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### 3D Object Retrieval

### Topology MatchingUsing MRGs

Presented by

Katz Sagi Leifman George

Based on:

”Topology Matching for Fully Automatic Similarity Estimation of 3D Shapes”

M. Hilaga, Y. Shinagawa, T. Kohmura, and TL Kunii,,SIGGRAPH 2001, pp. 203-212

“Matching 3D Models with Shape Distributions”

R.Osada, T.Funkhouser, B.Chazelle, D.Dobkin

3D Objects Retrieval – Why?

- Improved modeling tools
- Improved scanning devices
- Fast and cheap CPUs, Gfx HW
- Large databases
- E-commerce
- Medicine
- Entertainment
- Molecular biology
- Manufacturing

Object Boundaries

Features Occlusion

Camera dependent

Noise

Simple Contour representation

3D

Problematic Surface Representation

Ambiguous Triangulation

No Features Occlusion, shadows, noise

3D vs. 2D RetrievalCommon Approach

- Preprocess Stage:
- Objects Normalization (optional)
- Signature for each object
- Compact
- Capture the object properties
- Comparable

- Signature Comparison:
- Coarse-to-Fine (optional)
- Fast

DB

Object

signatures

Query

Queryprocess

Signatures

Comparison

Similarobjects

Object

Signature Properties

- What do we want from good signature?
- Robustness to resampling and simplification
- Translation, Orientation, Scale Invariance
- Possible Solutions
- Preprocess: Object Normalization
- Automatically embedded in the key by definition

- Possible Solutions

=

=

Object Normalization using Moments

Translation: (m100, m010, m001) – center of mass

Rotation, Scale:

∆(1,1) - main axis scale

U – Rotation Matrix

Different Methods

- Octrees
- Probability Shape Distributions
- Distances to Enclosing Sphere
- Reeb Graphs

Octrees

Signature:

- Each object is represented by Octree
- White, black, gray (gray level)
Signature Comparison:

- Coarse-to-Fine search (tree depth)

Probability Shape Distributions

Several types of signatures:

- A3: angle between three random surface points
- D1: distance from fixed point to random point
- D2: distance between two random surface points

X-axis: D2 distance

Y-axis: Probability of that distance

Signature Comparison: L1,L2,L∞for PDF and CDF

Distances to Enclosing Sphere

- Signature:
- Sphere is evenly sampled
- For each sphere sample min. distance to object calculated

Signature Comparison: L1,L2,L∞

The Idea

- The signature:
- Multiresolutional Reeb Graphs (MRGs)
- Represents the skeletal and topological structure of a 3D shape at various levels of resolution
- Constructed using a continuous function on the 3D shape.

- Correspondence between the parts of objects.
- Invariant to transformations and non-rigid deformations

- Multiresolutional Reeb Graphs (MRGs)
- The search:
- coarse to fine

How to “Reeb” an Object

- We’ll create a simple reeb graph using height function
- μ - height of the point V: μ(V(x,y,z))=z

MultiResolutional Reeb Graph(MRG)

- A series of Reeb graphs at various levels of detail

The Construction of the MRG

- Define the following notation:
- R-node: A node in an MRG.
- R-edge: An edge connecting R-nodes in an MRG.
- T-set: A connected component of triangles in a region
- µn -range: A range of the function µn concerning an R-node or a T-set.

The Construction of the MRG cont.

- The domain of µn is divided onto K µn-ranges:
- R0=[0,1/K),R1=[1/k,2/k)….Rk-1=[(K-1)/K,1)
- Note: The example uses the height function for the convenience of explanation

The Construction of the MRG cont.

- Subdivision
- Interpolate the position of two relevant vertices in the same proportion as their value of µn(v)

The Construction of the MRG cont.

- Calculate T-sets
- Connect R-nodes

The Construction of the MRG cont.

- Construct MRG
- fine-to-coarse (reverse)

Defining µ for Topology Matching

- Height function is not appropriate
- not invariant to transformations.

- Use a geodesic distance
- Not invariant to scale:
- Normalize [0,1]:

- Not invariant to scale:

Examples of the Distribution of the Function µ

- More asymmetric shapes have a wider range for µn(v)
- Sphere
- constant value of µn(v)=0

- Sphere

Matching

- Assign 2 attributes for each node (m) in the finest resolution
- Area
- Length

Matching cont.

- Define ‘+’:
- At coarse resolution
- Similarity (0<=w<=1)
- To satisfy:
- Define

“Topology Matching” Added Value

- Topology matching
can be used to find

correspondence

between meshes

- Problem:
The algorithm does

Not distinguish between

Left and right

Results

- 230 mesh objects

Future Work

- Use additional information
- Texture,color,curvature etc.

- Euclidean distance as the R-node attribute
- Use different µ functions
- Density for volumetric data

Appendix- MRG Construction

- When calculating the integral of geodesic distance the computational cost is high
- We employ a relatively simple method in which geodesic distance is approximated by Dijkstra’s algorithm based on edge length.
- We need to prepare the mesh for this approximation

Appendix- Preparing the Mesh

- The distribution of the vertices should be fine enough to represent the function µn(v) well.
- We need to resample the vertices until all edge length are less than a threshold p
- If edges of a mesh are uniform in a certain direction, the accuracy of the calculation of µ (v) is biased and results in an inaccurate calculation of µn(v)
- Special edges called “short-cut edges” may need to be added to the mesh to modify the uniformity by making the directions of edges isotropic.
- The algorithm for adding a short-cut edge:
- t1,t2 and t3 which are adjacent to the triangle tc are unfolded on the plane of tc
- New edges are generated between each of the vertex pairs but only if an edge is inside the unfolded polygon

Appendix- Calculating µn

- The calculation is done using Dijkstra’s algorithm:

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