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Guehring, Jens 
  
Gray Code Sequence 
012345 6 7 8 9 10111213141516171819202122232425262728 29 3031 
  
1 
2 
3 
4 
Phase Shift Sequence 
01234528657 8 910111213141516171819202122232425 2627 28 2930 31 
; Re c um | uw Ww 
6 ME. ME ——E . ME . ME. ME . ME Hi 
7 || mmu uw uw, ws. uw 
E HE —E —3HEE ME ME MH. ME ME 
Figure 5. Gray Code (top) and phase shift (bottom) of a 7 — 32 stripe code sequence. 
Note the pattern number four, which is used to resolve phase ambiguities (explained later). 
3.2.7 Combination of Gray Code and Phase Shift 
To obtain a resolution beyond the number of lines which can be switched by the projector, phase shifting can be 
applied. This uses the on/off intensity pattern generated by the switched projector lines as an approximation of a sine 
wave. The pattern is then shifted in steps of 7/2 for a total of N =4 pattern positions. Approximating the sampled 
values f(@;) at a certain fixed position (Figure 5, bottom) by 
C :cos(ó — à) 2 C -cos à, cos 9 t C -sinó, sinó 
= A-cos@+B sing 
where the coefficients A and B can be determined from Fourier analysis by 
2 N-1 2 N-1 
A-— 9 f ($,)cosó, and B ==>" f(¢)sing, 
N = Nis 
a phase shift ¢, =tan™ (B/A) is obtained, which in our case (N =4, ¢. ={0,7/2,7,37,37/2}) simplifies to 
ff. 
6 =tant A3 
f-h 
To estimate errors anticipated in phase computation, we assume that the intensity of the phase shift pattern can be 
approximated by 
f(C,D,%)=C -cos(@—@%)+D 
where C , à, and D are the unknown values for amplitude, phase shift and offset, respectively. 
Least squares parameter estimation uses four observation equations 
g, +v, =C -cos(ÿ —%)+ D 
where again @ = {0,7/2,7,37,37/2}. Assuming intensity measurements to be uncorrelated, we set the cofactor matrix 
Q, = diag(o3;.....07 ) Lo: being the variance of the grayvalue noise) and, since A” A =diag(4,2, 26° ) is a diagonal 
matrix, we finally obtain 
0, 
J2 
  
UE 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 331