3D Object Retrieval

536 Views

Download Presentation
## 3D Object Retrieval

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**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?**• Improved modeling tools • Improved scanning devices • Fast and cheap CPUs, Gfx HW • Large databases • E-commerce • Medicine • Entertainment • Molecular biology • Manufacturing**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**• 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 = =**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 • 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]:**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**• 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: