Extreme elevation on a 2-manifold

Abstract

Given a smoothly embedded 2-manifold in Ê ¿ , we define the elevation of a point as the height difference to a canonically defined second point on the same manifold. Our definition is invariant under rigid motions and can be used to define features such as lines of discontinuous or continuous but non-smooth elevation. We give an algorithm for finding points of locally maximum elevation, which we suggest mark cavities and protrusions and are useful in matching shapes as for example in protein docking.

Extending Persistence Using Poincaré and Lefschetz Duality

“…Examples include local curvature approximation functions, used, e.g., for protein docking [4] and global alignment [7], and the mean geodesic distance used for shape matching and indexing [8]. Particularly relevant to this paper is the elevation function introduced in [1] and applied to coarse protein docking in [11]. To define the elevation at a point p of a surface S, we consider the directional height function given by projecting S onto its normal line through p, and pair up the critical points of that function using persistence.…”

Toward Unsupervised Segmentation of Semi-Rigid Low-Resolution Molecular Surfaces

“…Given two input surfaces S A and S B , of n and m vertices respectively, the most time-consuming step is the PREPROCESSING step: compute the set of featurepairs of S A and S B using the elevation function [1]. The worst-case complexity for computing elevation maxima is O(n 4 ), although in practice it is much faster.…”

“…Our approach is based on the elevation function Elev: S → R over a surface S introduced in [1]. Roughly speaking, every point x ∈ S has a canonical pairing partner y that shares the same normal direction n x with x: the pair (x, y) describes a feature in direction n x , and Elev(x), defined as the height difference between x and y in this direction, indicates the size of this feature (see Figure 1(a) for an illustration in the plane).…”

“…Hence our algorithm first eliminates from those PFPs that are not consistent with LP[0]. It then chooses from the remaining PFPs the first one (thus with highest vote) that has a large Euclidean distance between corresponding points with respect to µ[0], and sets it as LP [1].…”

“…In order to identify the landmarks automatically, we exploit a voting idea, taking advantages of a set of potentially good rigid registrations. In particular, by exploiting the feature pairs computed from the so-called elevation function [1], we develop a scheme to vote for landmarks using landmark-based coordinates. This scheme is facilitated by the one-to-one correspondence between rigid transformations and pairs of feature-pairs existed in our framework.…”