SLIDE 1
Fractal 3D modeling of asteroids using wavelets on arbitrary meshes
Andre Jalobeanu
Automated Learning Group NASA Ames Research Center Moffett Field CA - USA
IAFA 2003 , Bucharest - Romania
SLIDE 2 Summary
- Fractal appearance of asteroids
- What tool to use to study them?
- Wavelets on meshes with arbitrary topology
– Topology, geometry and regularity – Subdivision meshes & wavelet transform
- Local scale / local direction
- Wavelet transform of 433 Eros – scale invariance
- A new multiscale model for fractal surfaces
- Applications
SLIDE 3
Fractal appearance of asteroids
SLIDE 4 What tool to use?
Check statistical self-similarity: first compute the power spectrum of the object.
- Quasi-planar surfaces: Fast Fourier Transform
- Spherical surfaces: spherical harmonics
- Arbitrary surfaces (highly irregular sampling): ?
Wavelets on subdivided meshes spectrum: feature size / scale
SLIDE 5
Topology, geometry and regularity Topological support Set of sites (vertices) + neighborhood system
Regularity = neighborhood regularity Semi-regular triangular mesh: 5 or 6 neighbors
Geometry 3D point associated to each site
The object can have an irregular geometry, but we define the wavelets on the semi-regular topological support.
geometry topology wavelets topology
SLIDE 6
Subdivision meshes
uniform subdivision Original mesh Subdivided mesh, level 1 …
A new topological vertex is created on each edge Each triangle is replaced by 4 smaller triangles ideal framework to define a multiresolution approach
SLIDE 7 Vertex prediction
Geometric subdivision : vertex creation by prediction m
Prediction of a vertex at level j+1 using 8 neighbor vertices at level j : interpolation scheme
[Sweldens-Schroeder]
SLIDE 8 Wavelets on subdivided meshes
Wavelet coefficients at level j+1 = vertices at level j+1 – prediction from level j represent the details at level j+1 Vertices even
approximation
prediction
– details
SLIDE 9
Multiresolution analysis of 433 Eros
Approximations : a1 … an Coarse approximations of the 3D surface (fine to coarse) Details : d1 … dn differences between two successive approximations
SLIDE 10 Local scale
1
local deformation topological support geometry
Details = absolute geometric variations (regardless of the local geometry) define a local scale
SLIDE 11
Local direction
Wavelet details are 3D vectors
actual vertex at level j+1
normal + parallel decomposition m
geometric detail sampling irregularity
SLIDE 12 Statistical self-similarity
Amplitude spectrum of 433 Eros (NEAR laser altimetry data)
log σ (mean amplitude
detail coefficients)
(local scale)
σ(r) = σ0 r -q
fractal exponent q=1.12 fractal dimension D=2.38
SLIDE 13 A new multiscale model
- Adaptive scale-invariant Gaussian model on :
acts as a smoothness prior (Bayesian inference)
geometric details = Gaussian random variables spatially adaptive parameters local scales
3D analog to the 2D fractional Brownian motion (using Fourier coefficients instead of wavelet coefficients) that efficiently describes the power spectrum of natural images
- Statistical model of w// : sampling regularity prior
SLIDE 14 Applications
- 3D object reconstruction from multiple images:
– Asteroids (~uniform albedo, spherical topology) – Planetary surfaces (unknown albedo, planar/spherical topo.)
- Simultaneous localization and mapping
- Spacecraft localization + recursive reconstruction of
the object surface (approach or flyby)
- Multi-sensor data fusion (optical, radar, altimetry)
- Fractal geometry and synthetic images:
– Generate synthetic photo-realistic surfaces – Compute reflectance functions for natural objects
SLIDE 15
General 3D surface reconstruction
Images (optical, radar, altimetry, etc.) Albedos Geometry Cameras Light sources
SLIDE 16 Asteroid surface reconstruction
- Many images available (various orientations, distances)
- The direction of the light source (Sun) is known
- ~ known camera parameters
- Albedo ~ constant
Infer the asteroid surface from N calibrated images
SLIDE 17 Conclusion
- The proposed wavelet transform has a linear
complexity, is critically sampled and works for meshes
- f arbitrary topology
- Coefficient decomposition and local scale estimate
- Statistical self-similarity of 433 Eros checked by
computing the amplitude spectrum of the geometry
- New model for natural surfaces, many applications
- Extension: support = irregular geometric mesh,
- data = scalar potential
⇒ study big spherical objects such as planets
