h(n) x(n) s(n) S (ω) = H(ω)X(ω) (5 1) H(ω) H(ω) = F[h(n)] (5 2) F X(ω) x(n) X(ω) = F[x(n)] (5 3) S (ω) s(n) S (ω) = F[s(n)] (5
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1 N. Wiener FFT Norbert Wiener MIT c /(12)
2 h(n) x(n) s(n) S (ω) = H(ω)X(ω) (5 1) H(ω) H(ω) = F[h(n)] (5 2) F X(ω) x(n) X(ω) = F[x(n)] (5 3) S (ω) s(n) S (ω) = F[s(n)] (5 4) (5 1) S (ω) Ŝ (ω) E(ω) = S (ω) Ŝ (ω) = S (ω) H(ω)X(ω) (5 5) E[ E(ω) 2 ] = E[ S (ω) H(ω)X(ω) 2 ] (5 6) E[ ] (5 6) H(ω) E[ (ω) 2 ] H(ω) = 2H(ω)P XX (ω) 2P XS (ω) (5 7) P XX (ω) P XS (ω) P XX (ω) = E[ X(ω) 2 ] (5 8) P XS (ω) = E[X(ω)S (ω)] (5 9) P XX (ω) c /(12)
3 P X S (ω) (5 7) 0 (5 6) H(ω) (5 7) 2H(ω)P XX (ω) 2P XS (ω) = 0 (5 10) H(ω) = P XS (ω) P XX (ω) (5 11) (5 11) X(ω) S (ω) W(ω) = F[w(n)] (5 12) S (ω) X(ω) P XX (ω) = P S S (ω) + P WW (ω) (5 13) X(ω) S (ω) P XS = E[(S (ω) + W(ω))S (ω)] = E[ S (ω) 2 ] = P S S (ω) (5 14) S (ω) (5 13) (5 14) (5 11) H(ω) = P S S (ω) P S S (ω) + P WW (ω) (5 15) (5 15) 2) [ ] (5 15) ˆP S S (ω) = S (ω) 2 (5 16) ( ) c /(12)
4 ˆP WW (ω) = W(ω) 2 (5 17) (5 16)(5 17) (5 15) S (ω) H(ω) = 2 (5 18) S (ω) 2 + W(ω) 2 (5 18) S (ω) 2 S (ω) 2 X(ω) 2 (5 19) 3) 1) S (ω) 2 2) S (ω) 2 = X(ω) 2 W(ω) 2 (5 20) W(ω) 2 X(ω) 2 (5 20) (5 18) H(ω) = X(ω) 2 W(ω) 2 X(ω) 2 (5 21) Spectral Subtraction : SS SS kHz 10kHz 3.4kHz 10kHz 51.2ms 1/2 SS X(ω) (5 1) H(ω) S (ω) (5 21) c /(12)
5 H(ω) = X(ω) 2 W(ω) 2 X(ω) 2 (5 22) (5 22) X(ω) 2 < W(ω) 2 (5 23) 0 H(ω) 1 (5 24) (5 22) H(ω) H R (ω) = H(ω) + H(ω) 2 (5 25) (5 23) SS 5 1 SS c /(12)
6 1) S.F. Boll, Suppression of Acoustic Noise in Speech Using Spectral Subtraction, IEEE Trans. Acoustics, Speech and Signal Processing, vol.assp-27, no.7, pp , ) S.V. Vaseghi, Advanced Digital Signal Processing and Noise Reduction, Second Edition, Wiley, ) J.S. Lim and A.V. Oppenheim, Enhancement and bandwidth cpmpression of noisy speech, Proc. IEEE, vol.67, no.12, pp , c /(12)
7 ) WF WF WF WF N x y y y = Bx + n (5 26) B n n x WF x ˆx ˆx = Ay (5 27) E[ x ˆx 2 ] E[( )] ( ) ( ) ( ) A 2) A = RB T (BRB T + Q) 1 (5 28) ( ) T ( ) ( ) 1 ( ) R Q R = E[xx T ] Q = E[nn T ] (5 29) (5 30) B Q (5 28) WF R R WF WF 1 WF x n WF 2-D DFT 2-D DCT c /(12)
8 2-D DFT B x n X Y 2-D DFT U U U H U H U U H U = UU H = I I (5 27) (5 28) 2-D DFT 3) ˆX = ΩY Ω = UAU H = ΛD H (DΛD H + Γ) 1 (5 31) (5 32) a 2-D DFT A = Ua D = UBU H Λ = URU H Γ = UQU H Ω WF ω DFT (k) ω DFT (k) = λ(k)d H (k) ; k = 1, 2,, N λ(k) d(k) 2 + γ(k) (5 33) d(k) 2 λ(k) γ(k) D 2 Λ Γ k ˆx = U H ˆX B X Y 2-D DFT 2-D DCT 2 2-D DWT WF 2-D DWT 2-D DWT 2-D DWT WF (5 33) WF WF 2-D DWT β(k) ω DWT (k) = β(k) + σ ; k = 1, 2,, N (5 34) 2 β(k) 2-D DWT σ 2 3 FIR-WF 4) x x S y S WF FIR-WF FIR-WF WF WF x ˆx ˆx = a T y S (5 35) a E[(x ˆx) 2 ] a 5) a = C 1 c (5 36) c /(12)
9 C c y S y S x C = E[y S y T S ] c = E[y S x] (5 37) (5 38) (5 36) a WF 1 a = C 1 c + C T C 1 1 (1 1T C 1 c) (5 39) WF WF WF 6) (5 28) WF (5 28) 0 Q B WF B B WF 1 WF 2 WF 1 WF WF Tichonov 2) d H (k) ω DFT (k) = d(k) 2 + ɛ ; k = 1, 2,, N (5 40) 2 ɛ 2 (5 33) WF λ(k) γ(k) 5 WF MMSE 5) ˆx = E[x y] (5 41) E[x y] y x x n MMSE WF 5) WF (4) WF WF c /(12)
