1. An M-ary KMP Classifier for Multi-aspect Target Classification
Xuejun Liao, Hui Li, and Balaji Krishnapuram
Department of Electrical and Computer Engineering
Duke University
International Conference on Acoustics, Speech, and Signal Processing, May l 7-21, 2004, Montreal, Quebec, Canada

2. Outline The M-ary Kernel Matching Pursuit (KMP)
Multi-aspect High Range Resolution (HRR) Target Classification
Results on MSTAR Dataset

3. The M-ary Classification Problem
data
Class 1 
Class 2
Class m
Class M
binary classifier
labeled as 0
One-versus-all rule:
labeled as 1

4. The M-ary Classification Problem (Cont.)
Training data set
yi is the class label of xi
is the importance weight of
One-versus-all (OVA) re-labeling
OVA labels

5. M-ary Kernel Matching Pursuit (KMP)
Decision functions
, m=1,2,…,M
basis functions
weight vectors
Squared error
Objective
Find the  and {w[m]} that minimize e((x),{w[m]})

6. Least-Square Solution of w[m]
where
Substitute
w[m]
back into

7. Selection of Basis Functions
Globally optimal selection
Locally optimal selection
Based on existing basis functions
select the next one
where
is the error decrease due to 

8. Selection of Basis Functions (Cont.)
Simple expression of the error decrease
where

9. Selection of Basis Functions (Cont.)
Update of the M-ary KMP
Time complexity of the M-ary KMP
Nc = number of candidate basis functions
which is independent of M

10. Multi-aspect High Range Resolution (HRR) Target Classification
Sensor-target configuration
Depression angle
Ground target
Airborne radar
sensor
Multi-aspect HRR data with pose information
xJ at J
... S2 S1 S3 S4 S7 S5 S6 S8
x2 at 2
x1 at 1
azimuth angle 

11. Probabilistic partition of azimuth and data
State 3
Azimuth partition
State 2
State 1
State 4 S2 S1 S3 S4 S7 S5 S6 S8
State 8
State 5
State 7
State 6
Data partition
Di is the data set of state i, i=1,2,…, L

12. Multi-aspect decision function
where
is the M-ary KMP trained with
the data in state i

13. The MSTAR dataset The ten MSTAR targets
The data set consists of the X-band HRR signatures of the ten MSTAR targets. The HRR signatures have a range resolution of approximately 0.3 meters.
Training data and testing data are distinct, each having a full coverage of 360 azimuth angles (depression angle fixed at 5), with 0.1 azimuth sampling

14. Results of the M-ary KMP
Confusion matrix of KMP, with L=120 states in azimuth, test sequence spanning 3 azimuth.
Sparsity:

15. Comparison to relevance vector machine (RVM)
Confusion matrix of RVM, with L=120 states in azimuth, test sequence spanning 3 azimuth.
Sparsity:

16. Classification performance versus azimuth span of observations

17. Convergence of the M-ary KMP
Shown for the M-ary KMP in a single state

18. Conclusions
The KMP is extended to the M-ary classification, with a time complexity
which is independent of M
The probabilistic state partition in multi-aspect HRR classification is handled in a natural way by the importance weights .
The results on ten MSTAR targets show that at a comparable classification rate, the M-ary KMP achieves greater sparsity and shorter training time than the RVM.

19. Acknowledgment
This work has been carried out under the supervision of Prof. Lawrence Carin of Duke University