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PhD Dissertation
Closed-form Blending of Local Symmetries
Closed-form Blending


We present a closed form solution for symmetrizing two and three-dimensional models, by solving for the optimal deformation that reconciles a set of local bilateral symmetries. Our main motivation is the symmetrization of digitized fossils, which often undergo affine compression and bending. Given a set of symmetric point-pairs, we compute local transformations which approximate bilateral symmetries over small neighborhoods. We then combine these local symmetries and solve for a global symmetry across the $y$-$z$ plane, while maintaining the shapes of the local neighborhoods. We present results on 2D and 3D objects that are deformed by compression and bending. We easily extend the technique to articulated models, for which the local symmetries are rotations.

Paper (pdf) in Proceedings of Symposium of Geometry Processing 2010


Feature-driven Deformation for Dense Correspondence


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, shape space 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.

Paper (pdf) in Proceedings of SPIE Medical Imaging 2009

Evolutionary Morphing


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 model which expresses the value of some continuous variable 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 de ne 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 scans of crania, we use our landmark editor to interactively place landmark points on the surface. We use these to compute a tree-morph that 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 blending them together. To do the blending, we compute a weighted average of their associated trivariate distance functions and then extract a
surface from the resulting function. We implement this idea using the squared distance function, rather than the usual signed distance function, in a novel way.

Paper (pdf) in Proceedings of IEEE Visualization, 2005

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