Draft for minimum length scale

[mawc2019]

Here is the first draft of the section of minimum length scale. The code still needs to be tweaked to improve the accuracy, and some curves in the figure will be replaced.

Paragraph in the main text
The minimum length scale (MLS) of a binary design pattern typically associates with the minimum diameter of circular features, as shown in Figures 2(a), (b), and (c). For a design pattern $A$ and a circular structuring element $B$ with its diameter not greater than the MLS of $A$, the morphological opening and closing of A by B satisfies
$A \circ B = A \bullet B = A$,
where $A \circ B = (A \ominus B) \oplus B$ is the morphological opening of $A$ by $B$, which is an erosion denoted as $\ominus$ followed by a dilation denoted as $\oplus$. Likewise, $A \bullet B = (A \oplus B) \ominus B$ is the morphological closing of $A$ by $B$, which is a dilation followed by an erosion. To identify the MLS, one can compare $A \circ B$ with $A \bullet B$ under a series of structuring elements $B$ with different diameters. Ideally, the minimum diameter that results in different $A \circ B$ and $A \bullet B$ or the maximum diameter that allows the same $A \circ B$ and $A \bullet B$ is the MLS. In principle, this process can be implemented with binary search, which starts from the maximum possible MLS, namely, the shortest edge of the design pattern. However, if this strategy based on comparison is carried out numerically, the discretization of $A$ and $B$ may cause large error in the calculated MLS. To reduce this side effect of discretization, we regard an interior solid pixel in $|A \circ B - A \bullet B|$ as a signature of significantly different $A \circ B$ and $A \bullet B$. As shown in Figure 2(d), in a 2d pattern, an interior solid pixel is a solid pixel with its all four neighbors being solid. Our stipulation on the significant difference excludes unwanted marginal effects and spurious results caused by discretization. Generally, the error of MLSs estimated by this method is at most the size of several pixels. If the MLS is much larger than the pixel size, the error would be relatively small; if the MLS is comparable to the pixel size, the error will be relatively large. In particular, as shown in Figure 2(b), if the design pattern contains sharp corners that are not small perturbations at the single-pixel level, the MLS should be zero but the method gives a nonzero value. On the other hand, we normally do not consider the sharp corners formed by the edges of a design pattern as an MLS feature.

Figure
Figures for minimum length scales.pdf

Caption
Morphological transformations as tools for searching the minimum length scale. (a) Square with rounded corners under morphological transformations. The diameters of the two structuring elements are 1.4 and 0.6, which are above and below the declared MLS 1.0, respectively. (b) Square under morphological transformations. Although the ideal MLS should be 0, the calculated MLS is 0.14, with the size of each pixel equal to 0.01. The diameters of the two structuring elements are 0.6 and 0.4. (c) Void disc under morphological transformations. The declared MLS, which is also the diameter in this case, is 2.0. The calculated MLS is 1.99. The diameters of the two structuring elements are 2.4 and 1.6, which are above and below the MLS. (d) Classification of pixels. The interior solid pixels signify the difference in $A \circ B$ and $A \bullet B$. (e) Variation of the numbers of interior solid pixels with the diameters of structuring elements. The red arrows indicate the declared MLSs.

[mawc2019]

Here is the second version of the main text and the figure.

Paragraph in the main text
The minimum length scale (MLS) of a binary design pattern typically associates with the minimum diameter of circular features, as shown in Figures 2(a), (f), and (k). For a design pattern $A$ and a circular structuring element $B$ with its diameter not greater than the MLS of $A$, the morphological opening and closing of A by B satisfies
$A \circ B = A \bullet B = A$,
where $A \circ B = (A \ominus B) \oplus B$ is the morphological opening of $A$ by $B$, which is an erosion denoted as $\ominus$ followed by a dilation denoted as $\oplus$. Likewise, $A \bullet B = (A \oplus B) \ominus B$ is the morphological closing of $A$ by $B$, which is a dilation followed by an erosion. To identify the MLS, one can compare $A \circ B$ with $A \bullet B$ under a series of structuring elements $B$ with different diameters. Ideally, the minimum diameter that results in different $A \circ B$ and $A \bullet B$ is the MLS. In principle, this process can be implemented with binary search, which starts from the maximum possible MLS, namely, the shortest edge of the design pattern. However, if this strategy based on comparison is carried out numerically, the discretization of $A$ and $B$ may cause large error in the calculated MLS. To reduce this side effect of discretization, we regard an interior solid pixel in $|A \circ B - A \bullet B|$ as a signature of significantly different $A \circ B$ and $A \bullet B$. Here, a solid pixel in a binary design pattern has the value 1, and if the pattern is 2d, we define an interior solid pixel as a solid pixel with its all four neighbors also being solid. Our stipulation on the significant difference between $A \circ B$ and $A \bullet B$ excludes unwanted marginal effects and spurious results caused by discretization. Generally, the error of MLSs estimated by this method is at most the size of several pixels. If the MLS is much larger than the pixel size, the error would be relatively small, as demonstrated in Figures 2(a-j); if the MLS is comparable to the pixel size, the error would be relatively large. In particular, as shown in Figures 2(k-o), if the design pattern contains sharp corners that are not small perturbations at the single-pixel level, the MLS should be zero but the method gives a nonzero value. On the other hand, we normally do not consider the sharp corners formed by the edges of a design pattern as an MLS feature.

