Surface Reconstruction Using RBF Reporter : Lincong Fang

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

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

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 ReconstructionIso-surfaceReconstruction

Introduction to RBF Interpolation problem

Introduction to RBF 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 , Berlin, Springer-Verlag.

Introduction to RBF An RBF is a weighted sum of translations of a radially symmetric basic function augmented by a polynomial term

Introduction to RBF Popular choices for include For fitting functions of three variables, good choices include

Introduction to RBF Matrix form

Introduction to RBF Matrix form

Reconstruction and representation of 3D objects with Radial Basis functions –J.C.Carr 1,2, R.K.Beatson 2, J.B.Cherrie 1, T.J.Mitchell 1,2 –W.R.Fright 1, B.C.McCallum 1, T.R.Evans 1 –1. Applied Research Associates NZ Ltd –2. University of Canterbury, New Zealand –Sig 2001

Off-surface Points

RBF Center Reduction

Greedy Algorithm Choose a subset from the interpolation nodes x i and fit an RBF only to these. Evaluate the residual, e i = f i – f(x i ), at all nodes. If max{e i } < fitting accuracy then stop. Else append new centers where e i is large. Re-fit RBF and goto 2

points, centers, accuracy of 5*10 -4

Noisy

Hole Filled & Non-uniformly

Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions –Bryan S. Morse 1, Terry S. Yoo 2, Penny Rheingans 3, –David T.Chen 2, K.R. Subramanian 4 –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 H. Wendland. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. AICM, 4: , 1995

Matrix Form

Choice of Support Size

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

Exterior 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

Single level Interpolation 35K points 6 seconds

Multi-level Interpolation

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 approximationLocal details

Adaptive PUNormalized RBF

Selection of Centers

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, p

Basic Function

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

Noisy

Summary

Thank you !!!