Figure 1
(from left to right) Image of a Krater (in this paper, K4), Kylix (Y3), and Pelike (P1), which have been analysed as part of this study. The objects are held by the Ashmolean Museum in Oxford under inventory numbers AN1944.15, AN1947.109, and AN1960.1200, respectively. Images © Ashmolean Museum, University of Oxford.
Figure 2
Lateral and top views of the Kraters studied, as represented by point clouds. The vessels have been arranged chronologically and labelled K1 (oldest object) to K4 (youngest). The extension in cm graphs represent the top views of the Kraters nestled within one another. The thicker lines enclose the outer edges of each vessel.
Figure 3
Lateral and top views of the Kylikes studied, as represented by point clouds. The vessels have been arranged chronologically and labelled Y1 (oldest object) to Y4 (youngest). The extension in cm graphs represent the top views of the Kylikes nestled within one another. The thicker lines enclose the outer edges of each vessel.
Figure 4
Lateral and top views of the Pelikai studied, as represented by point clouds. The vessels have been arranged chronologically and labelled P1 (oldest object) to P3 (youngest). The extension in cm graphs represent the top views of the Pelikai nestled within one another. The thicker lines enclose the outer edges of each vessel.
Figure 5
The mean Sinkhorn distance and computational time between models Y1 and Y2 as the number of points sampled from each model increases. The figure illustrates how, as the number of points sampled from a 3D model increases, the efficiency gains decrease. As the number of points sampled increases, the computational time (in blue) increases exponentially; at the same time, the mean value of the Sinkhorn score (in orange), after a quick convergence, remains stable.
Figure 6
Kylix model, displaying the total mesh vertices (blue) and sampled vertices (red). The figure displays the vertices of the mesh of a Kylix model (Y1) used in the dataset (light blue, high transparency) and the random subsample of one thousand points (red, larger).
Figure 7
A representation of the Wasserstein metric on a Euclidean space. The Wasserstein metric, W, joins two discrete distributions A and B by equally splitting masses (adding up to 1 in each case) on A and B and solving an optimal transport plan to redistribute them from A into B.
Figure 8
Transport plans, coloured by their normalised total cost, between two cloud points of Pelikai P1 and P2. One cloud of points is in blue directed towards the other, smaller, in red. The image clearly shows that the most extreme and different elements are those that need to be moved at a greater distance and thus have a larger cost (yellow-orange colour).
Table 1
Sinkhorn distances computed between each pair of objects of a particular type (Krater, Kylix and Pelike). The original values for vessels K1 to K4, Y1 to Y4, and P1 to P3 are the total pixel distances weighted by their split mass as defined by the optimal transport problem. For comparability, a normalised version is produced which displays the distances from 0 to 1, obtained by dividing each distance by the largest recorded distance within each vessel type.
| KRATER | KYLIX | PELIKE | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| SINKHORN ORIGINAL VALUES | |||||||||||||
| K1 | K2 | K3 | K4 | Y1 | Y2 | Y3 | Y4 | P1 | P2 | P3 | |||
| K1 | 2470 | 5878 | 13892 | Y1 | 10149 | 28013 | 10089 | P1 | 10297 | 19146 | |||
| K2 | 2470 | 3213 | 15413 | Y2 | 10149 | 4285 | 892 | P2 | 10297 | 3189 | |||
| K3 | 5878 | 3213 | 10687 | Y3 | 28013 | 4285 | 3200 | P3 | 19146 | 3189 | |||
| K4 | 13892 | 15413 | 10687 | Y4 | 10089 | 892 | 3200 | ||||||
| NORMALISED SINKHORN | |||||||||||||
| K1 | K2 | K3 | K4 | Y1 | Y2 | Y3 | Y4 | P1 | P2 | P3 | |||
| K1 | 0.16 | 0.38 | 0.90 | Y1 | 0.36 | 1.00 | 0.36 | P1 | 0.54 | 1.00 | |||
| K2 | 0.16 | 0.21 | 1.00 | Y2 | 0.36 | 0.15 | 0.03 | P2 | 0.54 | 0.17 | |||
| K3 | 0.38 | 0.21 | 0.69 | Y3 | 1.00 | 0.15 | 0.11 | P3 | 1.00 | 0.17 | |||
| K4 | 0.90 | 1.00 | 0.69 | Y4 | 0.36 | 0.03 | 0.11 | ||||||
Figure 9
Kylix point clouds superimposed at scale. The image displays Y1 (blue), Y2 (green), Y3 (red), and Y4 (orange), scaled to preserve the proportions of the originals and nested within each other. The scale is in cm, and the origin is in the centre of the vessels.
