ignal 
1ents 
(13) 
lata, 
er of 
(14) 
(15) 
1995 
251 
where ’j’ denotes the conjugate matrix transpose which equals the ordinary matrix transpose for real matrices, 
where vectors are treated as special matrices with a single column. 
APPENDIX B 
Let the matrix equation 
g=Af+n (16) 
describe a general problem where the original (possibly complex) signal vector f is transformed by a linear 
matrix operation, which is described by the (complex) matrix A, and is subsequently corrupted with additive 
noise. Let us assume that all variables have zero mean. Denote the covariance matrix of the signal by C, and 
the covariance matrix of the noise by C,. Their elements are given by 
C, FE[fi-fj and Crij = Eins n,): (17) 
Additionally, let us assume that signal and noise are uncorrelated 
Bif;-n;) — 0. (18) 
Let us find a restoration matrix W, reconstructing f — Wg, such the mean square error 
E{|Wg — £f) 2 E(Wg - £) (Wg —£)) (19) 
is minimum. "j' denotes the conjugate matrix transpose which equals the ordinary matrix transpose for real 
matrices, where vectors are treated as special matrices with a single column. 
Combining Eq. 16 with Eq. 19 and requiring that the derivatives with respect to the matrix elements W;; 
vanish leads to 5 
0= 5 |We~ f|?} = E{[(Af + n);[(WAf + Wn — f)*1;} (20) 
ij 
where '*' takes the conjugate complex and [x]; takes element i of a vector x. 
Eqs. 20 can be written in a compact matrix form 
E((Af 4- n(WAf 4- Wn -£f!1 20 (21) 
where 0 is the zero matrix. Taking into account that signal and noise are uncorrelated this leads to 
E(Aff' AUW!Y-- E[nn! W!] 2 E(Affi). (22) 
By substituting E(ff!) — C; and E{nn!} = C, and taking matrices out of the expectation operator this 
results in 
AC;A'W! + C,W! = AC;. (23) 
Taking the conjugate transpose on both sides and using ci-c + and CJ, = C,, leads to the generalized Wiener 
filtering matrix equation . 
W — C, A! (AC; A! -- C.) l. (24). 
ACKNOWLEDGEMENTS 
I would like to thank M. Hafner and T. Fromherz for carefully looking through this manuscript and for helping 
to improve it with a number of corrections, suggestions, and good questions. 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995