.. _example_applications_plot_morphology.py:


=======================
Morphological Filtering
=======================

Morphological image processing is a collection of non-linear operations related
to the shape or morphology of features in an image, such as boundaries,
skeletons, etc. In any given technique, we probe an image with a small shape or
template called a structuring element, which defines the region of interest or
neighborhood around a pixel.

In this document we outline the following basic morphological operations:

1. Erosion
2. Dilation
3. Opening
4. Closing
5. White Tophat
6. Black Tophat
7. Skeletonize
8. Convex Hull


To get started, let's load an image using ``io.imread``. Note that morphology
functions only work on gray-scale or binary images, so we set ``as_grey=True``.


.. code-block:: python

	
	import matplotlib.pyplot as plt
	from skimage.data import data_dir
	from skimage.util import img_as_ubyte
	from skimage import io
	
	phantom = img_as_ubyte(io.imread(data_dir+'/phantom.png', as_grey=True))
	fig, ax = plt.subplots()
	ax.imshow(phantom, cmap=plt.cm.gray)
	
	

.. image:: images/plot_morphology_1.png

Let's also define a convenience function for plotting comparisons:


.. code-block:: python

	
	def plot_comparison(original, filtered, filter_name):
	
	    fig, (ax1, ax2) = plt.subplots(ncols=2, figsize=(8, 4))
	    ax1.imshow(original, cmap=plt.cm.gray)
	    ax1.set_title('original')
	    ax1.axis('off')
	    ax2.imshow(filtered, cmap=plt.cm.gray)
	    ax2.set_title(filter_name)
	    ax2.axis('off')
	
	

Erosion
=======

Morphological ``erosion`` sets a pixel at (i, j) to the *minimum over all
pixels in the neighborhood centered at (i, j)*. The structuring element,
``selem``, passed to ``erosion`` is a boolean array that describes this
neighborhood. Below, we use ``disk`` to create a circular structuring element,
which we use for most of the following examples.


.. code-block:: python

	
	from skimage.morphology import erosion, dilation, opening, closing, white_tophat
	from skimage.morphology import black_tophat, skeletonize, convex_hull_image
	from skimage.morphology import disk
	
	selem = disk(6)
	eroded = erosion(phantom, selem)
	plot_comparison(phantom, eroded, 'erosion')
	
	

.. image:: images/plot_morphology_2.png

Notice how the white boundary of the image disappears or gets eroded as we
increase the size of the disk. Also notice the increase in size of the two
black ellipses in the center and the disappearance of the 3 light grey
patches in the lower part of the image.


Dilation
========

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.


.. code-block:: python

	
	dilated = dilation(phantom, selem)
	plot_comparison(phantom, dilated, 'dilation')
	
	

.. image:: images/plot_morphology_3.png

Notice how the white boundary of the image thickens, or gets dilated, as we
increase the size of the disk. Also notice the decrease in size of the two
black ellipses in the centre, and the thickening of the light grey circle in
the center and the 3 patches in the lower part of the image.


Opening
=======

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.


.. code-block:: python

	
	opened = opening(phantom, selem)
	plot_comparison(phantom, opened, 'opening')
	
	

.. image:: images/plot_morphology_4.png

Since ``opening`` an image starts with an erosion operation, light regions that
are *smaller* than the structuring element are removed. The dilation operation
that follows ensures that light regions that are *larger* than the structuring
element retain their original size. Notice how the light and dark shapes in the
center their original thickness but the 3 lighter patches in the bottom get
completely eroded. The size dependence is highlighted by the outer white ring:
The parts of the ring thinner than the structuring element were completely
erased, while the thicker region at the top retains its original thickness.


Closing
=======

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.

