Radial Basis Functions

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Felix Foster
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Radal Bass Functons Mesh Reconstructon Input: pont cloud Output: water-tght manfold mesh

Representaton of 3D Objects wth Radal Bass Functons by Carr et al.

F(x) < 0 nsde F(x) > 0 outsde Image from: www.farfeldtechnology.

1 Radal Bass Functons Mesh Reconstructon Input: pont cloud Output: water-tght manfold mesh Explct Connectvty estmaton Implct Sgned dstance functon estmaton Image from: Reconstructon and Representaton of 3D Objects wth Radal Bass Functons by Carr et al. Image from: Reconstructon and Representaton of 3D Objects wth Radal Bass Functons by Carr et al. Sgned Dstance Functon Surface Reconstructon F(x) = 0 Surface F(x) < 0 nsde F(x) > 0 outsde Image from: Gven a set of ponts P= { p, p,..., p } p R 1 2 Fnd a manfold surface S whch approxmates P n 3

2 Implct Surface Reconstructon Gven a set of ponts P= { p, p,..., p } p R 1 2 Fnd a manfold surface S whch approxmates P n 3 Implct Surface Reconstructon 1) Fnd sgned dstance functon d(x) from pont cloud 2) Evaluate d(x) on unform 3D grd S s zero level-set of a sgned dstance functon d S = { x d( x) = 0} 3) Extract mesh wth Marchng Cubes Fndng the Dstance Functon Many optons Dstance from tangent planes Radal bass functons Posson equaton... Scattered Data Interpolaton Problem Gven N ponts Reconstruct a functon S(x), such that Constrants S(x) contnuous S(x) s smooth ( x, f ) Sx ( ) = f

3 Radal Bass Functons Soluton Assume S(X) s weghted sum of bass functons ( ) Sx ( ) = wφ x x Radal Bass Functons Zoo Polyharmonc splne k φ ( r) = r log( r), k = 2,4,6,... k ( ), 1,3,5,... φ r = r k = = (,...) Radal Bass Functons Zoo Multquadratc () r φ = r +β 2 2 Gaussan 2 φ () r = e βr Compact support Fndng the Weghts Fnd weghts by solvng Solve a lnear system ( ) f = S( x ) = wφ x x j j j φ w = f, φ =φ x x ( ) j j j

4 Fndng the Weghts System s sparse when RBFs have compact support Gaussans vs. Polyharmoncs 2 Gaussans φ () r = e βr have compact support But parameter β depends on scale of data ponts requres tunng Non sparse system problematc for large number of ponts Fllng Holes - Extrapolaton Addng a polynomal to the RBF sum Allows to recreate polynomals exactly Improves extrapolaton Addng a Polynomal ( ) S( x) = wφ x x + P( x) P(x) low degree polynomal Wthout polynomal Wth polynomal

5 Reconstructon usng RBF Lookng for dstance functon, remember? For each pont p n nput we have d( p ) = 0 Use some RBF, solve and get d? Reconstructon usng RBF Problem, RBF system s Mw = 0 Wll get the trval soluton Add off-surface ponts Off-surface Ponts For each pont n data add 2 off-surface ponts on both sdes of surface Use normal data to fnd off-surface ponts Off-surface Ponts P= { p, p,..., p } p R 1 2 d( p ) = 0 d( p + cnˆ ) = c d( p cnˆ ) = c n 3 Reconstructon usng RBF needs orented ponts Image from: Reconstructon and Representaton of 3D Objects wth Radal Bass Functons by Carr et al. Images from: Reconstructon and Representaton of 3D Objects wth Radal Bass Functons by Carr et al.

Off-surface Ponts Estmated normal mght be ambguous Don t add off-surface ponts

Closest surface pont to projected pont should be generatng pont Reducng Number of

Constant projecton dstances Varyng projecton dstances Images from: Reconstructon

Images from: Reconstructon Reducng Number of Centers Greedy approach Start wth a

tolerance, and more centers Computatonal Complexty RBFs mght gve dense matrx O(N 2

Storage O(NlogN) Fttng O(1) - Evaluaton Images from: Reconstructon 544,000 ponts,

6 Off-surface Ponts Estmated normal mght be ambguous Don t add off-surface ponts there Project to whch dstance? Closest surface pont to projected pont should be generatng pont Reducng Number of Centers Not necessarly need all ponts to be centers to reconstruct surface Constant projecton dstances Varyng projecton dstances Images from: Reconstructon and Representaton of 3D Objects wth Radal Bass Functons by Carr et al. Images from: Reconstructon and Representaton of 3D Objects wth Radal Bass Functons by Carr et al. Reducng Number of Centers Greedy approach Start wth a subset of the ponts Repeat Ft RBFs Evaluate error at all ponts If larger than tolerance, and more centers Computatonal Complexty RBFs mght gve dense matrx O(N 2 ) - Storage O(N 3 ) Solvng the matrx O(N) Evaluatng a pont Fast methods exst O(N) Storage O(NlogN) Fttng O(1) - Evaluaton Images from: Reconstructon and Representaton of 3D Objects wth Radal Bass Functons by Carr et al. 544,000 ponts, 80,000 centers err =

7 Examples Images from: Reconstructon and Representaton of 3D Objects wth Radal Bass Functons by Carr et al.

Machine Learning 9. week

Machine Learning 9. week Machne Learnng 9. week Mappng Concept Radal Bass Functons (RBF) RBF Networks 1 Mappng It s probably the best scenaro for the classfcaton of two dataset s to separate them lnearly. As you see n the below