View
224
Download
0
Category
Preview:
Citation preview
An Optimization Approach to Improving Collections of Shape Maps
Andy Nguyen, Mirela Ben-Chen,Katarzyna Welnicka, Yinyu Ye, Leonidas Guibas
Computer Science Dept.Stanford University
Problem Statement
InputA collection of related shapes
A collection of maps between all pairs of shapes
A distance measure on each shape
6
Approach
Cycle consistency tells us something about accuracy
Remove the inaccuracies we find
Repeat using the better collection
9
Related WorkLearning Shape Metrics based on Deformations and Transport, Charpiat, NORDIA09
Dynamic time warping + matching energy for good shortest paths
Disambiguating Visual Relations Using Loop Constraints, Zach et al, CVPR2010
Use cycle consistency to remove incorrect correspondences
Pairwise mapping methodsMöbius Voting for Surface Correspondence, Lipman et al, SIGGRAPH 2009One Point Isometric Matching with the Heat Kernel, Ovsjanikov et al, SGP 2010Blended Intrinsic Maps, Kim et al, SIGGRAPH 2011
10
Definitions
Accuracy error:
Consistency error:
11
Eacc(mA ;B ) =1
jAj
X
p2A
dB (mA ;B (p); ~mA ;B (p))
Econs(°) =1
jAj
X
p2A
dA (p;m° (p))
A B
C
B
A
Relating Cycles to Edges
Call low error “good,” high error “bad”
Good and bad edges cause good and bad cycles
If we can only evaluate the cycles, what can we say about the edges? 12
Relating Cycles to Edges
Accuracy error of a path ° = {i1, …, in} is bounded*:
13
Eacc(°) ·n¡ 1X
j =1
Eacc(mi j ;i j + 1 )
*If ground-truth maps preserve the distortion measure
Proposal – Linear ProgramFor each 3-cycle ° in the graph, compute the distortion C°
Solve the following linear program to find weights for the edges:
Minimize
Subject to
Where14
X
e2E
wece
X
e2°
ce ¸ C° 8°
ce ¸ 0 8e2 E
we = 1=(X
° :e2°
C° )
Proposal
LP gives us a weighted graph
Weights give us shortest-path map compositions
But these are just like our input
Run the LP again?16
Proposal - CompleteRepeat the following:
Solve LP => obtain edge-weighted graph
Replace edges with shortest paths
Until one of the following is true:No edge replacements happen, or
No more 3-cycles are bad
18
Convergence - Theoretical
“Almost-accurate” collection: Each 3-cycle has at most 1 bad map
Every cycle’s distortion is either 0 or equal to the inaccuracy of the 1 bad map
LP weights are exactly the map accuracy errors
Guarantees consistency and accuracy after replacing maps with shortest paths
20
Results – 2D (DTW)
21
Max
con
sist
ency
err
or
Fraction of maps
Max
acc
urac
y er
ror
Fraction of cycles
Results – 2D (DTW)
22
Max
con
sist
ency
err
or
Fraction of maps
Max
acc
urac
y er
ror
Fraction of cycles
Results – 3D (Heat Kernel)
24Max
con
sist
ency
err
or
Fraction of maps
Max
acc
urac
y er
ror
Fraction of cycles
Future Work
Prove convergence in more general casesAllow for multiple maps between a given pair of shapesDiscover the structure of the collection using consistency information
29
Conclusions
Collections contain information that allow us to better evaluate mapsCycle consistency can be used to identify and remove bad mapsUsing an LP with 3-cycle constraints lets us do this efficientlyRepeating the process lets us incorporate longer cycles
30
Recommended