 0]], dtype=uint8)
>>> medial_axis(square).astype(np.uint8)
array([[0, 0, 0, 0, 0, 0, 0],
       [0, 0, 1, 0, 1, 0, 0],
       [0, 0, 0, 1, 0, 0, 0],
       [0, 0, 0, 1, 0, 0, 0],
       [0, 0, 0, 1, 0, 0, 0],
       [0, 0, 1, 0, 1, 0, 0],
       [0, 0, 0, 0, 0, 0, 0]], dtype=uint8)

Generates an octagon shaped structuring element.

For a given size of (m) horizontal and vertical sides and a given (n) height or width of slanted sides octagon is generated. The slanted sides are 45 or 135 degrees to the horizontal axis and hence the widths and heights are equal.

Generates a octahedron-shaped structuring element.

This is the 3D equivalent of a diamond. A pixel is part of the neighborhood (i.e. labeled 1) if the city block/Manhattan distance between it and the center of the neighborhood is no greater than radius.

Return greyscale morphological opening of an image.

The morphological opening on an image is defined as an erosion followed by a dilation. Opening can remove small bright spots (i.e. “salt”) and connect small dark cracks. This tends to “open” up (dark) gaps between (bright) features.

Examples

>>> # Open up gap between two bright regions (but also shrink regions)
>>> import numpy as np
>>> from skimage.morphology import square
>>> bad_connection = np.array([[1, 0, 0, 0, 1],
...                            [1, 1, 0, 1, 1],
...                            [1, 1, 1, 1, 1],
...                            [1, 1, 0, 1, 1],
...                            [1, 0, 0, 0, 1]], dtype=np.uint8)
>>> opening(bad_connection, square(3))
array([[0, 0, 0, 0, 0],
       [1, 1, 0, 1, 1],
       [1, 1, 0, 1, 1],
       [1, 1, 0, 1, 1],
       [0, 0, 0, 0, 0]], dtype=uint8)

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.

Notes

The algorithm is taken from [R780782]. Applications for greyscale reconstruction are discussed in [R781782] and [R782782].

References

Examples

>>> 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.

Generates a flat, rectangular-shaped structuring element.

Every pixel in the rectangle generated for a given width and given height belongs to the neighborhood.

Remove continguous holes smaller than the specified size.

Notes

If the array type is int, it is assumed that it contains already-labeled objects. The labels are not kept in the output image (this function always outputs a bool image). It is suggested that labeling is completed after using this function.

Examples

>>> from skimage import morphology
>>> a = np.array([[1, 1, 1, 1, 1, 0],
...               [1, 1, 1, 0, 1, 0],
...               [1, 0, 0, 1, 1, 0],
...               [1, 1, 1, 1, 1, 0]], bool)
>>> b = morphology.remove_small_holes(a, 2)
>>> b
array([[ True,  True,  True,  True,  True, False],
       [ True,  True,  True,  True,  True, False],
       [ True, False, False,  True,  True, False],
       [ True,  True,  True,  True,  True, False]], dtype=bool)
>>> c = morphology.remove_small_holes(a, 2, connectivity=2)
>>> c
array([[ True,  True,  True,  True,  True, False],
       [ True,  True,  True, False,  True, False],
       [ True, False, False,  True,  True, False],
       [ True,  True,  True,  True,  True, False]], dtype=bool)
>>> d = morphology.remove_small_holes(a, 2, in_place=True)
>>> d is a
True

Remove connected components smaller than the specified size.

Examples

>>> from skimage import morphology
>>> a = np.array([[0, 0, 0, 1, 0],
...               [1, 1, 1, 0, 0],
...               [1, 1, 1, 0, 1]], bool)
>>> b = morphology.remove_small_objects(a, 6)
>>> b
array([[False, False, False, False, False],
       [ True,  True,  True, False, False],
       [ True,  True,  True, False, False]], dtype=bool)
>>> c = morphology.remove_small_objects(a, 7, connectivity=2)
>>> c
array([[False, False, False,  True, False],
       [ True,  True,  True, False, False],
       [ True,  True,  True, False, False]], dtype=bool)
>>> d = morphology.remove_small_objects(a, 6, in_place=True)
>>> d is a
True

Return the skeleton of a binary image.

Thinning is used to reduce each connected component in a binary image to a single-pixel wide skeleton.

