pose-estimation data (‘volvo’; change in azimuth of 30 ...jmcauley/pdfs/poster_cvpr10.pdflinear...
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UnifiedGraphMatching inEuclideanSpacesJulian McAuley, Teófilo de Campos, Tibério Caetano
[email protected] [email protected] [email protected]
AbstractIn many applications of graph matching, one is in-terested in finding correspondences that are iso-morphic (i.e., the topology is preserved), as wellas isometric (i.e., distances are preserved). Whilethe subgraph isomorphism problem is known tobe NP-complete, the subgraph isometry problemis polynomial [1]. We show that [1] can be adaptedto find subgraphs that are simultaneously both iso-morphic and isometric in polynomial time. Fur-thermore, we use structured learning to determinethe extent to which different types of transforma-tions are present in our model.
Results
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∆(g,g
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CMU ‘house’ data (111 frames)
linear assignmentquadratic assignmentbalanced graph-matchingisometric matchingunified matching
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wall time (seconds)
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CMU ‘house’ timing (separation of 90 frames)
linear assignment
quadratic assignment
balanced graph-matching
isometric matching
unified matching
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∆(g,g
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Silhouette data (200 frames)
linear assignmentquadratic assignmentbalanced graph-matchingisometric matchingunified matching
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CMU ‘hotel’ data (101 frames)
linear assignmentquadratic assignmentbalanced graph-matchingisometric matchingunified matching
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Change in rotation
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Pose data (house, 70 frames)
linear assignmentquadratic assignmentisometric matchingunified matching
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Change in azimuth
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Pose data (volvo, 42 frames)
linear assignmentquadratic assignmentisometric matchingunified matching
The performance of our method compared to models from [2] and [3]. The im-ages at right compare our matching results to those from [2]; blue dots denotecorrespondences for which only our method was correct; red dots denote cor-respondences for which only our method was incorrect; gray dots denote thatboth methods were incorrect. Running times are shown at top-right.
Our Graphical Model
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(a) (b) (c)The graphical model (a) from [2] can be used to search for isometric instancesof the graph (b). However, it cannot identify isomorphic (or homeomorphic)instances, as it does not capture all of the topological constraints in (b). Byreplicating some of its nodes in (c), we are able to capture the missing topolog-ical constraints. The resulting model provably identifies the correct solutionsubject to zero noise.
Model ParametrizationWe use the structured learning approach of [4] to determine the importance of first-order features (such as Shape-Contexts and SIFT),second-order features (such as topological constraints and distances), and third-order features. Our graph-matching objective, map-ping the points in V to those in V ′ is defined as:
g = argminf :V→V ′
∑p1∈G]
⟨Φ1(p1, f(p1))︸ ︷︷ ︸
node features
, θnodes
⟩+
∑(p1,p2)∈G]
⟨Φ2(p1, p2, f(p1), f(p2))︸ ︷︷ ︸
edge features
, θedges
⟩+
∑(p1,p2,p3)∈G]
⟨Φ3(p1, p2, p3, f(p1), f(p2), f(p3))︸ ︷︷ ︸
triangle features
, θtri
⟩
Our approach is fully-supervised, i.e., we learn Θ = (θnodes; θedges; θtri) from manually labeled correspondences provided by the user.
Graph Isometry and Isomorphism
(a) (b) (c) (d)Isometry: b ⊆ a, b ⊆ cIsomorphism: b ⊆ d, c ⊆ a, d ⊆ bHomeomorphism: b ⊆ a, b ⊆ d, c ⊆ a, d ⊆ a, d ⊆ b
bibliography[1] T. S. Caetano and J. J. McAuley. Faster graphical models for
point-pattern matching. Spatial Vision, 2009.
[2] J. J. McAuley, T. S. Caetano, and A. J. Smola. Robustnear-isometric matching via structured learning of graph-ical models. NIPS, 2008.
[3] T. Cour, P. Srinivasan, and J. Shi. Balanced graph matching.In NIPS, 2006.
[4] I. Tsochantaridis, T. Hofmann, T. Joachims, and Y. Altun.Support vector machine learning for interdependent andstructured output spaces. In ICML, 2004.
Weight Vectors
wei
ght(
rela
tive)
CMU ‘house’ weights (separation of 90 frames):
wei
ght(
rela
tive)
Pose-estimation data (‘volvo’; change in azimuth of 30 degrees):
h(is
omor
phic
)d
(isom
etric
)h
(hom
eom
orph
ic)
sim
ilara
ngle
soc
clus
ion
cost
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ilart
riang
les
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ght(
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tive)
CMU ‘hotel’ weights (separation of 90 frames):
︸ ︷︷ ︸first-order features (shape-context)
We learn a different model for each dataset. The weight vectors we learndemonstrate the varying degrees of importance of isometry, isomorphism, andhomeomorphism in different applications.
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