Gaussian Process Implicit Surfaces Oliver Williams 1 Microsoft - PowerPoint PPT Presentation

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Gaussian Process Implicit Surfaces Oliver Williams 1 Microsoft Research and Trinity Hall Cambridge, UK Gaussian Processes in Practice Bletchley Park, 13 June 2006 1 Joint work with Andrew Fitzgibbon

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Talk outline Implicit surface modelling Spline regularization as a Gaussian process Covariance function 1D regression demonstration GPIS for 2D curves Covariance in 2D Probabilistic interpretation GPIS for 3D surfaces Covariance function Summary

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Implicit surface Scalar function f ( x ) defins a surface wherever it passes through a given value (e.g., 0) S 0 � { x ∈ R d | f ( x ) = 0 } .

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Implicit surface Scalar function f ( x ) defins a surface wherever it passes through a given value (e.g., 0) S 0 � { x ∈ R d | f ( x ) = 0 } . Example: Function f ( x ) for x ∈ R 2 defines a closed curve

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Fitting to data points Our setting (Turk and O’Brien 1999): ◮ Given a set of constraint points in 2D or 3D { x i } , fit f ( x ) ◮ Have constraints at f ( x i ) = 0 on the curve and at ± 1 off it e.g., ◮ Simple interior/exterior case ◮ Control normals to curve

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Fitting to data points Our setting (Turk and O’Brien 1999): ◮ Given a set of constraint points in 2D or 3D { x i } , fit f ( x ) ◮ Have constraints at f ( x i ) = 0 on the curve and at ± 1 off it e.g., ◮ Simple interior/exterior case ◮ Control normals to curve 0.5 0 −0.5 −1 −0.5 0 0.5

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Fitting to data points Our setting (Turk and O’Brien 1999): ◮ Given a set of constraint points in 2D or 3D { x i } , fit f ( x ) ◮ Have constraints at f ( x i ) = 0 on the curve and at ± 1 off it e.g., ◮ Simple interior/exterior case ◮ Control normals to curve 0.5 0 −0.5 −1 −0.5 0 0.5

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Fitting to data points Our setting (Turk and O’Brien 1999): ◮ Given a set of constraint points in 2D or 3D { x i } , fit f ( x ) ◮ Have constraints at f ( x i ) = 0 on the curve and at ± 1 off it e.g., ◮ Simple interior/exterior case ◮ Control normals to curve Alternative method: parametric surface : 0.5 x ( t ) , y ( t ) , [ z ( t )] ◮ What t to assign to data points? 0 ◮ How to handle different topologies? −0.5 ◮ Can represent non-closed −1 −0.5 0 0.5 curves/surfaces

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Topology change 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 0 0.5 1 3 2 1 0 −1 −2 −3 −2 0 2

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Regularization Find function passing through constraint points which minimizes thin-plate spline energy � � 2 � ∇ T ∇ f ( x ) E ( f ) = dx Ω

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Regularization Find function passing through constraint points which minimizes thin-plate spline energy � � 2 � ∇ T ∇ f ( x ) E ( f ) = dx Ω Fit f ( x ) with Gaussian process

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Regularization Find function passing through constraint points which minimizes thin-plate spline energy � � 2 � ∇ T ∇ f ( x ) E ( f ) = dx Ω Fit f ( x ) with Gaussian process Use covariance function equivalent to thin-plate spline

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Gaussian distribution ◮ Follow derivation in (MacKay 2003) in 1D

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Gaussian distribution ◮ Follow derivation in (MacKay 2003) in 1D ◮ consider the energy as a probability and define D as the linear differential operator � � 2 dx . D 2 f ( x ) � E ( f ) = − log P ( f ) + const = Ω

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Gaussian distribution ◮ Follow derivation in (MacKay 2003) in 1D ◮ consider the energy as a probability and define D as the linear differential operator � � 2 dx . D 2 f ( x ) � E ( f ) = − log P ( f ) + const = Ω ◮ Use f (Ω) as vector of function values for all points in Ω: − log P ( f (Ω)) = f (Ω) T [ D 2 ] T D 2 f (Ω) ,

