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3D Shape Matching and Retrieval
Computer Animation and Visualisation
Lecture 17
Taku Komura
Overview
● Introduction– Motivation, applications
– Challenges, key issues
● Global approaches– Image-based Methods
– Graph-based Methods
● Local approaches– Curvature
● Bag of visual words– Heat Kernel Signature
Many 3D objects in the world today to shape match
● Medical, CAD/CAM design, 3D scans, animation,
CAD/CAM
Manufacturing and production (You and Tsai, 2010)
Synthesizing new objects using existing objects [Kalogerakis et al.]
Recognizing and registering scanned data
● Recognizing objects
● Searching in the database for similar data and registering
Biometrics
3D video sequences
● Characterize a motion (Huang et al., 2010)
3D retrieval for navigation (Goodall et al. 2004)
Archeology
● Matching for reconstruction (Huang et al. 2006)
Challenges● The geometry can be very different
– Rigid objects
– Articulated objects
– Deformable objects
● The topology can be very different ● The data can contain a lot of noise
– Topological / geometric noise
Geometric and Topological Noise
● Noise can be added when scanning or processing the data
geometric noise topological noise
Key issues ● Input format
– Point clouds, meshes, volume data
● What type of features/descriptors are used?● Are the features/descriptors translation/rotation/scale
invariant? ● Abstraction
– Representation
– Whether the method can handle objects with different geometry and topology
Matching problems
● Non-rigid matching
● Partial matching
● Transform a 3D object into a numeric/symbolic
representation– Feature vectors
– Graphs
● Compare two objects through their representationsThe global approachThe global approachThe global approachThe global approach
The Global Approach
Depth-buffer descriptor
● Image-based descriptor (Vranic 2004)– Pose normalization
– Depth-buffer construction
– Fourier transformations
– Selection of coefficients
Depth-buffer descriptor
● Pose normalization - Typical procedure– Translate the center of mass to the origin of the
coordinate system
● Rotate according to the largest spread● Scale to common size
Depth-buffer descriptor
● Pose normalization– Done using principal component
analysis
Depth-buffer descriptor
● Construction– Project the object into the faces of a bounding
rectangle
Fourier Transform
Transform into the Frequency Domain
Spherical Harmonics
● The Fourier transform for a spherical function
Other approaches
● Ray-based feature vector (Vranic 2004)
Issues: Requires Alignment
● Difficult to align objects – whose points are equally distributed
– symmetric around an axis
● A rotation invariant feature is preferred
Rotation Invariant Spherical Harmonic Representation
Kazhdan et al SGP03
● Given a spherical depth map, convert to spherical harmonics
● Generate a vector of the power of each frequency
Issues with image-based methods
● Not very suitable for articulated / deformable objects well – The depth information is not consistent when the
pose changes
Graph-based Approaches
● Computing a graph structure out of the geometry and compare objects by graph matching
● Two types of graphs introduced here – Medial axis
– Reeb graph
Medial Axis
● A set of points that have two or more closest points on the surface
● The trajectory of the maximal circles/spheres
Computing the Medial Axis[Amenta et al.], Powercrust
● Approximate Method: – Using the Voronoi diagram
– Sample points on the surface and then compute the Voronoi diagram
– A Voronoi vertex is a point with three (2D)/four(3D) or more closest sample points.
– The corresponding Voronoi ball is the unique ball centred at the vertex that contains the closest sample points in its boundary.
– For every sample point there is an inner pole, namely the farthest Voronoi vertex in the intersection of the shape and the Voronoi cell of the sample point.
