Multilinear Principal Component Analysis of Tensor Objects for Recognition

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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
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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

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Overview

• MPCA natural extension of PCA
• Multilinear singular value eigentensor
• Input higher-order tensors
• Application gait recognition
• Sample data set 4th-order tensor
• Gait sample half gait cycle (normalized)
• Recognition outperforms baseline algorithm

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Notations

• Vector lowercase boldface
• Matrix uppercase boldface
• Tensor calligraphic letter
• n-mode product
• Scalar product
• Frobenius norm
• n-rank (n-mode vectors)

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Higher-order SVD

• Subtensors of the core tensor S
• All-orthogonality
• Ordered based on
• unitary

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PCA with tensor notation

• Basis vectors (PCs) columns of
• PCA subspace truncate ?
• Projection to feature space

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Multilinear PCA

• Centered input tensor samples
• HOSVD
• Keep columns of ?
• n-mode singular value
• Basis tensor (eigentensor)
• Projection
• MPCA features

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EigenTensorGait for recognition

• Gait sample half gait cycle (3rd-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 L1 norm)
• Sequence matching sum of min-dist

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Best rank approximation

• The original silhouettes

• Best rank-(10,10,3) approximation

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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

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Identification performance
Probe PI () at Rank 1 PI () at Rank 1 PI () at Rank 5 PI () at Rank 5
Probe Baseline MPCA Baseline MPCA
A(GAL) 79 94 96 99
B(GBR) 66 76 81 83
C(GBL) 56 66 76 81
D(CAR) 29 27 61 64
E(CBR) 24 36 55 52
F(CAL) 30 15 46 53
G(CBL) 10 19 33 48
Average 42 48 64 68
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MPCA CMC curves
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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

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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.

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Contact Information

• Haiping Lu
• Email haiping_at_dsp.toronto.edu
• Academic website
• http//www.dsp.toronto.edu/haiping/