250 
  
d) 
Figure 2: Wiener filtering reconstruction of a dummy, using c — 1 and p — —2. 
(a,b,c) Original images. 
(d) Recovered depth map. 
-(e) Shaded frontal view. 
(f) Shaded tilted view. 
and, as a consequence, good regularization parameters have to be found by trial and error. Finally, if signal 
and noise have a uncommon distribution, guessing the right stabilizer is almost impossible. 
Thus, it seems that the success of regularization largely stems from the fact that it implicitly implements 
the Wiener filtering approach for a typical class of SNR models. 
APPENDIX A 
Let us find the vector f which minimizes 
K M K N 
|Af — g? -- V^ |Ry£f? Z V [Af - g [Af — g]; - 3, À [Rf] [Ruf] (13) 
k=1 il k=11=1 
where, in the frame-work of regularization theory, g is interpreted as the vector containing the measured data, 
the M by N matrix A maps f to g, Ry represents the k-th stabilizer term, and K is the total number of 
stabilizer terms. ’x’ takes the conjugate complex and [x]; takes element : of a vector x. 
Eq. 13 takes its minimum if the derivatives with respect to the vector elements f; vanish. This leads to 
M K N 
0 = > Aij|Af — gli + S > Ry; [Rif]; (14) 
k=1i=1 
Eqs. 14 can be written in a compact matrix form 
K K 
F(ATA + S Ry'Ry) =glA= f= (ATA +) R'Ry)'Alg (15) 
k=1 : k=1 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop "From Pixels to Sequences", Zurich, March 22-24 1995