Practical issues for binary code pattern unwrapping in fringe projection method - Emerging Trends in Image Processing, Computer Vision, and Pattern Recognition - page 479

FIGURE 5 Shifted fringe patterns (120°) profiles in color.
FIGURE 6 Shifted fringe patterns a: 0 b: 120 c: − 120.
As it is clear, changing p 0 in relation P = ωt + p 0 causes to shift the fringe patern in x -axis. If
it supposes γ = 0 then for each pixel, after projecting three paterns, there are three intensities:
I1 = a + b*cos (φ-2π/3)
I2 = a + b*cos (φ)
I3 = a + b*cos (φ + 2π/3)
Solving this equation set with three unknowns a, b, and φ (i) gives us the output phase φ
for each pixel individually:fi
i = atan (sqrt(3)*(I1-I3)/(2*I2-I1-I3));
a = (I1 + I2 + I3)/3;
b = sqrt(3*(I1-I3) ∧ 2 + (2*I2-I1-I3) ∧ 2)/3;
a is the computed texture and b is data modulation. If b limits to infinity, then i will be un-
stable and highly effected from noise so is not validate. It means the quality of the extracted i
fidepends on how far are the value of (I1-I3) and (2*I2-I1-I3) from zero.
After computation of wrapped phase φ, the final phase Φ = φ + 2kπ in which k is unknown
ambiguity parameters. This ambiguity can be solved by different methods that will be de-
scribed in the next subject.
4 Binary code generation for phase ambiguity
resolution
Since phase modulation analysis uses the arctan function, it yields values in the range [− π,+π].
However, true phase values may extend over 2π range, resulting in discontinuities in the re-
covered phase and imprecision in the phase-unwrapping results. The procedure of determ-
ining discontinuities on the wrapped phase, resolving them, and achieving the unwrapped
phase is called phase unwrapping.
Considering the fact that in each row, the fringe paterns will be repeated every 20 pixels,
as a result, the phase ambiguity of each pixel will be increased with the amount 2*πi in each
20 pixels. In view of the above remark, if the phase ambiguity of each pixel is equal to k*2*π
(2kπ), therefore k values will be distributed in k = 0-1-2-3-4. Generally, k goes from 0 to ceil (n/
if − 1) giving k coefficients for phase ambiguity.
Assuming width of the projector equal to 1280 pixels (n = 1280) and 1280 pixels/20 pixels/
area = 64 = 2 6 areas to be coded. Consequently, six binary pictures (code paterns) will be
needed to code 64 areas.