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JOSHUA MIKE ET AL.
due to strong (pentagonal) symmetry; an extension of this work consists in taking specimen symmetry into consideration when applying our
framework.Nevertheless, this gain does not translate into an overall advantage in speed, as landmarks have to be marked under a microscope a priori, so that they can be identified, and their position recorded a poster- iori, because their position can be documented visually but is not evident in the scans (Atwood and Sumrall 2012). Consequently, the identifica- tion of landmarks on the thecal surface requires knowledge of blastoidmorphology to orient the specimens and identify the particular three plate junctions that serve as landmarks and manual dexterityto markthe
landmarks properly.
The process provides two points for user error to enter into the analysis—the initial marking of the landmarks and the recording of their position on the scans. In contrast, our CP distance–based method only requires marking the position of the stem facet, which is easily seen on the scans. Therefore, our method nearly eliminates sources of human error because its continuous description of shape is independent of how the observer records the data. The proposed CP-distance methodology
also provides advantages when compared with traditional methods by taking into consideration a more complete representation of specimen morphology. This is achieved by incorporating fine-scale shape measurements such as curvature. Indeed, a natural progres- sion is seen when going from the vault:pelvis and height:width ratios to summarize shape (encoded as a 2D rectangle) to the DP-distance landmark analysis based on 3D scans, in which 13 points represent the specimen’s shape. In this context, our methodology now uses the full geometric information of each specimen’s shape. Advances in computer imagery allow us to deploy this sophisticated technique on very-high-resolution scans, which represent an object’s shape accurately with thousands or even tens of thousands of points. CP distance–based analysis could poten-
FIGURE 7. Depiction of the amount of variance needed to agglomerate each cluster (as indicated in the horizontal axis) for each data set. EV, eigenvalues (of the variance matrix). We can consider this as the variance accounted for by each particular transition (e.g., from two to three clusters). The first datum is absent because it is much larger than the rest. After the transition from two to three clusters, the rest account for about the same amount of variance, so we conclude that there should be three clusters.
tially have a tremendous impact in the study of organisms that have a distinct shape but lack homologous landmarks, which makes morphometric analyses impossible. Such studies may include growth of colonial organisms or investigating ecophenotypic variation in organ- isms such as corals, sponges, and stromatolites.
TABLE 2. Reliability of our method’s ability to cluster and ultimately classify species of Pentremites. Left, each column depicts the results of our cross-validation for a particular resolution of data using DCP. Ninety percent of our data was sampled, and there are only two remaining specimens to be classified. The percentage of trials with each number of correct classifications is listed. The bottom row depicts the overall proportion of classifications that were correct. Right, analogous table using DAt on five remaining samples (10% of the total), no variation in resolution. Each cross-validation experiment was done with 10,000 sampling trials.
DCP resolution 100%
2 correct 1 correct 0 correct Overall
61.21% 32.40% 6.39%
77.41% 50%
30.83% 49.31% 19.86% 55.49%
20%
33.28% 50.04% 16.68% 58.30%
10%
25.58% 50.99% 23.43% 51.08%
5 correct 4 correct 3 correct 2 correct 1 correct 0 correct Overall
DAt
34.39% 32.27% 20.15% 9.98% 2.76% 0.45%
76.84%
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