Reflective Symmetry Detection in 3 Dimensions

Slides:



Advertisements
Similar presentations
Distinctive Image Features from Scale-Invariant Keypoints
Advertisements

Image Registration  Mapping of Evolution. Registration Goals Assume the correspondences are known Find such f() and g() such that the images are best.
Presented by Xinyu Chang
電腦視覺 Computer and Robot Vision I
Extended Gaussian Images
Robust Global Registration Natasha Gelfand Niloy Mitra Leonidas Guibas Helmut Pottmann.
3D Shape Histograms for Similarity Search and Classification in Spatial Databases. Mihael Ankerst,Gabi Kastenmuller, Hans-Peter-Kriegel,Thomas Seidl Univ.
Chapter 8 Content-Based Image Retrieval. Query By Keyword: Some textual attributes (keywords) should be maintained for each image. The image can be indexed.
Object Recognition using Invariant Local Features Applications l Mobile robots, driver assistance l Cell phone location or object recognition l Panoramas,
Introduction and point groups Stereographic projections Low symmetry systems Space groups Deformation and texture Interfaces, orientation relationships.
Xianfeng Gu, Yaling Wang, Tony Chan, Paul Thompson, Shing-Tung Yau
Frequency-Domain Range Data Registration for 3-D Space Modeling in Robotic Applications By Phillip Curtis.
Probabilistic Fingerprints for Shapes Niloy J. MitraLeonidas Guibas Joachim GiesenMark Pauly Stanford University MPII SaarbrückenETH Zurich.
Uncertainty Representation. Gaussian Distribution variance Standard deviation.
Semi-automatic Range to Range Registration: A Feature-based Method Chao Chen & Ioannis Stamos Computer Science Department Graduate Center, Hunter College.
Reverse Engineering Niloy J. Mitra.
Robust and large-scale alignment Image from
Small Codes and Large Image Databases for Recognition CVPR 2008 Antonio Torralba, MIT Rob Fergus, NYU Yair Weiss, Hebrew University.
CENG 789 – Digital Geometry Processing 06- Rigid-Body Alignment Asst. Prof. Yusuf Sahillioğlu Computer Eng. Dept,, Turkey.
Rotation Invariant Spherical Harmonic Representation of 3D Shape Descriptors Michael Kazhdan Thomas Funkhouser Szymon Rusinkiewicz Princeton University.
Robert Osada, Tom Funkhouser Bernard Chazelle, and David Dobkin Princeton University Matching 3D Models With Shape Distributions.
Iterative closest point algorithms
Matching and Recognition in 3D. Moving from 2D to 3D – Some Things are Easier No occlusion (but sometimes missing data instead)No occlusion (but sometimes.
Numerical geometry of objects
Object Recognition with Invariant Features n Definition: Identify objects or scenes and determine their pose and model parameters n Applications l Industrial.
Harmonic 3D Shape Matching Michael Kazhdan Thomas Funkhouser Princeton University Michael Kazhdan Thomas Funkhouser Princeton University.
Selecting Distinctive 3D Shape Descriptors for Similarity Retrieval Philip Shilane and Thomas Funkhouser.
The Princeton Shape Benchmark Philip Shilane, Patrick Min, Michael Kazhdan, and Thomas Funkhouser.
Shape Descriptors I Thomas Funkhouser CS597D, Fall 2003 Princeton University Thomas Funkhouser CS597D, Fall 2003 Princeton University.
1 Numerical geometry of non-rigid shapes In the Rigid Kingdom In the Rigid Kingdom Lecture 4 © Alexander & Michael Bronstein tosca.cs.technion.ac.il/book.
Matching and Recognition in 3D. Moving from 2D to 3D Some things harderSome things harder – Rigid transform has 6 degrees of freedom vs. 3 – No natural.
Scale Invariant Feature Transform (SIFT)