10 1 WF WF 7) D DWT 2-D DCT 2 1 WF a 2-D DWT 2-D DWT (5 34) β(k) 2 W 1 2-D DWT 2-D DWT X ˆX 1 W 2 X ˆX 2 = W 2 W1 T ˆX 1 (5 34) β(k) ˆX 2 2 (k) ˆX 2 (k) ˆX 2 k W 2 2-D DWT β(k) (5 34) WF 7) b 2-D DCT 2-D DCT 2-D DCT 8) 2 WF 1 2-D DCT WF 2 1 WF 8) 2 x f (x) y f (x y) x ˆx(y) L[x, ˆx(y)] L[x, ˆx(y)] L[x, ˆx(y)] = x ˆx(y) 2 (5 42) x x L[x, ˆx(y)] E[L[x, ˆx(y)] x] f (x) EE[L[x, ˆx(y)] x] EE[L[x, ˆx(y)] y] f (y) f (x y) E[L[x, ˆx(y)] y] 2 ˆx(y) = E[x y] MMSE 5) WF MMSE c /(12)
11 f (x) 9) 3 GMM 4) x L GMM M f (x L ) = P(s i )N(x L 0, R i ) i=1 (5 43) f ( ) M P(s) s N( µ, R) µ R ( ) GMM M E[ x ˆx 2 ] = N E[(x ˆx) 2 s i ]P(s i ) i=1 (5 44) E[(x ˆx) 2 s i ] WF FIR WF 10) GMM 11) WF WF a DWT DWT Λ L DWT Λ LO Λ L Γ L GMM [z i : i = 1, 2,, M] i z i Λ LO + Γ L GSM 10) b GMM EM 12) P(s i ), R i : i = 1, 2, M GMM (5) WF MMSE GMM WF MMSE 1) A. Jain, Fundamentals of Digital Image Processing. Englewood Cliffs, NJ: Prentice-Hall, c /(12)
12 2), II, vol.71, no.6, pp , June ) Richard A. Haddad, Thomas W. Parsons, Digital Signal Processing, NY: Computer Science Press, ) P.A. Maragos, R.W. Shafer and R.M. Mersereau, Two-Dimensional Linear Prediction and Its Application to Adaptive Predictive Coding of Images, IEEE Trans. Acoust. Speech & Signal Processing, vol. ASSP 32, no.6, pp , Dec ) Louis L. Scharf, Statistical Signal Processing, MA: Addison-Wesley Publishing Company, ) R. Neelamani, H. Choi, and R.G. Baraniuk, ForWaRD: Fourier wavelet regularized deconvolution for ill-conditioned systems, IEEE Trans. Signal Process., vol.52, no.2, pp , Feb ) S. Ghael, A. Sayeed, R. Baraniuk, Improved wavelet denoising via empirical wiener filtering, Proceedings of SPIE, San Diego, July ) Foi, A., V. Katkovnik, and K. Egiazarian, Pointwise Shape-Adaptive DCT for High-Quality Denoising and Deblocking of Grayscale and Color Images, IEEE Trans. Image Process., vol.16, no.5, pp , May ) Jose M. Bioucas-Dias, Bayesian Wavelet-Based Image Deconvolution:A GEM Algorithm Exploiting a Class of Heavy-Tailed Priors, IEEE Trans. Image Process., vol.15, no.4, April ) Javier Portilla, Vasily Strela, Martin J. Wainwright, and Eero P. Simoncelli, Image Denoising Using Scale Mixtures of Gaussians in the Wavelet Domain, IEEE Trans. Image Process, vol.12, no.11, Nov ) Yamane et. al., Image Restoration Using a Universal GMM Learning and Adaptive Wiener Filter, IEICE Trans. A, vol.92-a, no.10, Oct ) A. Dempster, N. Laird and D. Rubin, Maximum likelihood from incomplete data via the EM algorithm, J. Roy. Statist. Soc. B, vol.39, pp.1-38, c /(12)
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2 DS SS (SS+DS) Fig. 2 Separation algorithm for motorcycle sound by combining DS and SS (SS+DS). 3. [3] DS SS 2 SS+DS 1 1 B SS SS 4. NMF 4. 1 (NMF) Y
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a) Separation of Motorcycle Sound by Near Field Microphone Array and Nonnegative Matrix Factorization Chisaki YOSHINAGA, Nonmember, Yosuke TATEKURA a), Member, Kazuaki HAMADA, and Tetsuya KIMURA, Nonmembers
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On the Limited Sample Effect of the Optimum Classifier by Bayesian Approach he Case of Independent Sample Size for Each Class Xuexian HA, etsushi WAKA
Journal Article / 学術雑誌論文 ベイズアプローチによる最適識別系の有限 標本効果に関する考察 : 学習標本の大きさ がクラス間で異なる場合 (< 論文小特集 > パ ターン認識のための学習 : 基礎と応用 On the limited sample effect of bayesian approach : the case of each class 韓, 雪仙 ; 若林, 哲史
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