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Gaussian distribution ◮ Follow derivation in (MacKay 2003) in 1D ◮ consider the energy as a probability and define D as the linear differential operator � � 2 dx . D 2 f ( x ) � E ( f ) = − log P ( f ) + const = Ω ◮ Use f (Ω) as vector of function values for all points in Ω: − log P ( f (Ω)) = f (Ω) T [ D 2 ] T D 2 f (Ω) , ◮ This is a Gaussian disribution with mean zero and covariance: [ D 2 ] T D 2 � − 1 D 4 � − 1 . � � C = =

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Covariance function ◮ Entries of C indexed by u , v ∈ Ω ∂ 4 � D 4 ( u , w ) c ( w , v ) dw = δ ( u − v ) ⇒ ∂ r 4 c ( r ) = δ ( r ) Ω where we impose stationarity with r = u − v .

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Covariance function ◮ Entries of C indexed by u , v ∈ Ω ∂ 4 � D 4 ( u , w ) c ( w , v ) dw = δ ( u − v ) ⇒ ∂ r 4 c ( r ) = δ ( r ) Ω where we impose stationarity with r = u − v . ◮ Interpret as spectral density and solve ( ω ) = ω − 4 � � F c ( r ) 6 | r | 3 + a 3 r 3 + a 2 r 2 + a 1 r + a 0 . c ( r ) = 1 ⇒

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Covariance function 6 | r | 3 + a 3 r 3 + a 2 r 2 + a 1 r + a 0 c ( r ) = 1 Find constants using constraints on c ( · ) ◮ Symmetry: a 3 = a 1 = 0

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Covariance function 6 | r | 3 + a 3 r 3 + a 2 r 2 + a 1 r + a 0 c ( r ) = 1 Find constants using constraints on c ( · ) ◮ Symmetry: a 3 = a 1 = 0 ◮ Postive definiteness: simulate by making c ( r ) → 0 at ∂ Ω 2 | r | 3 − 3 Rr 2 + R 3 � c ( r ) = 1 � . 12 where R is the largest magnitude of r within Ω.

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary 1D regression demonstration GP predicts function values for set of points U ⊆ Ω P ( f ( U ) |X ) = Normal ( f ( U ) | µ, Q ) where µ = C T ux ( C xx + σ 2 I ) − 1 t Q = C uu − C T ux ( C xx + σ 2 I ) − 1 C ux . and The matrices are formed by evaluating c ( · , · ) between sets of points: i.e., C xx = [ c ( x i , x j )], C ux = [ c ( u i , x j )], and C uu = [ c ( u i , u j )].

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary 1D regression demonstration 2 2 1 1 0 0 −1 −1 −2 −2 −2 −1 0 1 2 −2 −1 0 1 2 (a) (b) Figure: Thin plate vs. squared exponential covariance. Mean (solid line) and 3 s.d. error bars (filled region) for GP regression (a) Thin- plate covariance; (b) Squared exponential covariance function c ( u i , u j ) = e − α � u i − u j � 2 with α = 2 , 10 and 100 ; error bars correspond to α = 10 .

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Covariance in 2D ◮ In 2D the Green’s equation is � 2 � ∇ T ∇ c ( r ) = δ ( r ) where now c ( u , v ) = c ( r ) with r � � u − v � .

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Covariance in 2D ◮ In 2D the Green’s equation is � 2 � ∇ T ∇ c ( r ) = δ ( r ) where now c ( u , v ) = c ( r ) with r � � u − v � . ◮ Solution (with similar constraints at the boundary of Ω) c ( r ) = 2 r 2 log | r | − r 2 + R 2 � � 1 + 2 log( R )

Implicit surface modelling Spline regularization as a Gaussian process GPIS for 2D curves GPIS for 3D surfaces Summary Demonstration ◮ Set constraint points ◮ 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −0.5 0 0.5 X : {⊙ = 0 , � = +1 , � = − 1 }