Reflective Symmetry Detection in 3 Dimensions Michael Kazhdan

Overview Introduction Related Work Definitions and Computation Results Future Work

Goal Present a shape-descriptor for model analysis Tasks: Registration Matching Properties: Parameterized over canonical domain Insensitive to noise Global

Overview Introduction Related Work Definitions and Computation Results Future Work

Related Work Alignment: Locally Parameterized Features: Generalized Hough Transform Ballard (1981) Geometric Hashing Lamdan (1989) Iterative Closest Point Besl, McKay (1992) Locally Parameterized Features: Spin Images Johnson, Hebert (1999) Harmonic Shape Images Zhang, Hebert (1999)

Related Work Canonically Parameterized Features: Extended Gaussian Images Horn (1984) Spherical Attribute Images Dellinguette, Hebert, Ikeuchi (1993) Orientation Histograms Sun, Si (1999) Moments Elad, Tal, Ar (2001) Shape Distributions Osada, Funkhouser, Chazelle, Dobkin (2001)

Overview Introduction Related Work Definitions and Computation Results Future Work

Reflective Symmetry Descriptor A function associating a measure of reflective symmetry to every plane through the origin Need to address: How do we measure symmetry? How do we compute the measure efficiently?

Overview Introduction Related Work Definitions and Computation Results Future Work

Measure of Symmetry Q: How close is a function f to be symmetric w.r.t to a reflection r? A: What is the distance to the nearest function g that is symmetric w.r.t. to r? f r g = ?

Measure of Symmetry Because the space of functions is a Hilbert space… Because reflection preserves the inner product… The closest symmetric function to f is the average of f with its reflection + = 2

Measure of Symmetry So that the measure of symmetry of f w.r.t. the reflection r is the (scaled) distance of f from its reflection: - 2 2

Overview Introduction Related Work Definitions and Computation Results Future Work

Functions on a Circle If f(t), t[0,2], is a function defined on a circle then the measure of symmetry of f with respect to reflection about the angle  is: t  2-t

Functions on a Disk f {fr1,fr2,fr3,…}

Functions on a Sphere Step 1: “North pole” symmetries by projection. Step 2: All symmetries by walking a great circle. f projected f

Voxel Grids Decompose the grid into concentric spheres, and apply the results for symmetry descriptors of spheres to the voxel grid.

Overview Introduction Related Work Definitions and Computation Results Future Work

Distinguishing Between Classes

Similarity Within Classes

Symmetry Within Classes

How Well is “Shape” Captured? Evaluate how well models can be: Registered Aligned by only using their symmetry descriptors

Registration Experiment (Ideal) Given a collection of models that are classified into groups and aligned: For each pair of models within a group: Find the rotation minimizing the L2-distance of the symmetry descriptors Evaluate how close the minimizing rotation is to the registering rotation

Evaluating the Rotation If M is the ideal registering rotation and N is the minimizing rotation found, how close is M to N? What is the angle of the rotation of MN-1?

Registration Experiment (Practice) Searching over the space of all rotations is computationally prohibitive: We know the axis about which the ideal aligning rotation occurs Search for best rotation about this axis

Registration Results Symmetry Principal Axes Model Database: % of Models % of Models Rotation Error Rotation Error Symmetry Principal Axes Model Database: Subset of Osada database that fully voxelized to 128x128x128 87 models, 24 Groups

Problems With Covariance Multi-Dimensional eigenspaces:

Registration Results

Matching Experiment (Ideal) Given a collection of models classified into groups: For each pair of models: Find rotation minimizing the L2-distance of the symmetry descriptors Use the minimum L2- distance as a measure of the match quality.

Matching Experiment (Heuristic) Searching over the space of all rotations is computationally prohibitive: Find the principal axis of symmetry of each of the models Search for minimizing rotation that maps principal axes of symmetry to each other

Matching Results First Tier First Two Tiers Nearest Neighbor Time Shape Distributions 47% 67% 62% 7.4 seconds Symmetries 48% 61% 69% ~4,500 seconds First Tier: If the query model belongs to a class with n models, how many of the top (n-1) matches are also in that class? Nearest Neighbor: How often is the top match in the same class as the query model? Model Database: Subset of Osada database that fully voxelized to 128x128x128 87 models, 24 Groups

Matching Results

Properties Parameterized over canonical domain: Insensitive to noise: Parameterized over the (projective) sphere Insensitive to noise: Integration scales down high frequency Fourier coefficients Global For functions f and g, and any reflection r:

Overview Introduction Related Work Definitions and Computation Results Future Work

Future Work Consider applications of the L properties of the symmetry descriptor Determine what information about shape can be easily extracted from the descriptor Explore the potential orthogonality of different matching methods Apply other alignment methods to the symmetry descriptors