Figure
Figures for minimum length scales v2 0920.pdf

Caption
Morphological transformations as tools for estimating the minimum length scale. (a) Void disc with diameter 2.0. (b-d) Absolute difference between morphological opening and closing, namely, $|A \circ B - A \bullet B|$, with structuring elements $B$ being 1.95, 2.0, and 2.09 in diameter, respectively. (e) Relative number of the interior pixels varying with the diameter of the structuring element. The ordinate represents the ratio of the number of interior solid pixels in $|A \circ B - A \bullet B|$ to the total number of solid pixels in the original pattern $A$, multiplied by $10^2$. The data points corresponding to Figures (b-d) are circled in blue. The true MLS, indicated by the red arrow, is 2.0, while the calculated MLS is 1.99. (f) Square with rounded corners. The circular region is 1.0 in diameter. (g-i) Absolute difference between morphological opening and closing with structuring elements being 0.95, 1.07, and 1.09 in diameter, respectively. (j) Same meaning as that in (b), but multiplied by $10^3$. The true MLS, indicated by the red arrow, is 1.0, while the calculated MLS is 1.06. (k) Square with edge length 2.0. (l-n) Absolute difference between morphological opening and closing with structuring elements being 0.05, 0.15, and 0.19 in diameter, respectively. (o) Same meaning as that in (b), but multiplied by $10^4$. The true MLS, indicated by the red arrow, is 0, while the calculated MLS is 0.14. The pixels size of all these design patterns is 0.01.

[stevengj]

Right now, I'm not sure the figures add much — they are mostly blank. Is there a way we can condense it to the most informative images?

[mawc2019]

I'm not sure the figures add much — they are mostly blank.

Some figures seem blank but are not blank. They contain a small number of solid pixels, but those solid-pixel clusters are too small. Only Figure (l) is completely blank.

Is there a way we can condense it to the most informative images?

Maybe we could replace some examples. To be specific, for Figures (g-i) and (l-n), we could use structure elements with diameters that are significantly larger than the MLSs. The resultant figures will contain larger solid-pixel clusters that are easily visible.
Another way is to add figures or insets to magnify some small solid-pixel clusters in the existing Figures (g-i) and (l-n).
In addition, we may delete some figures, such as Figures (a-e) and/or (k-o).

[mawc2019]

Here is a new draft for the figure. To make the figure more compact, the layout differs from what @stevengj suggested on Thursday.

Additionally, more length scales can be revealed if we investigate a larger range of opening/closing diameters.

[mawc2019]

The current version of Figure 2 in the manuscript is not suitable for demonstrating minimum length scales of both solid and void regions, because the void region of that rounded square has a trivial minimum length scale.
A tentative shape is constructed from coaxial circles. Both solid and void regions have nontrivial minimum length scales. The declared minimum length scales for solid and void regions are 30 and 40, respectively. The corresponding estimated values are 30.4 and 40.5. The shape and some morphological operations are as follows. We can select a few of them and create a new version of Figure 2.

[mawc2019]

Another difference from the previous version of Figure 2 is that, we do not count the number of interior solid pixels as we did before, but only evaluate whether there is interior solid pixels. Therefore the curve should not appear in the new version of Figure 2.

[mawc2019]

A tentative new version of the figure is here.
Figures for minimum length scales v8 0315.pdf

[mawc2019]

Here is a revised version. More text is added for clarity and the original design pattern is superimposed on violation patterns as a guide to the eye.
Figures for minimum length scales v9 0316.pdf

[mawc2019]

Here is a further revised version of Figure 2. While the previous version did not exclude interfacial pixels in the violation images, this new version excludes interfacial pixels, but the formulae become longer, which may not be suitable for the figure. So I removed those formulae and tweaked the figure accordingly. Those formulae are in the main text and can be cited in the caption.
Figures for minimum length scales v10 0327.pdf