Rigid Motion Invariant Classification of 3D-Texturessaurabh/AMS_FL_2009.pdfRigid Motion Invariant...
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Rigid Motion InvariantClassification of 3D-Textures
Saurabh JainDepartment of Mathematics
University of Houston
October 31st 20092009 Fall Southeastern Meeting, Boca Raton, Florida
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Acknowledgements
This work has been performed in collaboration with Prof. RobertAzencott and Prof. Manos Papadakis.
We are grateful to Simon Alexander for his suggestions and many fruitfuldiscussions.
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Outline
Background
Isotropic Multiresolution AnalysisDefinitionIsotropic Wavelet
Rotationally Invariant 3-D Texture ClassificationTexture ModelRotation of TexturesGaussian Markov Random FieldRotationally Invariant DistanceExperimental Results
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Textures
Figure: Examples of structural 2-D textures
Figure: Examples of stochastic 2-D textures
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Texture Examples from Biomedical Imaging
(a) 2D slice from 3D µCT x-raydata
(b) Slice from IntravascularUltra Sound data
Figure: Examples of medical 3D data sets.
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Outline
Background
Isotropic Multiresolution AnalysisDefinitionIsotropic Wavelet
Rotationally Invariant 3-D Texture ClassificationTexture ModelRotation of TexturesGaussian Markov Random FieldRotationally Invariant DistanceExperimental Results
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Definition
An IMRA is a sequence {Vj}j∈Z of closed subspaces of L2(Rd) satisfyingthe following conditions:
∀j ∈ Z, Vj ⊂ Vj+1,
(DM)jV0 = Vj ,
∪j∈ZVj is dense in L2(Rd),
∩j∈ZVj = {0},
V0 is invariant under translations by integers,
V0 is invariant under all rotations, i.e.,
O(R)V0 = V0 for all R ∈ SO(d),
where O(R) is the unitary operator given by O(R)f (x) := f (RT x).
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Definition
An IMRA is a sequence {Vj}j∈Z of closed subspaces of L2(Rd) satisfyingthe following conditions:
∀j ∈ Z, Vj ⊂ Vj+1,
(DM)jV0 = Vj ,
∪j∈ZVj is dense in L2(Rd),
∩j∈ZVj = {0},V0 is invariant under translations by integers,
V0 is invariant under all rotations, i.e.,
O(R)V0 = V0 for all R ∈ SO(d),
where O(R) is the unitary operator given by O(R)f (x) := f (RT x).
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2D IMRA refinable function and wavelet
(a) Fourier transform of the refinablefunction φ̂
(b) Fourier transform of the waveletψ̂1(2.)
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Outline
Background
Isotropic Multiresolution AnalysisDefinitionIsotropic Wavelet
Rotationally Invariant 3-D Texture ClassificationTexture ModelRotation of TexturesGaussian Markov Random FieldRotationally Invariant DistanceExperimental Results
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Texture Model
Let Xcont be a stationary Gaussian process on R3 and X be its sampleson Z3.
Let its autocovariance function ρcont be bandlimited to B2, the ball whereφ̂ is equal to 1.Note that
ρcont(k) = E[Xcont(k)Xcont(0)] = E[X(k)X(0)] = ρ(k)
Hence, ρcont =∑
k∈Z3 ρ(k)Tkφ.
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Texture Model
Let Xcont be a stationary Gaussian process on R3 and X be its sampleson Z3.Let its autocovariance function ρcont be bandlimited to B2, the ball whereφ̂ is equal to 1.
Note that
ρcont(k) = E[Xcont(k)Xcont(0)] = E[X(k)X(0)] = ρ(k)
Hence, ρcont =∑
k∈Z3 ρ(k)Tkφ.
