High-accuracy off-axis wavefront reconstruction from ... - OSA Publishing

Vol. 24, No. 22 | 31 Oct 2016 | OPTICS EXPRESS 25068

High-accuracy off-axis wavefront reconstruction from noisy data: local least square with multiple adaptive windows V LADIMIR K ATKOVNIK , 1 I GOR S HEVKUNOV, 2,* N IKOLAY V. P ETROV, 2 AND K AREN E GIAZARIAN 1 1 Department 2 Department

of Signal Processing, Technology University of Tampere, Finland of Photonics and Optical Information Technology, ITMO University, St. Petersburg, Russia

∗ [email protected]

Abstract: A variational algorithm to object wavefront reconstruction from noisy intensity observations is developed for the off-axis holography scenario with imaging in the acquisition plane. The algorithm is based on the local least square technique proposed in paper [J. Opt. Soc. Am. A, 21, 367 (2004)]. First, multiple reconstructions of the wavefront are produced for various size and various directional windows applied for localization of estimation. At the second stage, a special statistical rule is applied in order to select the best window size estimate for each pixel of the image and for each of the directional windows. At the third final stage the estimates of the different directions obtained for each pixel are aggregated in the final one. Simulation experiments and real data processing prove that the developed algorithm demonstrate the performance of the extraordinary quality and accuracy for both the phase and amplitude of the object wavefront. c 2016 Optical Society of America OCIS codes: (090.1995) Digital holography; (070.2025) Discrete optical signal processing.

References and links 1. W. Osten, A. Faridian, P. Gao, K. Körner, D. Naik, G. Pedrini, A. K. Singh, M. Takeda, and M. Wilke, “Recent advances in digital holography [invited],” Appl. Opt. 53, G44–G63 (2014). 2. D. P. Kelly, D. S. Monaghan, N. Pandey, T. Kozacki, A. Michałkiewicz, G. Finke, B. M. Hennelly, M. Kujawinska, and M. Kujawinska, “Digital Holographic Capture and Optoelectronic Reconstruction for 3D Displays,” Int. J. Digit. Multimed. Broadcast 2010, 1–14 (2010). 3. D. Karabacak, T. Kouh, and K. L. Ekinci, “Analysis of optical interferometric displacement detection in nanoelectromechanical systems,” J. Appl. Phys. 98, 124309 (2005). 4. M. Takeda, “Fourier fringe analysis and its application to metrology of extreme physical phenomena: a review [Invited],” Appl. Opt. 52, 20 (2013). 5. J. M. Huntley and H. O. Saldner, “Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithms,” Meas. Sci. Technol. 8, 986–992 (1997). 6. P. Memmolo, L. Miccio, M. Paturzo, G. D. Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7, 713 (2015). 7. T. Y. Nikolaeva and N. V. Petrov, “Characterization of particles suspended in a volume of optical medium at high concentrations by coherent image processing,” Opt. Eng. 54, 083101 (2015). 8. P. Langehanenberg, G. von Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Research 2, 1–11 (2011). 9. M. K. Kim, “Principles and techniques of digital holographic microscopy,” J. Photon. Energy 1, 018005 (2010). 10. Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal fourier analyses of a digital hologram sequence,” Appl. Opt. 46, 5719–5727 (2007). 11. Y. Zeng, X. Chang, H. Lei, X. Hu, and X. Hu, “Characteristics analysis of digital image-plane holographic microscopy,” Scanning 38, 288–292 (2015). 12. A. V. Belashov, N. V. Petrov, and I. V. Semenova, “Accuracy of image-plane holographic tomography with filtered backprojection: random and systematic errors,” Appl. Opt. 55, 81–88 (2016). 13. E. Sánchez-Ortiga, A. Doblas, M. Martínez-Corral, G. Saavedra, and J. Garcia-Sucerquia, “Aberration compensation for objective phase curvature in phase holographic microscopy: comment,” Opt. Lett. 39, 417 (2014). 14. H. O. Saldner, N.-E. Molin, and K. A. Stetson, “Fourier-transform evaluation of phase data in spatially phase-biased tv holograms,” Appl. Opt. 35, 332–336 (1996). 15. C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Exact complex-wave reconstruction in digital holography,” J. Opt. Soc. Am. A 28, 983–992 (2011).

#273991 Journal © 2016

http://dx.doi.org/10.1364/OE.24.025068 Received 19 Aug 2016; revised 7 Oct 2016; accepted 7 Oct 2016; published 18 Oct 2016


