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3D Object Retrieval 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

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3d object retrieval l.jpg

3D Object Retrieval

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 l.jpg
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


3d vs 2d retrieval l.jpg

2D

Object Boundaries

Features Occlusion

Camera dependent

Noise

Simple Contour representation

3D

Problematic Surface Representation

Ambiguous Triangulation

No Features Occlusion, shadows, noise

3D vs. 2D Retrieval


Common approach l.jpg
Common 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 l.jpg
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

=

=


Object normalization using moments l.jpg
Object Normalization using Moments

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

Rotation, Scale:

∆(1,1) - main axis scale

U – Rotation Matrix


Different methods l.jpg
Different Methods

  • Octrees

  • Probability Shape Distributions

  • Distances to Enclosing Sphere

  • Reeb Graphs


Octrees l.jpg
Octrees

Signature:

  • Each object is represented by Octree

  • White, black, gray (gray level)

    Signature Comparison:

  • Coarse-to-Fine search (tree depth)


Probability shape distributions l.jpg
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 l.jpg
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 l.jpg
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

  • The search:

    • coarse to fine


How to reeb an object l.jpg
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 l.jpg
MultiResolutional Reeb Graph(MRG)

  • A series of Reeb graphs at various levels of detail


The construction of the mrg l.jpg
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 l.jpg
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 cont17 l.jpg
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 cont18 l.jpg
The Construction of the MRG cont.

  • Calculate T-sets

  • Connect R-nodes


The construction of the mrg cont19 l.jpg
The Construction of the MRG cont.

  • Construct MRG

    • fine-to-coarse (reverse)


Defining for topology matching l.jpg
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]:


Examples of the distribution of the function l.jpg
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


Matching l.jpg
Matching

  • Assign 2 attributes for each node (m) in the finest resolution

    • Area

    • Length


Matching cont l.jpg
Matching cont.

  • Define ‘+’:

  • At coarse resolution

  • Similarity (0<=w<=1)

    • To satisfy:

    • Define



Topology matching added value l.jpg
“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 l.jpg
Results

  • 230 mesh objects


Future work l.jpg
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 l.jpg
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 l.jpg
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 n l.jpg
Appendix- Calculating µn

  • The calculation is done using Dijkstra’s algorithm: