1,a) 1 2 1 SIFT Bag-of-Visual-Words Bag-of-Visual-Words 1. BoVW [2] BoVW Dense [3] Interest Point [4] 1 376 8515 1 5 1 School of Science and Technology, Gunma University Tenjin-cho 1 5 1, Kiryu-shi, Gunma, 376 8515 Japan 2 980 8579 6 6 05 Graduate School of Engineering, Tohoku University, 6 6 05, Aramaki Aza Aoba, Aoba-ku, Sendai, 980 8579, Japan a) matsuzawa-tomoki@kato-lab.cs.gunma-u.ac.jp BoVW (Histogram Encoding) [2], [5], [6] [7], [8], (Fisher Encoding) [3] VLAD [9] Super Vector [10] Locality Constrained Linear Encoding [11] [11], [12], [13] (Average Pooling) (Max Pooling) [11], [12], [13] [5], [13] [12] Chatfield [14] [3] [6] [15] BoVW c 2015 Information Processing Society of Japan 1
[16], [17]. BoVW K- Visual Word [18] 2. : BoVW (FV) 2.1 Bag-of-Visual-Words (BoVW) BoVW [2] BoVW. (e.g. SIFT [19]), Visual Word [14]. Visual Word. [12] Visual Word K K Visual Word 2.2 (FV) (FV) θ = [θ 1,..., θ m ] p(x θ) [20] X FV T X := [x 1,..., x T ] FV f(x) := θ1 log p(x θ) EX (( θ1 log p(x θ)) 2 ). θm log p(x θ) EX (( θm log p(x θ)) 2 ). p(x θ) K p(x θ) = T K π k N (x t ; µ k, diag(σ k ) 2 ) π k k π k = 1 µ k R d σ k R d θ = [ π 1,..., π K, µ 1,..., µ K, σ1,..., σk ] (2d + 1)K FV (2d + 1)K K π 1,..., π K K FV 2dK [21], [22], [23] 2dK FV [24] E X (( θi log p(x θ)) 2 ) (a) (responsibility) [25] (b) T FV f euc (X) := [ f µ1 euc(x),..., f µk euc (X), f σ1 euc(x),..., f σk euc (X) ] k = 1,..., K c 2015 Information Processing Society of Japan 2
f µ k euc(x) 1 T πk diag(σ k ) 1 Y k,euc γ k,euc, f σ 1 ( k euc(x) diag(σk ) 2 Y k,euc Y k,euc 1 d 1 ) 2T T γk,euc πk Y k,euc := X µ k 1 T γ k,euc R T k γ k,euc t π k N (x t ; µ k, diag(σ k ) 2 ) k π k N (x t ; µ k, diag(σ k ) 2 ). BoVW FV N (x t ; µ k, diag(σ k ) 2 ) Visual Word FV 3. : BoVW FV [1], [26] x, x R d D(x, x ; A) := (x x ) A(x x ) A ( ) ( ) 3.1 BoVW [18] BoWV K Visual Word V := [v 1,..., v K ] R d K J cb-ho (V ; A) := T min D(x t, v kt ; A) k t {1,...,K} t=1 K Greedy T BoVW Bagof-Visual-Words(HoMahaBoVW) d Visual Word v k A k BoVW Bag-of-Visual-Words(HeMahaBoVW) HeMahaBoVW x k argmin D(x, v k ; A k ). k {1,...,K} Visual Word v k 3.2 FV [18] FV A = UΛU W := Λ 1/2 U x (i.e. D(x, x ; A) = W x W x ) [18] : p ho (X θ) := det(w ) T T K π k N (W x t ; µ k, diag(σ k ) 2 ). FV f ho (X) (HoMahaFV) HoMahaFV f ho (X) := [ f µ1 ho (X),..., f µk ho (X), f σ1 ho (X),..., f σk ho (X) ] E X (( θi log p ho (X θ)) 2 ) FV (2.2 ) f µ k ho (X) 1 T πk diag(σ k ) 1 Y k γ k, f σ k ho (X) 1 2T πk ( diag(σk ) 2 Y k Y k 1 d 1 T ) γk Y k := W X µ k 1 T γ k R T t π k N (W x t ; µ k, diag(σ k ) 2 ) k π k N (W x t ; µ k, diag(σ k ) 2 ). HoMahaFV W HoMahaFV (HeMahaFV) FV W 1,..., W K : c 2015 Information Processing Society of Japan 3
