基于有向点集的三维重建PPT

Algorithm Implementation
• Efficiency:
We show that the computation of the Fourier coefficients is actually a convolution, reducing the reconstruction complexity: O(R5) → O(R3logR).
1 V (x , y , z ) 0 if (x , y , z ) V otherw ise
V
V 1
V 0
Reduction to Volume Integration (Step 2)
Fourier Coefficients: The Fourier coefficients of the characteristic function give an expression of V as a sum of complex exponentials:
• Approach • Results
Resolution Sample Count Non-Uniformity Related Work
• Conclusion
Results (Resolution)
100,000 Points
100,000 Points
100,000 Points
2. Surface Fitting
Terzopoulos et al., 1991 Chen et al., 1995
3. Implicit Function Fitting
Hoppe et al., 1992 Whitaker, 1998 Davis et al., 2002 Turk et al., 2004 Curless et al., 1996 Carr et al., 2001 Ohtake et al., 2004 Shen et al., 2004
ˆV (l , m , n ) Flmn (p i ),ni
i 1 k

• •
Apply the inverse Fourier Transform to obtain the characteristic function. Extract the reconstruction an iso-surface of the characteristic function

Equivalence of Representations
There are many ways to represent a shape:
• • • • Point Set Polygon Soup Polygonal Mesh Solid Model
In one direction, the transition between representations is straight-forward The challenge is to transition in the other direction
Related Work
3. Implicit Function Fitting
Use the point samples to define an function whose values at the sample positions are zero.
<0
0
Sample Points
F(x,y)
•
Non-Uniformity:
We provide a simple heuristic for assigning weights to samples that may be non-uniformly distributed.
Outline
• Introduction
• Related Work
ˆV (l , m , n )
[ 0 ,1]3 2i lx my nz ( x , y , z ) e dxdydz V
e 2i lx my nz dxdydz
V
since the characteristic function is one inside of V and zero everywhere else.
Original Model 871,000 Triangles
Simplified Model 95,000 Triangles
Related Work
Three general approaches:
1. Computational Geometry
Boissonnat, 1984 Amenta et al., 1998 Edelsbrunner, 1984 Dey et al., 2003
•
•
Related Work
Approach
The Divergence Theorem Reduction to Volume Integration Implementation
• •
Results Conclusion
Divergence Theorem
Given a vector field F and a region V: The volume integral of F over V and the surface integral of F over V are equal: F (p ) dp F (p ),n (p ) dp
• Surface Blending
• Hole-Filling
• Compression
Geometry + Topology Representation
Geometry Representation
Applications
• Surface Blending
• Hole-Filling
• Compression • Simplification
>0
Related Work
3. Implicit Function Fitting
Use the point samples to define an function whose values at the sample positions are zero. Extract the iso-surface with iso-value equal to zero.
Reconstruction of Solid Models from Oriented Point Sets
Misha Kazhdan
Johns Hopkins University
Shape Spectrum
There are many ways to represent a shape:
• • • • Point Set Polygon Soup Polygonal Mesh Solid Model

?
Equivalence of Representations
There are many ways to represent a shape:
• • • • Point Set Polygon Soup Polygonal Mesh Solid Model
The goal of this work is to define a method for computing solid models from oriented point sets.

Applications
• Surface Blending
Disjoint Model
“Zippered” Model
Applications
• ole-Filling
Model with Hole
Water-Tight Model
Applications
Applying the Divergence Theorem
Surface Integration If Flmn(x,y,z) is any function whose divergence is equal to the (l,m,n)-th complex exponential:
F (x , y , z ) e

lmn
2i lx my nz

applying the Divergence Theorem, the volume integral can be expressed as a surface integral:
e V
2i lx my nz

dxdydz

V
F lmn ( p ), n ( p ) dp
V (x , y , z )
l ,m ,n i (lx my nz ) ˆ ( l , m , n ) e V
Reduction to Volume Integration (Step 3)
Volume Integration: The Fourier coefficients of the characteristic function V can be obtained by integrating:

Reconstruction Algorithm
Given an oriented point sample {(pi,ni)}: • Compute a Monte-Carlo approximation of the Fourier coefficients of the characteristic function:
Equivalence of Representations
There are many ways to represent a shape: