Surface Reconstruction Using RBF

Reporter : Lincong Fang

11.07.2007

Surface Reconstruction

sample

Reconstruction

Surface Reconstruction

• Delaunay/Voronoi– Alpha shape/Conformal alpha shape

– Crust/Power crust

– Cocone

– Etc.

• Implicit surfaces– Signed distance function

– Radial basis function(RBF)

– Poisson

– Fourier

– MPU

– Etc.

Implicit Surface

• Defined by implicit function– Such as

• Many topics within broad area of implicit surfaces

3( ) 1 | |, f X X X R

Implicit Surface

• Mesh independent representation - generate the desired mesh when you require it

• Compact representation to within any desired precision

• A solid model is guaranteed to produce manifold (manufacturable) surface

( ) 0f x

( ) 0f x

( ) 0f x

Implicit surface

• Tangent planes and normals can be determined analytically from the gradient of the implicit function

Implicit surface

• CSG operation

Implicit surface

• Morphing

Implicit surface reconstruction

( , , ) 0f x y z

Introduction to RBF

• Interpolation problem

31

31

Given n distinct points X={ } and a set of function

values { } , find an interpolant : such that

( ) , 1,..., .

ni i

ni i

i i

x R

f R f R R

f x f i n

Introduction to RBF

1

The smoothest interpolant has the simple form

( ) ( ) | |n

i ii

f x p x x x

J.Duchon. Splines minimizing rotation-invariant semi-norms in Sololev spaces. In W. Schempp and K.Zeller, editors, Constructive Theory of Functions of Several Variables, number 571 in Lecture Notes in Mathematics, pages 85-100, Berlin, 1977. Springer-Verlag.

3

2 2 22 2 2

2 2 2

2 2 22 2 2

( ) ( ) ( ) ( )

2( ) 2( ) 2( )

R

f f fE f

x y z

f f f

x y x z y z

Introduction to RBF

1

In general, an RBF is a function of the form

( ) ( ) (| |)n

i ii

f x p x x x

Basic function : R R 2( ) logr r r

2 2

( ) rr e

Introduction to RBF

• Popular choices for  include

• For fitting functions of three variables, good choices include

2( ) log( ) (for fitting smooth functions of two variables)r r r 2( ) exp( ) (mainly for neurall networks)r cr

2 2( ) (for various applications)r r c

The biharmonic ( ( ) ) linear polynomialr r 3The triharmonic ( ( ) ) quadratic polynomialr r

Introduction to RBF

• Matrix form

1

( ) ( ) (| |)n

i ii

f x p x x x

1 1 1 1

0n n n n

i i i i i i ii i i i

x y z

0 0T

A P fB

P c c

,

,

(| |), , 1,...,

( ), 1,..., , 1,..., .

i j i j

i j j i

A x x i j n

P p x i n j l

1 2 3 4( )p x c x c y c z c

( ) 0 1,...,jf x j n

Introduction to RBF

111 12 1 1 1 1 1

221 22 2 2 2 2 2

1 2

11 2

21 2

31 2

4

1

1

1

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1 1 1 1 0 0 0 0 0

n

n

nn n nn n n n n

n

n

n

x y z f

x y z f

x y z f

cx x x

cy y y

cz z z

c

• Matrix form

1

( ) ( ) (| |)n

i ii

f x p x x x

• Reconstruction and representation of 3D objects with Radial Basis functions– J.C.Carr1,2, R.K.Beatson2, J.B.Cherrie1, T.J.Mitchell1,2

– W.R.Fright1, B.C.McCallum1, T.R.Evans1

– 1. Applied Research Associates NZ Ltd

– 2. University of Canterbury, New Zealand

– Sig 2001

Off-surface Points

( , , ) 0, 1,...,i i if x y z i n

( , , ) 0, 1,...,i i i if x y z d i n N

RBF Center Reduction

Greedy Algorithm

• Choose a subset from the interpolation nodes xi and fit an RBF only to these.

• Evaluate the residual, ei = fi – f(xi), at all nodes.

• If max{ei } < fitting accuracy then stop.

• Else append new centers where ei is large.

