Figure 3

Illustration of the inner distance. The red dashed line denotes the inner distance, which is the shortest path within the shape boundary surface that connect two landmark points x and y. The right molecule is one deformation to the left one, and the relative change of the inner distances between the corresponding pair of points (e.g. x and y) during shape deformation are small. In contrast, the black bold line denotes the Euclidean distance defined as the length of the line segment between two landmark points x and y. Note that the Euclidean distance does not have the property of deformation invariant in contrast to the inner distance. This is because, the Euclidean distance does not consider whether the line segment crosses shape boundaries.