Generalized Barycentric Coordinates Kai Hormann Faculty of - PowerPoint PPT Presentation

Generalized Barycentric Coordinates Kai Hormann Faculty of Informatics Università della Svizzera italiana Lugano Università della Svizzera italiana, Lugano

Cartesian coordinates point (2,2) with � x -coordinate: 2 � y -coordinate: 2 di 2 y 3 mathematically: (2,2) 2 (2 2) (2,2) = 2 2 · (1,0) (1 0) (–3,1) 1 + 2 · (0,1) (0,0) –3 –2 –1 1 2 3 x in general: –1 ( x , y ) = x · (1,0) –2 (1,–2) + y · (0,1) –3 x - and y -coordinates w.r.t. base points René Descartes (1596–1650) (1,0) and (0,1) (1,0) and (0,1) 1 Generalized Barycentric Coordinates – Milano – 18 September 2012

Barycentric coordinates point ( a , b , c ) with 3 coordinates w.r.t. (1 0 0) (1,0,0) base points A B C base points A , B , C (0.5,0.5,0) mathematically: (0,1,0) ( a b c ) = ( a , b , c ) = a · A a · A + b · B (0.25,0.25,0.5) + c · C where where A = (1,0,0) (0.25,–0.25,1) B = (0,1,0) (0,0,1) C = (0,0,1) ( ) and August Ferdinand Möbius a + b + c = 1 (1790–1868) 2 Generalized Barycentric Coordinates – Milano – 18 September 2012

Barycentric coordinates � system of masses at positions � position of the system’s barycentre : position of the system s barycentre : � are the barycentric coordinates of � not unique � at least points needed to span 3 Generalized Barycentric Coordinates – Milano – 18 September 2012

Barycentric coordinates � Theorem [Möbius, 1827] : The barycentric coordinates The barycentric coordinates of with of with respect to are unique up to a common factor � example: 4 Generalized Barycentric Coordinates – Milano – 18 September 2012

Barycentric coordinates for triangles � normalized barycentric coordinates � properties properties � partition of unity � reproduction ep oduct o � positivity � Lagrange property g g p p y � application � linear interpolation of data � linear interpolation of data 5 Generalized Barycentric Coordinates – Milano – 18 September 2012

Generalized barycentric coordinates � finite-element-method with polygonal elements � convex convex [Wachspress 1975] [Wachspress 1975] � weakly convex [Malsch & Dasgupta 2004] � arbitrary arbitrary [Sukumar & Malsch 2006] [Sukumar & Malsch 2006] � interpolation of scattered data � interpolation of scattered data � natural neighbour interpolants [Sibson 1980] � – " – of higher order f h h d [Hiyoshi & Sugihara 2000] � Dirichlet tessellations [Farin 1990] 6 Generalized Barycentric Coordinates – Milano – 18 September 2012

Generalized barycentric coordinates � parameterization of piecewise linear surfaces � shape preserving coordinates shape preserving coordinates [Floater 1997] [Floater 1997] � discrete harmonic (DH) coordinates [Eck et al. 1995] � mean value (MV) coordinates mean value (MV) coordinates [Floater 2003] [Floater 2003] � other applications � other applications � discrete minimal surfaces [Pinkall & Polthier 1993] � colour interpolation l l [Meyer et al. 2002] � boundary value problems [Belyaev 2006] 7 Generalized Barycentric Coordinates – Milano – 18 September 2012

Arbitrary polygons � barycentric coordinates � normalized coordinates � normalized coordinates � properties linear precision � partition of unity � reproduction for all 8 Generalized Barycentric Coordinates – Milano – 18 September 2012

Convex polygons [Floater, H. & Kós 2006] [ , ] � Theorem: If all , then � positivity positivity � Lagrange property � linear along boundary linear along boundary � application � application � interpolation of data given at the vertices � inside the convex hull of the d h h ll f h � direct and efficient evaluation 9 Generalized Barycentric Coordinates – Milano – 18 September 2012

