of 
where, 2 5 z A c 
IU lim [HI Dum e muti] Int fua unir] 
OfLf.)- Jing ME Du CUIU 
The corresponding distortion of the processed image C (x,y) is, according 
to (19), given by, | 
  
Nv 2 2, t. "e 
dy = [fin Juri, J1-up| UF + (PF IA LINTE (22) 
S: Of 5l, (ru PR ) - (P1) ] npe 
‚where the functionalO(f4f2) in (20) and (22) is determined by the 
condition of minimizing R(dg) for a given value of dy, yielding: 
nFA AZ 2 ; 
SAALE (PH+PA)- Jer*} = & if ff:€S 
e(Lf.]z NUH AIM otherwise 
: . ns Leu = = 2 
where S is defined by the condition: [5 (PPh) -/PI*} zo 
(23) 
  
and with (20): 2 _ a iY 
: 2 HEN PUPA J-0IU( IP 
R(a,) 2 ein [Rldo)]=4 [max Jo, log C n jd f. df. 
S 
The significance of the expression of (20),(22) and (23) is that we can 
establish the relationship of R(dp) versus dp, by varying 
0 < O, < max [uEn*(PH + PA) —lunPr] , 
f.fz. 
for any combination of complex transfer functions H and P, given a 
specific stationary ground reflection signal characterized by its spectral 
density functionn Sy(£1f9), which in turn specifies the functionals UNI 2 
and 11 Fl( 2 by poste Ty (6) and (7). 
  
Optimum Processing Conditions 
  
  
The general expression for the rate distortion function given above can be 
further analyzed for the condition of minimum distortion (dy) as 
function of the transfer functions H and P. 
Straightforward algebra yields for dy — minimum: 
Fu A 
m Le WA 
EU *-+ NILE ff. fr) (24) 
which corresponds to the two dimensional Wiener filter. 
For this condition the rate distortion function yields: 
  
Ya. : 
LEI 
R (dra) = | max | 0, log 5 df, df, (25) 
< 
o 
de. i fex [ues à -«JIIFIl JH at. (26) 
—!/1. 
449