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This document explores the concept of deriving shapes from connectivity graphs and vice versa. It covers Hierarchical Methods, Applications, Graph Drawing, Compression, Connectivity Creatures, and more. Learn about the intricacies of connectivity shape computation and its applications in various fields.
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Connectivity Shapes Martin Isenburg University of North Carolina at Chapel Hill Stefan Gumhold University of Tübingen Craig Gotsman Technion - Israel Instituteof Technology
Overview • Shape from Connectivity • Connectivity from Shape • Hierarchical Methods • Applications • Graph Drawing • Compression • Connectivity Creatures • Discussion
Connectivity Shape Given a connectivity graph C = ( V, E ) consisting of a list verticesV = ( v1 ,v2 ,... ,vn ) and a set undirected edgesE = { e1 ,e2 ,... ,em } :ej = ( i1 ,i2 ) The connectivity shape CS ( C ) of C is alist of vectors ( x1 ,x2 ,x3 ,... ,xn ) :xi R3that 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 ||xi - xj || = 1 ( i , j ) E The number of unknowns is determined by Euler’s relation m = n + f + 2g - 1
Spring Energy ES Minimize ES = (|| xi - xj || - 1 )2 ( i , j ) E
Roughness Energy ER ER = L( xi )2
opt = argmax Volume( CS( C, )) [0,1] Optimal Smoothing opt
Modified Spring Energy E’S E’S = (|| xi - xj ||2- 1 )2 ( i , j ) E
Meshing / Re-meshing objective: generate a faithful approximation of a given shape, but use only edges of unit length we customized Turk method
