""" ======================= 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_gray=True``. """ import matplotlib.pyplot as plt import skimage as ski orig_phantom = ski.util.img_as_ubyte(ski.data.shepp_logan_phantom()) fig, ax = plt.subplots() ax.imshow(orig_phantom, cmap=plt.cm.gray) ###################################################################### # Let's also define a convenience function for plotting comparisons: def plot_comparison(original, filtered, filter_name): fig, (ax1, ax2) = plt.subplots(ncols=2, figsize=(8, 4), sharex=True, sharey=True) ax1.imshow(original, cmap=plt.cm.gray) ax1.set_title('original') ax1.set_axis_off() ax2.imshow(filtered, cmap=plt.cm.gray) ax2.set_title(filter_name) ax2.set_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, # ``footprint``, 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. footprint = ski.morphology.disk(6) eroded = ski.morphology.erosion(orig_phantom, footprint) plot_comparison(orig_phantom, eroded, 'erosion') ###################################################################### # 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 gray # 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. dilated = ski.morphology.dilation(orig_phantom, footprint) plot_comparison(orig_phantom, dilated, 'dilation') ###################################################################### # 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 center, and the thickening of the light gray 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. opened = ski.morphology.opening(orig_phantom, footprint) plot_comparison(orig_phantom, opened, 'opening') ###################################################################### # 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 retain 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 are 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: phantom = orig_phantom.copy() phantom[10:30, 200:210] = 0 closed = ski.morphology.closing(phantom, footprint) plot_comparison(phantom, closed, 'closing') ###################################################################### # Since ``closing`` an image starts with a dilation operation, dark regions # that are *smaller* than the structuring element are removed. The erosion # 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 # regions 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: phantom = orig_phantom.copy() phantom[340:350, 200:210] = 255 phantom[100:110, 200:210] = 0 w_tophat = ski.morphology.white_tophat(phantom, footprint) plot_comparison(phantom, w_tophat, 'white tophat') ###################################################################### # 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*. b_tophat = ski.morphology.black_tophat(phantom, footprint) plot_comparison(phantom, b_tophat, 'black tophat') ###################################################################### # 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. horse = ski.data.horse() sk = ski.morphology.skeletonize(horse == 0) plot_comparison(horse, sk, 'skeletonize') ###################################################################### # 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. hull1 = ski.morphology.convex_hull_image(horse == 0) plot_comparison(horse, hull1, 'convex hull') ###################################################################### # 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: horse_mask = horse == 0 horse_mask[45:50, 75:80] = 1 hull2 = ski.morphology.convex_hull_image(horse_mask) plot_comparison(horse_mask, hull2, 'convex hull') ###################################################################### # Additional Resources # ==================== # # 1. `MathWorks tutorial on morphological processing # `_ # # 2. `Auckland university's tutorial on Morphological Image Processing # `_ # # 3. https://en.wikipedia.org/wiki/Mathematical_morphology plt.show()