16. C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Local demodulation of holograms using the Riesz transform with application to microscopy,” J. Opt. Soc. Am. A 29, 2118–2129 (2012). 17. D. Kim, R. Magnusson, M. Jin, J. Lee, and W. Chegal, “Complex object wave direct extraction method in off-axis digital holography,” Opt. Express 21, 3658–3668 (2013). 18. D. J. Lee, C. A. Bouman, A. M. Weiner, “Single Shot Digital Holography Using Iterative Reconstruction,” Electronic Imaging 19, 1–6 (2016). 19. K. Khare, P. T. S. Ali,and J. Joseph, “Single shot high resolution digital holography,” Opt. Express 21, 2581 (2013). 20. V. Katkovnik, I. A. Shevkunov, N. V. Petrov, and K. Egiazarian, “Wavefront reconstruction in digital off-axis holography via sparse coding of amplitude and absolute phase,” Opt. Lett. 40, 2417–2420 (2015). 21. M. Liebling, T. Blu, and M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A 21, 367–377 (2004). 22. A. Belashov, N. Petrov, and I. Semenova, “Digital off-axis holographic interferometry with simulated wavefront,” Opt. Express 22, 28363–28376 (2014). 23. V. Katkovnik, I. A. Shevkunov, N. V. Petrov, and K. Egiazarian, “Sparse approximations of phase and amplitude for wave field reconstruction from noisy data,” Proc. SPIE 9508, 950802 (2015). 24. V. Katkovnik, “A new method for varying adaptive bandwidth selection,” IEEE Trans Sig. Process.47, 2567–2571 (1999). 25. V. Katkovnik, K. Egiazarian, and J. Astola, “Adaptive window size image de-noising based on intersection of confidence intervals (ICI) rule,” J. Math. Imaging Vision 16, 223–235 (2002). 26. V. Katkovnik, K. Egiazarian, and J. Astola, “Local approximation techniques in signal and image processing,” (SPIE, 2006). 27. Y.-L. Lee, Y.-C. Lin, H.-Y. Tu, and C.-J. Cheng, “Phase measurement accuracy in digital holographic microscopy using a wavelength-stabilized laser diode,” J. Opt. 15, 025403 (2013). 28. N. Levinson, “The Wiener (root mean square) error criterion in filter design and prediction,” J. Math. Phys. 24 261–278 (1946). 29. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Springer Ser. Opt. Sci. 22, 1268–1270 (1997). 30. E. Cuche, P. Marquet, and C. Depeursinge. “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).

1.

Introduction

Digital holography is a modern powerful technology became widespread at the beginning of the 1990s due to the possibility of wavefront registration on digital cameras. Nowadays it has many practical applications [1] including optical metrology, industry, medicine and science. One of the most significant advantages that holography has acquired in its digital guise are wide possibilities on automation of image processing procedures. It involves various algorithms for digital hologram reconstruction [2], noise filtering [3], interferometric calculation of phase difference [4], phase unwrapping [5] and solution for more specific tasks, such as particle tracking [6], characterization [7], autofocusing for live cell analysis [8], and many others. There are various configurations for digital hologram recording [9]: in-line, Gabor, offaxis Fresnel, Fourier, image plane and each of them as a rule imply special reconstruction algorithms. In the framework of this paper we restrict the perscrutation by the image-plane geometry and will consider the techniques for off-axis hologram reconstruction. Image-plane configuration finds application in digital holography [10], digital holographic microscopy (DHM) [11], holographic tomography [12] since it provides a convenient way to control the amplitude image of the object in the process of hologram recording. It is also important that telecentric image-plane optical system maintain the accuracy of DHM proving the spherical aberration compensation [13]. As regards of the algorithms used for reconstruction imageplane digital holograms, the standard method is a Fourier-Transform evaluation [14], which imply a windowed filtration one of the Fourier spectrum orders, shift to zero frequency with corresponding inverse Fourier transformation to yield a complex wave directly in the image plane. The disadvantage of this approach is an energy leakage and decreasing the resolution due to the limited spatial separation of the diffraction orders. There is a number of interesting extensions of this kind FT based filtering for off-axis holography. It is proved in [15], that provided some restrictions and assuming that a support of FT of the hologram belongs to a single quadrant a perfect reconstruction of the object wavefront can be produced for noiseless


observations. The effective steerable complex-domain 2D Riesz transform is developed in [16] for demodulation of the harmonic component in the hologram. In [17] the phase shifting is used for off-axis hologram and processing is produced for modify holograms calculated assuming the magnitude of the reference wave front is known. Variational formulations for off-axis holography is a comparatively novel and promising development. First, it allows to use the full hologram without filtering out the zero order term and to design algorithms targeted on optimal suppression of noise in observations. Second, priory used for phase and amplitude results in essential improvements of complex wave front reconstruction. As a rule this kind of the algorithms are iterative and computationally quite demanding. As examples of the variational formulations we wish to mention the recent paper [18], where the developed algorithm is targeted on Gaussian noise in observations and uses separate total variation regularization terms for amplitude and phase. The algorithm in [19] designed for the Gaussian noise uses for regularization local differences in the complex valued reconstructed wave fronts. In this way it supports smoothness of the wave front. The efficiency of the modern sparsity regularization in the variational formulation is demonstrated in [20], where the sparsity penalization is introduced separately for the absolute phase and amplitude. The absolute phase regularization is of a special interest for object with a wide range phase variation at least larger than 2π. One of the alternative approaches to the off-axis holography data processing is the local least square algorithm (LLS) [21], which is based on the solution of overdetermined system of linear equations for each pixel and thus combining a spatial phase shifting approach. The method became widespread and further been extended for the interferometric processing [22] and sparsity assumption [20] for more efficient noise filtering. But genuine automation capabilities of holographic image processing procedures is the big challenge for the near future. In this paper we introduce an advanced variational algorithm to object wavefront reconstruction from noisy intensity observations for the off-axis holography scenario with imaging in the acquisition plane. Our method is based on the original formulation of the problem, that takes into account the noise occurring at hologram registration, targeted on optimal reconstruction of wavefronts [23]. The proposed algorithm named Intersection of Confidence Intervals Local Least Square (ICI-LLS) uses multidirectional windowed estimates with different size, shape and direction of windows. For selection of the optimal window size the ICI rule is utilized [24–26]. This adaptability allows essentially improve the performance of the LLS algorithm. The paper organized as follows. Section 2 describes all necessary details to understand the proposed algorithm: subsection 2.1 clarifies the observation model, the LLS algorithm is explained at subsection 2.2. The accuracy of reconstruction by LLS algorithm is discussed in subsection 2.3. The necessity to find the balance between bias and variance of the possible estimates explained in subsection 2.4, where ICI algorithm is introduced. Further, in section 3 we present numerical and experimental results of validation of the introduced technique. 2. 2.1.

