Optimal spectral descriptors
GenericQ-dimensional spectral descriptor of the form
fτ(x) = X
k≥1
τ1(λk) ... τQ(λk)
φ2k(x)
| {z }
g(x)
=Ag(x)
parametrized by frequency responsesτ(λ) = (τ1(λ), . . . , τQ(λ))>
represented in some fixed basisβ1(λ), . . . , βM(λ)by anQ×M matrixA
Optimal spectral descriptors
GenericQ-dimensional spectral descriptor of the form
fA(x) = X
k≥1
A
β1(λk) ... βM(λk)
φ2k(x)
| {z }
g(x)
=Ag(x)
parametrized by frequency responsesτ(λ) = (τ1(λ), . . . , τQ(λ))>
represented in some fixed basisβ1(λ), . . . , βM(λ)by anQ×M matrixA
0 0.2 0.4 0.6 0.8 1
Optimal spectral descriptors
GenericQ-dimensional spectral descriptor of the form
fA(x) = AX
k≥1
β1(λk) ... βM(λk)
φ2k(x)
| {z }
g(x)
=Ag(x)
parametrized by linear combination coefficientsAofgeometry vectors g(x) = (g1(x), . . . , gM(x))>
Optimal spectral descriptors
GenericQ-dimensional spectral descriptor of the form
fA(x) = AX
k≥1
β1(λk) ... βM(λk)
φ2k(x)
| {z }
g(x)
=Ag(x)
parametrized by linear combination coefficientsAofgeometry vectors g(x) = (g1(x), . . . , gM(x))>
Optimal spectral descriptors
GenericQ-dimensional spectral descriptor of the form
fA(x) = AX
k≥1
β1(λk) ... βM(λk)
φ2k(x)
| {z }
g(x)
=Ag(x)
parametrized by linear combination coefficientsAofgeometry vectors g(x) = (g1(x), . . . , gM(x))>
Optimal Ain the spirit ofWiener filter:
attenuate frequencies with large noise content (deformation) pass frequencies with large signal content (discriminative geometric features)
Hard to model axiomatically... ...yet easy to learnfrom examples!
Optimal spectral descriptors
GenericQ-dimensional spectral descriptor of the form
fA(x) = AX
k≥1
β1(λk) ... βM(λk)
φ2k(x)
| {z }
g(x)
=Ag(x)
parametrized by linear combination coefficientsAofgeometry vectors g(x) = (g1(x), . . . , gM(x))>
Optimal Ain the spirit ofWiener filter:
attenuate frequencies with large noise content (deformation) pass frequencies with large signal content (discriminative geometric features)
Hard to model axiomatically...
...yet easy to learnfrom examples!
Optimal spectral descriptors
GenericQ-dimensional spectral descriptor of the form
fA(x) = AX
k≥1
β1(λk) ... βM(λk)
φ2k(x)
| {z }
g(x)
=Ag(x)
parametrized by linear combination coefficientsAofgeometry vectors g(x) = (g1(x), . . . , gM(x))>
Optimal Ain the spirit ofWiener filter:
attenuate frequencies with large noise content (deformation) pass frequencies with large signal content (discriminative geometric features)
Hard to model axiomatically...
...yet easy to learnfrom examples!
Optimal spectral descriptors
Given a set of pointsx, knowingly similar points (positive)x+, and knowingly dissimilar points (negative)x−, with respective geometry vectors g,g+,g−
Find optimal Aby
min
A:A>A=I
where C±=E(g−g±)(g−g±)> is covariance matrix
Optimal spectral descriptors
Given a set of pointsx, knowingly similar points (positive)x+, and knowingly dissimilar points (negative)x−, with respective geometry vectors g,g+,g−
Find optimal Aby minimizing the loss
min
A:A>A=I EγkfA(x)−fA(x+)k2−(1−γ)kfA(x)−fA(x−)k2
where C±=E(g−g±)(g−g±)> is covariance matrix
Optimal spectral descriptors
Given a set of pointsx, knowingly similar points (positive)x+, and knowingly dissimilar points (negative)x−, with respective geometry vectors g,g+,g−
Find optimal Aby minimizing the loss
min
A:A>A=I EγkAg−Ag+k2−(1−γ)kAg−Ag−k2
where C±=E(g−g±)(g−g±)> is covariance matrix
Optimal spectral descriptors
Given a set of pointsx, knowingly similar points (positive)x+, and knowingly dissimilar points (negative)x−, with respective geometry vectors g,g+,g−
Find optimal Aby minimizing the loss
min
A:A>A=I
trace A(γC+−(1−γ)C−)A>
where C±=E(g−g±)(g−g±)> is covariance matrix
Optimal spectral descriptors
Given a set of pointsx, knowingly similar points (positive)x+, and knowingly dissimilar points (negative)x−, with respective geometry vectors g,g+,g−
Find optimal AbyMahalanobis metric learning
min
A:A>A=I
trace A(γC+−(1−γ)C−)A>
Optimal spectral descriptor
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
−0.5 0 0.5 1
Optimal transfer functions learned from positive and negative examples
...
f f f
Descriptor robustness
Near-isometric deformation
Non-isometric deformation
Topo.
noise Geom.
noise
Uniform subsample
Nonunif.
subsample Missing
parts
Descriptors: Sun, Ovsjanikov, Guibas 2009; Aubry, Schlickewei, Cremers 2011; Litman
Descriptor robustness
Near-isometric deformation
Non-isometric deformation
Topo.
noise Geom.
noise
Uniform subsample
Nonunif.
subsample Missing
parts
HKS descriptor distance
Descriptors: Sun, Ovsjanikov, Guibas 2009; Aubry, Schlickewei, Cremers 2011; Litman Bronstein 2014; Evaluation: Masci, Boscaini, Bronstein, Vandergheynst 2015; data:
Descriptor robustness
Near-isometric deformation
Non-isometric deformation
Topo.
noise Geom.
noise
Uniform subsample
Nonunif.
subsample Missing
parts
WKS descriptor distance
Descriptors: Sun, Ovsjanikov, Guibas 2009; Aubry, Schlickewei, Cremers 2011; Litman
Descriptor robustness
Near-isometric deformation
Non-isometric deformation
Topo.
noise Geom.
noise
Uniform subsample
Nonunif.
subsample Missing
parts
Optimal Spectral descriptor distance
Descriptors: Sun, Ovsjanikov, Guibas 2009; Aubry, Schlickewei, Cremers 2011; Litman Bronstein 2014; Evaluation: Masci, Boscaini, Bronstein, Vandergheynst 2015; data:
Descriptor performance
CMC
0 20 40 60 80 100 0
0.2 0.4 0.6 0.8 1
number of matches
hitrate
ROC
0.001 0.01 0.1 1
0 0.2 0.4 0.6 0.8 1
false positive rate
truepositiverate
Corr. quality
0 0.1 0.2 0.3
0 0.2 0.4 0.6
geodesic radius
%correctcorrespondences
HKS WKS OSD
Descriptor performance (training and testing: FAUST)
