3D Object Retrieval

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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 Matching 3D Models with Shape
Distributions R.Osada, T.Funkhouser,
B.Chazelle, D.Dobkin
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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

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3D vs. 2D Retrieval

• 2D
• Object Boundaries
• Features Occlusion
• Camera dependent
• Noise
• Simple Contour representation

• 3D
• Problematic Surface Representation
• Ambiguous Triangulation
• No Features Occlusion, shadows, noise

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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
Signatures Comparison
Queryprocess
Query
Similarobjects
Object
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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

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Object Normalization using Moments
Translation (m100, m010, m001) center of mass
Rotation, Scale
?(1,1) - main axis scale U Rotation Matrix
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Different Methods

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

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Octrees

• Signature
• Each object is represented by Octree
• White, black, gray (gray level)
• Signature Comparison
• Coarse-to-Fine search (tree depth)

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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,L8 for PDF and CDF
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Distances to Enclosing Sphere

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

Signature Comparison L1,L2,L8
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Topology MatchingUsing MRGs
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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

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How to Reeb an Object

• Well create a simple reeb graph using height
  function
• µ - height of the point V µ(V(x,y,z))z

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MultiResolutional Reeb Graph(MRG)

• A series of Reeb graphs at various levels of
  detail

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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.

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The Construction of the MRG cont.

• The domain of µn is divided onto K µn-ranges
• R00,1/K),R11/k,2/k).Rk-1(K-1)/K,1)
• Note The example uses the height function for
  the convenience of explanation

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The Construction of the MRG cont.

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

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The Construction of the MRG cont.

• Calculate T-sets
• Connect R-nodes

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The Construction of the MRG cont.

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

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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

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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

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Matching

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

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Matching cont.

• Define
• At coarse resolution
• Similarity (0ltwlt1)
• To satisfy
• Define

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Matching cont.
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Topology Matching Added Value

• Topology matching
• can be used to find
• correspondence
• between meshes
• Problem
• The algorithm does
• Not distinguish between
• Left and right

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Results

• 230 mesh objects

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Future Work

• Use additional information
• Texture,color,curvature etc.
• Euclidean distance as the R-node attribute
• Use different µ functions
• Density for volumetric data

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Summary
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END
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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 Dijkstras
  algorithm based on edge length.
• We need to prepare the mesh for this approximation

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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

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Appendix- Calculating µn

• The calculation is done using Dijkstras
  algorithm