(1) 
atrix, 
enis 
yields two 
(7) 
(8) 
ter behaves 
(9) 
(10) 
(1) 
Marc Honikel 
  
Although Wiener filtering is theoretically the optimal method for the phase deconvolution in presence of noise, several 
problems limit its effectiveness. 
Firstly, the precise knowledge of the ideal image f and the noise term n can not be assumed for most applications 
including this one. Models, developed for the direct approximation of B, are too weak for the highly complex phase 
restoration task. Secondly, the assumption of spatially invariant degradations for the whole image is not valid for the 
spatial variant interferometric signature. Thirdly, the image and noise signals are assumed to be stationary, which is 
hardly valid in mixed terrain. 
In order to overcome limitations two and three, the proposed Wiener filtering will be applied locally, depending on the 
local SNR. The degradations behave spatially invariant within the borders of a limited window. In addition, the local 
power spectrum will hardly change within the window in comparison to the whole interferogram, thus fulfilling the 
stationarity requirement for the Wiener filtering. 
The absence of an ideal image will be overcome in this approach with a synthetic interferogram, simulated from a 
stereo-optical DEM. It serves as reference in those areas, where the interferometric phase determination fails, especially 
in layover areas. Though far from being perfect measurements, the simulated data will deliver here reliable information, 
as the optical DEM generation is much less error affected in steep terrain, due to the fact that layover does not occur 
with optical measurements. 
3 REALIZATION OF THE WIENER PHASE ESTIMATION 
31 Filter Development 
3.1.1 Bias estimation. The local offset between the interferometric phase measurement and the simulated phases is 
computed with (5) in a small estimation window, where both the correlation coefficients of the simulated and the 
interferometric phases are maximal. The bias estimation window is located as near as possible to the degraded region 
indicated by its low coherence, which would spoil the results, if it was used for bias estimation. 
In case of slightly degraded measurements, the matrix W becomes close to I and therefore 
b=E{f}-E{g} (12) 
For the local filtering approach pursued here, the resulting bias estimate can be assumed constant within the whole 
restoration window. 
3.1.2 Restoration function. In order to avoid the high computational and implementational cost for the numeric 
realization of (7) in the spatial domain, the validation of the assumptions made above is for the time performed in the 
Fourier domain, which allows a much faster processing. As the filtering is restricted locally, the results will suffer only 
little from the limitations of the Fourier domain processing. Andrews (1977) refers on the generalization of the filter 
equations of the vector-space domain from section 2 to the here implemented filter in the Fourier domain, which has 
been realized as a parameterized Fourier Wiener filter, with the transfer function 
W(k) ^ B*(k,D[|B(k D y P,(k D/P,(k, D] (13) 
With B(k,I) being the complex point spread function and P,(k,D/P.(k,l) the inverse of the signal to noise ratio and y is a 
filter design constant, which allows additional control over the filter behavior. If y is set to 1, the filter behaves like the 
traditional Wiener filter, else the parameter emphasizes (y > 1) or de-emphasizes (y < 1) the noise and signal statistics. 
Note that all terms of (13) are expressed in the Fourier domain, indicated with the k an pixel coordinates. 
3.2 Estimation of the filter parameters 
3.2.1 Synthetic interferogram computation. As input data for the proposed method, two measurements are used, the 
SAR interferogram and a stereo-optical DEM. First, a SAR interferogram (Fig. 2) has to be computed with the 
procedure given in detail in Prati (1994). 
The relation of the topographic phase component A¢ and the terrain elevation h in slant range is obtained, after 
compensation for the flat terrain contribution, by 
AG = Kh (14) 
where K is a scalar, depending on the sensor and orbit parameters (Prati, 1994). 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B1. Amsterdam 2000. 151