To illustrate this more clearly, let's add a small crack to the white border:


.. code-block:: python

	
	phantom = img_as_ubyte(io.imread(data_dir+'/phantom.png', as_grey=True))
	phantom[10:30, 200:210] = 0
	
	closed = closing(phantom, selem)
	plot_comparison(phantom, closed, 'closing')
	
	

.. image:: images/plot_morphology_5.png

Since ``closing`` an image starts with an dilation operation, dark regions
that are *smaller* than the structuring element are removed. The dilation
operation that follows ensures that dark regions that are *larger* than the
structuring element retain their original size. Notice how the white ellipses
at the bottom get connected because of dilation, but other dark region retain
their original sizes. Also notice how the crack we added is mostly removed.


White tophat
============

The ``white_tophat`` 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.

To make things interesting, we'll add bright and dark spots to the image:


.. code-block:: python

	
	phantom = img_as_ubyte(io.imread(data_dir+'/phantom.png', as_grey=True))
	phantom[340:350, 200:210] = 255
	phantom[100:110, 200:210] = 0
	
	w_tophat = white_tophat(phantom, selem)
	plot_comparison(phantom, w_tophat, 'white tophat')
	
	

.. image:: images/plot_morphology_6.png

As you can see, the 10-pixel wide white square is highlighted since it is
smaller than the structuring element. Also, the thin, white edges around most
of the ellipse are retained because they're smaller than the structuring
element, but the thicker region at the top disappears.


Black tophat
============

The ``black_tophat`` 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*.


.. code-block:: python

	
	b_tophat = black_tophat(phantom, selem)
	plot_comparison(phantom, b_tophat, 'black tophat')
	
	

.. image:: images/plot_morphology_7.png

As you can see, the 10-pixel wide black square is highlighted since it is
smaller than the structuring element.


Duality
-------

As you should have noticed, many of these operations are simply the reverse
of another operation. This duality can be summarized as follows:

1. Erosion <-> Dilation
2. Opening <-> Closing
3. White tophat <-> Black tophat


Skeletonize
===========

Thinning is used to reduce each connected component in a binary image to a
*single-pixel wide skeleton*. It is important to note that this is performed
on binary images only.



.. code-block:: python

	
	from skimage import img_as_bool
	horse = ~img_as_bool(io.imread(data_dir+'/horse.png', as_grey=True))
	
	sk = skeletonize(horse)
	plot_comparison(horse, sk, 'skeletonize')
	
	

.. image:: images/plot_morphology_8.png

As the name suggests, this technique is used to thin the image to 1-pixel wide
skeleton by applying thinning successively.


Convex hull
===========

The ``convex_hull_image`` is the *set of pixels included in the smallest
convex polygon that surround all white pixels in the input image*. Again note
that this is also performed on binary images.



.. code-block:: python

	
	hull1 = convex_hull_image(horse)
	plot_comparison(horse, hull1, 'convex hull')
	
	

.. image:: images/plot_morphology_9.png

As the figure illustrates, ``convex_hull_image`` gives the smallest polygon
which covers the white or True completely in the image.

If we add a small grain to the image, we can see how the convex hull adapts to
enclose that grain:


.. code-block:: python

	
	import numpy as np
	
	horse2 = np.copy(horse)
	horse2[45:50, 75:80] = 1
	
	hull2 = convex_hull_image(horse2)
	plot_comparison(horse2, hull2, 'convex hull')
	
	

.. image:: images/plot_morphology_10.png


Additional Resources
====================

1. `MathWorks tutorial on morphological processing
<http://www.mathworks.com/help/images/morphology-fundamentals-dilation-and-erosion.html>`_
2. `Auckland university's tutorial on Morphological Image Processing
<http://www.cs.auckland.ac.nz/courses/compsci773s1c/lectures/ImageProcessing-html/topic4.htm>`_
3. http://en.wikipedia.org/wiki/Mathematical_morphology



.. code-block:: python

	
	plt.show()
	

**Python source code:** :download:`download <plot_morphology.py>`
(generated using ``skimage`` |version|)



**IPython Notebook:** :download:`download <./notebook/plot_morphology.ipynb>`
(generated using ``skimage`` |version|)

