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Tutorial # 10Morphological Operations

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http://www.youtube.com/watch?v=ICtkU7I8oZE

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Mathematical morphology• Based on set theory

▫ In the context of image processing The objects/regions in an

image are the sets• For binary images

▫ Black region is one set▫ White region is one set

http://en.wikipedia.org/wiki/File:Venn_A_intersect_B.svg

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Morphological operations

•Local pixel transformations for transforming images based on region shapes and patterns

•Use logical operations •Often used on binary images•Examples:

▫Erosion▫Dilation▫Opening▫Closing

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Structuring element

•A simple, pre-defined shape• Used to apply the morphological operations

•Compares the shape with the neighbourhood of each pixel

•Examples:

http://www0.cs.ucl.ac.uk/staff/G.Brostow/classes/IP2008/L3_Morphology.pdf

Origin

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To use in MATLAB

•Need to specify structuring element – strel()

diamond disk

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XSet B

(structuring element)

Set A

For display purposes, the white background is equivalent to the black of a binary image while the purple region is a white region.

Example from Gonzalez et al. 2008

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X

Run B along A so that the origin of B touches each element of A.At each location, does B completely overlap with B?

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No there is not complete overlap. This pixel in A will not be included in our final solution.

Let’s move on to its neighbouring pixel.

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What happens if B overlaps the entire neighbouring region of a pixel in A?

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This pixel becomes part of the new solution.

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We can continue like this for every single pixel.

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The resulting solution gives us something like this. It’s a smaller version of the original set A. This is referred to as erosion.

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http://www.youtube.com/watch?v=fmyE7DiaIYQ

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Erosion

•Morphological operation that erodes or shrinks the boundaries of regions of some foreground pixels (ie: white pixels)

•This equation basically states:The erosion of A by B are the points z such

that the translation of B by z is contained in A (Gonzalez et al. 2008)

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Erosion Example

Original image Binary image

http://www.bristol-business.net/wp-content/uploads/2012/01/microchip.jpg

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Erosion Example

Original binary imageEroded image with structuring element

(square 3x3)

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Erosion Example

Original binary imageEroded image with structuring element

(square 5x5)

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Uses

•Can be used to filter out image details smaller than the structuring element (as seen in previous example

•Reduce thickness of objects

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Dilation

•Opposite of erosion•Causes expansion of foreground

boundaries

•This time, the structuring element B is reflected about its origin and is shifted by z

•The dilation is then all the elements of A which overlap with by at least one element.

•Since B is often symmetrical, and B are equal.

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X

Using the same example as before, let’s see what happens when B is moved along A. If any part of B is touching A, that pixel is added to the final solution.

In this case, the pixel is not added, since there is no overlap with A.

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On this second pixel, there is overlap with one pixel. Then we add the pixel in A at the origin.

X

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The dilation of A by B is now

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http://www.youtube.com/watch?v=xO3ED27rMHs

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Dilation Example

Original binary imageEroded image with structuring element

(square 3x3)

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What if we used a different structuring element?

Original binary image Eroded image with which structuring

element?

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What if we used a different structuring element?

Original binary image Eroded image with structuring element

(disk, r =10)Note the rounded edges

because a disk was used.

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Uses

•Can be used to bridge gaps between objects

•Instead of blurring an image using a lowpass filter, dilation results in a binary image

Image with weak or broken edges Dilation connects the gaps

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Zoomed in

Image with weak or broken edges Dilation connects the gaps (would require a large structuring

element for complete connection)

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How can I use this in MATLAB?

•Erosion – imerode()•Dilation – imdilate()•General (includes many of the different

morphological operations) – bwmorph()•Structuring element – strel()

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Combinations

•Erosion and dilation can then be combined to create two other morphological operations:▫Opening▫Closing

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Opening• Smoothes object contours• Removes thin protrusions• Erosion followed by dilation

• Opening can be thought of as the structuring element B, as a “rolling ball”

• The new boundary is found when B is “rolled” inside the boundary

• (Dark blue is the original, Red circles are the structuring elements and light blue is the resulting opening of A by B

http://en.wikipedia.org/wiki/File:Opening.png

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Advantages over erosion

Original noisy image Eroded image (disk, r = 1)

Opening of image (disk, r

= 1)Erosion can remove the undesired noise however also affects all of the desired foreground pixels. Opening retains the foreground pixels which the structuring element can roll around in.

Image courtesy of MATLAB

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Closing• Smoothes object contour• Fuses narrow breaks• Eliminates small holes• Fills gaps in the contour• Dilation followed by erosion

• Closing can be thought of as rolling B on the outside of A

• The new boundary is found when B is “rolled” outside the boundary

• (Dark blue is the original, Red circles are the structuring elements and all blue regions are the resulting closing of A by B

http://en.wikipedia.org/wiki/File:Closing.png

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Closing vs Dilation

Original image (want to remove

small holes)

Dilated image (disk, r = 10)

Closing of image (disk, r

= 10)http://homepages.inf.ed.ac.uk/rbf/HIPR2/close.htm

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Example: Segmentation

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Example

Binary Opened Dilated

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Example: Segmentation

Original image Closing the thresholded image

http://homepages.inf.ed.ac.uk/rbf/HIPR2/close.htm

Example: Border Irregularity

𝑓 1𝐵=

𝐴𝑐𝑙𝑜𝑠𝑒𝑑−𝐴𝑙𝑒𝑠𝑖𝑜𝑛

𝐴𝑙𝑒𝑠𝑖𝑜𝑛

+𝐴𝑙𝑒𝑠𝑖𝑜𝑛− 𝐴𝑜𝑝𝑒𝑛

𝐴𝑙𝑒𝑠𝑖𝑜𝑛

Idea: Capture “spiky” border irregularity

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