Proposed algorithm ICI-LLS (intersection of confidence intervals local least square) Observation model

The light emitted by a laser is split into two beams by a beam splitter: one beam is directed towards the surface of an object under study; the other one is the reference beam which is coming at the registration plane at the small angle to produce phase tilt. The spatial intensity distribution of these two beams (hologram) is then measured by a sensor array. The complexvalued wavefront at the sensor plane is given by u s = Bo exp( jϕo ) + Ar exp( jϕr ),

(1)


where uo = Bo exp( jϕo ) and ur = Ar exp( jϕr ) are the object and reference wavefronts, respectively. Here Bo and Ao are their amplitudes and ϕo and ϕr are phases. For the off-axis scenario where the plane reference wavefront is used the reference phase ϕr is defined in the form (2) ϕr = 2π(x sin α x + y sin α y )/λ. Here α x and α y are angles of the reference beam plane with respect to the optical axis z, (x, y) are coordinates in the sensor plane and λ is the wavelength. Thus, the reference beam at the sin α sensor plane is a 2D harmonic function of the frequencies sinλα x , λ y on x and y, respectively. The intensity of this beam (hologram) is defined as =

I

|Bo exp( jϕo ) + Ar exp( jϕr )| 2 = Bo2 + A2r + Bo Ar (exp( j (ϕo − ϕr )) + exp(− j (ϕo − ϕr ))).

(3)

Following [21] we use the auxiliary variables U and Z U = |Bo | 2 + | Ar | 2 and Z = uo Ar .

(4)

The key-point of this replacement is that the intensity I originally quadratic with respect to the amplitudes uo and Ar becomes linear function on the new variables U and Z. Then I can be rewritten as (5) I = U + (V ∗ Z + Z ∗ V ) , where the original unknown variables uo and Ar are replaced by the new variables U and Z and V = exp( jϕr ). If U and Z are given the amplitude Ar is computed according to the formula [21] ⎛ ⎞ 1/2 ⎜⎜⎜ U ± U 2 − 4| Z | 2 ⎟⎟⎟ ⎟⎠ , Ar = ⎜⎝ 2

(6)

where the plus is used for |uo | < Ar and the minus for |uo | > Ar , and Bo = | Z |/Ar .

(7)

Y = I + σε,

(8)

The observed data are noisy where I is the true intensity of the form Eq. (5), σ is the noisy standard deviation and ε stands for a random noise in observations. It is assumed in this paper that the noise is zero-mean i.i.d. standard Gaussian, ε ∼ N (0, 1). 2.2.

Local Least square reconstruction

The least square criterion proposed in [21] for this problem is natural for the observation model Eq. (8): w(t)[Y (t) − (U + (V ∗ (t) Z + Z ∗ V (t))]2 . (9) Jm =

t ∈ Xm

Here w(t) ≥ 0, q ∈ X m w(t) = 1, is a weight function (windows) applied to the variables in the neighborhood X m of the pixel x m . Note that the unknowns U and Z are assumed to be invariant in the neighborhood of x m while the observations Y (t) and the phase V (t) of the reference wave are varying in t.


Considered in the framework of this paper approach is based on multiple estimates applying in Eq. (9) a set of the non-symmetric windows (weight functions) different by the size (index h) and direction (index d). The minimum condition for Jm with respect to the variables U, Z, Z ∗ has the following form ∂Jm /∂Y = 0, ∂Jm /∂Z ∗ = 0 and ∂Jm /∂Z = 0. Calculation of these derivations results in the set of the equations defining the least square estimates of U, Z, Z ∗ [21]: ⎡ ⎤ ⎡ ⎤ ⎤⎡ ⎢⎢⎢ 1 α α ∗ ⎥⎥⎥ ⎢⎢⎢ Uˆ ⎥⎥⎥ ⎢⎢⎢ c1 ⎥⎥⎥ ⎢⎢⎢⎢ α ∗ 1 β ∗ ⎥⎥⎥⎥ ⎢⎢⎢⎢ Zˆ ⎥⎥⎥⎥ = ⎢⎢⎢⎢ c2 ⎥⎥⎥⎥ , (10) ⎣ ⎦ ⎦ ⎣ ˆ∗ ⎦ ⎣ α β 1 c3 Z where α

