During the visit, a new method for efficient retrieval of (watertight) 3D models was developed. The method consists of comparing the Fiedler vector of the Extended Reeb Graphs (ERGs). However, unlike most spectral methods, the geometry is taken into account as well.

Approach

Basing on the previous work, a shape is first segmented into large simple and small complex components by iteratively bisecting branching components and merging simple adjacent regions. As the result, the ERG constructed in this way has the branching nodes that are always connected only to simple nodes and never to other branching nodes. From here it follows that an edge is always adjacent to one simple node and one branching node representing the region between two critical points of the measuring function. At the second phase, the shape of simple components is analyzed according to two different shape characteristics. These are area of cross sections and bending of a component. We define the shape classification indices basing on the combination of these two shape characteristics, choosing the indices in the way that visually similar shapes have similar indices. For each simple component obtained after the segmentation we also compute its weight which is defined as the ratio of the surface area of the component to the surface area of the whole model. These shape indices together with segment weight are stored as the attributes of the corresponding edges of the ERG. In this way the ERG encodes both the topology and geometry of a 3D model.

We can encode all these edge properties in the Hermitian matrix and use its spectral decomposition for similarity evaluation. By obeying several constraints on the definition of the graph's topology and geometry measurements, and by encoding these values in a Hermitian matrix, the spectral behavior of the Laplacian matrix is mimicked. To find the similarity between to graphs we calculate the distance between the Fiedler vectors of their Hermitian matrices.