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Stochastic Control272 2. Random signal expansion 2.1 1-D discrete-time signals Let S be a zero mean, stationary, discrete-time random signal, made of M successive samples and let { s 1 , s 2 , . , s M } be a zero mean, uncorrelated random variable sequence, i.e.: E { s n s m } = E s 2 m δ n,m , (1) where δ n,m denotes the Kronecker symbol. It is possible to expand signal S into series of the form: S = M ∑ m=1 s m Ψ m , (2) where { Ψ m } m=1 M corresponds to a M-dimensional deterministic basis. Vectors Ψ m are linked to the choice of random variables sequence { s m } , so there are many decompositions (2). These vectors are determined by considering the mathematical expectation of the product of s m with the random signal S. It comes: Ψ m = 1 E s 2 m E { s m S } . (3) Classically and using a M-dimensional deterministic basis { Φ m } m=1 M , the random vari- ables s m can be expressed by the following relation: s m = S T Φ m . (4) The determination of these random variables depends on the choice of the basis { Φ m } m=1 M . We will use a basis, which provides the uncorrelation of the random variables. Using relations (1) and (4), we can show that the uncorrelation is ensured, when vectors Φ m are solution of the following quadratic form: Φ m T Γ SS Φ n = E s 2 m δ n,m , (5) where Γ SS represents the signal covariance. There is an infinity of sets of vectors obtained by solving the previous equation. Assuming that a basis { Φ m } m=1 M is chosen, we can find random variables using relation (4). Taking into account relations (3) and (4), we obtain as new expression for Ψ m : Ψ m = 1 E s 2 m Γ SS Φ m . (6) Furthermore, using relations (5) and (6), we can show that vectors Ψ m and Φ m are linked by the following bi-orthogonality relation: Φ m T Ψ n = δ n,m . (7) 2.2 Approximation error When the discrete sum, describing the signal expansion (relation (2)), is reduced to Q random variables s m , only an approximation S Q of the signal is obtained: S Q = Q ∑ m=1 s m Ψ m . (8) To evaluate the error induced by the restitution, let us consider the mean square error be- tween signal S and its approximation S Q : = E S − S Q 2 , (9) where . denotes the classical Euclidean norm. Considering the signal variance σ 2 S , it can be easily shown that: = σ 2 S − Q ∑ m=1 E s 2 m Ψ m 2 , (10) which corresponds to: = σ 2 S − Q ∑ m=1 Φ m T Γ SS 2 Φ m Φ m T Γ SS Φ m . (11) When we consider the whole s m sequence (i.e. Q equal to M), the approximation error is weak, and coefficients given by the quadratic form ratio: Φ m T Γ SS 2 Φ m Φ m T Γ SS Φ m are carrying the signal power. 2.3 Second order statistics The purpose of this section is the determination of the S Q autocorrelation and spectral power density. Let Γ S Q S Q be the S Q autocorrelation, we have: Γ S Q S Q [p] = E S Q [q] S ∗ Q [p −q] . (12) Taking into account relation (8) and the uncorrelation of random variables s m , it comes: Γ S Q S Q [p] = Q ∑ m=1 E s 2 m Ψ m [q]Ψ ∗ m [p −q], (13) which leads to, summing all elements of the previous relation: M ∑ q=1 Γ S Q S Q [p] = M ∑ q=1 Q ∑ m=1 E s 2 m Ψ m [q]Ψ ∗ m [p −q]. (14) The stochastic matched lter and its applications to detection and de-noising 273 2. Random signal expansion 2.1 1-D discrete-time signals Let S be a zero mean, stationary, discrete-time random signal, made of M successive samples and let { s 1 , s 2 , . . . , s M } be a zero mean, uncorrelated random variable sequence, i.e.: E { s n s m } = E s 2 m δ n,m , (1) where δ n,m denotes the Kronecker symbol. It is possible to expand signal S into series of the form: S = M ∑ m=1 s m Ψ m , (2) where { Ψ m } m=1 M corresponds to a M-dimensional deterministic basis. Vectors Ψ m are linked to the choice of random variables sequence { s m } , so there are many decompositions (2). These vectors are determined by considering the mathematical expectation of the product of s m with the random signal S. It comes: Ψ m = 1 E s 2 m E { s m S } . (3) Classically and using a M-dimensional deterministic basis { Φ m } m=1 M , the random vari- ables s m can be expressed by the following relation: s m = S T Φ m . (4) The determination of these random variables depends on the choice of the basis { Φ m } m=1 M . We will use a basis, which provides the uncorrelation of the random variables. Using relations (1) and (4), we can show that the uncorrelation is ensured, when vectors Φ m are solution of the following quadratic form: Φ m T Γ SS Φ n = E s 2 m δ n,m , (5) where Γ SS represents the signal covariance. There is an infinity of sets of vectors obtained by solving the previous equation. Assuming that a basis { Φ m } m=1 M is chosen, we