Marc Honikel 
  
Mathematically, this corruption is described in the vector-space model with 
2 =Bf + (1) 
where g is the interferometric phase observation vector, f is the vector of the ideal interferogram and B is a matrix, 
whose elements are points on the impulse response function, often denoted as point spread function. The noise n is 
assumed to be zero mean with known covariance matrix K, (Pratt, 1991). 
For the restoration of g, an estimate f of the ideal image f is sought, which is given by 
f = Wg+b Q) 
where W is the restoration matrix and b is a bias vector. W and b are chosen for Wiener estimation in such a way that 
they minimize the mean-square restoration error e, which is defined as 
em EFL GO G) 
Minimization of the least squares restoration error is achieved by applying the orthogonality principle, which yields two 
necessary and sufficient conditions for the determination of W and b. 
Firstly, the expectation values of the estimate and the image must be equal: 
EEE) (4) 
By substitution with (1) and (2), the bias b is given by 
be E[f) -WEf/g) E/1)—WBE{} + WE n} (5) 
Secondly, the restoration error must be orthogonal to the observation centered at its mean: 
E((f- f ( 8 -El g)) - 0 (6) 
By further substitution and simplification, this yields (Pratt, 1991) 
W — K,B! (BK,B! * K,)! (7) 
where K; and K, are the image and noise covariance matrices. 
Or, by expressing K, and K; by their energies 0,1 and ol, 
W - B' ( BB +o)’ (8) 
Certain aspects of the filter adaptation to the varying noise can directly be derived from (8). 
In case that the image signal-to-noise ratio reaches infinity, i.e. no noise corrupt the fringes, the Wiener filter behaves 
equivalent to an inverse deconvolution filter with 
W = B' (9) 
On the other hand, if the ratio approaches zero, W becomes 0 and therefore with (2) and (5) 
^ 
f Ef] (10) 
forcing a smooth solution in presence of extreme noise. 
Finally, if the image signal is not blurred, i.e. B = I, W becomes 
W=H1+a/o (1) 
indicating the low-pass properties of the filtering in the special case of only additive phase noise. 
  
150 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B1. Amsterdam 2000. 
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