Dilation algorithms—decomposing more shapes

In my previous post on dilation algorithms I discussed structuring element decomposition. We looked at the decomposition of structuring elements with rectangular domains. Today I want to follow up on the idea by looking at decompositions of other structuring element shapes.

The strel function offers a variety of flat structuring element shapes:

• diamond
• disk
• line
• octagon
• pair
• periodicline
• rectangle
• square

(The strel function also offers nonflat shapes, and shapes with arbitrary domains.)

Let's look at the 'diamond' shape. The doc says SE = strel('diamond', R) creates a flat, diamond-shaped structuring element, where R specifies the distance from the structuring element origin to the points of the diamond." For example:

se1 = strel('diamond', 5)

se1 =

Flat STREL object containing 61 neighbors.
Decomposition: 4 STREL objects containing a total of 17 neighbors

Neighborhood:
0     0     0     0     0     1     0     0     0     0     0
0     0     0     0     1     1     1     0     0     0     0
0     0     0     1     1     1     1     1     0     0     0
0     0     1     1     1     1     1     1     1     0     0
0     1     1     1     1     1     1     1     1     1     0
1     1     1     1     1     1     1     1     1     1     1
0     1     1     1     1     1     1     1     1     1     0
0     0     1     1     1     1     1     1     1     0     0
0     0     0     1     1     1     1     1     0     0     0
0     0     0     0     1     1     1     0     0     0     0
0     0     0     0     0     1     0     0     0     0     0

The display says the structuring element has 61 neighbors in its domain, but the four decomposed structuring elements have a total of only 17 neighbors.

What shapes makes up the decomposition? We can use getsequence and getnhood to find out.

se1_seq = getsequence(se1);
for k = 1:numel(se1_seq)
getnhood(se1_seq(k))
ans =

0     1     0
1     1     1
0     1     0

ans =

0     1     0
1     0     1
0     1     0

ans =

0     0     1     0     0
0     0     0     0     0
1     0     0     0     1
0     0     0     0     0
0     0     1     0     0

ans =

0     1     0
1     0     1
0     1     0

end

This decomposition comes from Rein van den Boomgard and Richard van Balen, "Methods for Fast Morphological Image Transforms Using Bitmapped Binary Images," Computer Vision, Graphics, and Image Processing: Models and Image Processing, vol. 54, no. 3, May 1992, pp. 252-254. (That journal name sure is a mouthful!)

Now let's look at the 'disk' structuring element. By default, strel('disk',R) actually gives you an approximation to a disk shape, where the approximation is based on a technique called radial decomposition using periodic lines. The reference sources are:

• Rolf Adams, "Radial Decomposition of Discs and Spheres," Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing, vol. 55, no. 5, September 1993, pp. 325-332.
• Ronald Jones and Pierre Soille, "Periodic lines: Definition, cascades, and application to granulometries," Pattern Recognition Letters, vol. 17, 1996, 1057-1063.
se2 = strel('disk', 5)

se2 =

Flat STREL object containing 69 neighbors.
Decomposition: 6 STREL objects containing a total of 18 neighbors

Neighborhood:
0     0     1     1     1     1     1     0     0
0     1     1     1     1     1     1     1     0
1     1     1     1     1     1     1     1     1
1     1     1     1     1     1     1     1     1
1     1     1     1     1     1     1     1     1
1     1     1     1     1     1     1     1     1
1     1     1     1     1     1     1     1     1
0     1     1     1     1     1     1     1     0
0     0     1     1     1     1     1     0     0

Similar to the 'diamond' shape above, the 'disk' shape is decomposed into a sequence of smaller structuring elements:

se2_seq = getsequence(se2);
for k = 1:numel(se2_seq)
getnhood(se2_seq(k))
ans =

1
1
1

ans =

1     0     0
0     1     0
0     0     1

ans =

1     1     1

ans =

0     0     1
0     1     0
1     0     0

ans =

1     1     1

ans =

1
1
1

end

I mentioned above that strel('disk',R) gives you an approximation to a disk. Let's see what that approximation looks like with a larger radius:

se3 = strel('disk', 21);
imshow(getnhood(se3), 'InitialMagnification', 'fit')

Well, that's definitely an approximation. It looks a lot more like an octagon than a circle. The syntax strel('disk',R,N) gives you more control over the approximation. N can be 0, 4, 6, or 8. The values 4, 6, and 8 indicate different levels of approximation. The default value is 4. With N equal to 8, the result looks closer to a circle, but dilation is a little slower because the decomposition isn't as efficient.

se4 = strel('disk', 21, 8);
imshow(getnhood(se4), 'InitialMagnification', 'fit')

Setting N = 0 asks strel to construct a disk shape with no decomposition approximation.

se5 = strel('disk', 21, 0);
imshow(getnhood(se5), 'InitialMagnification', 'fit')

Using getsequence we can see that there is no decomposition for se5:

numel(getsequence(se5))
ans =

1

Next time we'll do some timing experiments with the decomposed structuring elements.

Published with MATLAB® 7.6

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