10 20

X i a i (i, j) a ij (i, j, k) X x ijk j :j i i: R I J R K L IK JL a 11 a 12... a 1J a 21 a 22... a 2J = a I1 a I2... a IJ = [ 1 1 1 2 1 3... J L 1 J L ] R I K R J K IJ K = [ 1 1 2 2... K K ] I J

a 11 b 11 a 12 b 12... a 1J b 1J a 21 b 12 a 22 b 22... a 2J b 2J = a I1 b I1 a I2 b I2... a IJ b IJ ( )( ) = ( ) = = ( ) = ( ) ( ) T ( ) = T T ( ) = (( T ) ( T )) ( ) T

1 2 : : : 2 :: :: :: X R I 1 I2... I N X = I1 I2 i 1 =1 i 2 =1... IN x 2 i 1 i 2...i N i N =1 X R I I... I x ijk = x ikj = x jik = x jki = x kij = x kji i, j, k = 1,..., I

X R I 1 I2... I N 0 x i1 i 2...i N 0 i 1 = i 2 =... = i N X =

X R I 1 I 2... I N (n) 2 4 5 8 5 2 20 X R 3 4 2 1 4 7 10 13 16 19 22 1 = 2 5 8 11, 2 = 14 17 20 23 3 6 9 12 15 18 21 24 1 4 7 10 13 16 19 22 (1) = 2 5 8 11 14 17 20 23 3 6 9 12 15 18 21 24 1 2 3 13 14 15 (2) = 4 5 6 16 17 18 7 8 9 19 20 21 10 11 12 22 23 24 [ ] 1 2 11 12 (3) = 13 14 23 24 X R I 1 I 2... I n R J In X n I 1... I n 1 J I n+1... I N (X ) I1...I n 1 J I n+1... I N = I n i n =1 x i1 i 2...i n u jin. X Y (n) = (n).

X R I 1 I 2... I n R I n X ˆ n N 1 I 1 I 2... I n 1 I n+1... I N (X ˆ n i1...i n 1 i n+1...i n ) = I n i n=1 x i1 i 2...i N v in

n n = 1, 2,..., N n D M < D M M = 1 D u 1 T 1 1 = 1 n T 1 n 1ˆ = 1 N N n=1 n 1 N N { T 1 N T 1 } 2 = T 1 1 n=1 = 1 N N ( n )( n ) T n=1

t 1 1 1 1 T 1 1 = 1 λ 1 t 1 1 + λ 1 (1 T 1 u 1 ) 1 0 1 = λ 1 1 1 T 1 T 1 1 = 1 T 1 1 = λ 1 1 λ 1 M 1, 2,..., m M λ 1, λ 2,..., λ m M M

= [x 1, x 2,..., x n ] m = [s 1, s 2,..., s n ] = = = I(X; Y ) = H(X) H(X Y )

H(X) X H(X Y ) X Y H(X Y ) = H(X, Y ) H(Y ) H(X, Y ) X Y H(X) = x P (x)logp (x) H(X, Y ) = x,y P (x, y)logp (x, y) X Y ( + ) () + η(t)( f() T )() η(t) f(x) f(x) = (x) f(x) = x (x)

N() = H( Gaussian ) H() H( Gaussian ) N(V ) = E(ϕ(V )) E(ϕ(U)) 2 V U ϕ(.) i temp i = E(ϕ ( T i )) i E(ϕ( T i )) i = temp i temp i temp i = i i=1 j=1 wt i w jw j i = temp i temp i

[ ].5.5 M =.45.55 0.0012 0.0073 0.0003 0.1692

X R I J K R X a r b r c r r=1 a r R I b r R J c r R K r = 1,..., R X R x ijk a ir b jr c kr for i = 1,..., I j = 1,..., J k = 1,..., K r=1 r r r r X [,, ] λ R X λ r r r r [λ;,, ] r=1 X R r=1 [ λ r (1) r (2) r... (N) r λ; (1), (2),..., (N)]

R = T = r r Σ T = Σ = = Σ = R R X R I J K r=1 R X = r r r = [,, ] = [Π, Π, Π] r=1 Π R R

R ˆX X ˆX with ˆX = R λ r r r r = [λ;,, ] r=1

X R λ, (1),..., (N) (n) R In R n = 1,..., N λ (n) R In R n = 1,..., N n = 1,..., N = (1)T (1)... (n 1)T (n 1) (n+1)t (n+1)... (N)T (N) A (n) = X (n) (A (N)... A (n+1) A (n 1)... A (1) )V P Q R X G 1 2 3 = g pqr r r r = [[G;,, ]] p=1 q=1 r=1 R I P R J Q R K R X R I J K

P = Q = R G R P Q R P = Q = R X = G 1 (1) 2 (2)... (N) = [ G; (1), (2),..., (N)] R 1,..., R N R N < rank N (X ) X

X R G, (1),..., (N) n = 1,..., N (N) = R X (n) G = X 1 (1)T 2 (2)T... N (N)T 300 300

X R G, (1),..., (N) (n) R In R n = 1,..., N n = 1,..., N Y = X 1 (1)T... n 1 (n 1)T n+1 (n+1)t... N (N)T (n) = R Y (n) G = X 1 (1)T 2 (2)T... N (N)T R P P R Q Q R R R [G;,, ] = [ G 1 2 3 ;,, ] G Λ

X = G 1 2 = [G;,, ] X = G 1 = [G;,, ]

X(f) = x(t) = + + x(t)e 2jπft dt X(f)e 2jπft df

+ +

CW T ψ x (τ, s) = Ψ ψ x (τ, s) = 1 s x(t)ψ ( t τ ) dt s τ s 1 s

X[n] H n n d n G n a n

10 20 10 20 10 20 10 20 10 10 2, 4, 6 1, 3, 5 10 20 10 20 10 20

800 17

100 100

0.25 4 69 12 76 1

95 F rame i R K L i K L

T R N F R M V i R N+M V i F rame i 25 50 75

N N f = 2 N + 1 N 21 0.125 0.25 0.5 1 1.5 2 55 1155 1 3 5 7 9 1

0.125 0.25 0.5 1 1.5 2 85.17 87.51 90.55 90.81 90.14 86.06 87.36 89.50 94.38 95.25 96.56 95.52 88.38 90.58 93.02 92.88 94.02 89.55 88.53 91.28 93.82 93.77 95.82 90.54 1.5 1.5 1 1.5 100 300 25

100 94.85 89.10 98.26 200 95.51 92.05 98.30 300 97.47 94.83 98.31 400 97.23 95.45 98.30 500 96.98 94.77 98.31 600 96.97 94.15 98.31 700 97.22 93.58 98.30 800 97.22 93.58 98.30 900 96.98 93.54 99.14 1100 97.22 93.58 98.30 1155 97.22 93.58 98.30 3 3 1 96.56 3 99.25 5 98.96 7 98.66

0.5 1 3

scales T T (:, :, s) = wavelet(, scales(s)) scales(s) 2 scales(s) 2 σ 2

M T Method Method M T M T scales Method Method M T

1 1 3

[0 1]

1.4 1.6

1.7 1.5