Notes

The algorithm [Zha84785786] works by making successive passes of the image, removing pixels on object borders. This continues until no more pixels can be removed. The image is correlated with a mask that assigns each pixel a number in the range [0…255] corresponding to each possible pattern of its 8 neighbouring pixels. A look up table is then used to assign the pixels a value of 0, 1, 2 or 3, which are selectively removed during the iterations.

Note that this algorithm will give different results than a medial axis transform, which is also often referred to as “skeletonization”.

References

Examples

>>> X, Y = np.ogrid[0:9, 0:9]
>>> ellipse = (1./3 * (X - 4)**2 + (Y - 4)**2 < 3**2).astype(np.uint8)
>>> ellipse
array([[0, 0, 0, 1, 1, 1, 0, 0, 0],
       [0, 0, 1, 1, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 1, 1, 0, 0],
       [0, 0, 0, 1, 1, 1, 0, 0, 0]], dtype=uint8)
>>> skel = skeletonize(ellipse)
>>> skel.astype(np.uint8)
array([[0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 1, 0, 0, 0, 0],
       [0, 0, 0, 0, 1, 0, 0, 0, 0],
       [0, 0, 0, 0, 1, 0, 0, 0, 0],
       [0, 0, 0, 0, 1, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=uint8)

Compute the skeleton of a binary image.

Thinning is used to reduce each connected component in a binary image to a single-pixel wide skeleton.

Notes

The method of [Lee94787788] uses an octree data structure to examine a 3x3x3 neighborhood of a pixel. The algorithm proceeds by iteratively sweeping over the image, and removing pixels at each iteration until the image stops changing. Each iteration consists of two steps: first, a list of candidates for removal is assembled; then pixels from this list are rechecked sequentially, to better preserve connectivity of the image.

The algorithm this function implements is different from the algorithms used by either skeletonize or medial_axis, thus for 2D images the results produced by this function are generally different.

References

Generates a flat, square-shaped structuring element.

Every pixel along the perimeter has a chessboard distance no greater than radius (radius=floor(width/2)) pixels.

Generates a star shaped structuring element.

Start has 8 vertices and is an overlap of square of size 2*a + 1 with its 45 degree rotated version. The slanted sides are 45 or 135 degrees to the horizontal axis.

Perform morphological thinning of a binary image.

Notes

This algorithm [R790791] works by making multiple passes over the image, removing pixels matching a set of criteria designed to thin connected regions while preserving eight-connected components and 2 x 2 squares [R791791]. In each of the two sub-iterations the algorithm correlates the intermediate skeleton image with a neighborhood mask, then looks up each neighborhood in a lookup table indicating whether the central pixel should be deleted in that sub-iteration.

References

Examples

>>> square = np.zeros((7, 7), dtype=np.uint8)
>>> square[1:-1, 2:-2] = 1
>>> square[0, 1] =  1
>>> square
array([[0, 1, 0, 0, 0, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 1, 1, 1, 0, 0],
       [0, 0, 0, 0, 0, 0, 0]], dtype=uint8)
>>> skel = thin(square)
>>> skel.astype(np.uint8)
array([[0, 1, 0, 0, 0, 0, 0],
       [0, 0, 1, 0, 0, 0, 0],
       [0, 0, 0, 1, 0, 0, 0],
       [0, 0, 0, 1, 0, 0, 0],
       [0, 0, 0, 1, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0]], dtype=uint8)

Find watershed basins in image flooded from given markers.

Notes

This function implements a watershed algorithm [R794796] [R795796] 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

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.

Return white top hat of an image.

The white top hat of an image is defined as the image minus its morphological opening. This operation returns the bright spots of the image that are smaller than the structuring element.

Examples

>>> # Subtract grey background from bright peak
>>> import numpy as np
>>> from skimage.morphology import square
>>> bright_on_grey = np.array([[2, 3, 3, 3, 2],
...                            [3, 4, 5, 4, 3],
...                            [3, 5, 9, 5, 3],
...                            [3, 4, 5, 4, 3],
...                            [2, 3, 3, 3, 2]], dtype=np.uint8)
>>> white_tophat(bright_on_grey, square(3))
array([[0, 0, 0, 0, 0],
       [0, 0, 1, 0, 0],
       [0, 1, 5, 1, 0],
       [0, 0, 1, 0, 0],
       [0, 0, 0, 0, 0]], dtype=uint8)