1. International Conference on Pattern Recognition, Hong Kong, August 2006 1 Multilinear Principal Component Analysis of Tensor Objects for Recognition Haiping Lu, K.N. Plataniotis and A.N. Venetsanopoulos The Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto

2. 2 International Conference on Pattern Recognition, Hong Kong, August 2006 Motivation Real data in pattern recognition High-dimensional: dimensionality reduction Multidimensional: tensors PCA: reshape tensors into vectors Multilinear algebra 2DPCA, 3DPCA Multifactor analysis Objective: multilinear PCA for tensors

3. 3 International Conference on Pattern Recognition, Hong Kong, August 2006 Overview MPCA: natural extension of PCA Multilinear singular value & eigentensor Input: higher-order tensors Application: gait recognition Sample data set: 4 th -order tensor Gait sample: half gait cycle (normalized) Recognition: outperforms baseline algorithm

4. 4 International Conference on Pattern Recognition, Hong Kong, August 2006 Notations Vector: lowercase boldface Matrix: uppercase boldface Tensor: calligraphic letter n-mode product: Scalar product: Frobenius norm: n-rank (n-mode vectors):

5. 5 International Conference on Pattern Recognition, Hong Kong, August 2006 Higher-order SVD Subtensors of the core tensor S All-orthogonality Ordered based on : unitary

6. 6 International Conference on Pattern Recognition, Hong Kong, August 2006 PCA with tensor notation Basis vectors (PCs): columns of PCA subspace: truncate  Projection to feature space:

7. 7 International Conference on Pattern Recognition, Hong Kong, August 2006 Multilinear PCA Centered input tensor samples: HOSVD: Keep columns of  n-mode singular value: Basis tensor (eigentensor): Projection: MPCA features:

8. 8 International Conference on Pattern Recognition, Hong Kong, August 2006 EigenTensorGait for recognition Gait sample: half gait cycle (3 rd -order) To obtain samples: partition based on foreground pixels in silhouettes Noise removal: best rank approximation Temporal normalization: interpolation Feature distance: sum of the absolute differences (equivalent to L 1 norm) Sequence matching: sum of min-dist

9. 9 International Conference on Pattern Recognition, Hong Kong, August 2006 Best rank approximation The original silhouettes Best rank-(10,10,3) approximation

10. 10 International Conference on Pattern Recognition, Hong Kong, August 2006 Experiments Data: USF gait challenge data sets V.1.7 Different conditions: surface, shoe, view Sample size: 64x44x20 Best results: Performance measure: CMCs Results: better overall recognition rate compared with baseline algorithm

11. 11 International Conference on Pattern Recognition, Hong Kong, August 2006 Identification performance ProbeP I (%) at Rank 1P I (%) at Rank 5 BaselineMPCABaselineMPCA A(GAL)79949699 B(GBR)66768183 C(GBL)56667681 D(CAR)29276164 E(CBR)24365552 F(CAL)30154653 G(CBL)10193348 Average42486468

12. 12 International Conference on Pattern Recognition, Hong Kong, August 2006 MPCA CMC curves

13. 13 International Conference on Pattern Recognition, Hong Kong, August 2006 Conclusions MPCA: multilinear extension of PCA Application of MPCA: EigenTensorGait Half gait cycles as gait samples Best rank approximation to reduce noise Temporal normalization by interpolation Future works MPCA to other problems Other multilinear extensions

14. 14 International Conference on Pattern Recognition, Hong Kong, August 2006 Related work Haiping Lu, K.N. Plataniotis and A.N. Venetsanopoulos, "Gait Recognition through MPCA plus LDA", in Proc. Biometrics Symposium 2006 (BSYM 2006), Baltimore, US, September 2006.

15. 15 International Conference on Pattern Recognition, Hong Kong, August 2006 Contact Information Haiping Lu Email: haiping@dsp.toronto.edu Academic website: http://www.dsp.toronto.edu/~haiping/