Shape Matching and Anisotropy Michael Kazhdan, Thomas Funkhouser, and Szymon Rusinkiewicz Princeton University Michael Kazhdan, Thomas Funkhouser, and.
Blob detection.
The Planar-Reflective Symmetry Transform Princeton University.
Shape Classification Using the Inner-Distance Haibin Ling David W. Jacobs IEEE TRANSACTION ON PATTERN ANAYSIS AND MACHINE INTELLIGENCE FEBRUARY 2007.
Ashish Uthama EOS 513 Term Paper Presentation Ashish Uthama Biomedical Signal and Image Computing Lab Department of Electrical.
Yuping Lin and Gérard Medioni.  Introduction  Method  Register UAV streams to a global reference image ▪ Consecutive UAV image registration ▪ UAV to.
AdvisorStudent Dr. Jia Li Shaojun Liu Dept. of Computer Science and Engineering, Oakland University 3D Shape Classification Using Conformal Mapping In.
Alignment Introduction Notes courtesy of Funk et al., SIGGRAPH 2004.
Machine Vision for Robots
1 Faculty of Information Technology Generic Fourier Descriptor for Shape-based Image Retrieval Dengsheng Zhang, Guojun Lu Gippsland School of Comp. & Info.
Shape Matching for Model Alignment 3D Scan Matching and Registration, Part I ICCV 2005 Short Course Michael Kazhdan Johns Hopkins University.
Alignment and Matching
Intelligent Vision Systems ENT 496 Object Shape Identification and Representation Hema C.R. Lecture 7.
A 3D Model Alignment and Retrieval System Ding-Yun Chen and Ming Ouhyoung.
Shape Analysis and Retrieval Statistical Shape Descriptors Notes courtesy of Funk et al., SIGGRAPH 2004.
Axial Flip Invariance and Fast Exhaustive Searching with Wavelets Matthew Bolitho.
Signal Processing and Representation Theory Lecture 4.
Some Shape Descriptors for 3D Visual Objects
1 Multiple Classifier Based on Fuzzy C-Means for a Flower Image Retrieval Keita Fukuda, Tetsuya Takiguchi, Yasuo Ariki Graduate School of Engineering,
Shape Descriptors Thomas Funkhouser and Michael Kazhdan Princeton University Thomas Funkhouser and Michael Kazhdan Princeton University.
Action as Space-Time Shapes
CS 376b Introduction to Computer Vision 04 / 28 / 2008 Instructor: Michael Eckmann.
Geometric Modeling using Polygonal Meshes Lecture 3: Discrete Differential Geometry and its Application to Mesh Processing Office: South B-C Global.
CS654: Digital Image Analysis Lecture 36: Feature Extraction and Analysis.
CS848 Similarity Search in Multimedia Databases Dr. Gisli Hjaltason Content-based Retrieval Using Local Descriptors: Problems and Issues from Databases.
2004/03/03Sheun-Huei Guan, CML, NTU1 3D Model Retrieval After Shape Distributions.
Using simplified meshes for crude registration of two partially overlapping range images Mercedes R.G.Márquez Wu Shin-Ting State University of Matogrosso.
CSE 185 Introduction to Computer Vision Feature Matching.
Methods for 3D Shape Matching and Retrieval
1 Methods in Image Analysis – Lecture 3 Fourier CMU Robotics Institute U. Pitt Bioengineering 2630 Spring Term, 2004 George Stetten, M.D., Ph.D.
Partial Shape Matching. Outline: Motivation Sum of Squared Distances.
1 Overview representing region in 2 ways in terms of its external characteristics (its boundary)  focus on shape characteristics in terms of its internal.
CSCI 631 – Foundations of Computer Vision March 15, 2016 Ashwini Imran Image Stitching.
Scale Invariant Feature Transform (SIFT)
CAP 5415 Computer Vision Fall 2012 Dr. Mubarak Shah Lecture-5
Aim of the project Take your image Submit it to the search engine
CSE 185 Introduction to Computer Vision
Presentation transcript:

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