can find random variables using relation (4). Taking into account relations (3) and (4), we obtain as new expression for Ψ m : Ψ m = 1 E s 2 m Γ SS Φ m . (6) Furthermore, using relations (5) and (6), we can show that vectors Ψ m and Φ m are linked by the following bi-orthogonality relation: Φ m T Ψ n = δ n,m . (7) 2.2 Approximation error When the discrete sum, describing the signal expansion (relation (2)), is reduced to Q random variables s m , only an approximation S Q of the signal is obtained: S Q = Q ∑ m=1 s m Ψ m . (8) To evaluate the error induced by the restitution, let us consider the mean square error be- tween signal S and its approximation S Q : = E S − S Q 2 , (9) where . denotes the classical Euclidean norm. Considering the signal variance σ 2 S , it can be easily shown that: = σ 2 S − Q ∑ m=1 E s 2 m Ψ m 2 , (10) which corresponds to: = σ 2 S − Q ∑ m=1 Φ m T Γ SS 2 Φ m Φ m T Γ SS Φ m . (11) When we consider the whole s m sequence (i.e. Q equal to M), the approximation error is weak, and coefficients given by the quadratic form ratio: Φ m T Γ SS 2 Φ m Φ m T Γ SS Φ m are carrying the signal power. 2.3 Second order statistics The purpose of this section is the determination of the S Q autocorrelation and spectral power density. Let Γ S Q S Q be the S Q autocorrelation, we have: Γ S Q S Q [p] = E S Q [q] S ∗ Q [p −q] . (12) Taking into account relation (8) and the uncorrelation of random variables s m , it comes: Γ S Q S Q [p] = Q ∑ m=1 E s 2 m Ψ m [q]Ψ ∗ m [p −q], (13) which leads to, summing all elements of the previous relation: M ∑ q=1 Γ S Q S Q [p] = M ∑ q=1 Q ∑ m=1 E s 2 m Ψ m [q]Ψ ∗ m [p −q]. (14) Stochastic Control274 So, we have: Γ S Q S Q [p] = 1 M Q ∑ m=1 E s 2 m M ∑ q=1 Ψ m [q]Ψ ∗ m [p −q], (15) which corresponds to: Γ S Q S Q [p] = 1 M Q ∑ m=1 E s 2 m Γ Ψ m Ψ m [p]. (16) In these conditions, the S Q spectral power density is equal to: γ S Q S Q (ν) = 1 M Q ∑ m=1 E s 2 m γ Ψ m Ψ m (ν). (17) 3. The Stochastic Matched Filter expansion Detecting or de-noising a signal of interest S, corrupted by an additive or multiplicative noise N is a usual signal processing problem. We can find in the literature several processing meth- ods for solving this problem. One of them is based on a stochastic extension of the matched filter notion (Cavassilas, 1991; Chaillan et al., 2007; 2005). The signal of interest pattern is never perfectly known, so it is replaced by a random signal allowing a new formulation of the signal to noise ratio. The optimization of this ratio leads to design a bench of filters and regrouping them strongly increases the signal to noise ratio. 3.1 1-D discrete-time signals: signal-independent additive noise case Let us consider a noise-corrupted signal Z, made of M successive samples and corresponding to the superposition of a signal of interest S with a colored noise N. If we consider the signal and noise variances, σ 2 S and σ 2 N , we have: Z = σ S S 0 + σ N N 0 , (18) with E S 0 2 = 1 and E N 0 2 = 1. In the previous relation, reduced signals S 0 and N 0 are assumed to be independent, stationary and with zero-mean. It is possible to expand noise-corrupted signal Z into a weighted sum of known vectors Ψ m by uncorrelated random variables z m , as described in relation (2). These uncorrelated ran- dom variables are determined using the scalar product between noise-corrupted signal Z and deterministic vectors Φ m (see relation (4)). In order to determine basis { Φ m } m=1 M , let us describe the matched filter theory. If we consider a discrete-time, stationary, known input sig- nal s, made of M successive samples, corrupted by an ergodic reduced noise N 0 , the matched filter theory consists of finding an impulse response Φ, which optimizes the signal to noise ra- tio ρ. Defined as the ratio of the square of signal amplitude to the square of noise amplitude, ρ is given by: ρ = | s T Φ| 2 E |N 0 T Φ| 2 . (19) When the signal is not deterministic, i.e. a random signal S 0 , this ratio becomes (Cavassilas, 1991): ρ = E |S 0 T Φ| 2 E |N 0 T Φ| 2 , (20) which leads to: ρ = Φ T Γ S 0 S 0 Φ Φ T Γ N 0 N 0 Φ , (21) where Γ S 0 S 0 and Γ N 0 N 0 represent signal and noise reduced covariances respectively. Relation (21) corresponds to the ratio of two quadratic forms. It is a Rayleigh quotient. For this reason, the signal to noise ratio ρ is maximized when the impulse response Φ corresponds to the eigenvector Φ 1 associated to the greatest eigenvalue λ 1 of the following generalized eigenvalue problem: Γ S 0 S 0 Φ m = λ m Γ N 0 N 0 Φ m . (22) Let us consider the signal and noise expansions, we have: S 0 = M ∑ m=1 s m Ψ m N 0 = M ∑ m=1 η m Ψ m , (23) where the random variables defined by: s m = Φ m T S 0 η m = Φ m T N 0 (24) are not correlated: Φ m T Γ S 0 S 0 Φ n = E s 2 m δ n,m Φ m T Γ N 0 N 0 Φ n = E η 2 m δ n,m . (25) After a normalization step; it is possible to rewrite relations (25) as follows: Φ m T Γ S 0 S 0 Φ n = E s 2 m E η 2 m δ n,m Φ m T Γ N 0 N 0 Φ n = δ n,m . (26) Let P be a matrix made up of column vectors Φ m , i.e.: P = ( Φ 1 , Φ 2 , . . . , Φ M ) . (27) In these conditions, it comes: P T Γ N 0 N 0 P = I, (28) where I corresponds to the identity matrix. This leads to: P T Γ N 0 N 0 P −1 = I ⇔ P −1 Γ N 0 N 0 −1 P −T = I ⇔ P T = P −1 Γ N 0 N 0 −1 . (29) Let D be the following diagonal matrix: D = E s 2 1 /E η 2 1 0 . . . . . . 