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Accelerated single-beam wavefront reconstruction techniques based on relaxation and multiresolution strategies Konstantinos Falaggis,* Tomasz Kozacki, and Malgorzata Kujawinska Institute of Micromechanics and Photonics, Warsaw University of Technology, 8 Sw. A. Boboli St., Warsaw 02-525, Poland *Corresponding author:
[email protected] Received January 28, 2013; revised April 12, 2013; accepted April 16, 2013; posted April 18, 2013 (Doc. ID 184369); published May 10, 2013 A previous Letter by Pedrini et al. [Opt. Lett. 30, 833 (2005)] proposed an iterative single-beam wavefront reconstruction algorithm that uses a sequence of interferograms recorded at different planes. In this Letter, the use of relaxation and multiresolution strategies is investigated in terms of accuracy and computational effort. It is shown that the convergence rate of the conventional iterative algorithm can be significantly improved with the use of relaxation techniques combined with a hierarchy of downsampled intensities that are used within a preconditioner. These techniques prove to be more robust, to achieve a higher accuracy, and to overcome the stagnation problem met in the iterative wavefront reconstruction. © 2013 Optical Society of America OCIS codes: (090.0090) Holography; (100.5070) Phase retrieval; (070.7345) Wave propagation. http://dx.doi.org/10.1364/OL.38.001660
Single-beam wavefront reconstruction methods are of increased interest [1–3], as they allow reconstruction of the phase of a general volume speckle field in single-beam devices, such as optical microscopes. Transport of intensity equation (TIE) [3] based techniques enable phase reconstruction in such systems because TIE relates the phase of a wave to the axial intensity derivative. The latter is estimated using a series of intensity measurements, where the camera is stepped along the longitudinal axis. A TIE-based approach for a partially developed speckle field was demonstrated experimentally to enhance the calculation of the axial intensity derivative [3]. Unfortunately, the TIE solution is paraxial, and errors or noise in the estimation of the axial derivative directly propagate into the calculated phase. A reconstruction method that applies the constraints of the intensity recorded at multiple camera planes via wave-propagation algorithms is the so-called single-beam multiple-intensity reconstruction (SBMIR) technique, which was reported by Pedrini et al. [2] and extended by Almoro et al. [1]. This technique assumes a value of the phase at the first plane and propagates the optical wave field to the next, subsequent, and following planes, while the intensity of the calculated field at a given plane is replaced by the measured intensity. At the last plane, the same procedure is applied in the backward direction until the solver reaches the first plane and a single iteration cycle is completed [1]. This Letter proposes two techniques that accelerate the SBMIR. The first strategy overestimates the solution at each iteration step, using the parameter β as U n1 1 − βL fU n g βAn1 expi∠L fU n gg
forward and backward wave-propagation operators, respectively. This technique reduces to the conventional SBMIR of [1] for the special case of β 1. The rules in Eqs. (1) and (2) are similar to the adaptive additive algorithm [4] or the successive over-relaxation technique [5], which apply an overestimation to the solution using a so-called relaxation parameter β. Therefore, the procedures in Eqs. (1) and (2) are referred to as relaxation strategies for singlebeam multiple-intensity reconstruction (R-SBMIR). The advantage of R-SBMIR is shown by means of a simulation example. Consider the case of the random phase object in Fig. 1, which has a mean feature size of 13.8 μm and a maximum phase difference of 8π, and is illuminated by a plane wave. A total of 25 intensity measurements are recorded at a distance of 6.33 mm with a plane separation of 1 mm. The sample size is equal to 3.45 μm, and the total number of pixels is 512 × 512. In accordance with [2], the wave propagation is implemented using a modified angular spectrum (AS) method [6], as it requires low computer memory and enables parallel computation on a “GeForce GTX560” graphics processing unit (GPU). The AS simulation with a zero-padding factor of 2 [6] has
(1)
and U n 1 − βL− fU n1 g βAn expi∠L− fU n1 gg;
(2)
in forward and backward directions, respectively, where U n is the calculated field at the nth plane, An I n −1∕2 , I n is the measured intensity, and L f·g and L− f·g are the 0146-9592/13/101660-03$15.00/0
Fig. 1. RMS error of the R-SBMIR for various values of β using a random phase object with a maximum phase difference of 8π for the case of 25 intensities captured at 6.33 mm with an SNR of 35 dB. © 2013 Optical Society of America