Table 2
Krater, Kylix, and Pelike alternative distances. The Chamfer distance is the average distance between each vessel’s point and the nearest point from a second vessel and vice-versa. The Hausdorff distance is the greatest of the distances obtained from a point in one vessel to the closest point in a second vessel and vice-versa. The partial Hausdorff or Chamfer distances, not mentioned here, are the intermediate step: they do not consider the other direction and are therefore non-symmetric. For comparability, a normalised version is produced which displays the distances from 0 to 1, obtained by dividing each distance by the largest recorded distance per vessel type.
| KRATER | KYLIX | PELIKE | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| CHAMFER ORIGINAL VALUES | |||||||||||||
| K1 | K2 | K3 | K4 | Y1 | Y2 | Y3 | Y4 | P1 | P2 | P3 | |||
| K1 | 5.21 | 7.81 | 8.66 | Y1 | 9.27 | 15.00 | 10.78 | P1 | 10.87 | 14.41 | |||
| K2 | 5.21 | 5.50 | 10.22 | Y2 | 9.27 | 6.61 | 3.18 | P2 | 10.87 | 5.44 | |||
| K3 | 7.81 | 5.50 | 10.12 | Y3 | 15.00 | 6.61 | 6.13 | P3 | 14.41 | 5.44 | |||
| K4 | 8.66 | 10.22 | 10.12 | Y4 | 10.78 | 3.18 | 6.13 | ||||||
| NORMALISED CHAMFER | |||||||||||||
| K1 | K2 | K3 | K4 | Y1 | Y2 | Y3 | Y4 | P1 | P2 | P3 | |||
| K1 | 0.51 | 0.76 | 0.85 | Y1 | 0.62 | 1.00 | 0.72 | P1 | 0.75 | 1.00 | |||
| K2 | 0.51 | 0.54 | 1.00 | Y2 | 0.62 | 0.44 | 0.21 | P2 | 0.75 | 0.38 | |||
| K3 | 0.76 | 0.54 | 0.99 | Y3 | 1.00 | 0.44 | 0.41 | P3 | 1.00 | 0.38 | |||
| K4 | 0.85 | 1.00 | 0.99 | Y4 | 0.72 | 0.21 | 0.41 | ||||||
| HAUSDORFF ORIGINAL VALUES | |||||||||||||
| K1 | K2 | K3 | K4 | Y1 | Y2 | Y3 | Y4 | P1 | P2 | P3 | |||
| K1 | 3.21 | 4.54 | 8.60 | Y1 | 5.62 | 8.84 | 4.23 | P1 | 4.82 | 7.09 | |||
| K2 | 3.21 | 4.15 | 8.82 | Y2 | 5.62 | 4.17 | 2.64 | P2 | 4.82 | 3.79 | |||
| K3 | 4.54 | 4.15 | 6.44 | Y3 | 8.84 | 4.17 | 5.49 | P3 | 7.09 | 3.79 | |||
| K4 | 8.60 | 8.82 | 6.44 | Y4 | 4.23 | 2.64 | 5.49 | ||||||
| NORMALISED HAUSDORFF | |||||||||||||
| K1 | K2 | K3 | K4 | Y1 | Y2 | Y3 | Y4 | P1 | P2 | P3 | |||
| K1 | 0.36 | 0.51 | 0.98 | Y1 | 0.64 | 1.00 | 0.48 | P1 | 0.68 | 1.00 | |||
| K2 | 0.36 | 0.47 | 1.00 | Y2 | 0.64 | 0.47 | 0.30 | P2 | 0.68 | 0.53 | |||
| K3 | 0.51 | 0.47 | 0.73 | Y3 | 1.00 | 0.47 | 0.62 | P3 | 1.00 | 0.53 | |||
| K4 | 0.98 | 1.00 | 0.73 | Y4 | 0.48 | 0.30 | 0.62 | ||||||
Figure 10
Points where the partial Hausdorff distance between Y2 and Y3 lays. The image displays the sub-samples of Y2 in green and Y3 in red. The Hausdorff distance is represented in blue (left handle of the objects). As can be seen in this example, the feature that tends to define the Hausdorff distance in Greek vessels is the handle. The models are centred so that the origin of the 3D space is in the centroid of the cloud of points.