0 0 E s 2 2 /E η 2 2 0 . . . 0 . . . . . . . . . . . . . . . 0 . . . 0 E s 2 M −1 /E η 2 M −1 0 0 . . . . . . 0 E s 2 M /E η 2 M . (30) The stochastic matched lter and its applications to detection and de-noising 275 So, we have: Γ S Q S Q [p] = 1 M Q ∑ m=1 E s 2 m M ∑ q=1 Ψ m [q]Ψ ∗ m [p −q], (15) which corresponds to: Γ S Q S Q [p] = 1 M Q ∑ m=1 E s 2 m Γ Ψ m Ψ m [p]. (16) In these conditions, the S Q spectral power density is equal to: γ S Q S Q (ν) = 1 M Q ∑ m=1 E s 2 m γ Ψ m Ψ m (ν). (17) 3. The Stochastic Matched Filter expansion Detecting or de-noising a signal of interest S, corrupted by an additive or multiplicative noise N is a usual signal processing problem. We can find in the literature several processing meth- ods for solving this problem. One of them is based on a stochastic extension of the matched filter notion (Cavassilas, 1991; Chaillan et al., 2007; 2005). The signal of interest pattern is never perfectly known, so it is replaced by a random signal allowing a new formulation of the signal to noise ratio. The optimization of this ratio leads to design a bench of filters and regrouping them strongly increases the signal to noise ratio. 3.1 1-D discrete-time signals: signal-independent additive noise case Let us consider a noise-corrupted signal Z, made of M successive samples and corresponding to the superposition of a signal of interest S with a colored noise N. If we consider the signal and noise variances, σ 2 S and σ 2 N , we have: Z = σ S S 0 + σ N N 0 , (18) with E S 0 2 = 1 and E N 0 2 = 1. In the previous relation, reduced signals S 0 and N 0 are assumed to be independent, stationary and with zero-mean. It is possible to expand noise-corrupted signal Z into a weighted sum of known vectors Ψ m by uncorrelated random variables z m , as described in relation (2). These uncorrelated ran- dom variables are determined using the scalar product between noise-corrupted signal Z and deterministic vectors Φ m (see relation (4)). In order to determine basis { Φ m } m=1 M , let us describe the matched filter theory. If we consider a discrete-time, stationary, known input sig- nal s, made of M successive samples, corrupted by an ergodic reduced noise N 0 , the matched filter theory consists of finding an impulse response Φ, which optimizes the signal to noise ra- tio ρ. Defined as the ratio of the square of signal amplitude to the square of noise amplitude, ρ is given by: ρ = | s T Φ| 2 E |N 0 T Φ| 2 . (19) When the signal is not deterministic, i.e. a random signal S 0 , this ratio becomes (Cavassilas, 1991): ρ = E |S 0 T Φ| 2 E |N 0 T Φ| 2 , (20) which leads to: ρ = Φ T Γ S 0 S 0 Φ Φ T Γ N 0 N 0 Φ , (21) where Γ S 0 S 0 and Γ N 0 N 0 represent signal and noise reduced covariances respectively. Relation (21) corresponds to the ratio of two quadratic forms. It is a Rayleigh quotient. For this reason, the signal to noise ratio ρ is maximized when the impulse response Φ corresponds to the eigenvector Φ 1 associated to the greatest eigenvalue λ 1 of the following generalized eigenvalue problem: Γ S 0 S 0 Φ m = λ m Γ N 0 N 0 Φ m . (22) Let us consider the signal and noise expansions, we have: S 0 = M ∑ m=1 s m Ψ m N 0 = M ∑ m=1 η m Ψ m , (23) where the random variables defined by: s m = Φ m T S 0 η m = Φ m T N 0 (24) are not correlated: Φ m T Γ S 0 S 0 Φ n = E s 2 m δ n,m Φ m T Γ N 0 N 0 Φ n = E η 2 m δ n,m . (25) After a normalization step; it is possible to rewrite relations (25) as follows: Φ m T Γ S 0 S 0 Φ n = E s 2 m E η 2 m δ n,m Φ m T Γ N 0 N 0 Φ n = δ n,m . (26) Let P be a matrix made up of column vectors Φ m , i.e.: P = ( Φ 1 , Φ 2 , . . . , Φ M ) . (27) In these conditions, it comes: P T Γ N 0 N 0 P = I, (28) where I corresponds to the identity matrix. This leads to: P T Γ N 0 N 0 P −1 = I ⇔ P −1 Γ N 0 N 0 −1 P −T = I ⇔ P T = P −1 Γ N 0 N 0 −1 . (29) Let D be the following diagonal matrix: D = E s 2 1 /E η 2 1 0 . . . . . . 0 0 E s 2 2 /E η 2 2 0 . . . 0 . . . . . . . . . . . . . . . 0 . . . 0 E s 2 M −1 /E η 2 M −1 0 0 . . . . . . 