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been carried out in MATLAB using the JACKET library, and a single iteration for 25 planes requires 0.96 s. The plots in Fig. 1 illustrate the trends for how the root-meansquare (RMS) error of the phase of the wave at the first camera plane evolves with time for the R-SBMIR for various values of β having additional additive white Gaussian intensity noise with a signal-to-noise ratio (SNR) of SNR 35 dB. The SNR is defined as the ratio between the square of the mean value and the square of the standard deviation of the measured intensity at the camera plane. A property of the R-SBMIR is that the convergence stagnates after a relatively short time period. However, the various choices of β do not only control the time where the phase RMS error stagnates, but they also affect the level of the stagnated RMS error. It is shown that for an SNR of 35 dB, the conventional SBMIR (R-SBMIR with β 1) stagnates after ∼110 s at an RMS error level of 0.0359λ. More importantly, other choices of β improve the convergence rate, e.g., the choice of β 0.15 reaches within 30 s the same accuracy as the SBMIR and additional iterations can further reduce the error: the algorithm stagnates after ∼500 s near an RMS error of 0.0107λ. The main difference is that the SBMIR has a fixed constraint for the field amplitude that in presence of noise hinders the algorithm from converging to the correct solution. The R-SBMIR relaxes this constraint and enables the field information of the other planes to be use in the field calculation at a given plane. In essence, a value of β < 1 imposes the field amplitude constraints as a weighted sum of the individual measured intensities. The lower the value of β, the stronger the influence of the previously calculated fields on current field calculation. In the presence of noise, a lower value of β is preferable as the calculated value of the field is less affected by the uncertainty in the measured intensity, as the errors are averaged out. Notably, all R-SBMIRs have a similar impact on the spectral components of the error: they eliminate after a few iterations the high-frequency errors, but prove to have slow convergence—or even to stagnate—when it comes to the elimination of low-frequency errors. The remaining error is a phase tilt, bell-shaped error, or another low-frequency artifact. A similar behavior is also observed in the area of linear algebra [5] for, e.g., Jacobi or Gauss–Seidel iterations that solve iteratively linear systems. Multigrid solvers [5] are strategies that overcome this problem utilizing a series of downsampled computational grids suitable for global error elimination. Motivated by that, we propose the use of a multiresolution (MR) strategy. The steps are described as follows: (1) the captured intensities are downsampled by a factor of qP , where q is an integer >1, P is the number of reduction levels, and the hierarchy stage is referred to as level P. (2) At level P the R-SBMIR is applied using the downsampled intensities, where the initial phase estimate can be a random or a constant phase [1]. (3) The estimated phase of the R-SBMIR is upsampled by a factor of q, so it can serve as an estimate of the initial phase at the (by a factor of q) finer grid of the next higher level. (4) At the next higher level, the R-SBMIR is carried out using the phase obtained in step 3 and downsampled intensities with same grid size. (5) Steps 3 and 4 are repeated until the MR solver reaches the top level (level 0). (6) At level 0, the estimated phase of level 1 is upsampled by a factor
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of q and is used as an input to the R-SBMIR with the nondownsampled captured intensities. Here, the up- and downsampling procedure is implemented via nearest integer interpolation. Notably, at the coarser grid the low-frequency error has a larger frequency relative to the sample size and is thereby eliminated effectively by R-SBMIR. The use of MR strategies requires less computational effort, because of the reduced number of pixels and the fact that the low-frequency error is tackled effectively at the downsampled grids. The object properties affect the number of iterations needed and can be estimated empirically for a given a class of objects. This information allows adjusting the iteration at each level to the experimental needs. It is possible to stop the calculation when the solver stagnates, i.e., by comparing the phase calculated at the nth iteration with the result of an earlier iteration. The notation of this work refers to an MR strategy with a constant number of X iterations and P downsampled grids for q 2 as the C-X-P algorithm. If a strategy should focus more on the absolute accuracy, a linear (L-X-P) or a quadratic (Q-X-P) increase of the iterations with the number of levels is preferable. For instance, a C-25-4 algorithm has five levels (levels 0–4), and a total of 25 iterations are carried out at each level. The L-35-2 algorithm computes a total of 105 iterations at the lowest level (level 2), 70 iterations at the next higher level (level 1), and 35 iterations at the top level (level 0). Figure 2 shows a comparison of various C-X-P, L-X-P, and Q-X-P strategies, where there is a multitude of MR strategies available, which reach the same or an improved accuracy compared to the basic R-SBMIR, but in a significantly shorter computation time, as, e.g., the C-15-4, C-35-4, or L-15-4 strategies. The best candidates from this range of solvers are C-15-4 with β 0.75 and C-35-4 with β 0.5. A significantly increased accuracy can be obtained if strategies having more computation time are applied. This is shown in Fig. 2 for multiple L-X-P and Q-X-P with 35 and 55 iterations, which carry out more iterations than the C-15-4, C-35-4, or L-15-4 strategies. As expected, for these sets of strategies a rather low value of β is preferred: the results in Fig. 2 reflect the trend in Fig. 1, where a relaxation strategy with a low value of β plays out its strength for a higher number of iterations, and a larger value of β
Fig. 2. Comparison of the RMS error of the R-SBMIR for the case of Fig. 1 and various MR strategies with P 4. The plots of Fig. 1 are replotted for comparison.