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Texture Model
Let Xcont be a stationary Gaussian process on R3 and X be its sampleson Z3.Let its autocovariance function ρcont be bandlimited to B2, the ball whereφ̂ is equal to 1.Note that
ρcont(k) = E[Xcont(k)Xcont(0)] = E[X(k)X(0)] = ρ(k)
Hence, ρcont =∑
k∈Z3 ρ(k)Tkφ.
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Texture Model
Let Xcont be a stationary Gaussian process on R3 and X be its sampleson Z3.Let its autocovariance function ρcont be bandlimited to B2, the ball whereφ̂ is equal to 1.Note that
ρcont(k) = E[Xcont(k)Xcont(0)] = E[X(k)X(0)] = ρ(k)
Hence, ρcont =∑
k∈Z3 ρ(k)Tkφ.
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Rotation of Textures
Let Rα be the operator induced on L2(R3) by the rotation α ∈ SO(3).
The autocovariance function of RαXcont is given by Rαρcont :
E[RαXcont(s)RαXcont(0)] = E[Xcont(αT s)Xcont((αT 0)]
= ρcont(αT s) = Rαρcont(s).
Now, the sequence of samples, 〈Rαρcont ,Tkφ〉}k∈Z3 is denoted by Rαρ.
〈Rαρcont ,Tkφ〉 = 〈ρcont ,R∗αTkφ〉 = 〈ρcont ,Tαkφ〉
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Rotation of Textures
Let Rα be the operator induced on L2(R3) by the rotation α ∈ SO(3).The autocovariance function of RαXcont is given by Rαρcont :
E[RαXcont(s)RαXcont(0)] = E[Xcont(αT s)Xcont((αT 0)]
= ρcont(αT s) = Rαρcont(s).
Now, the sequence of samples, 〈Rαρcont ,Tkφ〉}k∈Z3 is denoted by Rαρ.
〈Rαρcont ,Tkφ〉 = 〈ρcont ,R∗αTkφ〉 = 〈ρcont ,Tαkφ〉
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Rotation of Textures
Let Rα be the operator induced on L2(R3) by the rotation α ∈ SO(3).The autocovariance function of RαXcont is given by Rαρcont :
E[RαXcont(s)RαXcont(0)] = E[Xcont(αT s)Xcont((αT 0)]
= ρcont(αT s) = Rαρcont(s).
Now, the sequence of samples, 〈Rαρcont ,Tkφ〉}k∈Z3 is denoted by Rαρ.
〈Rαρcont ,Tkφ〉 = 〈ρcont ,R∗αTkφ〉 = 〈ρcont ,Tαkφ〉
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Rotation of Textures
Let Rα be the operator induced on L2(R3) by the rotation α ∈ SO(3).The autocovariance function of RαXcont is given by Rαρcont :
E[RαXcont(s)RαXcont(0)] = E[Xcont(αT s)Xcont((αT 0)]
= ρcont(αT s) = Rαρcont(s).
Now, the sequence of samples, 〈Rαρcont ,Tkφ〉}k∈Z3 is denoted by Rαρ.
〈Rαρcont ,Tkφ〉 = 〈ρcont ,R∗αTkφ〉 = 〈ρcont ,Tαkφ〉
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Gaussian Markov Random Field
A stochastic process X on Z3 is a stationary GMRF if a realizationsatisfies the following difference equation:
xk = µ+∑r∈η
θr(xk−r − µ) + ek.
where the correlated Gaussian noise, e = (e1, . . . , eNT), has the following
structure:
E[ekel] =
σ2, k = l,−θk−lσ
2, k− l ∈ η,0, else.
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Auto-covariance function
For a stationary random process X on Z3, the auto-covariance function isgiven by
ρ(l) = E[X(l)X(0)]
Given a realization x on Λ ⊂ Z3, ρ can be approximated by
ρ0(l) =1
NT
∑r∈Λ
xrxr+l, for all l ∈ Λ
for a sufficiently large Λ; NT := |Λ|.The parameters of the GMRF model fitted to the ‘rotated texture’,denoted by Rαx, can be calculated using Rαρ.
