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+#import "/common.typ": post, link-self, static, img
+
+#show: post.with(
+    title: "Otsu et al. (2018), part 3",
+    date: datetime(year: 2020, month: 7, day: 26),
+)
+
+In the #link-self("gsoc/otsu2018_2")[previous post] I explained how PCA can be used to create a convenient basis for representing real-world reflectivities. The procedure uses a set of measured spectra to compute an orthonormal basis, sorted by variance or "statistical significance." All but the three most significant vectors are then truncated. I called the difference between the actual spectrum and the reconstructed error term $bold(epsilon)$.
+
+Naturally the smaller the error, the better. Because $bold(epsilon)$ is a vector, the authors chose the Euclidean norm (2-norm or L2) to represent the (scalar) error:
+$
+  epsilon = sqrt(sum_i bold(epsilon)_i^2)
+$
+To get an idea of how well the method works in the general case, the authors would sum up reconstruction errors of *all* spectra in the dataset. I call this the total error $E$:
+$
+  E = sum_i epsilon |_(bold(s)_i)
+$
+where $epsilon |_(bold(s)_i)$ denotes the reconstruction error of $bold(s)_i$, the $i$-th measured spectrum.
+
+Using just one basis for everything isn't ideal. The picture below shows reconstructions of a few colorchecker spectra. While by definition XYZ tristimulus values (and perceived colors) match perfectly, the spectra aren't that close.
+
+#figure(
+  img(static("gsoc/otsu_clustering1.png")),
+  caption: [Reconstructions of a few colorchecker spectra.],
+)
+
+= Clustering
+
+The authors @Otsu2018 decided to subdivide the CIE 1931 xy space using a binary space partitioning tree. Every node that isn't a leaf divides its subspace in two with a horizontal or vertical line. Leaves, called clusters in the article, contain spectra that lie in their subspace. PCA is done for every cluster separately, creating a set of bases.
+
+#figure(
+  img(static("gsoc/chromaticity_diagram.png")),
+  caption: [CIE 1931 xy chromaticity diagram],
+)
+
+A quick note about CIE 1931 xy chromaticity coordinates. They're derived from CIE 1931 XYZ tristimulus values:
+$
+  x &= X / (X + Y + Z) \
+  y &= Y / (X + Y + Z)
+$
+They're often used where only the chromaticity matters and not the lightness $Y$. When plotted, they give rise to the familiar "color triangle" or "horseshoe" diagram, pictured on the right.
+
+The algorithm for creating an optimized clustering is described as follows.
+- Search every possible new partitioning in the tree. For every color of every leaf, try partitions with a horizontal and a vertical line going through the color, assigning the color itself to the "lesser" part (to the left or below the line).
+- Discard partitions that would create very small nodes. More on this later.
+- Compute how each partition would affect the total error of the entire tree (the sum of $E$ of every leaf). Discard those that would increase it.
+- Realize the partition that decreases the error the most.
+- Repeat the entire process.
+The algorithm terminates when no more partitions are possible or there are enough clusters. Authors chose 8 clusters, because of diminishing returns and dataset size concerns. You can look at #link("http://lightmetrica.org/h-otsu/download/rgb2spec.zip")[the original implementation] or #link("https://github.com/enneract/otsu2018/tree/d218ef0097174897634477bcc180b6268ba6c337")[my code]. The article also has many more plots and comparisons than I post here, so it's definitely worth looking at too.
+
+Below are the results of running my code. I used the entire dataset of 1,279 spectra and optimized for 8 clusters. This takes about 3 hours on my dated i7 3770.
+
+#figure(
+  img(static("gsoc/otsu_clustering2b.png")),
+  caption: [A visualization of the clustering. Each cluster has its own symbol for the colors that belong to it. (I actually used the cross symbol twice, my bad.)],
+)
+
+#figure(
+  img(static("gsoc/otsu_clustering2a.png")),
+  caption: [Reconstructions of the same colorchecker spectra as before, this time using clustering. The discrepancies are notably smaller and some reconstructed spectra look exactly like the measurements.],
+)
+
+= Minimum node size
+
+I mentioned that nodes cannot be allowed to be split too finely. First of all, there have to be at least three spectra in a node for PCA to even work. Fewer will result in a singular $M$ and the algorithm simply won't be able to continue. In practice, having few spectra is no good either. Below is a result of PCA decomposition done with just three spectra.
+
+#figure(
+  img(static("gsoc/otsu2018_eigenstates_3_colors.png")),
+  caption: [PCA decomposition of the first three spectra in Otsu et al.\'s dataset. An explanation of what is plotted here is in the #link-self("gsoc/otsu2018_2")[previous post].],
+)
+
+Only *two* eigenvectors are meaningful and the rest are basically noise. Trying to use this for reconstruction will create completely random-looking spectra, usually going way outside the [0, 1] range. Clearly a lot more data needs to be used to get useful results. In my program I (arbitrarily) set the minimum cluster size to the average cluster size divided by two and it seems to work pretty well.
+$
+  "min. cluster size" = floor(1 / 2 (("num. of spectra") / ("num. of clusters")))
+$
+
+But can reconstructed spectra exceed the allowed [0, 1] range even if the basis is good? Unfortunately yes, and this will be the topic of the next (and the last) post in this series.
+
+#bibliography("/refs.bib")