=

w(t)V ∗ (t)

β=

t ∈ Xm

c1

=

w(t)(V ∗ (t)) 2 ,

t ∈ Xm

w(t)Y (t)

c2 =

t ∈ Xm

w(t)V (t)Y (t)

t ∈ Xm

c3 =

w(t)V ∗ (t)Y (t).

t ∈ Xm

Using the vector-matrix notation we rewrite Eq. (10) in the compact form: ⎡ ⎢⎢⎢ 1 ˆ Ab = c, A = ⎢⎢⎢⎢ α ∗ ⎣ α 2.3.

α 1 β

α∗ β∗ 1

⎡ ⎡ ˆ ⎤ ⎤ ⎤ ⎢⎢ c1 ⎥ ⎢ U ⎥ ⎥⎥⎥ ⎥⎥⎥ , bˆ = ⎢⎢⎢⎢⎢ Zˆ ⎥⎥⎥⎥⎥ , c = ⎢⎢⎢⎢ c ⎥⎥⎥⎥⎥ ⎢⎣ 2 ⎥⎦ ⎢⎣ ⎥⎦ ⎥⎦ c3 Zˆ ∗

(11)

Accuracy of reconstruction

−1 The solution of bˆ = A c defines an estimate (point-wise estimate) for a single pixel x m . Due to noise in observations Y (t) these estimates are random. What follows for selection of the optimal window size essentially uses confidence intervals for these point-wise estimates. For the Gaussian observations an confidence interval is given by expectation and variance of the estimate. The bias is calculated as the expectation of the random error of estimation ˆ i.e. Δbˆ = b − b,

ˆ E{ Δb} E{c}

ˆ = b − A −1 E{c}, = b − E{b} w(t)I (t), w(t)V (t)I (t), w(t)V ∗ (t)I (t)]T , = [ t ∈ Xm

t ∈ Xm

(12)

t ∈ Xm

where E{·} stands for mathematical expectation of random variables. ˆ is given by the equation Then, the centered random error e0 = b−E{b} Ae0 = Δc,

(13)

where Δc = Δc1

=

( Δc1 , Δc2 , Δc3 )T w(t)ε(t), Δc2 = w(t)V (t)ε(t), t ∈ Xm

Δc3 =

t ∈ Xm

(14) ∗

w(t)V (t)ε(t).

t ∈ Xm

The 3 × 3 covariance matrix of the random error e0 is defined as cove0 = E{e0 e0H } = A −1 E{ΔcΔc H }A −1 = −1

−1

A cov Δc A .

(15)


Here the subindex ’H’ denotes the Hermitian conjugate, i.e. transpose and complex conjugation as simultaneous operations. The formula for the covariance covΔc follows immediately from Eq. (14) as

cov Δc

⎡ ⎢⎢⎢ q ∈ X m w 2 (t) ⎢ w 2 (t)V (t) = σ ⎢⎢⎢ ⎣ q ∈ Xm 2 ∗ q ∈ X m w (t)V (t) 2

w 2 (t)V ∗ (t) 2 q ∈ X m w (t)

2 ∗ 2 q ∈ X m w (t)(V (t)) q

∈ Xm

w 2 (t)V (t)

q ∈ Xm 2 2 q

∈ X m w (t)V (t) 2 q ∈ X m w (t)

⎤ ⎥⎥⎥ ⎥⎥⎥ . ⎥⎦

(16)

Inserting the last expression in Eq. (15) defines the covariance matrix cove0 of the random error of reconstruction. The diagonal elements of this matrix are of a special interest as they give the variances of the estimates of U, Z, and Z ∗ : cove0 (1, 1) cove0 (2, 2) cove0 (3, 3)

=

2 σU ,

=

σ 2Z , σ 2Z ∗

=

(17) = σ 2Z .

However, the obtained formulas give the estimation bias Eq. (12) and covariance Eq. (15) for the intermediate optimization variables U and Z, while the goal is to obtain the corresponding characteristics for the initial variables of the problem : the object phase ϕ0 , the object amplitude B0 and the reference amplitude Ar . The links between these two groups of the variable are defined by Eq. (4). Due to nonlinearity of these equations an analytical passage from U, Z, and Z ∗ to ϕ0 , B0 and Ar is possible only using the linearization of these equations. It means that it can be done only for small errors of reconstruction. The adaptive algorithm presented in the next section is heavily dependent on the variances of the estimates. Omitting quite bulky calculations we present the final results showing the formulas for the variances for ϕ0 , B0 and Ar provided that the random component of the errors are small. The link between the small deviations of the variables can be presented in the form ⎡ ⎤ ⎤ ⎡ ⎢⎢⎢ Δϕo ⎥⎥⎥ ⎢⎢⎢ ΔU ⎥⎥⎥ ⎢ ⎢⎢⎢⎢ ΔZ ⎥⎥⎥⎥ = ΔA ⎢⎢⎢ ΔBo ⎥⎥⎥⎥ , ⎣ ⎦ ⎦ ⎣ ΔZ ∗ ΔAr where