• Re-fit RBF and goto 2

• 544000 points, 80000 centers, accuracy of 5*10-4

Noisy

2 2

1

1min || || ( ( ) )

n

ii

f f x fN

8

0 0T

A n I P f

P c

Hole Filled & Non-uniformly

• Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions– Bryan S. Morse1, Terry S. Yoo2, Penny Rheingans3,

– David T.Chen2, K.R. Subramanian4

– 1. Department of CS, Brigham Young University

– 2. National Library of Medicine

– 3. Department of CS and EE, University of Maryland Baltimore County

– 4. Department of CS, University of North Carolina at Charlotte

– Proceeding of the International Conference on Shape Modeling and Applications 2001

Compactly-supported RBF

(1 ) ( ) if 1( )

0 otherwise

pr P r rr

H. Wendland. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. AICM, 4:389-396, 1995

4( ) (1 ) (4 1)r r r

2 2

( ) rr e

Matrix Form

111 12 1 1 1 1 1

221 22 2 2 2 2 2

1 2

11 2

21 2

31 2

4

1

1

1

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1 1 1 1 0 0 0 0 0

n

n

nn n nn n n n n

n

n

n

x y z f

x y z f

x y z f

cx x x

cy y y

cz z z

c

Choice of Support Size

Comparison

Comparison

• Compactly supported basic functions is much more efficient.

• Non-compactly supported basic functions are better suited to extrapolation and interpolation of irregular, non-uniformly sampled data.

• Modeling with implicit surfaces that interpolate– Greg Turk

• GVU Center, College of Computing

• Georgia Institute of Technology– James F.O’Brien

• EECS, Computer Science Division

• University of California, Berkeley

Modeling

Interior Constraints

Matrix Form

111 12 1 1 1 1 1

221 22 2 2 2 2 2

1 2

11 2

21 2

31 2

4

1

1

1

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1 1 1 1 0 0 0 0 0

n

n

nn n nn n n n n

n

n

n

x y z f

x y z f

x y z f

cx x x

cy y y

cz z z

c

Exterior Constraints

Normal Constraints

Normal Constraints

Example

Polygonal surface The interpolating implicit surface defined by the 800 vertices and their normals

• A Multi-scale Approach to 3D Scattered Data Interpolation with Compactly Supported Basis Functions– Yutaka Ohtake, Alexandaer Belyaev, Hans-Peter Seidel

– Computer Graphics Group, Max-Planck-Institute for informatics

– Germany

– Proceedings of the Shape Modeling International 2003

Construct RBF

( ) [ ( ) ] || ||i

i i ic C

f x g x x c

4( ) (1 ) (4 1)r r r

( ) ( / )r r

( ) 0 ( )i i

j i ij i j ijc C c C

f p g p

(|| ||)ij i jp p

2 2( , ) 2w h u v Au Buv Cv

( ) ( , )ig x w h u v

Single level Interpolation

• 35K points

• 6 seconds

0.02

Multi-level Interpolation

Multi-level Interpolation

1 2{ , ,..., }MP P P P 0 ( ) 1f x

1( ) ( ) ( )k k kf x f x o x

( ) [ ( ) ] (|| ||)k

k ki

k k k ki i i

p P

o x g x x p

1( ) ( ) 0k k k ki if p o p

1 / 2k k 1 cL

Coarse to Fine

• 3D Scattered Data Approximation with Adaptive Compactly Supported Radial Basis Functions– Yutaka Ohtake, Alexandaer Belyaev, Hans-Peter Seidel

– Computer Graphics Group, Max-Planck-Institute for informatics

– Germany

– Proceedings of the Shape Modeling International 2004

Construct RBF

Base approximation Local details

( ) [ ( ) ] || ||

( ) || || || ||

i

i

i i

i i

i i ic C

i i i ic C c C

f x g x x c

g x x c x c

2 2( , ) 2w h u v Au Buv Cv Du Ev F

( ) ( , )ig x w h u v

Adaptive PU Normalized RBF

( ) || || || || 0i i

i i

i i i ic C c C

g x x c x c

|| ( ) ||(|| ||)

|| ( ) ||i

i

ii

j jj

x cx c

x c

( ) 0f x

Selection of Centers

100 500 1000 2000

1

(|| ||)i

M

j j ii

v p c

Example

Compare With Multi-scale

• Reconstructing Surfaces Using Anisotropic Basis Functions– Huong quynh Dinh, Greg Turk

• Georgia Institute of Technology

• College of Computing– Greg Slabaugh

• Georgia Institute of Technology

• Scholl of Electrical and Computer Engineering

• Center for Signal and Image Processing– Computer Vision, Vol 2, 2001, p606-613.

Basic Function

( ) (| ( ) |)B x M x c : scaling functionM

Direction of Anisotropy

• Covariance matrix

– Corner point : all three eigenvalues are nearly equal

– Edge point : one strong eigenvalue

– Plane point : two eigenvalues are nearly equal and larger than the third

1

( ) ( )( )n

Ti i i i

i

C p v c v c

Noisy

1

1

( )'( )

n

ij jji n

ijj

C pC p

2| |

2i jp p

rij e

Summary

Summary

Thank you !!!