Surface editing operations commonly require geometric details of the surface to be preserved as much as possible. We argue that geometric detail is an intrinsic property of a surface and that, consequently, surface editing is best performed by operating over an intrinsic surface representation. We provide such a representation of a surface, based on the Laplacian of the mesh, by encoding each vertex relative to its neighborhood. The Laplacian of the mesh is enhanced to be invariant to locally linearized rigid transformations and scaling. Based on this Laplacian representation, we develop useful editing operations: interactive free-form deformation in a region of interest based on the transformation of a handle, transfer and mixing of geometric details between two surfaces, and transplanting of a partial surface mesh onto another surface. The main computation involved in all operations is the solution of a sparse linear system, which can be done at interactive rates. We demonstrate the effectiveness of our approach in several examples, showing that the editing operations change the shape while respecting the structural geometric detail.

This paper describes a fully automatic pipeline for finding an intrinsic map between two non-isometric, genus zero surfaces. Our approach is based on the observation that efficient methods exist to search for nearly isometric maps (e.g., Möbius Voting or Heat Kernel Maps), but no single solution found with these methods provides low-distortion everywhere for pairs of surfaces differing by large deformations. To address this problem, we suggest using a weighted combination of these maps to produce a "blended map." This approach enables algorithms that leverage efficient search procedures, yet can provide the flexibility to handle large deformations. The main challenges of this approach lie in finding a set of candidate maps { m i } and their associated blending weights { b i ( p )} for every point p on the surface. We address these challenges specifically for conformal maps by making the following contributions. First, we provide a way to blend maps, defining the image of p as the weighted geodesic centroid of m i ( p ). Second, we provide a definition for smooth blending weights at every point p that are proportional to the area preservation of m i at p . Third, we solve a global optimization problem that selects candidate maps based both on their area preservation and consistency with other selected maps. During experiments with these methods, we find that our algorithm produces blended maps that align semantic features better than alternative approaches over a variety of data sets.

Inferred dietary preference is a major component of paleoecologies of extinct primates. Molar occlusal shape correlates with diet in living mammals, so teeth are a potentially useful structure from which to reconstruct diet in extinct taxa. We assess the efficacy of Dirichlet normal energy (DNE) calculated for molar tooth surfaces for reflecting diet. We evaluate DNE, which uses changes in normal vectors to characterize curvature, by directly comparing this metric to metrics previously used in dietary inference. We also test whether combining methods improves diet reconstructions. The study sample consisted of 146 lower (mandibular) second molars belonging to 24 euarchontan taxa. Five shape quantification metrics were calculated on each molar: DNE, shearing quotient, shearing ratio, relief index, and orientation patch count rotated (OPCR). Statistical analyses were completed for each variable to assess effects of taxon and diet. Discriminant function analysis was used to assess ability of combinations of variables to predict diet. Values differ significantly by diets for all variables, although shearing ratios and OPCR do not distinguish statistically between insectivores and folivores or omnivores and frugivores. Combined analyses were much more effective at predicting diet than any metric alone. Alone, relief index and DNE were most effective at predicting diet. OPCR was the least effective alone but is still valuable as the only quantitative measure of surface complexity. Of all methods considered, DNE was the least methodologically sensitive, and its effectiveness suggests it will be a valuable tool for dietary reconstruction.

No abstract

We describe approaches for distances between pairs of two-dimensional surfaces (embedded in three-dimensional space) that use local structures and global information contained in interstructure geometric relationships. We present algorithms to automatically determine these distances as well as geometric correspondences. This approach is motivated by the aspiration of students of natural science to understand the continuity of form that unites the diversity of life. At present, scientists using physical traits to study evolutionary relationships among living and extinct animals analyze data extracted from carefully defined anatomical correspondence points (landmarks). Identifying and recording these landmarks is time consuming and can be done accurately only by trained morphologists. This necessity renders these studies inaccessible to nonmorphologists and causes phenomics to lag behind genomics in elucidating evolutionary patterns. Unlike other algorithms presented for morphological correspondences, our approach does not require any preliminary marking of special features or landmarks by the user. It also differs from other seminal work in computational geometry in that our algorithms are polynomial in nature and thus faster, making pairwise comparisons feasible for significantly larger numbers of digitized surfaces. We illustrate our approach using three datasets representing teeth and different bones of primates and humans, and show that it leads to highly accurate results.homology | Mobius transformations | morphometrics | Procrustes T o document and understand physical and biological phenomena (e.g., geological sedimentation, chemical reactions, ontogenetic development, speciation, evolutionary adaptation, etc.), it is important to quantify the similarity or dissimilarity of objects affected or produced by the phenomena under study. The grain size or elasticity of rocks, geographic distances between populations, or hormone levels and body masses of individuals-these can be readily measured, and the resulting numerical values can be used to compute similarities/distances that help build understanding. Other properties like genetic makeup or gross anatomical structure cannot be quantified by a single number; determining how to measure and compare these is more involved (1-4). Representing the structure of a gene (through sequencing) or quantification of an anatomical structure (through the digitization of its surface geometry) leads to more complex numerical representations. Even though such representations are not measurements allowing direct comparison among samples of genes or anatomical structures, they form an essential initial step for such quantitative comparisons. The one-dimensional, sequential arrangement of genomes and the discrete variation (four nucleotide base types) for each of thousands of available correspondence points help reduce the computational complexity of determining the most likely alignment between genomes; alignment procedures are now increasingly automated (5). The resulting, rapid...