Topological Persistence - Mathematical Tools for Shape Analysis and Description - page 87

method traditionally employed in statistical shape analysis [ 64 ], and demonstrate that this helps
distinguishing clinically relevant treatment effects.
More in detail, a collection of ND240 jaw bones is represented by kD22 landmark points
chosen by an expert for their clinical relevance. After aligning the jaw bones in R 3 , the so-called
mean shape is computed, that is, a set of k points obtained by averaging the landmarks over the
N bones. ese points are used to build an abstract simplicial complex through a Delaunay tri-
angulation.
For each of the N jaws, define the weight of an edge connecting two points as the ratio
between the Euclidean distance between the landmarks corresponding to those points, and the
average over all subjects of the Euclidean distance between those landmarks. e weights produce
a Rips filtration (cf. section 11.1 ) of the Delaunay triangulation: all the nodes join at time zero, the
edge between two points joins at a time corresponding to the edge weight, and higher-dimensional
simplices join once all of their faces have. When this filtration is performed on an individual
landmark configuration, the edges that join the filtration first are those that are smallest in that
subject, relative to the entire group.
is process produces N filtrations, and, corresponding, N persistence diagrams, one for
each jaw. Actually, 3N persistence diagrams are computed, namely the 0th, 1st, and 2nd persis-
tence diagram. e Wasserstein distance between persistence diagrams is computed to obtain four
NN matrices of pairwise distances between pairs of jaws: one NN for each dimension (0,
1, 2), and a cumulative matrix. Multidimensional scaling of the matrices is performed to embed
the data as points in R 2 .
A close analysis of these data reveals a correlation between one of the embedding coordi-
nates and the treatment used on the patients (jaw expansion), enabling one to distinguish between
the control group and the two treatment groups as the treatment evolves over time. is demon-
strates that the persistent homology method is able to distinguish clinically relevant treatment
effects.
Another example of application of persistence diagrams and spaces for textured 3D shape
retrieval will be presented in chapter 12 .