Correcting biased envelope estimation of sinusoidal signals with random noise

7/23/2019 Correcting biased envelope estimation of sinusoidal signals with random noise

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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


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



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Introduction


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


Results

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References



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


Results

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x

y

Typical intensity pattern

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200 400 600 800 10000

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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


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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.


I t d ti
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Introduction


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Biasing envelope estimation

Correcting biased envelope estimation

Variance of the noise


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.


Introduction


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Introduction


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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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Introduction
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Introduction


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1.AI a, 2.B(I a)2 =b2/2, 3.(I a)/

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Introduction


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1.AI a, 2.B(I a)2 =b2/2, 3.(I a)/

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Introduction


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1.AI a, 2.B(I a)2 =b2/2, 3.(I a)/

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Results

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Results

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Introduction
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Results

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Intensity pattern normalization noisy pattern

Noisy pattern: I=a+bcos+ FAILS!!!

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Introduction

I t it tt li ti Bi i l ti ti


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Results

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References






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Results

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Introduction

Intensity pattern normalization Biasing envelope estimation


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Results

Conclusions

References






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Introduction

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Results

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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

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Results

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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!


Introduction

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y p

Results

Conclusions

References

g p



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

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Results

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References



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.


IntroductionIntensity pattern normalization
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Results

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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.



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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Results

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References

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Correcting biased envelope estimation of sinusoidal signals with random noise