0 E s 2 M /E η 2 M . (30) Stochastic Control276 It comes: P T Γ S 0 S 0 P = D, (31) which corresponds to, taking into account relation (29): P −1 Γ N 0 N 0 −1 Γ S 0 S 0 P = D ⇔ Γ N 0 N 0 −1 Γ S 0 S 0 P = PD ⇔ Γ S 0 S 0 P = Γ N 0 N 0 PD, (32) which leads to: Γ S 0 S 0 Φ m = E s 2 m E η 2 m Γ N 0 N 0 Φ m . (33) This last equation shows, on the one hand, that λ m equals E s 2 m /E η 2 m and, on the other hand, that the only basis { Φ m } m=1 M allowing the simultaneous uncorrelation of the random variables coming from the signal and the noise is made up of vectors Φ m solution of the generalized eigenvalue problem (22). We have E η 2 m = 1 and E s 2 m = λ m when the eigenvectors Φ m are normalized as follows: Φ m T Γ N 0 N 0 Φ m = 1, (34) In these conditions and considering relation (6), the deterministic vectors Ψ m of the noise- corrupted signal expansion are given by: Ψ m = Γ N 0 N 0 Φ m . (35) In this context, the noise-corrupted signal expansion is expressed as follows: Z = M ∑ m=1 (σ S s m + σ N η m )Ψ m , (36) so that, the quadratic moment of the m th coefficient z m of the noise-corrupted signal expansion is given by: E z 2 m = E (σ S s m + σ N η m ) 2 , (37) which corresponds to: σ 2 S λ m + σ 2 N + σ S σ N Φ m T ( Γ S 0 N 0 + Γ N 0 S 0 ) Φ m (38) Signal and noise being independent and one of them at least being zero mean, we can assume that the cross-correlation matrices, Γ S 0 N 0 and Γ N 0 S 0 , are weak. In this condition, the signal to noise ratio ρ m of component z m corresponds to the native signal to noise ratio times eigenvalue λ m : ρ m = σ 2 S σ 2 N λ m . (39) So, an approximation S Q of the signal of interest (the filtered noise-corrupted signal) can be built by keeping only those components associated to eigenvalues greater than a certain threshold. In any case this threshold is greater than one. 3.2 Extension to 2-D discrete-space signals We consider now a M × M pixels two-dimensional noise-corrupted signal, Z, which corre- sponds to a signal of interest S disturbed by a noise N. The two-dimensional extension of the theory developed in the previous section gives: Z = M 2 ∑ m=1 z m Ψ m , (40) where { Ψ m } m=1 M 2 is a M 2 -dimensional basis of M × M matrices. Random variables z m are determined, using a M 2 -dimensional basis { Φ m } m=1 M 2 of M × M matrices, as follows: z m = M ∑ p,q=1 Z[p, q]Φ m [p, q]. (41) These random variables will be not correlated, if matrices Φ m are solution of the two- dimensional extension of the generalized eigenvalue problem (22): M ∑ p 1 ,q 1 =1 Γ S 0 S 0 [p 1 − p 2 , q 1 −q 2 ]Φ m [p 1 , q 1 ] = λ n M ∑ p 1 ,q 1 =1 Γ N 0 N 0 [p 1 − p 2 , q 1 −q 2 ]Φ m [p 1 , q 1 ], (42) for all p 2 , q 2 = 1, . . . , M. Assuming that Φ m are normalized as follows: M ∑ p 1 ,p 2 ,q 1 ,q 2 =1 Γ N 0 N 0 [p 1 − p 2 , q 1 −q 2 ]Φ m [p 1 , q 1 ]Φ m [p 2 , q 2 ] = 1, (43) the basis { Ψ m } m=1 M 2 derives from: Ψ m [p 1 , q 1 ] = M ∑ p 2 ,q 2 =1 Γ N 0 N 0 [p 1 − p 2 , q 1 −q 2 ]Φ m [p 2 , q 2 ]. (44) As for the 1-D discrete-time signals case, using such an expansion leads to a signal to noise ratio of component z m equal to the native signal to noise ratio times eigenvalue λ m (see relation (39)). So, all Φ m associated to eigenvalues λ m greater than a certain level - in any case greater than one - can contribute to an improvement of the signal to noise ratio. 3.3 The white noise case When N corresponds to a white noise, its reduced covariance is: Γ N 0 N 0 [p −q] = δ[p − q]. (45) Thus, the generalized eigenvalue problem (22) leading to the determination of vectors Φ m and associated eigenvalues is reduced to: Γ S 0 S 0 Φ m = λ m Φ m . (46) The stochastic matched lter and its applications to detection and de-noising 277 It comes: P T Γ S 0 S 0 P = D, (31) which corresponds to, taking into account relation (29): P −1 Γ N 0 N 0 −1 Γ S 0 S 0 P = D ⇔ Γ N 0 N 0 −1 Γ S 0 S 0 P = PD ⇔ Γ S 0 S 0 P = Γ N 0 N 0 PD, (32) which leads to: Γ S 0 S 0 Φ m = E s 2 m E η 2 m Γ N 0 N 0 Φ m . (33) This last equation shows, on the one hand, that λ m equals E s 2 m /E η 2 m and, on the other hand, that the only basis { Φ m } m=1 M allowing the simultaneous uncorrelation of the random variables coming from the signal and the noise is made up of vectors Φ m solution of the generalized eigenvalue problem (22). We have E η 2 m = 1 and E s 2 m = λ m when the eigenvectors Φ m are normalized as follows: Φ m T Γ N 0 N 0 Φ m = 1, (34) In these conditions and considering relation (6), the deterministic vectors Ψ m of the noise- corrupted signal expansion are given by: Ψ m = Γ N 0 N 0 Φ m . (35) In this context, the noise-corrupted signal expansion is expressed as follows: Z = M ∑ m=1 (σ S s m + σ N η m )Ψ m , (36) so that, the quadratic moment of the m th coefficient z m of the noise-corrupted signal expansion is given by: E z 2 m = E (σ S s m + σ N η m ) 2 , (37) which corresponds to: σ 2 S λ m + σ 2 N + σ S σ N Φ m T ( Γ S 0 N 0 + Γ N 0 S 0 ) Φ m (38) Signal and noise being independent and one of them at least being zero mean, we can assume that the cross-correlation matrices, Γ S 0 N 0 and Γ N 0 S 0 , are weak. In this condition, the signal to noise ratio ρ m of component z m corresponds to the native signal to noise ratio times eigenvalue λ m : ρ m = σ 2 S σ 2 N λ m . (39) So, an approximation S Q of the signal of interest (the filtered noise-corrupted signal) can be built by keeping only those components associated to eigenvalues greater than a certain threshold. In any case this threshold is greater than one. 3.2 Extension to 2-D discrete-space signals We consider now a M × M pixels two-dimensional noise-corrupted signal, Z, which corre- sponds to a signal of interest S disturbed by a noise N. The two-dimensional extension of the theory developed in the previous section gives: Z = M 2 ∑ m=1 z m Ψ m , (40) where { Ψ m } m=1 M 2 is a M 2 -dimensional basis of M × M matrices. Random variables z m are determined, using a M 2 -dimensional basis { Φ m } m=1 M 2 of M × M matrices, as follows: z m = M ∑ p,q=1 Z[p, q]Φ m [p, q]. (41) These random variables will be not correlated, if matrices Φ m are solution of the two- dimensional extension of the generalized eigenvalue problem (22): M ∑ p 1 ,q 1 =1 Γ S 0 S 0 [p 1 − p 2 , q 1 −q 2 ]Φ m [p 1 , q 1 ] = λ n M ∑ p 1 ,q 1 =1 Γ N 0 N 0 [p 1 − p 2 , q 1 −q 2 ]Φ m [p 1 , q 1 ], (42) for all p 2 , q 2 = 1, . . . , M. Assuming that Φ m are normalized as follows: M ∑ p 1 ,p 2 ,q 1 ,q 2 =1 Γ N 0 N 0 [p 1 − p 2 , q 1 −q 2 ]Φ m [p 1 , q 1 ]Φ m [p 2 , q 2 ] = 1, (43) the basis { Ψ m } m=1 M 2 derives from: Ψ m [p 1 , q 1 ] = M ∑ p 2 ,q 2 =1 Γ N 0 N 0 [p 1 − p 2 , q 1 −q 2 ]Φ m [p 2 , q 2 ]. (44) As for the 1-D discrete-time signals case, using such an expansion leads to a signal to noise ratio of component z m equal to the native signal to noise ratio times eigenvalue λ m (see relation (39)). So, all Φ m associated to eigenvalues λ m greater than a certain level - in any case greater than one - can contribute to an improvement of the signal to noise ratio. 3.3 The white noise case When N corresponds to a white noise, its reduced covariance is: Γ N 0 N 0 [p −q] = δ[p − q]. (45) Thus, the generalized eigenvalue problem (22) leading to the determination of vectors Φ m and associated eigenvalues is reduced to: Γ S 0 S 0 Φ m = λ m Φ m . (46) Stochastic Control278 In this context, we can show that basis vectors Ψ m and Φ m are equal. Thus, in the particular case of a white noise, the stochastic matched filter theory is identical to the Karhunen-Loève expansion (Karhunen, 1946; Loève, 1955): Z = M ∑ m=1 z m Φ m . (47) One can show that when the signal covariance is described by a decreasing exponential func- tion (Γ S 0 S 0 (t 1 , t 2 ) = e −α|t 1 −t 2 | , with α ∈ R +∗ ), basis { Φ m } m=1 M corresponds to the Fourier basis (Vann Trees, 1968), so that the Fourier expansion is a particular case of the Karhunen- Loève expansion, which is a particular case of the stochastic matched filter expansion. 3.4 The speckle noise case Some airborne SAR (Synthetic Aperture Radar) imaging devices randomly generate their own corrupting signal, called the speckle noise, generally described as a multiplicative noise (Tur et al., 1982). This is due to the complexity of the techniques developed to get the best resolution of the ground. Given experimental data accuracy and quality, these systems have been used in sonars (SAS imaging device), with similar characteristics. Under these conditions, we cannot anymore consider the noise-corrupted signal as described in (18), so its expression becomes: Z = S. ∗N, (48) where . ∗ denotes the term by term product. In order to fall down in a known context, let consider the Kuan approach (Kuan et al., 1985). Assuming that the multiplicative noise presents a stationary mean ( ¯ N = E{N}), we can define the following normalized observation: Z norm = Z/ ¯ N. (49) In this condition, we can represent (49) in terms of signal plus signal-dependent additive noise: Z norm = S + N − ¯ N ¯ N . ∗S. (50) Let N a be this signal-dependent additive colored noise: N a = ( N/ ¯ N −1 ) . ∗S. (51) Under these conditions, the mean quadratic value of the m th component z m of the normalized observation expansion is: E z 2 m = σ 2 S λ n + σ 2 N a + σ S σ N a Φ m T Γ S 0 N a 0 + Γ N a 0 S 0 Φ m , (52) where N a 0 corresponds to the reduced noise N a . Consequently, the signal to noise ratio ρ m becomes: ρ m = σ 2 S λ m σ 2 N a + σ S σ N a Φ m T Γ S 0 N a 0 + Γ N a 0 S 0 Φ m . (53) As S 0 and ( N/ ¯ N −1 ) are independent, it comes: Γ S 0 N a 0 [p 1 , p 2 , q 1 , q 2 ] = E { S 0 [p 1 , q 1 ]N a 0 [p 2 , q 2 ] } , (54) which is equal to: 1 σ N a E { S 0 [p 1 , q 1 ]S[p 2 , q 2 ] } E {N[p 2 , q 2 ]} ¯ N −1 =0 = 0. (55) So that, the cross-correlation matrices between signal S 0 and signal-dependent noise N a 0 van- ishes. For this reason, signal to noise ratio in a context of multiplicative noise like the speckle noise, expanded into the stochastic matched filter basis has the same expression than in the case of an additive noise. 