Automatic Target Recognition Using Algebraic Function of Views

1. Automatic Target Recognition Using Algebraic Function of Views Computer Vision Laboratory Dept. of Computer Science University of Nevada, Reno

2. Outline • Background of algebraic functions of views • Frame work • Imposing rigid constraint • Indexing scheme • Geometric manifold • Mixture of Gaussians • Modified Framework • Future work UNR - Computer Vision Laboratory

3. Orthographic Projection • Case: • 3D rigid • transformations • (3 ref. views) UNR - Computer Vision Laboratory

4. Orthographic Projection • Case: 3D linear transformations (2 ref views) UNR - Computer Vision Laboratory

5. Training stage Recognition stage Images from various viewpoints New image Convex grouping Convex grouping Image groups Model groups Access Compute index Selection of reference views Index Structure Retrieve Using SVD & IA Estimate the range of parameters of AFoVs Predict the parameters Using NN/SVD Predict groups by sampling parameter space Verify hypotheses Using constraints Validated appearances Evaluate match Compute index UNR - Computer Vision Laboratory

6. How to generate the appearances of a group? • Estimate each parameter’s range of values • Sample the space of parameter values • Generate a new appearance for each sample of values UNR - Computer Vision Laboratory

7. Estimate the Range of Values of the Parameters or and Using SVD: and UNR - Computer Vision Laboratory

8. Estimate the Interval values of the Parameters (cont’d) • Assume normalized coordinates: • Use Interval Arithmetic(Moore, 1966) • (note that the solutions will be the identical) UNR - Computer Vision Laboratory

9. Models UNR - Computer Vision Laboratory

10. 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. UNR - Computer Vision Laboratory

11. Unrealistic Views without the constraints UNR - Computer Vision Laboratory

12. 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 UNR - Computer Vision Laboratory

13. Realistic Views UNR - Computer Vision Laboratory

14. P4 P3 P4 P1 P2 P3 P2 P1 K-d Tree K-d tree is a data structure which partitions space using hyperplanes. UNR - Computer Vision Laboratory

15. 5 nearest neighbor query (a) Query View MSE=0.0015 MSE=0.0014 MSE=0.0016 MSE=0.0015 MSE=0.0022 UNR - Computer Vision Laboratory

16. 5 nearest neighbor query (b) Query view MSE=2.0134e-4 MSE=6.3495e-4 MSE=5.0291e-4 MSE=9.3652e-4 MSE=0.0017 UNR - Computer Vision Laboratory

17. 5-nearest-neighbor query (c ) Query view MSE=3.1926e-4 MSE=5.0356e-4 MSE=8.6303e-4 MSE=0.0010 MSE=0.0013 UNR - Computer Vision Laboratory

18. Geometric manifold • By applying PCA, each object can be represented as a parametric manifold in two different eigenspaces: the universal eigenspace and the object’s 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. UNR - Computer Vision Laboratory

19. Eigenspace of the car model Without the rigid constraints With the rigid constraints UNR - Computer Vision Laboratory

20. The 5-nearest neighbor query results in universal eigenspace (m=3) UNR - Computer Vision Laboratory

21. The 5-nearest neighbor query results in universal eigenspace (m=4) UNR - Computer Vision Laboratory

22. Parameters Prediction UNR - Computer Vision Laboratory

23. Training process UNR - Computer Vision Laboratory

24. Actual and predicted parameters UNR - Computer Vision Laboratory

25. 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  UNR - Computer Vision Laboratory

26. EM algorithm • Initialization • Expectation step • Maximization step UNR - Computer Vision Laboratory

27. 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. UNR - Computer Vision Laboratory

28. Recognition results by mixture models The no. of eigenvectors m=8, then apply random projection to 3 dimensional space UNR - Computer Vision Laboratory

29. 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 UNR - Computer Vision Laboratory

30. A coarse k-d tree Totally, 2242 groups with group size 8 of 10 models has been used to construct the k-d tree. UNR - Computer Vision Laboratory

31. 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. UNR - Computer Vision Laboratory

32. Mixture model for Rocket-g2 (Point 8—Point 16) UNR - Computer Vision Laboratory

33. Mixture model for Tank-g3 (Point 16—Point 24) UNR - Computer Vision Laboratory

34. Mixture model for Car-g1 (Point 1—Point 8) UNR - Computer Vision Laboratory

35. 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 UNR - Computer Vision Laboratory

36. Ranking by probabilities-Car view The first 4 nearest neighbor are chosen in the coarse k-d tree. UNR - Computer Vision Laboratory

37. Ranking by probabilities-Tank view UNR - Computer Vision Laboratory

38. Ranking by probabilities-Rocket view Both of them are correct, because of the symmetry of the data UNR - Computer Vision Laboratory

39. Some verification results Group 1, MSE=8.0339e-5 Group 2, MSE=4.3283e-5 UNR - Computer Vision Laboratory

40. Some verification results Group 2, MSE=5.3829e-5 Group 1, MSE=2.5977e-5 Group 3, MSE=5.9901e-5 UNR - Computer Vision Laboratory

41. Some verification results Group 1, MSE=3.9283e-5 Group 1 (Shift 4), MSE=3.3383e-5 UNR - Computer Vision Laboratory

42. Some verification results Group 2, MSE=4.4156e-5 Group 3, MSE=3.3107e-5 UNR - Computer Vision Laboratory

43. Future work • Apply the system to the real data • Integrate AFoVs with convex grouping • Optimal selection of reference views UNR - Computer Vision Laboratory