Introduction Sparse Representation Experiments Discussion Robust Face Recognition via Sparse Representation Allen Y. Yang <yang@eecs.berkeley.edu> April 18, 2008, NIST Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion Face Recognition: “Where amazing happens” Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion Face Recognition: “Where amazing happens” Figure: Steve Nash, Kevin Garnett, Jason Kidd. Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion Sparse Representation Sparsity A signal is sparse if most of its coefficients are (approximately) zero. Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion Sparse Representation Sparsity A signal is sparse if most of its coefficients are (approximately) zero. Sparsity in frequency domain 1 Figure: 2-D DCT transform. Sparsity in spatial domain 2 Figure: Gene microarray data. Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion Sparsity in human visual cortex [Olshausen & Field 1997, Serre & Poggio 2006] Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion Sparsity in human visual cortex [Olshausen & Field 1997, Serre & Poggio 2006] Feed-forward : No iterative feedback loop. 1 Redundancy : Average 80-200 neurons for each feature representation. 2 Recognition : Information exchange between stages is not about individual neurons, but 3 rather how many neurons as a group fire together. Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion Problem Formulation Notation 1 Training: For K classes, collect training samples { v 1 , 1 , · · · , v 1 , n 1 } , · · · , { v K , 1 , · · · , v K , nK } ∈ R D . Test: Present a new y ∈ R D , solve for label( y ) ∈ [1 , 2 , · · · , K ]. Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion Problem Formulation Notation 1 Training: For K classes, collect training samples { v 1 , 1 , · · · , v 1 , n 1 } , · · · , { v K , 1 , · · · , v K , nK } ∈ R D . Test: Present a new y ∈ R D , solve for label( y ) ∈ [1 , 2 , · · · , K ]. Data representation in (long) vector form via stacking 2 Figure: Assume 3-channel 640 × 480 image, D = 3 · 640 · 480. Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion Problem Formulation Notation 1 Training: For K classes, collect training samples { v 1 , 1 , · · · , v 1 , n 1 } , · · · , { v K , 1 , · · · , v K , nK } ∈ R D . Test: Present a new y ∈ R D , solve for label( y ) ∈ [1 , 2 , · · · , K ]. Data representation in (long) vector form via stacking 2 Figure: Assume 3-channel 640 × 480 image, D = 3 · 640 · 480. Mixture subspace model for face recognition [Belhumeur et al. 1997, Basri & Jocobs 2003] 3 Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion Classification of Mixture Subspace Model Assume y belongs to Class i 1 y = α i , 1 v i , 1 + α i , 2 v i , 2 + · · · + α i , n 1 v i , n i , = A i α i , where A i = [ v i , 1 , v i , 2 , · · · , v i , n i ]. Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion Classification of Mixture Subspace Model Assume y belongs to Class i 1 y = α i , 1 v i , 1 + α i , 2 v i , 2 + · · · + α i , n 1 v i , n i , = A i α i , where A i = [ v i , 1 , v i , 2 , · · · , v i , n i ]. Nevertheless, Class i is the unknown variable we need to solve: 2 α 1 α 2 = A x ∈ R 3 · 640 · 480 . . Sparse representation y = [ A 1 , A 2 , · · · , A K ] . . α K Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion Classification of Mixture Subspace Model Assume y belongs to Class i 1 y = α i , 1 v i , 1 + α i , 2 v i , 2 + · · · + α i , n 1 v i , n i , = A i α i , where A i = [ v i , 1 , v i , 2 , · · · , v i , n i ]. Nevertheless, Class i is the unknown variable we need to solve: 2 α 1 α 2 = A x ∈ R 3 · 640 · 480 . . Sparse representation y = [ A 1 , A 2 , · · · , A K ] . . α K 0 ··· 0 ] T ∈ R n . x 0 = [ 0 ··· 0 α T 3 i Sparse representation encodes membership! Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion Dimensionality Redunction Construct linear projection R ∈ R d × D , d is the feature dimension . 1 y . = R y = RA x 0 = ˜ A x 0 ∈ R d . ˜ ˜ A ∈ R d × n , but x 0 is unchanged. Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion Dimensionality Redunction Construct linear projection R ∈ R d × D , d is the feature dimension . 1 y . = R y = RA x 0 = ˜ A x 0 ∈ R d . ˜ ˜ A ∈ R d × n , but x 0 is unchanged. Holistic features 2 Eigenfaces [Turk 1991] Fisherfaces [Belhumeur 1997] Laplacianfaces [He 2005] Partial features 3 Unconventional features 4 Downsampled faces Random projections Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion ℓ 1 -Minimization Ideal solution: ℓ 0 -Minimization 1 x ∗ = arg min y = ˜ ( P 0 ) � x � 0 s.t. ˜ A x . x � · � 0 simply counts the number of nonzero terms. However, generally ℓ 0 -minimization is NP-hard . Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion ℓ 1 -Minimization Ideal solution: ℓ 0 -Minimization 1 x ∗ = arg min y = ˜ ( P 0 ) � x � 0 s.t. ˜ A x . x � · � 0 simply counts the number of nonzero terms. However, generally ℓ 0 -minimization is NP-hard . Compressed sensing : Under mild condition, ℓ 0 -minimization is equivalent to 2 x ∗ = arg min y = ˜ ( P 1 ) � x � 1 s.t. ˜ A x , x where � x � 1 = | x 1 | + | x 2 | + · · · + | x n | . Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion ℓ 1 -Minimization Ideal solution: ℓ 0 -Minimization 1 x ∗ = arg min y = ˜ ( P 0 ) � x � 0 s.t. ˜ A x . x � · � 0 simply counts the number of nonzero terms. However, generally ℓ 0 -minimization is NP-hard . Compressed sensing : Under mild condition, ℓ 0 -minimization is equivalent to 2 x ∗ = arg min y = ˜ ( P 1 ) � x � 1 s.t. ˜ A x , x where � x � 1 = | x 1 | + | x 2 | + · · · + | x n | . ℓ 1 -Ball 3 ℓ 1 -Minimization is convex. Solution equal to ℓ 0 -minimization. Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion ℓ 1 -Minimization Routines Matching pursuit [Mallat 1993] Find most correlated vector v i in A with y : i = arg max � y , v j � . 1 A ← A ( i ) , x i ← � y , v i � , y ← y − x i v i . 2 Repeat until � y � < ǫ . 3 Basis pursuit [Chen 1998] Start with number of sparse coefficients m = 1. 1 2 Select m linearly independent vectors B m in A as a basis x m = B † m y . Repeat swapping one basis vector in B m with another vector not in B m if improve � y − B m x m � . 3 If � y − B m x m � 2 < ǫ , stop; Otherwise, m ← m + 1, repeat Step 2. 4 Quadratic solvers : y = A x 0 + z ∈ R d , where � z � 2 < ǫ x ∗ = arg min {� x � 1 + λ � y − A x � 2 } [LASSO, Second-order cone programming]: Much more expensive. Matlab Toolboxes for ℓ 1 -Minimization ℓ 1 -Magic by Candes SparseLab by Donoho cvx by Boyd Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
Introduction Sparse Representation Experiments Discussion Sparse Representation Classification Solve ( P 1 ) ⇒ x 1 . Project x 1 onto face subspaces: 1 0 0 α 1 0 0 α 2 , δ 2 ( x 1 ) = , · · · , δ K ( x 1 ) = . δ 1 ( x 1 ) = . . (1) . . . . . . . 0 0 α K Allen Y. Yang <yang@eecs.berkeley.edu> Robust Face Recognition via Sparse Representation
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