– Connect the inner poles according to the adjacency of the samples to approximate the medial axis
Medial Axis: Noise filtering
● The medial axis is sensitive geometric noise (bumps at the surface)
● We need filter the noise to produce a medial axis that represents the shape well
Scale Axis Transform
● Removing the inner poles whose maximal spheres are absorbed by nearby larger maximal spheres when their sizes are scaled
https://www.youtube.com/watch?v=-VsFTXfFXHQ#t=82
Reeb Graph
● Named after Georges Reeb (A French mathematician)
● A graph structure that represents the object geometry
● Used a lot in computer graphics and pattern recognition
Reeb Graph
● Given a function that we call Morse function, that defines a scalar value on the surface of a manifold, the Reeb graph is a graph whose nodes represent the isolines that have the same values, and the edges represent the connection of the isolines between different values
Geodesic Distance
● The shortest distance over the surface
● An approximation can be computed using graph distance
● Geodesic distance can be used as the Morse function
Geodesic Integral[Hilaga et al. 2001]
● We can use the integral of the geodesic distance to resolve the ambiguity of setting the start point
● µ(v) = ∫p g(v,p) dS● Normalize
– µn(v) = (µ(v) – min(µ)) / min(µ)
Multiresolution Reeb Graph
● Graph matching can be costly so can use the multiresolution Reeb graph
● Binary discretization preserve parent-child relationships
● Exploit them for matching
[Hilaga et al. 2001]
Matching process
● Calculate similarity● Match nodes
– Find pairs with maximal similarity
– Preserve multires hierarchy topology
● Sum up similarity
[Hilaga et al. 2001]
Content-based Retrieval[Hilaga et al. 2001]
https://www.youtube.com/watch?v=xA0S_oEWD1Y
Summary for Graph-based Approaches
● Good for computing the similarity based on the articulated structures
● Rotation / scale invariant (if normalized) ● Medial axis is sensitive to geometric noise but
can do robust matching after filtering● Not robust to topological noise
geometric noise topological noise
Overview
● Introduction– Motivation, applications
– Challenges, key issues
● Global approaches– Image-based Methods
– Graph-based Methods
● Local approaches– Curvature
● Bag of visual words– Heat Kernel Signature
Local Features and Partial Matching
● Features based on local geometry ● Good for finding saliency ● Useful for partial shape matching● Use curvature
Curvature
● Amount by which a geometric object deviates from being flat, or straight
● The curvature is the magnitude of the rate of change of the tangent vector T:
● Or can be computed by
Segmenting and Matching based on Curvature
● Curvature has been used for segmenting 2D contours and matching
● Need multiscale analysis for robustness
(conduct smoothing, and compute the curvature at different levels)
Curvature on Surfaces : Principal Curvature (1)
● At a point p on the surface, we can define a normal plane that contains the normal vector
● It also contains a unique direction tangent to the surface and cut the surface in a plane curve
Curvature on Surfaces: Principal Curvature (2)
● The plane curve have different curvature for different normal planes
● The principal curvatures at p, denoted and , are the maximum and minimum values of this curvature.
● Special Curvature – Mean curvature :
– Gaussian curvature
Discrete Laplace-Beltrami [Meyer et al.]
Discrete Curvatures
Voronoi Region Area
● Sum the A below for all the triangles of 1-ring(v i)
If the triangle T is non-obtuse● A + ●
● If obtuse and vi is the obtuse angle,
A += area(T) / 2
– Else A += area (T) / 4
Extracting Saliency using Curvature
● The salient points can be extracted using the curvature
● Can use the curvature as an attribute at the surface for matching
Overview
● Introduction– Motivation, applications
– Challenges, key issues
● Global approaches– Image-based Methods
– Graph-based Methods
● Local approaches– Curvature
● Bag of visual words– Heat Kernel Signature
Heat Kernel Signature● A vector that represents the temperature at a
heated point as time passes
[t1, t2, t3, ...., tn]● The temperature after a short time describes
the local geometry● The temperature after a long time describes the
global geometry
Local Geometric Feature by HKS
● The heat in short time has a similar distribution to the curvature
Comparison of the shapes
● Using the heat kernel signature vector, conduct a K-means clustering and produce a geometric vocabulary P = {p1 , . . . , pV } of size V.
● For each point x, we can compute the feature distribution of the vocabulary
θ (x) = (θ1(x), . . . , θV(x)) where
Bag of Features
● Integrating the feature distribution over the entire shape X yields a V × 1 vector
● For comparing two shapes, we can compute the distance between bag of features
Example of Bag of Features
Summary: Heat Kernel Signature
● Can obtain both the local and global features● Robust against topological noise
– The heat does not diffuse much when there is a connection with less volume
● The bag of features provide a quick scheme to compare the objects
References● Kazhdan et al. “Rotation Invariant Spherical Harmonic Representation of 3D Shape
Descriptors”, SGP03
● Ivan Sipiran and Benjamin Bustos, “Shape Matching for 3D Retrieval and Recognition”, SIBGRAPI 2013 Tutorial , Arequipa - Perú, August 5, 2013
● ALEXANDER M. BRONSTEIN et al., Shape Google: Geometric Words and Expressions for Invariant Shape Retrieval, Transactions on Graphics 2011, 30(1)
● Mark Meyer et al. Discrete Differential-Geometry Operators for Triangulated 2-Manifolds
● Amenta et al. “The Power Curst”
● Joachim Giesen, Balint Miklos, Mark Pauly, Camille Wormser: The Scale Axis Transform, ACM Symposium on Computational Geometry 2009
●
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