Answers.Body: <p>I think by writing left to right people create clockwise and counterclockwise patterns in the rounded parts of their typography.</p>
  <p>For example, I think it'd be pretty unusual to write a 9 like this --&gt; <a href="https://i.sstatic.net/33E0i.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/33E0i.png" alt="digit example from OP image" /></a></p>
  <p>I bet you could also detect the original orientation of the 8s if you wanted. The circle that isn't continuous/smooth is most likely at the top.</p>
  
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Answers.Text: Great explanation and great examples, @pip.pip!!  Using your insight, I think I can pick out the 6's and 9's in many examples in the post.  (I think I can also pick out the top and bottom of the 8's.)  Having said that, it seems like the separation between 6's and 9's in the paper is perhaps a little too good, no?
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Answers.Body: <blockquote>
  <p>&quot;... since from the view of human perception 6 rotated by 180 degrees is equivalent to 9 and vice versa.&quot;.</p>
  </blockquote>
  <p>Only with some typefaces, and almost never with handwritten text; someone would have to make an extra effort to make their text ambiguous when rotated (or mirrored).</p>
  <p>A &quot;6&quot; has a bent <em>tail</em>, while a &quot;9&quot; tends to be straighter.</p>
  <p>Similarly with a &quot;4&quot;, tending to have a closed top and a flattened midline, while an &quot;h&quot; can never have a closed bottom and usually has a curved midline.</p>
  <p>It's how the character is written in its <em>natural</em> orientation that determines its appearance, rotation (and scaling) has no effect on that.</p>
  <p>With a <a href="https://en.wikipedia.org/wiki/Dot_matrix_printing" rel="nofollow noreferrer">9-pin Dot Matrix Printer</a> all bets are off, since the usual font was not designed to differentiate between rotated characters.</p>
  <p><a href="https://i.sstatic.net/aPChW.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/aPChW.png" alt="Dot Matrix Printer font" /></a></p>
  <p>If anything, the fontographer <em>cheated</em> and rotated the characters whenever possible to avoid the effort of creating a new character from scratch.</p>
  <p>Later fonts such as <a href="https://www.myfonts.com/collections/data-70-font-itc?tab=individualStyles" rel="nofollow noreferrer">Data-70</a> were able to avoid rotation ambiguities at the cost of readability, and increased geekiness.</p>
  <p><a href="https://i.sstatic.net/xIGiD.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/xIGiD.png" alt="Data 70 Font" /></a></p>
  
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Body: <p><a href="https://www.researchgate.net/publication/200744481_Exploring_Strategies_for_Training_Deep_Neural_Networks" rel="nofollow noreferrer">Rotated MNIST</a> is a popular dataset for benchmarking models equivariant to rotations on <span class="math-container">$\mathbb{R}^2$</span>, described by <span class="math-container">$SO(2)$</span> group or its discrete subgroups like <span class="math-container">$\mathbb{Z}^{n}$</span>:</p>
  <ul>
  <li><a href="https://arxiv.org/abs/1602.07576" rel="nofollow noreferrer">Group equivariant convolutional networks</a></li>
  <li><a href="https://arxiv.org/abs/1612.04642" rel="nofollow noreferrer">Harmonic networks</a></li>
  </ul>
  <p>It consists of <strong>all</strong> digits from 0 to 9 rotated on an arbitrary angle from <span class="math-container">$[0, 2 \pi)$</span>. However, what makes me a bit puzzled is that digits <span class="math-container">$6$</span> and <span class="math-container">$9$</span> seem to be confused by any learning algorithms, since from the view of human perception <span class="math-container">$6$</span> rotated by 180 degrees is equivalent to <span class="math-container">$9$</span> and vice versa.</p>
  <p>The original paper in the description of Rotated MNIST doesn't comment on this point at all, which is strange, since it is a very natural question to ask.</p>
  <p><a href="https://i.sstatic.net/LLZ16.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/LLZ16.png" alt="enter image description here" /></a></p>
  <p>In the paper <a href="https://www.researchgate.net/figure/Visualization-of-features-encoding-of-digit-class-6-and-9-from-MNIST-rot-Each-point_fig3_312170273" rel="nofollow noreferrer">Oriented Response Networks</a> - authors plot embeddings of rotated digits projected via t-SNE on a 2d plane. There is a clear separation between all rotated versions of 6 and the rotated version of 9 for ORN.</p>
  <p><a href="https://i.sstatic.net/yd1rB.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/yd1rB.png" alt="enter image description here" /></a></p>
  <p>I do not understand how it can be achieved? Probably, the networks understand much more in writing the digit, there are some subtle features, inaccessible to humans, but recognizable by powerful classifier?</p>
  
Tags: <neural-networks><classification><papers><features><mnist>
Title: How can a neural network distinguish a rotated 6 and 9 digits?
UserDisplayName: spiridon_the_sun_rotator
ThreadText: # How can a neural network distinguish a rotated 6 and 9 digits?
  
  (Asked by: username_0 on 2021-12-09)
  
  [Rotated MNIST](https://www.researchgate.net/publication/200744481_Exploring_Strategies_for_Training_Deep_Neural_Networks) is a popular dataset for benchmarking models equivariant to rotations on $\mathbb{R}^2$, described by $SO(2)$ group or its discrete subgroups like $\mathbb{Z}^{n}$:
  
  * [Group equivariant convolutional networks](https://arxiv.org/abs/1602.07576)
  * [Harmonic networks](https://arxiv.org/abs/1612.04642)
  
  It consists of **all** digits from 0 to 9 rotated on an arbitrary angle from $[0, 2 \pi)$. However, what makes me a bit puzzled is that digits $6$ and $9$ seem to be confused by any learning algorithms, since from the view of human perception $6$ rotated by 180 degrees is equivalent to $9$ and vice versa.
  
  The original paper in the description of Rotated MNIST doesn't comment on this point at all, which is strange, since it is a very natural question to ask.
  
  [![enter image description here](https://i.sstatic.net/LLZ16.png)](https://i.sstatic.net/LLZ16.png)
  
  In the paper [Oriented Response Networks](https://www.researchgate.net/figure/Visualization-of-features-encoding-of-digit-class-6-and-9-from-MNIST-rot-Each-point_fig3_312170273) - authors plot embeddings of rotated digits projected via t-SNE on a 2d plane. There is a clear separation between all rotated versions of 6 and the rotated version of 9 for ORN.
  
  [![enter image description here](https://i.sstatic.net/yd1rB.png)](https://i.sstatic.net/yd1rB.png)
  
  I do not understand how it can be achieved? Probably, the networks understand much more in writing the digit, there are some subtle features, inaccessible to humans, but recognizable by powerful classifier?
  
  **Tags:** neural-networks, classification, papers, features, mnist
  
  
  ---
  
  ## 2 answers
  
  ### Answer by username_1 on 2022-10-26. Score: 3
  
  I think by writing left to right people create clockwise and counterclockwise patterns in the rounded parts of their typography.
  
  For example, I think it'd be pretty unusual to write a 9 like this --> [![digit example from OP image](https://i.sstatic.net/33E0i.png)](https://i.sstatic.net/33E0i.png)
  
  I bet you could also detect the original orientation of the 8s if you wanted. The circle that isn't continuous/smooth is most likely at the top.
  
  > **Comments:**
  > **username_2**: Great explanation and great examples, @pip.pip!!  Using your insight, I think I can pick out the 6's and 9's in many examples in the post.  (I think I can also pick out the top and bottom of the 8's.)  Having said that, it seems like the separation between 6's and 9's in the paper is perhaps a little too good, no?  
  
  ---
  
  ### Answer by username_3 on 2022-10-26. Score: 1
  
  > "... since from the view of human perception 6 rotated by 180 degrees is equivalent to 9 and vice versa.".
  
  Only with some typefaces, and almost never with handwritten text; someone would have to make an extra effort to make their text ambiguous when rotated (or mirrored).
  
  A "6" has a bent *tail*, while a "9" tends to be straighter.
  
  Similarly with a "4", tending to have a closed top and a flattened midline, while an "h" can never have a closed bottom and usually has a curved midline.
  
  It's how the character is written in its *natural* orientation that determines its appearance, rotation (and scaling) has no effect on that.
  
  With a [9-pin Dot Matrix Printer](https://en.wikipedia.org/wiki/Dot_matrix_printing) all bets are off, since the usual font was not designed to differentiate between rotated characters.
  
  [![Dot Matrix Printer font](https://i.sstatic.net/aPChW.png)](https://i.sstatic.net/aPChW.png)
  
  If anything, the fontographer *cheated* and rotated the characters whenever possible to avoid the effort of creating a new character from scratch.
  
  Later fonts such as [Data-70](https://www.myfonts.com/collections/data-70-font-itc?tab=individualStyles) were able to avoid rotation ambiguities at the cost of readability, and increased geekiness.
  
  [![Data 70 Font](https://i.sstatic.net/xIGiD.png)](https://i.sstatic.net/xIGiD.png)
  
  
  ---
  
  