Research in Shape Modeling by Kathryn Leonard & Sibel Tari

Author:Kathryn Leonard & Sibel Tari
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(5.3)

This criterion does not favor preserving large distances over small ones. The algorithms starts with a set of random points in the Poincaré disk. In each iteration, it moves each of the points along the gradient direction of the energy function shown in Eq. 5.3 with a Mobius transform until one of the stopping tolerances is met or the maximum iteration number is reached.

5.4.1 Experiments on Real and Synthetic Data

While much of tree-space looks locally like a Euclidean space, there are two local features which are decidedly not Euclidean: corners and open books. A corner is point concentration of negative curvature (see Fig. 5.1b), while an open book is a set of Euclidean half-space attached together along their axes, or “spine” (see Fig. 5.2). These two features, as well as the high dimension of the local Euclidean space, are the sources of error for the low-distortion embedding. We generate synthetic datasets that isolate the two features to determine how hyperbolic MDS (HMDS) and hyperbolic isomap (HIsomap) treat them. We compare the results both qualitatively and quantitatively with embeddings done with classical MDS and isomap. More specifically, the datasets are CORNER, in which 250 points are generated by sampling the distance from the origin from a Gaussian distribution and sampling an angle with one of the orthant boundaries uniformly from the interval ; 3SHEETS_2D, in which 50 points are generated in each of 3 2-dimensional sheets; 3SHEETS_3D, in which 50 points are generated in each of 3 3-dimensional sheets; 5SHEETS_2D, in which 50 points are generated in each of 5 2-dimensional sheets; 5SHEETS_3D, in which 50 points are in each of 5 3-dimensional sheets; and COPD, in which the lung airway trees of 125 healthy patients and 125 patients with COPD are randomly selected. Within each sheet, the 50 points were generated by sampling from a symmetric normal distribution in the underlying Euclidean space that is centered at the origin.

The multiplicative distortion for each embedding approach is summarized in Table 5.6. The multiplicative distortion for a single distance between two points in the dataset is original_distance∕embedded_distance. The distortion for the whole dataset is max_distortion∕min_distortion, where max_distortion is the maximum distortion of any two points and min_distortion is the minimum distortion for any two points. HMDS and HIsomap perform the best for almost all of the datasets. The embedded visualizations and the histograms for each dataset are found in Figs. 5.5 and 5.6.

Fig. 5.5The embedded datasets. For the CORNER, 3SHEETS_2D, 3SHEETS_3D, 5SHEETS_2D, and 5SHEETS_3D dataset embeddings, points have the same color if they are located in the same quadrant or sheet. For the COPD dataset embeddings, the class of healthy patients is colored in red, and the class of patients with COPD are colored in blue

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