⎡ ⎢⎢⎢ ΔA = ⎢⎢⎢⎢ ⎣

0 j Bo exp( jϕo ) Ar − j Bo exp(− jϕo ) Ar

2Bo exp( jϕo ) Ar exp(− jϕo ) Ar

2Ar Bo exp( jϕo ) Bo exp(− jϕo )

⎤ ⎥⎥⎥ ⎥⎥⎥ . ⎥⎦

Then, the covariance matrix for the random errors in ϕ0 , B0 and Ar is of the form covϕ, B o , A r = ΔA−1 · cove0 · ΔA− H

(18)

where cove0 is defined in Eq. (15). These covariance matrix can be calculated numerically using also numerically obtained cove0 . However, if the covariance matrix cove0 is diagonal, our experiments confirm this assumption, the links between the variances of U, Z, and Z ∗ and ϕ0 , B0 and Ar can be given in quite simple form.


σϕ2 o (x)

=

σ 2B o (x)

=

σ 2A r (x)

=

σ 2Z (x)

, 2Bo2 (x) A2r (x) 2 (x) 2A2r (x) · σ 2Z (x) + Bo2 (x) · σU 4( A2r (x) − Bo2 (x)) 2 2 (x) + 2B 2 (x) · σ 2 (x) A2r (x) · σU o Z 4( A2r (x) − Bo2 (x)) 2

(19) , .

These formulas are valid provided we reconstruct all three unknowns ϕ0 , B0 and Ar . However, if Ar is known and only two variables ϕ0 , B0 are reconstructed the variances of their estimates are of the form:

2.4.

σϕ2 o (x)

=

σ 2B o (x)

=

σ 2Z (x) Bo2 (x) A2r (x) σ 2Z (x)

+

2 (x) σU

4Bo4 (x)

,

(20)

4Bo2 (x)

Bias-variance compromise and ICI algorithm

Let the estimates generated by minimization of Eq. (3) use the windows wh , where h is a parameter defining the size of the window function. Then the all estimates depends on this h. The object phase estimate can be denoted as ϕˆ o,h where the subscript emphasizes this depends of the estimate on h. In a similar way dependence on h can be shown for all other estimates. Let us call the estimate ϕˆ o,h 0 optimal on h if the parameter h+ is minimizing the mean squared error criterion (21) h+ = arg min E{( ϕˆ o,h − ϕo ) 2 }. h

It is well known that l ϕˆ o ,h (x, h) E{( ϕˆ o,h (x) − ϕo (x)) 2 } = |mϕˆ o ,h (x, h)| 2 + σϕ2ˆ o ,h (x, h), where mϕˆ o ,h and σϕ2ˆ o ,h indicate the bias and the variance of the estimate ϕˆ o,h . The bias and the variance with respect to the window size parameter h is well studied for the local polynomial estimates [24–26]. Their typical behavior is illustrated in Fig. 1. In general, larger h means smoother estimates with a larger bias and smaller variance, because larger h means stronger averaging of observation random errors. Then, the least squared criterion l ϕˆ o ,h (x, h) has an optimal value on h o pt compromising variance and bias values. It is quite obvious that the estimates defined by the least square criterion Eq. (9) nonlinear with respect to h demonstrate quite similar behavior. Thus, the optimal value of the smoothing parameter h exists and if found can improve the quality and the accuracy of estimation. The formulas of the previous subsection in principal allow to calculate both the bias and the variance and then to find the optimal h. As we can see from Eq. (19) the variance does not depends on the object phase ϕo and can be calculated at least approximately if the estimates of 2 (x) and σ 2 (x) are obtained. The situation with the the amplitudes as well as the variances σU Z bias is much more demanding because it requires that the all unknown parameters including the phase should be given or well estimated. There is a technique, Intersection of Confidence Intervals (ICI) algorithm, which is able to give good estimates of the optimal h without calculation of the bias at all. This algorithm is developed and shown to be efficient for the local polynomial approximation estimates [24– 26]. Here we show that this approach overall as well the developed algorithm are completely applicable to the considered nonlinear least square estimates for off-axis holography.


Fig. 1. Bias-variance balance with respect to the window size h.

The algorithm is simple in implementation. Let H be a set of the ordered window size values H = {h1 < h2 < ... < h J }. The estimates yˆh (x) are calculated for h ∈ H and compared. A special multiple hypotheses testing is exploited in order to identify a window size close to the optimal one. This statistic needs only the estimates yˆh (x) and the variances of these estimates σ y2ˆ h (x, ) both calculated for h ∈ H. For the given input sets of the estimates and their variance we obtain the optimal window size h o pt , the optimal estimate and the variance of this estimate.

Fig. 2. Block-diagram of the ICI-LLS algorithm.