4. The Stochastic Matched Filter in a de-noising context In this section, we present the stochastic matched filtering in a de-noising context for 1-D discrete time signals. The given results can easily be extended to higher dimensions. 4.1 Bias estimator Let Z be a M-dimensional noise corrupted observed signal. The use of the stochastic matched filter as a restoring process is based on the decomposition of this observation, into a random variable finite sequence z m on the { Ψ m } m=1 M basis. An approximation S Q is obtained with the z m coefficients and the Q basis vectors Ψ m , with Q lower than M: S Q = Q ∑ m=1 z m Ψ m . (56) If we examine the M-dimensional vector E S Q , we have: E S Q = E Q ∑ m=1 Ψ m z m = Q ∑ m=1 Ψ m Φ T m E { Z } (57) Using the definition of noise-corrupted signal Z, it comes: E S Q = Q ∑ m=1 Ψ m Φ T m ( E { S } + E { N } ) . (58) Under these conditions, the estimator bias B S Q can be expressed as follows: B S Q = E S Q −S = Q ∑ m=1 Ψ m Φ T m −I E { S } + Q ∑ m=1 Ψ m Φ T m E { N } , (59) The stochastic matched lter and its applications to detection and de-noising 279 In this context, we can show that basis vectors Ψ m and Φ m are equal. Thus, in the particular case of a white noise, the stochastic matched filter theory is identical to the Karhunen-Loève expansion (Karhunen, 1946; Loève, 1955): Z = M ∑ m=1 z m Φ m . (47) One can show that when the signal covariance is described by a decreasing exponential func- tion (Γ S 0 S 0 (t 1 , t 2 ) = e −α|t 1 −t 2 | , with α ∈ R +∗ ), basis { Φ m } m=1 M corresponds to the Fourier basis (Vann Trees, 1968), so that the Fourier expansion is a particular case of the Karhunen- Loève expansion, which is a particular case of the stochastic matched filter expansion. 3.4 The speckle noise case Some airborne SAR (Synthetic Aperture Radar) imaging devices randomly generate their own corrupting signal, called the speckle noise, generally described as a multiplicative noise (Tur et al., 1982). This is due to the complexity of the techniques developed to get the best resolution of the ground. Given experimental data accuracy and quality, these systems have been used in sonars (SAS imaging device), with similar characteristics. Under these conditions, we cannot anymore consider the noise-corrupted signal as described in (18), so its expression becomes: Z = S. ∗N, (48) where . ∗ denotes the term by term product. In order to fall down in a known context, let consider the Kuan approach (Kuan et al., 1985). Assuming that the multiplicative noise presents a stationary mean ( ¯ N = E{N}), we can define the following normalized observation: Z norm = Z/ ¯ N. (49) In this condition, we can represent (49) in terms of signal plus signal-dependent additive noise: Z norm = S + N − ¯ N ¯ N . ∗S. (50) Let N a be this signal-dependent additive colored noise: N a = ( N/ ¯ N −1 ) . ∗S. (51) Under these conditions, the mean quadratic value of the m th component z m of the normalized observation expansion is: E z 2 m = σ 2 S λ n + σ 2 N a + σ S σ N a Φ m T Γ S 0 N a 0 + Γ N a 0 S 0 Φ m , (52) where N a 0 corresponds to the reduced noise N a . Consequently, the signal to noise ratio ρ m becomes: ρ m = σ 2 S λ m σ 2 N a + σ S σ N a Φ m T Γ S 0 N a 0 + Γ N a 0 S 0 Φ m . (53) As S 0 and ( N/ ¯ N −1 ) are independent, it comes: Γ S 0 N a 0 [p 1 , p 2 , q 1 , q 2 ] = E { S 0 [p 1 , q 1 ]N a 0 [p 2 , q 2 ] } , (54) which is equal to: 1 σ N a E { S 0 [p 1 , q 1 ]S[p 2 , q 2 ] } E {N[p 2 , q 2 ]} ¯ N −1 =0 = 0. (55) So that, the cross-correlation matrices between signal S 0 and signal-dependent noise N a 0 van- ishes. For this reason, signal to noise ratio in a context of multiplicative noise like the speckle noise, expanded into the stochastic matched filter basis has the same expression than in the case of an additive noise. 4. The Stochastic Matched Filter in a de-noising context In this section, we present the stochastic matched filtering in a de-noising context for 1-D discrete time signals. The given results can easily be extended to higher dimensions. 