Automatic Target Recognition Using Algebraic Function of Views

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Automatic Target Recognition Using Algebraic
Function of Views

• Computer Vision Laboratory
• Dept. of Computer Science
• University of Nevada, Reno

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Outline

• Background of algebraic functions of views
• Frame work
• Imposing rigid constraint
• Indexing scheme
• Geometric manifold
• Mixture of Gaussians
• Modified Framework
• Future work

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Orthographic Projection

• Case
• 3D rigid
• transformations
• (3 ref. views)

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Orthographic Projection

• Case 3D linear transformations

(2 ref views)
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(No Transcript)
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How to generate the appearances of a group?

• Estimate each parameters range of values
• Sample the space of parameter values
• Generate a new appearance for each sample of
  values

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Estimate the Range of Values of the Parameters
or
and
Using SVD
and
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Estimate the Interval values of the Parameters
(contd)

• Assume normalized coordinates
  
• Use Interval Arithmetic (Moore, 1966)
• (note that the solutions
  will be the identical)

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Models
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Impose rigidity constraints

• For general linear transformations of the object,
  without additional constraints, it is impossible
  to distinguish between rigid and linear but not
  rigid transformation of the object. To impose
  rigidity, additional constraints must be met.

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Unrealistic Views without the constraints
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View generation

• Select two model views
• Move the centroids of the views to (0, 0) to make
  the translation parameters become zeros, such
  that there are no needs to sample them later
• Computer the range of parameters by SVD and IA
• For each sampling step of the parameters (a1, a2,
  a3, b1, b2, b3), generate the novel views if the
  novel view satisfies both the interval constraint
  and the rigidity constraints, store this view as
  a valid view

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Realistic Views
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K-d Tree
K-d tree is a data structure which partitions
space using hyperplanes.
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5 nearest neighbor query (a)
Query View
MSE0.0015
MSE0.0014
MSE0.0016
MSE0.0015
MSE0.0022
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5 nearest neighbor query (b)
Query view
MSE2.0134e-4
MSE6.3495e-4
MSE5.0291e-4
MSE9.3652e-4
MSE0.0017
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5-nearest-neighbor query (c )
Query view
MSE3.1926e-4
MSE5.0356e-4
MSE8.6303e-4
MSE0.0010
MSE0.0013
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Geometric manifold

• By applying PCA, each object can be represented
  as a parametric manifold in two different
  eigenspaces the universal eigenspace and the
  objects own eigenspace. The universal eigenspace
  is computed using the generated transformed views
  of all objects of interest to the recognition
  system, the object eigenspace is computed using
  generated views of an object only. Therefore the
  geometric manifold is parameterized by the
  parameters of the algebraic functions.

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Eigenspace of the car model
Without the rigid constraints
With the rigid constraints
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The 5-nearest neighbor query results in universal
eigenspace (m3)
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The 5-nearest neighbor query results in universal
eigenspace (m4)
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Parameters Prediction
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Training process
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Actual and predicted parameters
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Mixture of Gaussians
A mixture is defined as a weighted sum of K
components where each component is a parametric
density function
Each mixture component is a Gaussian with mean ?
and covariance matrix ?
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EM algorithm

• Initialization
• Expectation step
• Maximization step

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Random projection

• A random projection from n dimensions to d
  dimensions is represented by a d?n matrix. It
  does not depend on the data and can be chosen
  rapidly.
• Data from a mixture of k Gaussians can be
  projected into just O(logk) dimensions while
  still retaining the approximate level of
  separation between clusters.
• Even if the original clusters are highly
  eccentric (i.e. far from spherical), random
  projection will make them more spherical.

S. Dasgupta, Experiments with random
projection, In proc. Of 16th conference on
uncertainty in artificial intelligence, 2001.
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Recognition results by mixture models
The no. of eigenvectors m8, then apply random
projection to 3 dimensional space
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Training stage
Recognition stage
Images from various viewpoints
New image
Convex grouping
Convex grouping
Image groups
Model groups
A coarse k-d tree
Access
Compute index
Selection of reference views
Compute probabilities of Gaussian mixtures
Index Structure
Retrieve
Using SVD IA
Ranking the candidates by probabilities
Estimate the range of parameters of AFoVs
Predict groups by sampling parameter space
Predict the parameters Using NN/SVD
Using constraints
Validated appearances
Verify hypotheses
Compute index
Evaluate match
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A coarse k-d tree
Totally, 2242 groups with group size 8 of 10
models has been used to construct the k-d tree.
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Mixture of Gaussians

• A universal eigenspace has been built by more
  dense views in the transformed space.
• 28 Gaussian mixture models have been built for
  all the groups in the universal eigenspace
  offline.

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Mixture model for Rocket-g2
(Point 8Point 16)
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Mixture model for Tank-g3
(Point 16Point 24)
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Mixture model for Car-g1
(Point 1Point 8)
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Test view

• Test view are generated by applying any
  orthographic projection on the 3D model.
• 2 noise has been added to the test view.
• Assume we can divide the test view into same
  groups as the reference views
• Assume the correspondences are not known, a
  circular shift has been applied to the points in
  the groups of test view to try all the possible
  situations

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Ranking by probabilities-Car view
The first 4 nearest neighbor are chosen in the
coarse k-d tree.
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Ranking by probabilities-Tank view
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Ranking by probabilities-Rocket view
Both of them are correct, because of the symmetry
of the data
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Some verification results
Group 1, MSE8.0339e-5
Group 2, MSE4.3283e-5
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Some verification results
Group 2, MSE5.3829e-5
Group 1, MSE2.5977e-5
Group 3, MSE5.9901e-5
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Some verification results
Group 1, MSE3.9283e-5
Group 1 (Shift 4), MSE3.3383e-5
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Some verification results
Group 2, MSE4.4156e-5
Group 3, MSE3.3107e-5
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Future work

• Apply the system to the real data
• Integrate AFoVs with convex grouping
• Optimal selection of reference views