Transcript Laplace-Beltrami Eigenfunctions for Deformation Invariant

TEMPLATE BASED
SHAPE DESCRIPTOR
Raif Rustamov
Department of Mathematics and
Computer Science
Drew University, Madison, NJ, USA
Components of descriptors in general



Selection of surface feature
Mapping
Signal Processing

Need this discussion to set up the context for our
approach
Selection of surface feature

A function on the surface that captures a property
relevant to shape description:





constant function (restriction of the surface's
characteristic function to the surface itself)
distance to the center of mass
curvature
components of the normal vector
We refer to the selected function as the feature
function.
Mapping

The feature function is used to construct a new function
defined on some predetermined domain



Common mapping domains:





The new domain called the “mapping domain”
the new function the “mapped feature function”
Spheres
Planes
the 3D space (surface's bounding volume)
surface itself
Mapping procedures:


projection
Identity, if mapping domain = surface itself
Signal Processing


Extract concise noise-robust numerical descriptor
from the mapped feature function.
Depends on the mapping domain:





Sphere – Spherical Harmonic Transform
Plane or box volume – 2D or 3D Fourier transform
Ball volume – 3D Zernike Transform.
Mapped feature function is expanded in a series in
terms of the relevant basis
Expansion coefficients are used as the shape
descriptor
Example I: Saupe, Vranic 2001



Shoot rays from the origin (center of mass),
determine the distance to the farthest intersection
point with the bounding mesh
Parameterize the rays by the unit sphere to obtain
a function on the sphere
Use spherical harmonic transform on this function to
extract the numerical shape descriptor.
Example I: Saupe, Vranic 2001

Surface feature


Mapping



Distance to the origin
Mapping domain: the unit sphere
Mapping procedure: project onto the sphere, resolve
collisions by selecting the larger function value
Signal processing

Spherical harmonic transform
Example II: Depth Buffer



Heczko, Keim, Saupe, Vranic 2002
Place a normalized mesh into a unit cube
Generate six gray-scale images on each face of
the cube by parallel projection


The grayness value is the distance from the cube face to
the model
Apply 2D Fourier transform to each of the six grayscale images
Example II: Depth Buffer


For each cube face:
Surface feature


Mapping



distance from mesh point to the face
Mapping domain: cube face
Mapping procedure: project onto the face, resolve
collisions by selecting the smaller function value
Signal processing

2D Fourier transform
Generality

More examples easily generated



Compare to classification in Bustos et al. survey
Mapping ≈ object abstraction
Signal processing ≈ numerical transformation
Observations

Mapping:

Mapping domain:



Mapping procedure:


≠ original surface
a primitive geometry: sphere, plane etc
Projection
Signal processing


well established: Fourier, Zernike, Spherical Harmonics
limits possible mapping domains
Contributions

Mapping:

Mapping domain:


Mapping procedure:


any fixed surface – template
interpolation: mean-value coordinates, Shepard
Signal processing

via manifold harmonics – eigenfunctions of LaplaceBeltrami operator
Why templates?

Mademlis, Daras, Tzovaras, Strintzis 2008:
 Since
ellipses approximate elongated shapes better
than spheres:
Mapping domain: ellipsoid
 Signal processing: ellipsoidal harmonics

 Showed
experimentally better retrieval results than
sphere + spherical harmonics

We take this idea further:
 Mapping
domain: any fixed surface
Why template ≠surface itself?


Expand the feature function in terms of the manifold
harmonics of the original surface?
Problem: notoriously difficult to match the harmonics
coming from different surfaces
 Sign
flipping
 Eigenfunction switching
 Linear combinations

Fixed template: extracted expansion coefficients
are in direct correspondence
Why interpolation?

Projection
 Mapped feature function can be discontinuous at
overlaps
 Gibbs effect may render low-frequency expansion
coefficients used as the shape descriptor
inadequate for representing the function
Feature function is distance to the origin
Jump discontinuity
Why interpolation?

Projection
 Redundancy
 the
value sets of the mapped feature functions on various
templates will be almost the same
 limits the gains of concatenating descriptors obtained from
different templates
Why interpolation?

Interpolation
 No
Gibbs effect
 mapped
 Less
feature function is smooth
redundancy
 the
value sets of the mapped feature functions on various
templates depend on relative positions
 mean-value coordinates can inject more shape information
into the mapped feature function

a mesh can be reconstructed given the mean-value coordinates
Construction of the descriptor



Selection of surface feature
Mapping
Signal Processing
 Now
discuss details
Selection of surface feature


All models are normalized using shift, continuous
PCA, isotropic scaling
Many possibilities, but not:
 the
characteristic function
 nor linear function of coordinates

To focus discussion
f
= distance from a mesh point to the origin
 Similar to Saupe, Vranic 2001
Mapping





Model surface S, Template surface T
Given
Construct
Shepard interpolation
Mean-value interpolation
Mapping: Shepard

Model surface S, Template surface T
Given
Construct

0th order precision: constant functions reproduced


Mapping: Barycentric





Model surface S, Template surface T
Given
Construct
are barycentric coordinates of point p with
respect to vertex
1st order precision: linear functions reproduced
Mapping: Barycentric

A few different kinds of barycentric coordinates
 Mean-value,
positive mean-value
 Harmonic
 Maximum
Entropy
 Green coordinates, Complex in 2D

We use mean-value coordinates
 Closed
formula
 Fastest to evaluate
Signal processing


We have a function
Need a compact representation
 Expand
the function into series
 Use low-frequency coefficients


Need a function basis on template surface T
Manifold harmonics = Laplace-Beltrami
eigenfunctions
Signal Processing



Manifold harmonics generalize Fourier basis to
Riemannian manifolds
Spherical harmonics = manifold harmonics on the
sphere
Have similar properties
 Orthogonal
 Concept
of frequency
 low-frequency
coefficients are noise-robust
 convey essential information about function
Signal Processing

Egenvalues, eigenfunctions solve
 Evaluation
procedures well known
 Solve symmetric eigenvalue problem for a matrix
 We use cotangent Laplacian with voronoi point-areas
 Pre-compute for the given templates and store
 Our templates have about 500 vertices, the process
takes less than 3 seconds
 The storage for each template = 10,500 floats=
=20*500+500 = #eigs * #vertices + #vertices
to store eigenvectors
to store point areas
Resulting shape descriptor





Feature function
Mapped feature function
The template’s feature function
Quotient function
Expand into series
Resulting shape descriptor

Feature vector, N=20

For template surface T,


Normalization: scale to get a = 1
Use L2 distance
Experiments: Benchmark

Models: watertight benchmark
 400
closed surface models
 20 equal object classes
Experiments: Implementation



Implemented in MATLAB
Use C++ for mean-value coordinate computation
Timing
 About
1 minute per model when mean-value
coordinates used
 Could make faster if simplified the models
Experiments: Templates

Templates: randomly chosen models
 Simplified
using Qslim
 Makes mean-value computation faster
Compare Mapping Methods


Template = sphere
Projection vs. Shepard (a=1,2,3) vs. Mean-value
 At
long distance behavior of mean-value interpolant is
similar to that of Shepard with a=2
Compare templates

Mapping via mean-value interpolation

M
Compare templates



Beneficial to combine – relative independence
All templates are normalized as objects in the
benchmark – span similar spatial regions
The descriptors could have been made even more
independent if the templates were differently
posed.
Future work





Investigate dependency between the nature of the
template and the produced retrieval results
No “ideal" template for all kinds of shapes
Flexibility of our approach – choose optimal
templates based on the shape database at hand
How to choose?
Can we design a rotationally invariant descriptor?