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Activity 7: Morphological Operations

When talking about morphology, what immediately comes to mind are the forms and structures or shapes of objects. Hence, performing morphological operations imply that the shape or form of an object is altered. 
     In this activity, we will perform morphological operations on binary images. In particular, we make use of erosion and dilation.

Erosion and dilation were performed on the following:
1. A 5×5 square
2. A triangle, base = 4 boxes, height = 3 boxes
3. A hollow 10×10 square, 2 boxes thick
4. A plus sign, one box thick, 5 boxes along each line

Using each of the structuring elements below:
1. 2×2 ones
2. 2×1 ones
3. 1×2 ones
4. cross, 3 pixels long, one pixel thick.
5. A diagonal line, two boxes long, i.e. [[0 1],[1 0] ].

     When performing these operations, it is important to note the “anchor” or “origin” of the structuring element in order to give an accurate prediction of the result. For the 2x2 ones, 2x1 ones, and 1x2 ones, the origin is set at [1,1]. For the diagonal line, it was set at [2,1] while for the cross it was set at the center [2,2].
     The results of the erosion and dilation processes were predicted and drawn in a graphing paper. These were then verified using the erode() and dilate() commands in Scilab 4.1.2 SIP toolbox.

A. Dilation
If you have an image A, and a structuring element B, the dilation of A by B results to the expansion of A by the shape of B as shown in Figure A1.
Figure A1. Dilation of A by B

     This may be described by the following mathematical expression

which includes all z’s that are translations of a reflected B, which, when intersected with A does not give an empty set.
     The figures below show the hand-drawn predictions and the results using Scilab. The shaded regions indicate the areas that were added to the original image. Also, the broken lines represent the boundaries of the original image and the solid lines represent the sides of the new image. 
     The first column shows the original binary image. The 2nd to 6th columns are the dilated images using the structuring elements 1 to 5 (above), respectively.
As shown, all of the predictions are in agreement with the results from Scilab.

1. 5x5 Square   

2. Triangle

3. Plus sign

4. Hollow 10x10 square

B. Erosion
Unlike dilation, erosion causes the shape of the image to be reduced. The erosion of A by B results to the reduction of A by the shape of B as shown in Figure B1.
Figure B1. Erosion of A by B
     The following mathematical expression describes the process of erosion.
i.e. the erosion of A by B is the set of all  z’s, such that B, translated by z is contained in A.
     The figures below show the hand-drawn predictions and the results using Scilab. This time, the shaded regions indicate the areas that were removed from the original image. The broken lines  still represent the boundaries of the original image and the solid lines represent the sides of the new image. 
     The first column shows the original binary image. The 2nd to 6th columns are the eroded images using the structuring elements 1 to 5 (above), respectively. As shown, all of the predictions are in agreement with the results from Scilab.

1. 5x 5 Square 

2. Triangle 

3. Plus sign

4. Hollow 10 x 10 square

Finally, I give myself a grade of 10/10 for being able to do all the required tasks and for understanding the concepts of morphological operations, particularly erosion and dilation.
Thank you to Ms. Mabel Saludares for the graphing paper.

Reference:
1. M. Soriano, "Activity 7: Morphological Operations," App Phy 186 Activity sheet, 2012                              

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