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Fig. 3. RMS error of the L-55-4 algorithm with β 0.25 for various sets of phase objects and SNR. One exemplary phase object for each set is shown.
is preferable if the number of iterations is small. An R-SBMIR with a large value of β stagnates earlier than for cases with a small value of β. Hence, the larger values of β for C-15-4 give a higher accuracy. Similarly, for β 0.75, the Q-35-4 strategy is as accurate as the C-35-4. Further, for a moderate number of iterations as for C-35-4 or L-15-4, the MR strategy gives the best results for a value of β near β 0.5. Moreover, for a higher number of iterations, a good choice of β is near β 0.15 or β 0.25, as for a L-35-4, L-55-4, Q-35-4, or Q-55-4. For the example in Fig. 2, the most accurate result has been obtained for L-55-4 with β 0.25. Notably, instabilities in the form of small vortices may appear in the estimated phase for larger β. The spatial dimensions of those vortices increase with the number of iterations, and may even cause the whole solution to diverge. Some simulations showed that these singularities disperse and disappear after upsampling to the next higher level of the MR strategy within a few iterations. Although this is an advantage in comparison to SBMIR, this measure is not always successful, as for the cases of C-15-4, Q-15-4, and C-35-4, where the solution diverges. Nevertheless, in any case it has been observed that those vortices can be effectively suppressed when choosing β < β0 with β0 < 1, where β0 depends on the phase object, the number of planes, and the plane separation. We conclude that a relaxation strategy combined with an MR strategy allows overcoming the instability within SBMIR. Nevertheless, despite this robustness, many MR strategies give a higher accuracy, i.e., L-35-4, L-55-4, and Q-55-4 (with β 0.25), as well as Q-35-4 and Q-55-4 (with β 0.15), give an error below 0.006λ. The applicability of these techniques for a wide range of phase objects has been carried out via simulation. The results for the L-55-4 algorithm with β 0.25 are shown in Fig. 3 for various SNR. Each set in Fig. 3 consists of 50 random phase objects, where the object feature size for each set 1–5 is 13.8, 27.6, 55.2,
110.4, and 220.8 μm, respectively. These examples allow studying the performance for band-limited objects. The results indicate that the MR and R-SBMIR strategies can be robustly applied for a wide range of phase objects with spatial frequencies similar to those of sets 1–3. R-SBMIR exploits the diversity within the recorded intensities. From the theoretical point of view, only those spatial frequencies of the wavefront can be reconstructed, which causes considerable variation in the intensities. High-frequency objects provide diversity at all planes, as the intensities are not similar. Low spatial frequencies have, however, due to defocusing, only a minor impact on the dissimilarity between recorded intensities. Hence, the slowly varying objects as in sets 4 and 5 are not reconstructed properly with the L-55-4 algorithm, unless a “lucky” phase estimate is provided as in input to the R-SBMIR. An alternative is to increase the number of iterations, or the diversity within the recorded intensities. The latter can be accomplished by changing the position and separation of the planes, or by using a speckle illumination [1]. This measure expands the applicability of R-SBMIR to slowly varying objects. In conclusion, we propose the use of a relaxation strategy within SBMIR [1], in order to accelerate the convergence. The use MR strategies further accelerates the convergence by tackling low-frequency errors using multiple downsampled intensities. The combined MR and R-SBMIR strategies permit us to achieve the same accuracy with a lower computation time than for the conventional SBMIR (factor 3 for C-15-4) or—more importantly—a higher accuracy (factor 6.5 for L-55-4, β 0.25). The choice of β depends on the number of iterations. As a rule of thumb, for a number of 15, 35, and 55 iterations, the value of β should be chosen near β 0.75, β 0.5, and β 0.25, respectively. The use of an MR strategy with a different value of β at each level, or one in which the value of β varies as the number of iterations increases, is the subject of future work. The research leading to the described results was realized within program TEAM/2011-7/7 of the Foundation for Polish Science, cofinanced from European Funds of Regional Development. We acknowledge the support of the statutory funds of Warsaw University of Technology. References 1. P. F. Almoro, G. Pedrini, P. N. Gundu, W. Osten, and S. G. Hanson, Opt. Lasers Eng. 49, 252 (2011). 2. G. Pedrini, W. Osten, and Y. Zhang, Opt. Lett. 30, 833 (2005). 3. P. F. Almoro, L. Waller, M. Agour, C. Falldorf, G. Pedrini, W. Osten, and S. G. Hanson, Opt. Lett. 37, 2088 (2012). 4. V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002). 5. G. Strang, Computational Science and Engineering (Wellesley-Cambridge, 2007). 6. T. Kozacki, K. Falaggis, and M. Kujawinska, Appl. Opt. 51, 7080 (2012).
