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Figure 3. A3D model of an archaeological figure and its voxel representation Once the voxel grid has been labeled in this way, the descriptor can be calculated. Broadly speaking, what it is done is to assume that when passing a cutting plane through the voxel grid there is perfect reflective symmetry between the two halves, so that one of the two halves can be replaced with the other if that property were fulfilled. Then, the reflective symmetry distance is calculated in the real model and compare it with the assumed model to measure any difference. If both representations are equal (zero distance), then there is perfect symmetry and a radius of | is assigned to the corresponding plane. If not, a value less than 1 is assigned according to how different these figures are.

Figure 3 A3D model of an archaeological figure and its voxel representation Once the voxel grid has been labeled in this way, the descriptor can be calculated. Broadly speaking, what it is done is to assume that when passing a cutting plane through the voxel grid there is perfect reflective symmetry between the two halves, so that one of the two halves can be replaced with the other if that property were fulfilled. Then, the reflective symmetry distance is calculated in the real model and compare it with the assumed model to measure any difference. If both representations are equal (zero distance), then there is perfect symmetry and a radius of | is assigned to the corresponding plane. If not, a value less than 1 is assigned according to how different these figures are.

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Figure 1. Three figurines found in the Sacred Precinct of Tenochtitlan. They are similar in their overall geometry, but differ in some details, such as the design of the head and the position of the arms. A recognition system must be able to find the 3D models of these objects, despite their partial differences in shape.
Figure 1. Three figurines found in the Sacred Precinct of Tenochtitlan. They are similar in their overall geometry, but differ in some details, such as the design of the head and the position of the arms. A recognition system must be able to find the 3D models of these objects, despite their partial differences in shape.
Figure 2. Frequency probability histograms resulting from obtaining measurements of angles between vectors for four objects from the Templo Mayor museum collection. Notice the great difference between the histogram of the serpentiform scepter (first object on the left), which shows the mode close to zero, and the histogram of the flat anthropomorphic figure (second from left to right). On the other hand, the great similarity of the histograms corresponding to the Tlaloc vessels located on the right side can be appreciated. Comparing these histograms allows the recognition system to determine whether or not two objects belong to the same class.
Figure 2. Frequency probability histograms resulting from obtaining measurements of angles between vectors for four objects from the Templo Mayor museum collection. Notice the great difference between the histogram of the serpentiform scepter (first object on the left), which shows the mode close to zero, and the histogram of the flat anthropomorphic figure (second from left to right). On the other hand, the great similarity of the histograms corresponding to the Tlaloc vessels located on the right side can be appreciated. Comparing these histograms allows the recognition system to determine whether or not two objects belong to the same class.
Figure 4. Graphs of the reflexive symmetry descriptors obtained for three objects from the Templo Mayor collection. As with the histograms shown in Figure 2, reflexive symmetry descriptors offer another method for calculating dissimilarity between 3D models.
Figure 4. Graphs of the reflexive symmetry descriptors obtained for three objects from the Templo Mayor collection. As with the histograms shown in Figure 2, reflexive symmetry descriptors offer another method for calculating dissimilarity between 3D models.
Figure 5. Graphical representation of harmonic functions calculated to obtain shape descriptors from 3D models. The simplest function is the sphere and from there higher order functions are derived. To obtain the mathematical description of a 3D model it is necessary to decompose the general form into constituent elements. The more complex the shape of an object, the more harmonic functions will be needed to describe it. The results of the decomposition are used to compare the shape of 3D models. Image by Inigo.quilez - Own work, CC BY-SA 3.0, (available here).
Figure 5. Graphical representation of harmonic functions calculated to obtain shape descriptors from 3D models. The simplest function is the sphere and from there higher order functions are derived. To obtain the mathematical description of a 3D model it is necessary to decompose the general form into constituent elements. The more complex the shape of an object, the more harmonic functions will be needed to describe it. The results of the decomposition are used to compare the shape of 3D models. Image by Inigo.quilez - Own work, CC BY-SA 3.0, (available here).
Figure 6. Graphs of the decomposition coefficients obtained by calculating harmonic functions for three objects of the collection of the Templo Mayor collection.
Figure 6. Graphs of the decomposition coefficients obtained by calculating harmonic functions for three objects of the collection of the Templo Mayor collection.
Figure 7. Illustration of the software developed for the search and recovery of 3D models of archaeological objects. Above, on the left, the consultation model (i.e. anthropomorphic sculpture) is shown; below left illustrates obtaining the reflexive symmetry descriptor for that query model; on the right are the search results obtained when the user requests to compare the query model with the models stored in the repository. It can be seen that the system retrieves all objects similar to the query model.
Figure 7. Illustration of the software developed for the search and recovery of 3D models of archaeological objects. Above, on the left, the consultation model (i.e. anthropomorphic sculpture) is shown; below left illustrates obtaining the reflexive symmetry descriptor for that query model; on the right are the search results obtained when the user requests to compare the query model with the models stored in the repository. It can be seen that the system retrieves all objects similar to the query model.
Figure 8. Another example of searching and retrieving 3D models. In this case, all copies of anthropomorphic figures were requested from the system applying the descriptor of harmonic functions. Figures 7, 8 and 9 illustrate the user interface during three query examples. The first window (top-left) shows the query model. The icons on the left side allow the user to select one of the three shape descriptors that have been implemented, while several options on the right allow the user to change the appearance of the model (e.g. visualise the query models as
Figure 8. Another example of searching and retrieving 3D models. In this case, all copies of anthropomorphic figures were requested from the system applying the descriptor of harmonic functions. Figures 7, 8 and 9 illustrate the user interface during three query examples. The first window (top-left) shows the query model. The icons on the left side allow the user to select one of the three shape descriptors that have been implemented, while several options on the right allow the user to change the appearance of the model (e.g. visualise the query models as
Figure 9. Athird example of how the software works. In this case stone masks were recovered. It should be noted that, despite the fact that the query model lacks a fragment, the system was able to produce the expected results, even recovering a mask fragment that clearly belongs to the class of the objects that the user expected.
Figure 9. Athird example of how the software works. In this case stone masks were recovered. It should be noted that, despite the fact that the query model lacks a fragment, the system was able to produce the expected results, even recovering a mask fragment that clearly belongs to the class of the objects that the user expected.
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