VIEWS: 7 PAGES: 48 POSTED ON: 7/19/2011
Signal Subspace Speech Enhancement Page 0 of 43 Presentation Outline Introduction Principals Orthogonal Transforms (KLT Overview) Papers Review Page 1 of 47 Introduction Two major classes of speech enhancement – By modeling of noise/speech: like HMM Highly dependent on speech signal syntax and noise characteristics – Based on transformation: Spectral Subtraction Musical noise Signal Subspace belongs to the second class (nonparametric) Page 2 of 47 Schematic Diagram Noisy signal Orthogonal Modifying Inverse (time domain) Transform Coefficients Transform Estimated Clean Signal Page 3 of 47 Schematic Diagram Noisy signal (time domain) Signal+Noise subspace Estimating Clean signal Estimating from Framing Orthogonal Transform Dimensions of Gs Signal+Noise overlapping Subspaces subspace Inverse Clean Producing two Gn orthogonal Noise Transform Signal subspaces subspace Page 4 of 47 Principals Procedure – Estimate the dimension of the signal+noise subspace in each frame – Estimate clean signal from (S+N) subspace by considering some criteria (main part) energy of the residual noise energy of the signal distortion – Nulling the coefficients related to the noise subspace Page 5 of 47 Principals Assumptions – Noise & speech are uncorrelated – Noise is additive & white (whitened) – Covariance matrix of the noise in each frame is positive definite and close to a Toeplitz matrix – Signal is more statistically structured than noise process Page 6 of 47 Principals Key Factor in Signal Subspace method – Covariance matrices of the clean signal have some zero eigenvalues. The improvement in SNR is proportional to the number of those zeros. Nullifying the coefficients of the noise subspace corresponds to that of weak spectral components in spectral subtraction. Page 7 of 47 Orthogonal Transforms Signal Subspace decomposition can be achieved by applying: – KLT via Eigenvalue Decomposition (ED) of signal covariance matrix via Singular Value Decomposition (SVD) of data matrix SVD approximation by recursive methods – DCT as a good approximation to the KLT – Walsh, Haar, Sine, Fourier,… Page 8 of 47 Orthogonal Transforms: Karhunen-Loeve Transform (KLT) Also known as “Hotelling”, “Principal Component” or “Eigenvector" Transform Decorrelates the input vector perfectly – Processing of one component has no effect on the others Applications – Compression, Pattern Recognition, Classification, Image Restoration, Speech Recognition, Speaker Recognition,… Page 9 of 47 KLT Overview Let R be the N N correlation matrix of a random complex sequence x ( x1 , x2 ,..., xN )T then x1 R E xxH x2 E x1 x2 xN x N Where E is the expectation operator and R is Hermitian matrix. Page 10 of 47 KLT Overview Let be N N unitary matrix which diagonalizes R 1 H R H Diag 1 , 2 ,..., N i , i 1,2,..., N are the eigenvalues of R. H is called the KLT matrix. Page 11 of 47 KLT Overview Property of H : •Consider the following transform: y x H sequence y is uncorrelated because : E yy H E H xxH H E xxH H R y has no cross-correlation Page 12 of 47 KLT Overview What is ? R R R H H where 12 N and `s are ith column of i Ri ii , i 1,2,..., N Thus i ' s are eigenvectors corresponding to i ' s Page 13 of 47 KLT Overview Comments – The arrangement of y auto-correlations is the same as that of i s ' – KLT can be based on Covariance matrix – Using largest eigenvalues to reconstruct sequence with negligible error – KLT is optimal Page 14 of 47 KLT Overview Difficulties – Computational Complexity (no fast algorithm) – Dependency on the statistics of the current frame – Make uncorrelated not independent Utilize KLT as a Benchmark in evaluating the performance of the other transforms. Page 15 of 47 Papers Review 1. A Signal Subspace Approach for S.E. [Ephraim 95] 2. On S.E. Algorithms based on Signal Subspace Methods [Hansen] 3. Extension of the Signal Subspace S.E. Approach to Colored Noise [Ephraim] 4. An Adaptive KLT Approach for S.E. [Gazor] 5. Incorporating the Human Hearing Properties in Signal Subspace Approach for S.E. [Jabloun] 6. An Energy-Constrained Signal Subspace Method for S.E. [Huang] 7. S.E. Based on the Subspace Method [Asano] Page 16 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] Principal – Decompose the input vector of the noisy signal into a signal+noise subspace and a noise subspace by applying KLT Enhancement Procedure – Removing the noise subspace – Estimating the clean signal from S+N subspace – Two linear estimators by considering: Signal distortion Residual noise energy Page 17 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] Notes – Keeping the residual noise below some threshold to avoid producing musical noise – Since DFT & KLT are related, SS is a particular case of this method – if # of basis vectors (for linear combination of a vector) are less than the dim of the vector, then there are some zero eigenvalues for its correlation matrix Page 18 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] Basics – speech signal : z=y+w , K-dimensional M – y smVm , M K m 1 s1 ,, sM Are zero mean complex variables – y Vs – If M=K, representation is always possible. – Else ―damped complex sinusoid model‖ can be used. – Span( V ): produces all vector y Page 19 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] When M<K, all vectors y lie in a subspace of RK spanned by the columns of V SIGNAL+NOISE SUBSPACE Covariance matrix of clean signal y y Vs Ry Eyy VRsV # # ; K M, M M, M K Rank ( R y ) M has K M zero eigenvalues Page 20 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] Covariance matrix of noise w : (K-Dim) RK Rw E ww I # 2 w n S n n Rank ( Rw ) K – White noise vectors fill the entire Euclidean space RK – Thus the noise exists in both S+N subspace and complementary subspace NOISE SUBSPACE Page 21 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] The discussion indicates that Euclidean space of the noisy signal is composed of a signal subspace and a complementary noise subspace This decomposition can be performed by applying KLT to the noisy signal : Let z Vs w The covariance matrix of z is: Rz E zz VRsV Rw # # Page 22 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] Noise is additive Rz Ry Rw Let Rz U zU # be the eigendecomposition of Rz Where U u1 ,, uk are eigenvectors of Rz and z diag z 1,, z K Eigenvalues of Rw are 2 w y k if k 1,, M 2 z k 2 w w if k M 1,, K Page 23 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] Estimating Dimensions of Signal Subspace M U u1 ,, uk Let U U1 ,U 2 U1 uk : z k 2 w : principal eigenvectors Because span(U1 ) span(V ) ,Hence U1U1# is the orthogonal projector onto the S+N subspace Page 24 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] Thus a vector z of noisy signal can be decomposed as UU I U1U U 2U I # # 1 # 2 z U1U1 z U 2U 2 z # # U 1# is the Karhunen-Loeve Transform Matrix. # The vector U 2U z does not contain signal information 2 and can be nulled when estimating the clean signal. However, M (dim of S+N subspace) must be calculated precisely Page 25 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] Linear Estimation of the clean signal – Time Domain Constrained Estimator Minimize signal distortion while constraining the energy of residual noise in every frame below a given threshold – Spectral Domain Constrained Estimator Minimize signal distortion while constraining the energy of residual noise in each spectral component below a given threshold Page 26 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] Time Domain Constrained Estimator – Having z=y+w Let y Hz be a linear estimator of y ˆ where H is a K*K matrix – The residual signal is r y y ( H I ) y Hw ry rw ˆ Representing signal distortion and residual noise respectively Page 27 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] Defining Criterion ry ( H I ) y Energy: y2 trE ry ry# rw Hw Energy: w trE rw rw 2 # Solving : min y2 H subject to : ε α 1 K 2 w 2 w 0 M K Minimize signal distortion while constraining the energy of residual noise in the entire frame below a given threshold Page 28 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] After solving the Constrained minimization by „„Kuhn- Tucker‟‟ necessary conditions we obtain HTDC Ry Ry I 2 w 1 Where is the Lagrange multiplier that must satisfy 1 K tr R Ry I 2 y 2 w 2 Eigendecomposition of HTDC G 0 # H TDC U U 0 0 Page 29 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] In order to null noisy components G 0 # H TDC U U 0 0 G y y 2 1 w HTDC U1GU # 1 If ( max M K ) then HTDC=I, which means minimum distortion and maximum noise Page 30 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] Spectral Domain Constrained Estimator – Minimize signal distortion while constraining the energy of residual noise in each spectral component below a given threshold. Results: H UQU # Q diag (q11 , , qKK ) 12 k 1, , M qKK k 0 k M 1, , K k exp{v / y (k )} 2 w Page 31 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] Notes – The most computational complexity is in Eigendecomposition of the estimated covariance. – Eigendecomposition of Toeplitz covariance matrix of the noisy vector is used as an approximate to KLT – Compromise between large T in estimating Rz ,and large K to satisfy M<K, while KT can not be too large Page 32 of 47 A Signal Subspace Approach for S.E. [Ephraim 95] Implementation Results – The improvement in SNR is proportional to K /M – The SDC estimator is