Tutorial # 10 Morphological Operations 1

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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 3 rsect_B.svg

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 4

Structuring element A simple, pre-defined shape Used to apply the morphological operations Compares the shape with the neighbourhood of each pixel Examples: 5 Origin

To use in MATLAB Need to specify structuring element – strel() 6 diamonddisk

7 X Set 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

8 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?

9 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.

10 What happens if B overlaps the entire neighbouring region of a pixel in A? X This pixel becomes part of the new solution.

11 We can continue like this for every single pixel.

12 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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Erosion 14 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)

Erosion Example 15 Original image Binary image content/uploads/2012/01/microchip.jpg

Erosion Example 16 Original binary image Eroded image with structuring element (square 3x3)

Erosion Example 17 Original binary image Eroded image with structuring element (square 5x5)

Uses Can be used to filter out image details smaller than the structuring element (as seen in previous example Reduce thickness of objects 18

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. 19

20 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.

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

22 The dilation of A by B is now

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Dilation Example 24 Original binary image Eroded image with structuring element (square 3x3)

What if we used a different structuring element? 25 Original binary image Eroded image with which structuring element?

What if we used a different structuring element? 26 Original binary image Eroded image with structuring element (disk, r =10) Note the rounded edges because a disk was used.

Uses Can be used to bridge gaps between objects Instead of blurring an image using a lowpass filter, dilation results in a binary image 27 Image with weak or broken edgesDilation connects the gaps

Zoomed in 28 Image with weak or broken edges Dilation connects the gaps (would require a large structuring element for complete connection)

How can I use this in MATLAB? Erosion – imerode() Dilation – imdilate() General (includes many of the different morphological operations) – bwmorph() Structuring element – strel() 29

Combinations Erosion and dilation can then be combined to create two other morphological operations: ▫Opening ▫Closing 30

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 31 le:Opening.png

Advantages over erosion 32 Original noisy imageEroded 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

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 33 le:Closing.png

Closing vs Dilation 34 Original image (want to remove small holes) Dilated image (disk, r = 10) Closing of image (disk, r = 10)

Example: Segmentation 35

Example 36 BinaryOpenedDilated

Example: Segmentation 37 Original imageClosing the thresholded image /close.htm

Example: Border Irregularity Idea: Capture “spiky” border irregularity 36