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Introduction
Intensity pattern normalization
Results
Conclusions
References
Correcting biased envelope estimation of sinusoidalsignals with random noise
Rigoberto Juarez-Salazar
Universidad Tecnologica de la MixtecaInstituto de Fsica y Matematicas
XXV Escuela Nacional de Optimizacion y Analisis NumericoMexico D.F., Septiembre, 2015.
1 / 1 6 Rigoberto Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
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Introduction
Intensity pattern normalization
Results
Conclusions
References
Content
1 IntroductionOptical 3D imaging system
2 Intensity pattern normalization
Biasing envelope estimationCorrecting biased envelope estimationVariance of the noise
3 Results
4 Conclusions
5 References
2 / 1 6 Rigoberto Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
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Introduction
Intensity pattern normalization
Results
Conclusions
References
Optical 3D imaging system
Introduction Sinusoidal signals processing
There are many applications in communications and engineering where the informationof interest is encoded as phase in a time, space, or spatiotemporal signal.
Figura :Communication antennas and the principle ofphase modulation.
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Introduction
Intensity pattern normalization
Results
Conclusions
References
Optical 3D imaging system
Introduction Sinusoidal signals processing
Figura :Laser interferometer and two-dimensional phasemodulation.4 / 1 6 Rigoberto Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
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Introduction
Intensity pattern normalization
Results
Conclusions
References
Optical 3D imaging system
Introduction Sinusoidal signals processing
x
y
Typical intensity pattern
200 400 600 800 1000
100
200
300
400
500
600
700
200 400 600 800 10000
50
100
150
200
250
x
Instensity
Intensity pattern profile
Figura :Typical profile of a intensity pattern.5 / 1 6 Rigoberto Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
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Introduction
Intensity pattern normalization
Results
Conclusions
References
Optical 3D imaging system
Introduction Sinusoidal signals processing
Interferometric patterns
Interferometric intensity patterns are examples of two-dimensional signals where theinformation of interest is encoded as a phase distribution.
In all cases, since the information of interest is encoded as a phase distribution, weneed to apply efficient techniques to extract the desired information (the phase) byprocessing sinusoidal signals.
6 / 1 6 Rigoberto Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
I t d ti
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Introduction
Intensity pattern normalization
Results
Conclusions
References
Biasing envelope estimation
Correcting biased envelope estimation
Variance of the noise
Intensity pattern normalization
An intensity pattern can be described as
I(p) =a(p) +b(p) cos(p) +n(p), (1)
where
pis a two-dimensional spatial variable,
ais thebackground light,bis themodulation light,
is the phase of interest, and
nis random noise.
Because of the phase is the desired information, both the functions aand bareundesired as well as the noise.
The process of to suppress the background and modulation lights is known as intensitypattern normalization.
7 / 1 6 Rigoberto Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
Introduction
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Introduction
Intensity pattern normalization
Results
Conclusions
References
Biasing envelope estimation
Correcting biased envelope estimation
Variance of the noise
Intensity pattern normalization noise-free
For noise-free pattern: I=a+bcos
1.AI a, 2.B(I a)2 =b2/2, 3.(I a)/
2 cos.
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8 / 1 6 Rigoberto Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
Introduction
http://find/7/23/2019 Correcting biased envelope estimation of sinusoidal signals with random noise
9/28
Introduction
Intensity pattern normalization
Results
Conclusions
References
Biasing envelope estimation
Correcting biased envelope estimation
Variance of the noise
Intensity pattern normalization noise-free
For noise-free pattern: I=a+bcos
1.AI a, 2.B(I a)2 =b2/2, 3.(I a)/
2 cos.
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8 / 1 6 Rigoberto Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
Introduction
http://find/7/23/2019 Correcting biased envelope estimation of sinusoidal signals with random noise
10/28
Introduction
Intensity pattern normalization
Results
Conclusions
References
Biasing envelope estimation
Correcting biased envelope estimation
Variance of the noise
Intensity pattern normalization noise-free
For noise-free pattern: I=a+bcos
1.AI a, 2.B(I a)2 =b2/2, 3.(I a)/
2 cos.