4.1 Bias estimator Let Z be a M-dimensional noise corrupted observed signal. The use of the stochastic matched filter as a restoring process is based on the decomposition of this observation, into a random variable finite sequence z m on the { Ψ m } m=1 M basis. An approximation S Q is obtained with the z m coefficients and the Q basis vectors Ψ m , with Q lower than M: S Q = Q ∑ m=1 z m Ψ m . (56) If we examine the M-dimensional vector E S Q , we have: E S Q = E Q ∑ m=1 Ψ m z m = Q ∑ m=1 Ψ m Φ T m E { Z } (57) Using the definition of noise-corrupted signal Z, it comes: E S Q = Q ∑ m=1 Ψ m Φ T m ( E { S } + E { N } ) . (58) Under these conditions, the estimator bias B S Q can be expressed as follows: B S Q = E S Q −S = Q ∑ m=1 Ψ m Φ T m −I E { S } + Q ∑ m=1 Ψ m Φ T m E { N } , (59) Stochastic Control280 where I denotes the M × M identity matrix. Furthermore, if we consider the signal of interest expansion, we have: S = M ∑ m=1 Ψ m Φ T m S, (60) so that, by identification, it comes: M ∑ m=1 Ψ m Φ T m = I. (61) In this condition, relation (59) can be rewritten as follows: B S Q = − M ∑ m=Q+1 Ψ m Φ T m E { S } + Q ∑ m=1 Ψ m Φ T m E { N } . (62) This last equation corresponds to the estimator bias when no assumption is made on the signal and noise mean values. In our case, signal and noise are both supposed zero-mean, so that the stochastic matched filter allows obtaining an unbiased estimation of the signal of interest. 4.2 De-noising using a mean square error minimization 4.2.1 Problem description In many signal processing applications, it is necessary to estimate a signal of interest disturbed by an additive or multiplicative noise. We propose here to use the stochastic matched filtering technique as a de-noising process, such as the mean square error between the signal of interest and its approximation will be minimized. 4.2.2 Principle In the general theory of stochastic matched filtering, Q is chosen so as the Q first eigenvalues, coming from the generalized eigenvalue problem, are greater than one, in order to enhance the m th component of the observation. To improve this choice, let us consider the mean square error between the signal of interest S and its approximation S Q : = E S − S Q T S − S Q . (63) It is possible to show that this error, function of Q, can be written as: (Q) = σ 2 S 1 − Q ∑ m=1 λ n Ψ m 2 + σ 2 N Q ∑ m=1 Ψ m 2 . (64) The integer Q is chosen so as to minimize the relation (64). It particularly verifies: ( (Q) −(Q −1) ) < 0 & ( (Q + 1) −(Q) ) > 0, let us explicit these two inequalities; on the one hand: (Q + 1) −(Q ) = σ 2 N −σ 2 S λ Q+1 Ψ Q+1 2 > 0 and on the other hand: (Q) −(Q −1 ) = σ 2 N −σ 2 S λ Q Ψ Q 2 < 0. Hence, integer Q verifies: σ 2 S λ Q > σ 2 N > σ 2 S λ Q+1 . The dimension of the basis { Ψ m } m=1 Q , which minimizes the mean square error between the signal of interest and its approximation, is the number of eigenvalues λ m verifying: σ 2 S σ 2 N λ m > 1, (65) where σ 2 S σ 2 N is the signal to noise ratio before processing. Consequently, if the observation has a high enough signal to noise ratio, many Ψ m will be considered for the filtering (so that S Q tends to be equal to Z), and in the opposite case, only a few number will be chosen. In these conditions, this filtering technique applied to an obser- vation Z with an initial signal to noise ratio S N Z substantially enhances the signal of interest perception. Indeed, after processing, the signal to noise ratio S N S Q becomes: S N S Q = S N Z Q ∑ m=1 λ m Ψ m 2 Q ∑ m=1 Ψ m 2 . (66) 4.2.3 The Stochastic Matched Filter As described in a forthcoming section, the stochastic matched filtering method is applied us- ing a sliding sub-window processing. Therefore, let consider a K-dimensional vector Z k cor- responding to the data extracted from a window centered on index k of the noisy data, i.e.: Z k T = Z k − K −1 2 , . . . , Z [k], . . . , Z k + K −1 2 . (67) This way, M sub-windows Z k are extracted to process the whole observation, with k = 1, . . . , M. Furthermore, to reduce the edge effects,the noisy data can be previously completed with zeros or using a mirror effect on its edges. According to the sliding sub-window processing, only the sample located in the middle of the window is estimated, so that relation (56) becomes: S Q[k] [k] = Q[k] ∑ m=1 z m,k Ψ m K + 1 2 , (68) with: z m,k = Z k T Φ m (69) [...]... 