Examples � Wachspress (WP) coordinates � mean value (MV) coordinates � discrete harmonic (DH) coordinates � discrete harmonic (DH) coordinates 10 Generalized Barycentric Coordinates – Milano – 18 September 2012

Normal form [Floater, H. & Kós 2006] [ , ] � Theorem: All barycentric coordinates can be written as with certain real functions � three-point coordinates � with � Theorem: Such a generating function g g exists for all three point coordinates exists for all three-point coordinates 11 Generalized Barycentric Coordinates – Milano – 18 September 2012

Three-point coordinates � Theorem: if and only if is � positive positive � monotonic � convex convex � sub-linear � examples � WP coordinates � MV coordinates � DH coordinates 12 Generalized Barycentric Coordinates – Milano – 18 September 2012

Non-convex polygons Wachspress mean value discrete harmonic � poles, if , because 13 Generalized Barycentric Coordinates – Milano – 18 September 2012

Star-shaped polygons � Theorem: if and only if is � positive positive � super-linear � examples � MV coordinates � DH coordinates � Theorem: Th f for some if is if i � strictly super-linear 14 Generalized Barycentric Coordinates – Milano – 18 September 2012

Mean value coordinates [H. & Floater 2006] [ ] � Theorem: MV coordinates have no poles in 15 Generalized Barycentric Coordinates – Milano – 18 September 2012

Mean value coordinates � properties � well-defined everywhere in � Lagrange property � linear along boundary g y � linear precision for � smoothness at , otherwise , � similarity invariance for � application � direct interpolation of data 16 Generalized Barycentric Coordinates – Milano – 18 September 2012

Implementation � Mean Value coordinates 17 Generalized Barycentric Coordinates – Milano – 18 September 2012

Implementation � efficient and robust evaluation of the function 18 Generalized Barycentric Coordinates – Milano – 18 September 2012

Colour interpolation 19 Generalized Barycentric Coordinates – Milano – 18 September 2012

Vector fields 20 Generalized Barycentric Coordinates – Milano – 18 September 2012

Smooth shading 21 Generalized Barycentric Coordinates – Milano – 18 September 2012

Rendering of quadrilateral elements 22 Generalized Barycentric Coordinates – Milano – 18 September 2012

Transfinite interpolation mean value coordinates radial basis functions 23 Generalized Barycentric Coordinates – Milano – 18 September 2012

Smooth distance function � Function approximates the distance function � and along the boundary � smooth, except at the vertices 24 Generalized Barycentric Coordinates – Milano – 18 September 2012

Mesh animation 25 Generalized Barycentric Coordinates – Milano – 18 September 2012

Image warping original image mask warped image 26 Generalized Barycentric Coordinates – Milano – 18 September 2012

Mesh warping � MV coordinates in 3D [Ju et al. 2005] � negative inside the domain MVC PMVC � positive MV coordinates [Lipman et al. 2007] � only C 0 -continuous � no closed form MVC PMVC 27 Generalized Barycentric Coordinates – Milano – 18 September 2012

Harmonic coordinates � define normalized coordinate as solution of PDE subject to subject to � Lagrange property � well-defined � smooth � � linear precision � positivity � efficient � � animation for Ratatouille [Joshi et al. 2007] 28 Generalized Barycentric Coordinates – Milano – 18 September 2012

Positive barycentric coordinates � drawbacks so far … � mean value coordinates � mean value coordinates � negative � positive mean value coordinates � not smooth (only C 0 ) ( y ) � harmonic coordinates � rather expensive to compute � not smooth in practice 29 Generalized Barycentric Coordinates – Milano – 18 September 2012

Maximum entropy coordinates [H. & Sukumar 2008] [ ] � based on maximizing the Shannon-Jaynes entropy � Lagrange property � well defined � smooth ( � ) � Lagrange property � well-defined � smooth ( � ) � linear precision � positivity � efficient ( � ) 30 Generalized Barycentric Coordinates – Milano – 18 September 2012