Graphics Radial Basis Function

8/13/2019 Graphics Radial Basis Function

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STNY BRKSTATE UNIVERSITY OF NEW YORK

Department of Computer Science

Center for Visual Computing

CSE528 Lectures

Radial Basis Functions for

Computer Graphics (A BriefIntroduction)


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Contents

1. Introduction to radial basis functions

2. Mathematics

3. How to fit a 3D surface4. Applications


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Radial Basis Functions andTheir Applications in Graphics


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Scattered Data Modeling


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Smooth Surfaces


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Scattered Data Interpolation Radial Basis Functions (RBFs) are a powerful solution to the

Problem of Scattered Data Fitting

N point samples are given as data inputs, we want to interpolate,extrapolate, and/or approximate

This problem occurs in many areas:

Mesh repair and model completion

Surface reconstruction

Range scanning, geographic surveys, medical data

Field visualization (2D and 3D) Image warping, morphing, registration

Artificial intelligence

Etc.


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2D Radial Basis Functions

Implicit Curve

Parametric Height Field


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A Very Brief History Discovered by Duchon in 1977

Applications to Computer Graphics:

Savchenko, Pasko, Okunev, Kunii1995 Basic RBF, complicated topology bits

Turk & OBrien 1999

Variational implicit surfaces

Interactive modeling, shape transformation

Carr et al

1997Medical Imaging 2001Fast Reconstruction


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3D Radial Basis Functions

Implicit Surface

Scalar Field


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Extrapolation (Hole-Filling Capability)

Mesh repair and model completion

Fit surface to vertices of mesh

RBF will fill holes

if it minimizes curvature !!


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Smoothing

Smooth out noisy range scan data

Repair the rough segmentation


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Now a bit of math

(dont panic)


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The Scattered Data Fitting Problem We wish to reconstruct a function S(x), given N samples (xi, fi),

such that S(xi)=fi xiare thepoints from measurement

Reconstructed function is denoted S(x)

Infinite number of solutions We have specific constraints:

S(x) should be continuous over the entire domain

We want a smooth surface


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The Generic Form of RBF Solution

)()(1

xxxx PsN

i

ii

normEuclideantheis

componentpolynomialdegree-lowais)(theis(r)

centeroftheis

x

x

x

Pfunctionbasic

weight ii


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Terminology: Support Support is the footprint

of the function

Two types of support matters most:

CompactorFinitesupport:

function value is zero outside

of a certain interval

Non-CompactorInfinitesupport:

not compact (no interval, goes all the way to )

3xy

2xey


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Basic Functions ( )

Essentially, they can be of any functional type However, very difficult to define properties of the RBFs for

an arbitrary basic function

Support of function has major implications A non-compactly supported basic function implies aglobal

solution, dependent on allpoints!

Allows extrapolation (hole-filling)


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Standard (Commonly-Used) BasicFunctions

Polyharmonics (Cncontinuity)

2D:

3D:

Multiquadric:

Gaussian: compact support, used in artificial intelligence field

)log()( 2 rrr n

12)( nrr

22)( crr

2

)( crer


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Polyharmonics 2D Biharmonic:

Thin-Plate Spline

3D Biharmonic: C1continuity, Polynomial is degree 1

Node Restriction: nodes not colinear

3D Triharmonic: C2continuity, Polynomial is degree 2

Important Bit: Can provide Cncontinuity

)log()( 2 rrr

3)( rr

rr )(


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Guaranteeing Smoothness RBFs are members of , theBeppo-Levi

space of distributions onR3withsquare integrablesecond derivatives

has a rotation-invariant semi-norm:

Semi-norm is a measure of energy of s(x)

Functions with smaller semi-norm are smoother

Smoothest function is the RBF (Duchon proved this)

xdsssssss yzxzxyzzyyxx222222 222

)(BL 3)2( R

)(BL 3)2( R


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What about P(x ) ? P(x) ensures minimization of the curvature

3D Biharmonic: P(x) = a + bx + cy+ dz

Must solve for coefficients a,b,c,d Adds 4 equations and 4 variables to the linear system

Additional solution constraints:

0111 1

i

N

i

ii

N

i

i

N

i

i

N

i

ii zyx


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Finding an RBF Solution The weights and polynomial coefficients are unknowns

We know N values of s(x):

We also have 4 side conditions

)()(0

j

N

i

ijij Ps xxxx

jjjNjNjj dzcybxaf xxxx 11

0111 1

i

N

i

ii

N

i

i

N

i

i

N

i

ii zyx


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The Linear System Ax = b

0

0

0

0

0000

0000

0000

000011

1

1 11

1

1

1

1

111111

NN

N

N

N

NNNNNN

N

f

f

d

c

b

a

zz

yy

xx

zyx

zyx

ij

jii

fxP

)(

0 i

0 iix

0 iiy

0 iiz

ijji xx where


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Properties of the Matrix Depends heavily on the basic function

Polyharmonics:

Diagonal elements are zeronot diagonally dominant Matrix issymmetricandpositive semi-definite

Ill-conditioned if there are near-coincident centers

Compactly-supported basic functions have a sparse matrix Introduce surface artifacts

Can be numerically unstable


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Analytic Gradients Easy to calculate

Continuous depending on basic function

Partial derivatives for biharmonic gradient canbe calculated in parallel:

b

xxc

x

s N

i

ii

1 ixx

c

yyc

y

s N

i

ii

1 ixx

d

zzc

z

s N

i

ii

1 ixx


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Fitting 3D RBF Surfaces

(its tricky)


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Basic Procedure1. Acquire N surface points

2. Assign them all the value 0(This will be the iso-value for the surface)

3. Solve the system, polygonize, and render:


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Off-Surface Points Why did we get a blank screen?

Matrix was Ax = 0

Trivial solution is s(x) = 0

We need to constrain the system

Solution: Off-Surface Points

Points inside and outside of surface

Project new centers alongpoint normals

Assign values: 0 outside

Projection distance has a large effect on smoothness


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Invalid Off-Surface Points

Have to make sure that

off-surface points stay

inside/outside surface!

Nearest-Neighbor test


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Point Normals?

Easy to get from polygonal meshes

Difficult to get from anything else

Can guess normal by fitting a plane to local

neighborhood of pointsNeed outward-pointing vector to determine orientation

Range scanner position, black pixels

For ambiguous cases, dont generate off-surface point


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Computational Complexity How long will it take to fit 1,000,000 centers?

Forever (more or less)

3.6 TB of memory to hold matrix

O(N3) to solve the matrix

O(N) to evaluate a point Infeasible for more than a few thousandcenters

Fast Multipole Methods make it feasible

O(N) storage, O(NlogN) fitting and O(1) evaluation Mathematically complex


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Center Reduction

Remove redundant centers

Greedy algorithm

Buddha Statue: 543,652 surface points

80,518 centers

5 x 10-4accuracy


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FastRBF FarFieldTechnology (.com)

Commercial implementation

3D biharmonic fitter with Fast Multipole Methods

Adaptive Polygonizer that generates optimized triangles

Grid and Point-Set evaluation

Expensive They have a free demo limited to 30k centers

Use iterative reduction to fit surfaces with more points


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Applications

(and eye candy)


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Cranioplasty (Carr 97)


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Molded Cranial Implant


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Morphing

Turk99 (SIGGRAPH)

4D Interpolation between two surfaces


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Morphing With Influence Shapes


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Statue of Liberty

3,360,300 data points

402,118 centers

0.1m accuracy


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Credits

Pictures copied from:

Papers by J.C. Carr and Greg Turk

FastRBF.com

References:


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Graphics Radial Basis Function

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