The block-diagram of this algorithm is shown in Fig. 2 and the input-out relations can be shown by the formula: [h+ (x), ϕˆ o,h + (x), σϕ2ˆ o ,h + (x, )] = ICI ( ϕˆ o,h (x), σϕ2ˆ o ,h (x), h ∈ H),

(22)

where ICI denotes the ICI algorithm In completely the same style the ICI algorithm can be applied for reconstruction of the amplitudes Bo and Ar and the adaptive h+ for these variables can be different from the one selected for the phase.


2.5.

Multiple directional windows

The considered local least square estimates and ICI rule can be used with arbitrary symmetric and non-symmetric weight functions. The multiple window reconstruction is an efficient development of this approach [25, 26]. An idea of this multidirectional sectorial estimation is demonstrated in Fig. 3, where x denotes the coordinate where the variable of interest should be estimated, for instance for the phase ϕo . The neighborhood of x is split in non-overlapping sectorial areas. Let the LLS be applied for estimation of ϕo for each of this sectors for all h ∈ H and further the ICI algorithm selects the best h+ for each of these directions. Figure 3 illustrates the idea of this type sectorial estimation. The sector of the direction θ i has an adaptive h+

length hθ+i and Sθ iθ i denotes the area used for estimation in this direction. In principal, ICI gives different h+ for different directions and the total area used for estimation may have a star shaped neighborhood shown in Fig. 3. Each of this sectors gives its own estimates for the point x. The final estimates is calculated as an aggregation of these partial estimates.

Fig. 3. Adaptive windows.

Formally, the multiple window estimation can be given as follows. The criterion Eq. (9) is replaced by wh,d (t)[Y (t) − (U + (V ∗ (t) Z + Z ∗ V (t))]2 , (23) Jm = t ∈ Xm

where wh,d are weights of different size, h ∈ H, and different directions d ∈ D. Here D is a set of the directions used in estimation. Then the estimates, in particular the phase estimates, are different for each h and d, ϕˆ o,h,d (x). Applying the ICI algorithm we select the optimal length h+d with the estimate ϕˆ o,h +d ,d (x) for each of the directions. The aggregation of this partial directional estimate into the final estimate is produced according to the formula [25, 26]

ϕˆ o (x) =

d ∈D

ϕˆ o,h +d ,d (x) σ 2

1

ϕ o ,h + , d d

d ∈D σ 2

1

ϕ o ,h + , d d

(x)

.

(24)

(x)

This is an aggregation with the weights defined by the variances σϕ2 h+ .

+ o ,h d ,d

(x) of the

corresponding directional estimates with the selected sizes In this way we obtain a highly efficient anisotropic adaptive scale estimator with the angular resolution depending on a number of sectors in D. 3.

Results and discussion

For the correct demonstration of ICI-LLS algorithm advantages, comparison of ICI-LLS and LLS methods for different objects was made. All hologram simulations were made in off-axis configuration with conditions that are close to real. Laser wavelength λ = 532 nm, pixel size


was 2.8 μm, angles between reference and object wavefronts α x and α y were taken 1.8725 and 0.0671 degrees, respectively. For the ICI-LLS algorithm to obtain good reconstruction we have to adjust the number of parameters: window type (invariant or Gaussian);window shape (square, half-plane, four quadrant, sectorial); number of directions N for sectorial windows; a set of window sizes denoted as H; and for Gaussian window in LLS - size and parameter σ for the weight function. 3.1.

Numerical simulation

In the given paper, as in our previous works [20, 22, 23] we imply image-plane scenario where a wavefront from the object plane to the image plane transferred by means bitelecentric 4 f optical system. Since numerical wave propagation procedure can be omitted in simulation we use dimensionless characterization of image size in pixels. By analogy with the paper [27], let us start from reconstructions of a simple step-wise phase object, 256 × 256, with invariant amplitude Bo = 1 and the phase taking two values, 0.2 and 2 rad. The following parameter were taken for the LLS algorithm: Gaussian window 7×7, σ = 1 for noiseless case and for the noisy case SNR=40; Gaussian window 10 × 10, σ = 3 for SNR=5. Here SNR stands for Signal to Noise Ratio. These parameters enables the best performance for LLS algorithm. The parameters for ICI-LLS are given in Table 1, column "Phase step". The synthesized holograms are shown in Fig. 4. The influence of the step-wise phase can be seen as shifts of the fringes in the middle of these images. Reconstruction results for the step-wise phase without noise, and with SNR=40 and SNR=5 are shown in Figs. 5, 6 and 7, respectively.

Fig. 4. Noisy holograms of different SNR simulated for the step-wise phase object.

Table 1. Window parameters for ICI-LLS.

Type Shape Directions number Windows sizes, H Gaussian σ Computation time, sec

no noise Invariant Half-plane 2 [2 42 82] 20

Phase step SNR=40 Invariant Half-plane 2 [2:5:42] 21

SNR=5 Gaussian Half-plane 2 [4:5:44] H/3 22

Cameraman SNR=40 Gaussian Sectorial 6 [2:5:27] H/6 37

C60 film SNR=40 Gaussian Sectorial 4 [6:2:16] H/12 21

The first column in Fig. 5-Fig. 7 shows longitudinal cross-sections of LLS and ICI-LLS reconstructions, the second column shows phase and amplitude reconstructed by LLS, third phase and amplitude reconstructed by ICI-LLS. In these figures the top rows give results for phase reconstruction and the bottom rows - results for amplitude. It is clear for all cases that the ICI-LLS algorithm demonstrates the better results and reconstructs the object wavefront more precisely and with better noise suppression than LLS algorithm.