p he (X θ) := T K π k det(w k ) N (W k x t ; µ k, diag(σ k ) 2 ). W k k A k Y k := W k X µ k 1 T p he(x θ) µ k σ k FV f µ k he (X) 1 T πk diag(σ k ) 1 Y kγ k, f σ k he (X) 1 2T πk ( diag(σk ) 2 Y k Y k 1 d 1 T ) γ k T γ k t π k det(w k ) N (W k x t ; µ k, diag(σ k ) 2 ) k π k det(w k ) N (W k x t ; µ k, diag(σ k ) 2 ). 4. HeMahaFV 4.1 HeMahaBoVW HeMahaBoVW K Visual Word A 1,..., A K HeMahaBoVW c A (c) c A (c) 0.05 Visual Word Visual Word K K 2 HeMahaBoVW A (c) W (c) K 2 K 2 K p he (X θ) Greedy EM EM E-step M-step c K µ 1,c,..., µ K,c R d, σ 1,c,..., σ K,c R d, π c R K T x 1,..., x T : L(θ (c) ) := det(w (c) ) T π k,c N (W (c) x t ; µ k,c, diag(σ k,c ) 2 ) T K θ (c) := {µ 1,c,..., µ K,c, σ 1,c,..., σ K,c, π c } EM Algorithm 1 EM Algorithm for HeMahaFV Method Input: Observation x 1,..., x T R d and initial values µ (0) 1,c,..., µ(0) K,c Rd, σ (0) 1,c,..., σ(0) K,c Rd, π c R K 1: for l = 1, 2,... do 2: E step Compute γ (l) t,k := π(l 1) k,c k π(l 1) k,c N (W (c) x t ; µ (l 1) k,c, diag(σ (l 1) k,c ) 2 ) N (W (c)x t ; µ (l 1) k,c, diag(σ(l 1) k,c )2 ). for (t, k) {1,..., T } {1,..., K }; 3: M step Update the parameter values by µ (l) k,c := t γ(l) t,k W (c)x t. (σ (l) k,c )2 := π (l) k,c := 1 T t γ(l) t,k t γ(l) for k = 1,..., K ; 4: end for t k,c )) t,k ((W (c)x t µ (l) k,c ) (W (c)x t µ (l) γ (l) t,k. t γ(l) t,k. 4.2 HeMahaFV HeMahaFV K 2 K 2 5. HeMahaBoVW, HeMahaFV 6 c 2015 Information Processing Society of Japan 4
(a) FMD 2 LSP15 15, 200 30 10 (b) LSP15 1 Categorization Performance. BoVW, FV (HoEucBoVW, HoEucFV) HeMahaBoVW (HeEucBoVW) HeMahaFV θ (HeEucFV) [18] (HoMahaBoVW, HoMahaFV) 5.1 3 Dense SIFT Visual Word BoVW K = 1024 FV K = 256 FV Power L2 [27] HeEucFV HeMahaFV L2 Flickr Material Database(FMD)[28] LSP15[5] One-vs-rest SVM FMD Flickr.com 10 100 5.2 1 FMD LSP15 BoVW FV HeMahaFV 5.3 101 CalTech 101(Cal 101)[29] (Cal 10 Cal 20... Cal 50) (Cal 10 Cal 20... Cal 50) Cal 101 10 20 10 50 Cal 101 Top-N Accuracy 2 N 10 1 2 HeMahaFV FV(EucHoFV) 6. BoVW FV HeMahaBoVW HeMahaFV HeMac 2015 Information Processing Society of Japan 5
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