Establishing reliable correspondences between object surfaces is a fundamental operation, required in many contexts such as cleaning up and completing imperfect captured data, texture and deformation transfer, shapespace analysis and exploration, and the automatic generation of realistic distributions of objects. We present a method for matching a template to a collection of possibly target meshes. Our method uses a very small number of user-placed landmarks, which we augment with automatically detected feature correspondences, found using spin images. We deform the template onto the data using an ICP-like framework, smoothing the noisy correspondences at each step so as to produce an averaged motion. The deformation uses a differential representation of the mesh, with which the deformation can be computed at each iteration by solving a sparse linear system. We have applied our algorithm to a variety of data sets. Using only 11 landmarks between a template and one of the scans from the CEASAR data set, we are able to deform the template, and correctly identify and transfer distinctive features, which are not identified by user-supplied landmarks. We have also successfully established correspondences between several scans of monkey skulls, which have dangling triangles, non- manifold vertices, and self intersections. Our algorithm does not require a clean target mesh, and can even generate correspondence without trimming our extraneous pieces from the target mesh, such as scans of teeth.

We introduce a technique to visualize the gradual evolutionary change of the shapes of living things as a morph between known three-dimensional shapes. Given geometric computer models of anatomical shapes for some collection of specimens - here the skulls of the some of the extant members of a family of monkeys - an evolutionary tree for the group implies a hypothesis about the way in which the shape changed through time. We use a statistical method which expresses the value of some variable - in this case the shape - at an internal point in the tree as a weighted average of the values at the leaves. The framework of geometric morphometrics can then be used to define a shape-space, based on the correspondences of landmark points on the surfaces, within which these weighted averages can be realized as actual surfaces. Our software provides tools for performing and visualizing such an analysis in three dimensions. Beginning with laser range surfaces scans of skulls, we use our landmark editor to interactively place landmark points on the surface. We use these to compute a treemorph which smoothly interpolates the shapes across the tree. Each intermediate shape in the morph is a linear combination of all of the input surfaces. We create a surface model for an intermediate shape by warping all the input meshes towards the correct shape and then merging them together. Our merging procedure is novel. Given several similar surface meshes, we compute a weighted average between them by averaging their associated trivariate squared distance functions, and then extract the extremal surface which traces out the ''valleys'' along which the averaged function is nearly zero.

Diagenetic distortion can be a major obstacle to collecting quantitative shape data on paleontological specimens, especially for three-dimensional geometric morphometric analysis. Here we utilize the recently-published algorithmic symmetrization method of fossil reconstruction and compare it to the more traditional reflection & averaging approach. In order to have an objective test of this method, five casts of a female cranium of Papio hamadryas kindae were manually deformed while the plaster hardened. These were subsequently "retrodeformed" using both algorithmic symmetrization and reflection & averaging and then compared to the original, undeformed specimen. We found that in all cases, algorithmic retrodeformation improved the shape of the deformed cranium and in four out of five cases, the algorithmically symmetrized crania were more similar in shape to the original crania than the reflected & averaged reconstructions. In three out of five cases, the difference between the algorithmically symmetrized crania and the original cranium could be contained within the magnitude of variation among individuals in a single subspecies of Papio. Instances of asymmetric distortion, such as breakage on one side, or bending in the axis of symmetry, were well handled, whereas symmetrical distortion remained uncorrected. This technique was further tested on a naturally deformed and fossilized cranium of Paradolichopithecus arvernensis. Results, based on a principal components analysis and Procrustes distances, showed that the algorithmically symmetrized Paradolichopithecus cranium was more similar to other, less-deformed crania from the same species than was the original. These results illustrate the efficacy of this method of retrodeformation by algorithmic symmetrization for the correction of asymmetrical distortion in fossils. Symmetrical distortion remains a problem for all currently developed methods of retrodeformation.