Martin Isenburg University of North Carolina at Chapel Hill Craig Gotsman Technion - Israel Institute of Technology Stefan Gumhold University of Tübingen Connectivity Shapes
Introduction
Overview Shape from Connectivity Connectivity from Shape Hierarchical Methods Applications –Graph Drawing –Compression –Connectivity Creatures Discussion
Shape from Connectivity
Connectivity Shape Given a connectivity graph C = ( V, E ) consisting of a list vertices V = ( v 1, v 2,..., v n ) and a set undirected edges E = { e 1, e 2,..., e m } : e j = ( i 1, i 2 ) The connectivity shape CS ( C ) of C is a list of vectors ( x 1, x 2, x 3,..., x n ) : x i R 3 that satisfy some “natural” property.
Some “Natural” Property “all edges have unit length” Equilibrium state of spring system. The connectivity shape is the solution to a set of m equations of the form || x i - x j || = 1 ( i, j ) E The number of unknowns is determined by Euler’s relation m = n + f + 2g - 1
Spring Energy E S Minimize E S = ( || x i - x j || - 1 ) 2 ( i, j ) E
Roughness Energy E R E R = L( x i ) 2
Final equation
Family of Connectivity Shapes
Optimal Smoothing opt opt = argmax Volume ( CS ( C, ) ) [ 0,1 ]
Iterative Solver
Modified Spring Energy E’ S E’ S = ( || x i - x j || ) 2 ( i, j ) E
Connectivity from Shape
Meshing / Re-meshing objective: generate a faithful approximation of a given shape, but use only edges of unit length we customized Turk method
Smoothing Parameter dev
Example Run
Hierarchical Methods
Constructing the Hierarchy
Applications
Mesh Compression
Connectivity Creatures
End
Bloopers