The PlanarReflective Symmetry Transform Princeton University Motivation Symmetry
- Slides: 65
The Planar-Reflective Symmetry Transform Princeton University
Motivation Symmetry is everywhere
Motivation Symmetry is everywhere Perfect Symmetry [Blum ’ 64, ’ 67] [Wolter ’ 85] [Minovic ’ 97] [Martinet ’ 05]
Motivation Symmetry is everywhere Local Symmetry [Blum ’ 78] [Mitra ’ 06] [Simari ’ 06]
Motivation Symmetry is everywhere Partial Symmetry [Zabrodsky ’ 95] [Kazhdan ’ 03]
Goal A computational representation that describes all planar symmetries of a shape ? Input Model
Symmetry Transform A computational representation that describes all planar symmetries of a shape ? Input Model Symmetry Transform
Symmetry Transform A computational representation that describes all planar symmetries of a shape ? Perfect Symmetry = 1. 0
Symmetry Transform A computational representation that describes all planar symmetries of a shape ? Zero Symmetry = 0. 0
Symmetry Transform A computational representation that describes all planar symmetries of a shape ? Local Symmetry = 0. 3
Symmetry Transform A computational representation that describes all planar symmetries of a shape ? Partial Symmetry = 0. 2
Symmetry Measure Symmetry of a shape is measured by correlation with its reflection
Symmetry Measure Symmetry of a shape is measured by correlation with its reflection Symmetry = 0. 7
Symmetry Measure Symmetry of a shape is measured by correlation with its reflection Symmetry = 0. 3
Outline • Introduction • Algorithm – Computing Discrete Transform – Finding Local Maxima Precisely • Applications – Alignment – Matching – Segmentation – Viewpoint Selection • Summary
PRST
Outline • Introduction • Algorithm – Computing Discrete Transform – Finding Local Maxima Precisely • Applications – Alignment – Segmentation • Summary – Matching – Viewpoint Selection
Computing Discrete Transform n planes • Brute Force • Convolution • Monte-Carlo
Computing Discrete Transform O(n 3) planes X = O(n 6) O(n 3) dot product O(n 6) n planes • Brute Force • Convolution • Monte-Carlo
Computing Discrete Transform O(n 6) O(n 5 Log n) n planes • Brute Force • Convolution • Monte-Carlo
Computing Discrete Transform • Brute Force • Convolution • Monte-Carlo O(n 6) O(n 5 Log n) O(n 4) For 3 D meshes – Most of the dot product contains zeros. – Use Monte-Carlo Importance Sampling.
Offset Monte Carlo Algorithm Angle Input Model Symmetry Transform
Monte Carlo Algorithm Offset Monte Carlo sample for single plane Angle Input Model Symmetry Transform
Offset Monte Carlo Algorithm Angle Input Model Symmetry Transform
Offset Monte Carlo Algorithm Angle Input Model Symmetry Transform
Offset Monte Carlo Algorithm Angle Input Model Symmetry Transform
Offset Monte Carlo Algorithm Angle Input Model Symmetry Transform
Offset Monte Carlo Algorithm Angle Input Model Symmetry Transform
Weighting Samples Need to weight sample pairs by the inverse of the distance between them P 2 d P 1
Weighting Samples Need to weight sample pairs by the inverse of the distance between them Two planes of (equal) perfect symmetry
Weighting Samples Need to weight sample pairs by the inverse of the distance between them Votes for vertical plane…
Weighting Samples Need to weight sample pairs by the inverse of the distance between them Votes for horizontal plane.
Outline • Introduction • Algorithm – Computing Discrete Transform – Finding Local Maxima Precisely • Applications – Alignment – Segmentation • Summary – Matching – Viewpoint Selection
Finding Local Maxima Precisely Motivation: • Significant symmetries will be local maxima of the transform: the Principal Symmetries of the model Principal Symmetries
Finding Local Maxima Precisely Approach: • Start from local maxima of discrete transform • Finding the candidate plane by using threshold(1 -r/R)
Finding Local Maxima Precisely Approach: • Start from local maxima of discrete transform • Refine iteratively to find local maxima precisely ………. Initial Guess Final Result
Outline • Introduction • Algorithm – Computing discrete transform – Finding Local Maxima Precisely • Applications – Alignment – Segmentation • Summary – Matching – Viewpoint Selection
Application: Alignment Motivation: • Composition of range scans • Feature mapping PCA Alignment
Application: Alignment Approach: • Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.
Application: Alignment Approach: • Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.
Application: Alignment Approach: • Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.
Application: Alignment Approach: • Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.
Application: Alignment Results: PCA Alignment Symmetry Alignment
Application: Matching Motivation: • Database searching = Query Database Best Match
Application: Matching Observation: • All chairs display similar principal symmetries
Application: Matching Approach: • Use Symmetry transform as shape descriptor = Query Transform Database Best Match
Application: Matching Results: • The PRST provides orthogonal information about models and can therefore be combined with other shape descriptors
Application: Matching Results: • The PRST provides orthogonal information about models and can therefore be combined with other shape descriptors
Application: Matching Results: • The PRST provides orthogonal information about models and can therefore be combined with other shape descriptors
Application: Segmentation Motivation: • Modeling by parts [Chazelle ’ 95][Li ’ 01] [Mangan ’ 99][Garland ’ 01] [Katz ’ 03]
Application: Segmentation Observation: • Components will have strong local symmetries not shared by other components
Application: Segmentation Observation: • Components will have strong local symmetries not shared by other components
Application: Segmentation Observation: • Components will have strong local symmetries not shared by other components
Application: Segmentation Observation: • Components will have strong local symmetries not shared by other components
Application: Segmentation Observation: • Components will have strong local symmetries not shared by other components
Application: Segmentation Approach: • Cluster points on the surface by how well they support different symmetries …. . Support = 0. 1 Support = 0. 5 Support = 0. 9 Symmetry Vector = { 0. 1 , 0. 5 , …. , 0. 9 }
Application: Segmentation Results:
Application: Viewpoint Selection Motivation: • Catalog generation • Image Based Rendering [Blanz ’ 99][Vasquez ’ 01] [Lee ’ 05][Abbasi ’ 00] Picture from Blanz et al. ‘ 99
Application: Viewpoint Selection Approach: • Symmetry represents redundancy in information.
Application: Viewpoint Selection Approach: • Symmetry represents redundancy in information • Minimize the amount of visible symmetry • Every plane of symmetry votes for a viewing direction perpendicular to it Best Viewing Directions
Application: Viewpoint Selection Results: Viewpoint Function
Application: Viewpoint Selection Results: Viewpoint Function Best Viewpoint
Application: Viewpoint Selection Results: Viewpoint Function Best Viewpoint Worst Viewpoint
Application: Viewpoint Selection Results:
The End
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