80 2 2 3 3 60 4 40 5 6 20 7 8 0 9 Azimuth (m) Azimuth (m) 50 (b) Result of the detection algorithm Noisy data (dB magnitude) 0 Result of the detection algorithm (U152>T152 in white) 0 Noisy data (dB magnitude) 0 4 5 6 7 8 9 10 68 70 72 74 76 78 Sight (m) 80 82 84 86 88 −20 (c) SAS image representing two mine like objects: a sphere and a cylinder lying on its side 10 68 70 72 74 76 78 Sight (m) 80 82 ... λ, p(z/H0 ) D0 (87 ) 290 Stochastic Control where λ is the convenient threshold Taking into account relations (83 ), (84 ) and (85 ), it comes: 2 M > D1 σN λm 2 2 < D0 2 2 (ln λ − M ln σN ) + ∑ ln σS λm + σN 2 + σN σS m =1 M ∑ 2 m=1 σS λm UM (88 ) TM In these conditions, the detection and the false alarm probabilities are equal to: ∞ Pd = TM ∞ pUM (u/H1 )du and Pf a = TM pUM (u/H0 )du (89 ) So, the detection... Z = −12 dB) The stochastic matched filter and its applications to detection and de-noising 293 1 0.9 ρ0= − 12 dB Detection probability (Pd) 0 .8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.2 0.4 0.6 False alarm probability (Pfa) 0 .8 1 Fig 6 ROC curves for a SNR taking values in [−12 dB; 12 dB] 1 3 0 .8 0.6 2 0.4 0.2 Magnitude Magnitude 1 0 0 −0.2 −1 −0.4 −2 −3 0 −0.6 −0 .8 200 400 600 Time (sec) 80 0 1000 1200 (a)... (sec) 80 0 1000 1200 (b) Result of the detection algorithm −3 7 x 10 1400 6 pU (u/H1) in black lines, pU (u/H0) in red lines 1200 Magnitude 95 1000 80 0 95 600 400 5 4 3 2 1 200 0 200 400 600 Time (sec) 80 0 1000 1200 (c) U M and TM according to (88 ) Fig 7 1st experiment: Whale echoes detection 0 0 500 1000 u values 1500 2000 2500 (d) Probability density functions pUM (u/H0 ) and pUM (u/H1 ) 294 Stochastic. .. (75) The result power spectral density is presented on figure 2.b −1 −1 2.2 −0 .8 5.5 −0 .8 2 5 1 .8 −0.6 4.5 −0.4 1.6 −0.4 4 −0.2 1.4 −0.2 3.5 1.2 0 1 0.2 0 .8 0.4 0.6 0.6 0.4 0 .8 1 −1 0.2 −0.5 0 Normalized frequencies 0.5 (a) Estimated noise PSD 1 Normalized frequencies Normalized frequencies −0.6 3 0 2.5 0.2 2 0.4 1.5 0.6 1 0 .8 1 −1 0.5 −0.5 0 Normalized frequencies 0.5 1 (b) Modeled signal PSD Fig 2 Signal... Vol 6, N o 11, 2002, pp 481 - 483 Ph Courmontagne, N Vergnes and C Jauffret, An optimal subspace projection for signal detection in noisy environment, Proceedings of OCEANS’07, September 2007, Vancouver, Canada 2 98 Stochastic Control Wireless fading channel models: from classical to stochastic differential equations 299 16 X Wireless fading channel models: from classical to stochastic differential equations... no echo is assumed to be present and takes into account the hilly seabed The stochastic matched filter and its applications to detection and de-noising 295 Fig 8 Signal model: the wave is blocked by objects lying on seafloor and a shadow is generated 100 3 70 4 60 5 50 6 40 7 Azimuth (m) Azimuth (m) 3 2 80 2 1 90 1 8 5 6 8 20 9 4 7 30 10 50 Sight (m) 55 60 9 0 10 45 10 45 (a) SAS image representing... AUVfest 20 08) 1 Courtesy of AUVfest 20 08: http://oceanexplorer.noaa.gov 288 Stochastic Control The same process than for the previous example has been applied to this image to reduce the speckle level The main differences between the two experiments rest on the computation of the signal and noise statistics As the speckle noise is a multiplicative noise (see relation ( 48) ), the noise covariance, the noise... (m) 80 82 84 86 88 −20 (c) SAS image representing two mine like objects: a sphere and a cylinder lying on its side 10 68 70 72 74 76 78 Sight (m) 80 82 84 86 (d) Result of the detection algorithm Fig 9 2nd experiment: Mine detection on SAS images 88 296 Stochastic Control The data set used for this study has been recorded in 1999 during a joint experiment between GESMA (Groupe d’Etudes Sous-Marines de... Van Nostrand, 1955 J.-F Cavassilas, Stochastic matched filter, Proceedings of the Institute of Acoustics (International Conference on Sonar Signal Processing), pp 194-199, Vol 13, Part 9, 1991 F Chaillan, C Fraschini and P Courmontagne, Speckle Noise Reduction in SAS Imagery, Signal Processing, Vol 87 , N o 4, 2007, pp 762- 781 F Chaillan, C Fraschini and P Courmontagne, Stochastic Matched Filtering Method . 0.5 1 −1 −0 .8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0 .8 1 0.2 0.4 0.6 0 .8 1 1.2 1.4 1.6 1 .8 2 2.2 (a) Estimated noise PSD Normalized frequencies Normalized frequencies −1 −0.5 0 0.5 1 −1 −0 .8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0 .8 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 (b). 0.5 1 −1 −0 .8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0 .8 1 0.2 0.4 0.6 0 .8 1 1.2 1.4 1.6 1 .8 2 2.2 (a) Estimated noise PSD Normalized frequencies Normalized frequencies −1 −0.5 0 0.5 1 −1 −0 .8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0 .8 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 (b). 20 08) 1 Courtesy of AUVfest 20 08: http://oceanexplorer.noaa.gov The stochastic matched lter and its applications to detection and de-noising 287 An analysis of the figure 3.b shows that the stochastic