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8 / 1 6 Rigoberto Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
Introduction
http://find/7/23/2019 Correcting biased envelope estimation of sinusoidal signals with random noise
11/28
Introduction
Intensity pattern normalization
Results
Conclusions
References
Biasing envelope estimation
Correcting biased envelope estimation
Variance of the noise
Intensity pattern normalization noise-free
For noise-free pattern: I=a+bcos
1.AI a, 2.B(I a)2 =b2/2, 3.(I a)/
2 cos.
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8 / 1 6 Rigoberto Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
Introduction
http://find/7/23/2019 Correcting biased envelope estimation of sinusoidal signals with random noise
12/28
Intensity pattern normalization
Results
Conclusions
References
Biasing envelope estimation
Correcting biased envelope estimation
Variance of the noise
Intensity pattern normalization noise-free
For noise-free pattern: I=a+bcos
1.AI a, 2.B(I a)2 =b2/2, 3.(I a)/
2 cos.
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8 / 1 6 Rigoberto Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
Introduction
http://find/7/23/2019 Correcting biased envelope estimation of sinusoidal signals with random noise
13/28
Intensity pattern normalization
Results
Conclusions
References
Biasing envelope estimation
Correcting biased envelope estimation
Variance of the noise
Intensity pattern normalization noise-free
For noise-free pattern: I=a+bcos
1.AI a, 2.B(I a)2 =b2/2, 3.(I a)/
2 cos.
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8 / 1 6 Rigoberto Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
Introduction
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Intensity pattern normalization
Results
Conclusions
References
Biasing envelope estimation
Correcting biased envelope estimation
Variance of the noise
Intensity pattern normalization noisy pattern
Noisy pattern: I=a+bcos+ FAILS!!!
1.AI a, 2.B(I a)2 =b2/2, 3.(I a)/
2 cos.
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9 / 1 6 Rigoberto Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
http://find/7/23/2019 Correcting biased envelope estimation of sinusoidal signals with random noise
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7/23/2019 Correcting biased envelope estimation of sinusoidal signals with random noise
16/28
Introduction
I t it tt li ti Bi i l ti ti
7/23/2019 Correcting biased envelope estimation of sinusoidal signals with random noise
17/28
Intensity pattern normalization
Results
Conclusions
References
Biasing envelope estimation
Correcting biased envelope estimation
Variance of the noise
Intensity pattern normalization noisy pattern
Noisy pattern: I=a+bcos+ FAILS!!!
1.AI a, 2.B(I a)2 =b2/2, 3.(I a)/
2 cos.
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7/23/2019 Correcting biased envelope estimation of sinusoidal signals with random noise
18/28
Intensity pattern normalization
Results
Conclusions
References
Biasing envelope estimation
Correcting biased envelope estimation
Variance of the noise
Intensity pattern normalization noisy pattern
Noisy pattern: I=a+bcos+ FAILS!!!
1.AI a, 2.B(I a)2 =b2/2, 3.(I a)/
2 cos.
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9 / 1 6 Rigoberto Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
Introduction
Intensity pattern normalization Biasing envelope estimation
http://find/http://goback/7/23/2019 Correcting biased envelope estimation of sinusoidal signals with random noise
19/28
Intensity pattern normalization
Results
Conclusions
References
Biasing envelope estimation
Correcting biased envelope estimation
Variance of the noise
Intensity pattern normalization noisy pattern
Noisy pattern: I=a+bcos+ FAILS!!!
1.AI a, 2.B(I a)2 =b2/2, 3.(I a)/
2 cos.
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9 / 1 6 Rigoberto Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
Introduction
Intensity pattern normalization Biasing envelope estimation
http://find/7/23/2019 Correcting biased envelope estimation of sinusoidal signals with random noise
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Intensity pattern normalization
Results
Conclusions
References
Biasing envelope estimation
Correcting biased envelope estimation
Variance of the noise
Biasing envelope estimation
Consider a noisy fringe pattern of the form
I=a+bcos+, (2)
where is random Gaussian noise with zero mean and standard deviation .
