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TEMPLATE BASED SHAPE DESCRIPTORRaif RustamovDepartment of Mathematics and Computer ScienceDrew 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 The new domain called the “mapping domain” the new function the “mapped feature function”
Common mapping domains: 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 Distance to the origin
Mapping 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 gray-scale images
Example II: Depth Buffer
For each cube face: Surface feature
distance from mesh point to the face Mapping
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:
≠ original surface a primitive geometry: sphere, plane etc
Mapping procedure: Projection
Signal processing well established: Fourier, Zernike, Spherical
Harmonics limits possible mapping domains
Contributions
Mapping: Mapping domain:
any fixed surface – template Mapping procedure:
interpolation: mean-value coordinates, Shepard Signal processing
via manifold harmonics – eigenfunctions of Laplace-Beltrami 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 originJump 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 feature function is smooth Less 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?
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