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Abstract
Given a graph between N high-dimensional nodes, can we faithfully visualize it in just a few dimensions? We present an algorithm that improves the state-of-the art in dimensionality reduction and visualization by extending the Maximum Variance Unfolding method. Visualizations are shown for social networks, species trees, image datasets and human activity.
If the connectivity between N nodes is unknown, can we link them to build a graph? The space to explore is daunting with 2^(N^2) choices but two interesting subfamilies are tractable: matchings and b-matchings. We place distributions over these families and recover the optimal graph or perform Bayesian inference over graphs efficiently using belief propagation algorithms. Higher order distributions over matchings can also be handled efficiently via fast Fourier algorithms. Applications are shown in tracking, classification, and clustering.
Abstract
The last few years have brought a more concrete understanding of the mathematical relationship between strokes in drawings, linear features in images, and lines on 3D shapes. However, fundamental questions remain unanswered about how our perceptual system resolves these lines as giving evidence about shape. These questions need to be addressed to assemble mathematically defined lines into clear and compelling drawings. I'll discuss my ongoing research on line drawings from this perspective.
Abstract
Deforming objects in a "shape preserving" manner is a challenging problem with various applications in geometric modeling and computer graphics. In this talk I will examine two geometric-driven approaches to the problem.
First, reformulating the problem using the notion of Cartan's moving frames reveals an intriguing relationship between harmonic maps into the group of rotations and shape preserving deformations. The moving frames are known in differential geometry for their ability to simplify some surface theory argumentation. Their employment leads to rather simple algorithms for shape deformation and shape blending of discrete surfaces (meshes). I will present the moving frames as rigid-motion invariant surface representation which is suited for deformations and shape blending. Then I will discuss the optimal rotation field between two isometric surfaces and its use in the context of surface deformation.
Second, I will present a recent result demonstrating a closed-form solution to shape preserving space deformations. The new scheme guarantees a pure conformal mapping in 2D and quasi-conformal mappings in 3D. This generalizes recent interesting affine-invariant free form deformation techniques and provides an extremely fast algorithm for shape preserving deformations. Bio
Yaron Lipman is a PhD candidate at Tel-Aviv University. He received the BSc degree in mathematics and computer science from Tel Aviv University in 2003. His research interests are in applied geometry, approximations and computer graphics
Abstract
Digital 3D content is ubiquitous in numerous areas, such as research, medicine, entertainment and other commerce. Yet, for the creation and modification of 3D models we rely on highly specialized tools, which unfortunately limits the user base to trained professionals. In this talk I will outline and demonstrate two of my research projects that are designed to alleviate this problem. The goal is to provide the user with intuitive and familiar "2D to 3D" interfaces without compromising the underlying robust mathematical framework, but rather hiding it from the user.
For the simple creation of surface meshes, I will present an interface for designing freeform surfaces with a collection of 3D curves. The user first creates a rough 3D model by using a sketching interface, where user-drawn strokes stay on the model surface and serve as handles for controlling the geometry. These curves can be
added, removed, and deformed easily, as if working with a 2D line drawing. For a given set of curves, the system automatically constructs a smooth surface embedding by applying real-time functional optimization.
For further surface modification, I will present a silhouette over-sketching interface. The user sketches a stroke that is the suggested position of part of a silhouette of the displayed surface.
The system then segments all image-space silhouettes of the projected surface, identifies among all silhouette segments the best matching part, derives vertices in the surface mesh corresponding to the silhouette part, selects a sub-region of the mesh to be modified, and feeds appropriately modified vertex positions together with the sub-mesh into a mesh deformation tool.
Overall, these algorithms have been designed to enable interactive creation and modification of the surface -- yielding a surface modeling and editing system that strives to come close to the experience of sketching 3D models on paper. I will conclude with an overview on ongoing and future research. Bio
Andrew Nealen is a postdoctoral researcher at the computer graphics lab of TU Berlin, Germany. He holds M.Sc. degrees in computer science, structural engineering and architecture from TU Darmstadt, Germany. In computer graphics he has previously explored and presented algorithms for texture synthesis, physically based modeling and animation of elastoplastic solids, and adaptive raytracing of point based models. Currently, his research deals with interfaces and algorithms for the construction, modification, structural nalysis and animation of surface meshes, as well as triangle mesh optimization.
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