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Connectivity Shapes. Martin Isenburg University of North Carolina at Chapel Hill. Stefan Gumhold University of Tübingen. Craig Gotsman Technion - Israel Institute of Technology. Introduction. Overview. Shape from Connectivity Connectivity from Shape Hierarchical Methods Applications
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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