Fig. 5. Noiseless step-wise phase object reconstructions by LLS and ICI-LLS algorithms. RMSE values show the accuracy of the corresponding reconstructions. Top row – phases, bottom row – amplitudes.

Fig. 6. Noisy (low noise level, SNR=40) step-wise phase object reconstructions by LLS and ICI-LLS algorithms. RMSE values show the accuracy of the corresponding reconstructions. Top row – phases, bottom row – amplitudes.

It is seen that the LLS reconstructions for amplitude are severely damaged by artifacts appeared in the area where phase changes values from 0.2 to 2.0 rad. These artifacts are successfully diminished by ICI-LLS. For the numerical comparison of reconstructions we use RMSE (Root Mean Square Error) criterion [28]. This comparison is done in Table 2. It is seen that RMSE values for ICI-LLS much smaller than for LLS. Even in the case of the very noisy data with SNR=5 the ICI-LLS algorithm shows RMSE values much smaller than those the LLS algorithm shows for the noiseless case. Next, we show results and performance of the algorithms for much more complex phase object. It is assumed that the phase variations are defined by the cameraman test-image. This phase is scaled to the interval [0, π/2] and the amplitude B0 is equal to 1. ICI-LLS parameters are given in Table 1, column "Cameraman", and for LLS we take window size 5 × 5 and σ = 1. Here we demonstrate performance of the ICI-LLS algorithm with the multidirectional sectorial windows. The number of directions is taken equal to 6 in order to enable an improved


Fig. 7. Noisy (high noise level, SNR=5 dB) step-wise phase object reconstructions by LLS and ICI-LLS algorithms. RMSE values show the accuracy of the corresponding reconstructions. Top row – phases, bottom row – amplitudes.

Table 2. Root Mean Square Errors (RMSE) for phase step object.

SNR no noise 40 5

LLS Phase Ampl 0.0618 0.0518 0.0653 0.0869 0.137 0.141

ICI-LLS Phase Ampl 0.000014 0.0000059 0.00154 0.0325 0.044 0.0836

SPAR Phase Ampl 0.053 0.003 0.053 0.0062 0.069 0.061

FT Phase 0.123 0.124 0.188

Ampl 0.0563 0.0588 0.141

reconstruction of the small details of this complex phase object. Angular size of these directional sectors is equal to 60◦ .

25

20

15

10

5

Fig. 8. Adaptive varying window sizes obtained by the ICI algorithm for the cameraman phase object. These images of pixel-wise optimal window sizes correspond to the directional sectorial windows of the directions: first row left-to-right 0, π/3, 2π/3, second row left-to-right π, 4π/3, 5π/3.


Fig. 9. Cameraman phase object reconstruction results:images (first row) and horizontal middle line cross-sections (second row).The cross-section of the initial phase is omitted because the difference with ICI-LLS is too small.

The crucial feature of the ICI-LLS algorithm is varying adaptive window sizes. What kind of window sizes appeared in reconstruction is shown in Fig. 8 for 6 sectorial estimates. Figure 8 shows 6 images corresponding to different direction. From left to right in the fist row there are directions 0, π/3, 2π/3 and the second row the directions π, 4π/3, 5π/3. Small and large window sizes are shown by black and white, respectively. Images in Fig. 8 show that the window sizes delineate the true image of the cameraman and the variations of the window sizes provides a shadowing of the image edges from different directions in full agreement with the directional windows used for wavefront reconstruction.

Fig. 10. LLS phase reconstructions for cameraman phase object with different sizes of the processing windows.

Noise in the observations naturally influence the ICI-LLS algorithm, here noise effects are seen in the images as isolated black spikes. It deserves to be mentioned that these isolated errors of ICI-LLS have different locations into 6 different images and nearly do not influence the final reconstructed wavefront shown in Fig. 9. That Fig. 9 shows top row: true phase (left image) and reconstructions by LLS (middle) and ICI-LLS (right); bottom row: cross-sections of these reconstructions. These results are obtained for quite noisy data with SNR=40. These cross-sections clearly show the advantage of ICI-LLS: noise is well filtered on smooth parts of the cross-section while the peaks and jumps are accurately reconstructed. In this image we do not show the cross-section of the true image, because the difference with ICI-LLS is too small. For comparison with the ICI-LLS reconstruction in Fig. 10 we show the reconstruction given by LLS algorithm with the different and fixed window sizes. It is seen that small window (2 × 2) leads to very noisy reconstruction while larger windows oversmooth the reconstruction in such


way that the important details are lost. 3.2.