more powerful than the TDC estimator – SNR improvements in Signal Subspace and SS are similar – Subjective Test 83.9 preferred Signal Subspace over noisy signal 98.2 preferred Signal Subspace over SS Page 33 of 47 On S.E. Algorithms based on Signal Subspace Methods [Hansen] The dimension of the signal subspace is chosen at a point with almost equal singular values Gain matrices for different estimators – SDC Less sensitive to errors in – TDC the noise estimation – MV Musical noise Lowest residual noise – LS G=I Lowest signal distortion and highest residual noise M K 2 noise K /M improvement in SNR SDC improves the SNR in the range 0-20 db Page 34 of 47 Extension of the Signal Subspace S.E. Approach to Colored Noise [Ephraim] Whitening approach is not desirable for SDC estimator. Obtaining gain matrix H for SDC estimator min 2 d H subject to : E viN αi i 1,...,m 2 12 ~ H Rw UHU Rw1 2 ~ H is not diagonal when the input noise is colored Whitening Orthogonal Transformation U‟ modify ~ components by H Page 35 of 47 An Adaptive KLT Approach for S.E. [Gazor] Goal – Enhancement of speech degraded by additive colored noise Novelty – Adaptive tracking based algorithm for obtaining KLT components – A VAD based on principle eigenvalues Page 36 of 47 An Adaptive KLT Approach for S.E. [Gazor] Objective – Minimize the distortion when residual noise power is limited to a specific level Type of colored noise – Have a diagonal covariance matrix in KLT domain G y y 2 1 w Replaced by G y y n 1 Page 37 of 47 An Adaptive KLT Approach for S.E. [Gazor] Adaptive KLT tracking algorithm – named ―projection approximation subspace tracking‖ – reducing computational time – Eigendecomposition is considered as a constrained optimization problem – Solving the problem considering quasi-stationarity of speech – Then a recursive algorithm is planned to find a close approximation of eigenvectors of the noisy signal Page 38 of 47 An Adaptive KLT Approach for S.E. [Gazor] Voice activity detector – When the current principle components’ energy is above 1/12 its past minimum and maximum Implementation Results SNR Non- Noise Ephraim‟s (dB) Processed Type 10 85% 55% white 5 75% 69% white 0 64% 89% white 10 75% 73% office 5 85% 79% office 0 68% 89% office Page 39 of 47 Incorporating the Human Hearing Properties in the Signal Subspace Approach for S.E. [Jabloun] Goal – Keep the residual noise as much as possible, in order to minimize signal distortion Novelty – Transformation from Frequency to Eigendomain for modeling masking threshold. eigendomain eigendomain IFET Masking FET Many masking models were introduced in frequency domain; like Bark scale Page 40 of 47 Incorporating the Human Hearing Properties in the Signal Subspace Approach for S.E. [Jabloun] Use noise prewhitening to handle the colored noise Implementation results Compared with Compared with Input SNR noisy signal Signal Subspace 20 dB 92% 71% 10 dB 85% 78% 5 dB 85% 92% Page 41 of 47 An Energy-Constrained Signal Subspace Method for S.E. [Huang] Novelty – The colored noise is modelled by an AR process. – Estimating energy of clean signal to adjust the speech enhancement Prewhitening filter is constructed based on the estimated AR parameters. – Optimal AR coeffs is given by [Key 98] Page 42 of 47 An Energy-Constrained Signal Subspace Method for S.E. [Huang] Implementation Results Word Recognition Accuracy for noisy digits Input SNR 0 dB 5 dB 10 dB 20 dB Baseline 40 % 70 % 90 % 100 % ECSS 90 % 100 % 100 % 100 % SNR improvement for isolated noisy digits Input SNR 0 dB 5 dB 10 dB 20 dB Improvement 7.6 6.4 5.2 2.9 Page 43 of 47 S.E. Based on the Subspace Method [Asano]—Microphone Array The input spectrum observed at the mth microphone D X m k Am,d k .S d k N m k d 1 Vector notation for all microphones Ambient x k Aksk n k Directional Sources Noise (spatial) correlation matrix for xk is R k E[x k x H ] k Microphone array Then Eigenvalue Decomposition is applied to R k Page 44 of 47 S.E. Based on the Subspace Method [Asano]—Microphone Array Procedure – Weighting the eigenvalues of spatial correlation matrix Energy of D directional sources is concentrated on D largest eigenvalues Ambient noise is reduced by weighting eigenvalues of the noise-dominant subspace discarding M-D smallest eigenvalues when direct-ambient ratio is high – Using MV beamformer to extract directional component from modified spatial correlation matrix Page 45 of 47 S.E. Based on the Subspace Method [Asano]—Microphone Array Implementation results – Two directional speech signals + Ambient noise Recognition Rate: MV MV-NSR SNR A B1 A B1 5 dB 66.9% 71.5% 72.3% 78% 10 dB 81.1% 86.6% 81.5% 87.2% Page 46 of 47 Thanks For Your Attention The End Page 47 of 47