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Auto-covariance function
For a stationary random process X on Z3, the auto-covariance function isgiven by
ρ(l) = E[X(l)X(0)]
Given a realization x on Λ ⊂ Z3, ρ can be approximated by
ρ0(l) =1
NT
∑r∈Λ
xrxr+l, for all l ∈ Λ
for a sufficiently large Λ; NT := |Λ|.
The parameters of the GMRF model fitted to the ‘rotated texture’,denoted by Rαx, can be calculated using Rαρ.
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Auto-covariance function
For a stationary random process X on Z3, the auto-covariance function isgiven by
ρ(l) = E[X(l)X(0)]
Given a realization x on Λ ⊂ Z3, ρ can be approximated by
ρ0(l) =1
NT
∑r∈Λ
xrxr+l, for all l ∈ Λ
for a sufficiently large Λ; NT := |Λ|.The parameters of the GMRF model fitted to the ‘rotated texture’,denoted by Rαx, can be calculated using Rαρ.
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Rotationally Invariant Distance
We define the texture signature Γx, via
Γx(α) =[θ̂(Rαρ), σ̂2(Rαρ)
]
Now, we define a distance between two textures by the followingexpression:
minα0∈SO(3)
∫SO(3)
KLdist (Γx1(α), Γx2(αα0)) dα.
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Rotationally Invariant Distance
We define the texture signature Γx, via
Γx(α) =[θ̂(Rαρ), σ̂2(Rαρ)
]Now, we define a distance between two textures by the followingexpression:
minα0∈SO(3)
∫SO(3)
KLdist (Γx1(α), Γx2(αα0)) dα.
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Experimental Results
T1,0 T1,π2
T2,0 T2,π2
T1,0 0.0007 0.0005 0.0072 0.0137
T1,π2
0.0010 0.0007 0.0101 0.0182
T2,0 0.0123 0.0128 0.0006 0.0004
T2,π2
0.0093 0.0101 0.0012 0.0009
Table: Distances between two rotations of two distinct textures using the
rotationally invariant distance and autocovariance resampled on Z3
4 .
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Experimental Results
T1,0 T1,π2
T2,0 T2,π2
T1,0 0.0006 0.0006 0.0073 0.0136
T1,π2
0.0013 0.0007 0.0100 0.0164
T2,0 0.0125 0.0203 0.0010 0.0004
T2,π2
0.0119 0.0082 0.0007 0.0008
Table: Distances between two rotations of two distinct textures using the
rotationally invariant distance and autocovariance resampled on Z3
2 .
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Experimental Results
T1,0 T1,π2
T2,0 T2,π2
T1,0 0.0026 0.0812 0.0330 0.1750
T1,π2
0.1118 0.0010 0.0852 0.0562
T2,0 0.0454 0.0694 0.0016 0.0108
T2,π2
0.0607 0.0473 0.0246 0.0018
Table: Distances between two rotations of two distinct textures using therotationally invariant distance and autocovariance sampled on the originalgrid Z3.
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Experimental Results
T1 T2 T3 T4 T5
T1 0.0006 0.0073 0.4232 2.3180 1.7724
T2 0.0125 0.0010 0.4894 2.5227 1.8381
T3 0.4466 0.5134 0.0004 0.5208 0.4563
T4 2.4314 2.6315 0.5605 0.0021 0.3533
T5 1.8200 1.9227 0.4318 0.2540 0.0043
Table: Distances between five distinct textures using the rotationally invariant
distance and autocovariance resampled on the grid Z3
2 .
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Experiments with 2-D Textures
(c) Grass (d) Sand
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Experiments with 2-D Textures
grass sand
grass 0.0200 0.0806
sand 0.0032 0.0443
grass sand
grass 0.0107 0.3418
sand 0.7174 0.0223
Table: Distances between the sand and grass textures for the original data (left)for the low pass component (right).
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