Comparison with other techniques

In this section we briefly discuss a comparison of ICI-LLS versus the standard Fourier transform (FT) [30] and the recent variational Sparse Phase and Amplitude Reconstruction (SPAR) algorithm [20]. The latter one is especially designed for the noisy additive Gaussian data and uses the sparse approximations for phase and amplitude. This iterative algorithm can be treated as one of the best in the field. The LLS algorithm is used with the Gaussian window 10 × 10 with σ = 3, the best parameters for the case. The results shown in Fig. 11 are obtained for the step-wise phase object and very noisy data (SNR=5). FT gives the worst result both visually and numerically in RMSE values. LLS demonstrates a better performance, and the best results are achieved by ICI-LLS and SPAR. Visually the SPAR reconstruction looks better without peakwise errors appeared in ICI-LLS, while RMSE value for ICI-LLS is smaller than RMSE for SPAR. Comparison of the accuracy reconstruction for both phase and amplitude and for various Table 3. Root Mean Square Errors (RMSE) for cameraman phase object.

SNR 40

LLS Phase Ampl 0.098 0.106

ICI-LLS Phase Ampl 0.069 0.042

SPAR Phase Ampl 0.085 0.0115

FT Phase Ampl 0.137 0.034

SNR is presented in Table 2. The worst accuracy results are shown for FT and LLS with the clear advantage of ICI-LLS and SPAR. The comparison of ICI-LLS and SPAR shows that the phase reconstruction by ICI-LLS is more accurate, while the amplitude reconstruction for the noise data is more accurate for SPAR. In Table 3 similar results are shown for the comparison of the algorithms for the cameraman test-image with the best results for ICI-LLS and SPAR. The most important conclusion of this comparison is that the accuracy of the proposed ICILLS is much better than the accuracy of FT and LLS even with the best fixed window size. It is important also that ICI-LLS is able to demonstrate the accuracy which is better than the accuracy of the advanced variation SPAR algorithm. The computational complexity of ICILLS measured by time required for calculation of reconstructions is illustrated in Table 1, last row. The computational time is different for different experiments because in ICI-LLS the LLS estimates are calculated for all window sizes and all window directions independently. For larger number of the used windows the computational time is higher. In this comparison LLS is the fastest algorithm as it requires calculations for a single window only. The computational time of ICI-LLS is measured assuming that the variances of the estimates are already calculated. It can be done in advance because these variances depend only on window sizes and their shape and do not depend on measurements. SPAR is an iterative algorithm and takes more time for reconstruction and its computational time was about 47 sec. for the step-wise phase object for 20 iterations, FT is very fast and takes only 0.34 sec. All calculations are produced in MATLAB on PC with Intel Core i5-3570K CPU 3.40GHz. 3.3.

Experimental validation

For the experimental data reconstructions C60 film was taken as an object under study, this film is transparent and has a refractive index nr = 2.2. Experimental setup based on Mach-Zehnder interferometer [23] was utilized. Off-axis and on-axis holograms were recorded consequently by changing an incident angle of the reference beam. System parameters of the experiment were described in section 3.1. The in-line hologram reconstruction by Phase Shifting Digital holography (PSDH) [29] was taken as reconstruction standard because it has a highest resolution and highest accuracy among these methods. ICI-LLS parameters are given in Table 1, column


Fig. 11. Comparative reconstructions of the step-wise phase from very noisy hologram (SNR=5): FT, ICI-LLS and SPAR algorithms. The advantage of ICI-LLS is obvious.

Fig. 12. Comparison of PSDH, LLS and ICI-LLS reconstructions for real data experiment: images (first row) and horizontal middle line cross-sections (second row).

“C60 film”. LLS parameters are window size 12 × 12 and σ = 2. In Fig. 12 phase reconstructions are shown. Top row: PSDH (left), LLS (middle) and ICI-LLS (right); bottom row - crosssections of these reconstructions. As can be seen, the ICI-LLS image is essentially sharper than LLS image. Visually quality of phase images by ICI-LLS and PSDH are nearly identical. LLS method shows smooth reconstructed structure, that is caused by the size of the LLS window. That smoothening can be a reason of the information lost in areas with wrapped phases. As for cross-sections, graph illustrates that ICI-LLS method repeats the shape of the PSDH reconstruction, the shape of the object has 3 picks and ICI-LLS reconstruct each of them with quite sharp shape, while LLS algorithm smooths the sharp details and even does not resolve peak that is located in region of 59-th pixel. It demonstrates that the proposed algorithm is able to achieve for the off-axis setup the accuracy of phase reconstruction close to the accuracy of


PSDH, which potentially is one of the best algorithms for pixel-by-pixel phase reconstruction. 4.

Conclusion

In this paper, the high-accuracy data adaptive algorithm for wavefront reconstruction from noisy intensity observations for the off-axis holography scenario with imaging in the acquisition plane is presented. It is based on a multiple use of different shapes, types, sizes and directions of the operating windows. This method shows an essentially improved performance in comparison with LLS algorithm. It is shown by simulations as well as by experimental results. Funding Academy of Finland (project no. 287150, 2015-2019); Russian Ministry of Education and Science (project within the state mission for institutions of higher education, agreement 2014/190).