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

1 -- 5 5 2011 2 1940 N. Wiener FFT 5-1 5-2 Norbert Wiener 1894 1912 MIT c 2011 1/(12)

1 -- 5 -- 5 5--1 2008 3 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 2011 2/(12)

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 2011 3/(12)

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 5 1 3.4kHz 10kHz 3.4kHz 10kHz 51.2ms 1/2 SS X(ω) (5 1) H(ω) S (ω) (5 21) c 2011 4/(12)

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) 0 5 1 SS 5 1 SS c 2011 5/(12)

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.113-120, 1979. 2) S.V. Vaseghi, Advanced Digital Signal Processing and Noise Reduction, Second Edition, Wiley, 2000. 3) J.S. Lim and A.V. Oppenheim, Enhancement and bandwidth cpmpression of noisy speech, Proc. IEEE, vol.67, no.12, pp.1586-1604, 1979. c 2011 6/(12)

1 -- 5 -- 5 5--2 2009 7 1) WF WF WF WF 5--2--1 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 2011 7/(12)

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 2011 8/(12)

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 1 1 1 T C 1 1 (1 1T C 1 c) (5 39) 1 1 4 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) 5--2--2 WF 5--2--1(4) WF WF c 2011 9/(12)

1 WF WF 7) 2 1 2-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 2011 10/(12)

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--2--1(5) WF MMSE GMM WF MMSE 1) A. Jain, Fundamentals of Digital Image Processing. Englewood Cliffs, NJ: Prentice-Hall, 1989. c 2011 11/(12)

2), II, vol.71, no.6, pp.593-601, June 1988. 3) Richard A. Haddad, Thomas W. Parsons, Digital Signal Processing, NY: Computer Science Press, 1991. 4) 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.1213-1228, Dec. 1988. 5) Louis L. Scharf, Statistical Signal Processing, MA: Addison-Wesley Publishing Company, 1991. 6) 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.418-433, Feb. 2004. 7) S. Ghael, A. Sayeed, R. Baraniuk, Improved wavelet denoising via empirical wiener filtering, Proceedings of SPIE, San Diego, July 1997. 8) 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.1395-1411, May 2007. 9) 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 2006. 10) 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. 2003. 11) Yamane et. al., Image Restoration Using a Universal GMM Learning and Adaptive Wiener Filter, IEICE Trans. A, vol.92-a, no.10, Oct. 2009. 12) 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, 1977. c 2011 12/(12)