Background light estimation
The first estimator is unbiased because it converges to the expected value of thesignals and the noise have zero mean. Then
A
I a. (3)
10 / 16 Rigober to Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
Introduction
Intensity pattern normalization Biasing envelope estimation
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Intensity pattern normalization
Results
Conclusions
References
Biasing envelope estimation
Correcting biased envelope estimation
Variance of the noise
Biasing envelope estimation
Now, the recovered background light is subtracted from the frame and squared toobtain
(I a)2 = (bcos+)2 = b2
2+
b2
2cos+2b cos+2. (4)
In this case, when an estimatorBis applied to(I a)2 we have
B(I a)2 = b2
2+E[2] =
b2
2+2, (5)
whereE[]is the expected value.
From equation (??) we have that the envelope is recovered as
b=
2( 2). (6)
However, we need to know the variance 2 of the noise!
11 / 16 Rigober to Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
Introduction
Intensity pattern normalization Biasing envelope estimation
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y p
Results
Conclusions
References
g p
Correcting biased envelope estimation
Variance of the noise
Determining the variance of the noise
Sinceb>>22, we can use the approximation
2 =
b2 +22 b+
2
b, (7)
then
b
2
2
b. (8)
Therefore
I= I a
2=
1
2
b2 +2
cos+
12
. (9)
Thus, if the function cosis known, we have that
I cos= 2
b2 +2cos+
12
12
. (10)
Finally, we extract the variance of from the variance of
(I cos)2. (11)12 / 16 Rigober to Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
Introduction
Intensity pattern normalization Biasing envelope estimation
http://find/7/23/2019 Correcting biased envelope estimation of sinusoidal signals with random noise
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Results
Conclusions
References
Correcting biased envelope estimation
Variance of the noise
Phase demodulation
The proposed approach requires to know the function cos . For this, we use the
Fourier fringe analysis method to extract the encoded phase and then to compute itscosine.
13 / 16 Rigober to Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
IntroductionIntensity pattern normalization
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Results
Conclusions
References
Results
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Results
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References
Results
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Results
Conclusions
References
Conclusions
The process of normalization of intensity sinusoidal patterns was described andthe biasing effect in the envelope estimation by noise was highlighted.
A method to obtain the statistical properties of the noise in sinusoidal signals wasproposed.
The feasibility of this approach was tested by showing that the proposed methodavoids the bias by noise of envelope estimation.
15 / 16 Rigober to Juarez Salazar [email protected] Correcting envelope estimation of noisy sinusoidal signals
IntroductionIntensity pattern normalization
Results
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Results
Conclusions
References
R. Juarez-Salazar et al, Intensity normalization of additive and multiplicativespatially multiplexed patterns with n encoded phases,Optics and Lasers in
Engineering, p.Accepted for publication, Aug. 2015.
R. Juarez-Salazar et al, Theory and algorithms of an efficient fringe analysistechnology for automatic measurement applications,Appl. Opt., vol. 54,pp. 53645374, Jun 2015.
R. Juarez-Salazar et al, Generalized phase-shifting algorithm for inhomogeneousphase shift and spatio-temporal fringe visibility variation,Opt. Express, vol. 22,
pp. 47384750, Feb 2014.
R. Juarez-Salazar et al, Phase-unwrapping algorithm by arounding-least-squares approach,Optical Engineering, vol. 53, no. 2, p. 024102,2014.
R. Juarez-Salazar et al, Generalized phase-shifting interferometry by parameterestimation with the least squares method,Optics and Lasers in Engineering,vol. 51, no. 5, pp. 626 632, 2013.
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IntroductionIntensity pattern normalization
Results
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Results
Conclusions
References
Thank you very much for your attention
Any question?
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