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Sparse representation utilizing tight frame for phase retrieval
- Baoshun Shi^{1},
- Qiusheng Lian^{1}Email author and
- Shuzhen Chen^{1}
https://doi.org/10.1186/s13634-015-0288-9
© Shi et al. 2015
- Received: 25 June 2015
- Accepted: 13 November 2015
- Published: 19 November 2015
Abstract
We treat the phase retrieval (PR) problem of reconstructing the interest signal from its Fourier magnitude. Since the Fourier phase information is lost, the problem is ill-posed. Several techniques have been used to address this problem by utilizing various priors such as non-negative, support, and Fourier magnitude constraints. Recent methods exploiting sparsity are developed to improve the reconstruction quality. However, the previous algorithms of utilizing sparsity prior suffer from either the low reconstruction quality at low oversampled factors or being sensitive to noise. To address these issues, we propose a framework that exploits sparsity of the signal in the translation invariant Haar pyramid (TIHP) tight frame. Based on this sparsity prior, we formulate the sparse representation regularization term and incorporate it into the PR optimization problem. We propose the alternating iterative algorithm for solving the corresponding non-convex problem by dividing the problem into several subproblems. We give the optimal solution to each subproblem, and experimental simulations under the noise-free and noisy scenario indicate that our proposed algorithm can obtain a better reconstruction quality compared to the conventional alternative projection methods, even outperform the recent sparsity-based algorithms in terms of reconstruction quality.
Keywords
- Phase retrieval
- Tight frame
- Sparse representation
- Signal processing
1 Introduction
In science and engineering fields, such as crystallography, neutron radiography, astronomy, signal processing, and optical imaging [1, 2], it is difficult to design sophisticated measuring setups to allow direct recording of the phase, which carries the critical structural information of the test object or signal [1]. Interestingly, an alternative mean called algorithmic phase retrieval is arising in these fields. The goal of phase retrieval (PR) algorithms is to retrieve the signal only through its Fourier spectrum magnitude that can be obtained by the sensors. However, since the global phase shift, conjugate inversion, spatial shift on the interest signal can lead to the same Fourier magnitude, the PR problem is ill-posed. Therefore, prior information on the underlying signal is incorporated into the recovery process to enable its recovery.
In the past decades, alternative projection strategy pioneered by Gerchberg and Saxton [3] for PR is popular. The object magnitude constraint and Fourier magnitude constraint are utilized in the Gerchberg-Saxton (GS) algorithm, which addresses the problem of recovering a complex object from its Fourier magnitude via projecting onto the constrained sets alternatively. Instead of the magnitude constraint of the GS algorithm in object domain, Fienup [4] in 1978 suggested a PR algorithm called hybrid-input output (HIO) algorithm, which incorporates the non-negativity and support constraint into the PR process. Further study of alternative projection strategy [5–7] can be regarded as the modification or extension of the HIO algorithm and the GS algorithm.
Recently, the sparsity prior for PR is focused by researchers [8–12]. Theoretically, the sparsity prior can be incorporated into the object constraint of any alternative projection algorithm to improve the performance. Mukherjee et. al. proposed the so-called Max-K algorithm [8], which incorporates sparsity into the object constraint of alternative projection strategy via solving the sparse coding subproblem; Loock et. al. [9] incorporated the sparsity constraint into relaxed averaged alternating reflectors (RAAR) algorithm and proposed a shearlet soft thresholding procedure for PR from near-field sampled data, namely Fresnel magnitude. Another sparsity-based strategy for PR is based on greedy strategy, including greedy sparse phase retrieval (GESPAR) [10] and nonlinear basis pursuit [11]. It has been shown that GESPAR could achieve lower computational complexity compared to the alternative projection algorithm with sparsity constraints [10].
In the image PR field, the image regularization, such as l _{1} regularization [13, 14], is focused by researchers. They often formulate the non-convex l _{1} minimization problem and solve the problem by alternating directions method of multipliers (ADMM) [15], which can obtain a suboptimal solution to the non-convex problem. Inspired by this idea, in this paper, we extend the spatial sparsity prior to transform sparsity prior based on translation invariant Haar pyramid (TIHP) [16] for PR. The proposed regularization is based on the assumption that the underlying image can be represented sparsely in TIHP tight frame. The assumption is natural for a wide class of natural images. Indeed, TIHP tight frame have been shown to provide suitable results for image restoration [16, 17]. We formulate the sparse representation regularization term and incorporate it into the PR optimization problem combining with the support and Fourier magnitude constraint. Due to the non-convexity of the objective function, the optimal solution to the corresponding problem is difficult to obtain. Nevertheless, ADMM technique, which can obtain a satisfied solution to the PR problem [13, 14], is utilized in this paper.
- 1.
We propose a sparse representation regularization term based on the TIHP tight frame for phase retrieval. We combine the sparse representation regularization term with the data consistency term and object constraint term of utilizing the indicator function to formulate a new phase retrieval problem. The sparse representation regularization term of utilizing TIHP tight frame is helpful to retrieve the missing phase as well as recover the image at low oversampled factors. Moreover, additional spatial priors of the image can be incorporated into the object constraint via enforcing the spatial priors in the constraint set, specially support prior and the intensity constraint of the underlying image are utilized in this paper;
- 2.
The alternative iterative algorithm of utilizing ADMM technique for solving the formulated optimization problem is proposed via dividing the formulated problem into several subproblems. We give the optimal solution to each subproblem theoretically, and experimental results demonstrated the better convergence behavior of this approach;
- 3.
We demonstrate the sparsity measure of utilizing l _{1} norm can obtain better reconstruction than l _{0} norm for our framework heuristically. Experimental results indicate that our proposed algorithm can obtain better reconstruction quality compared with the alternative projection algorithms of utilizing the same sparsity prior. Additionally, our algorithm is robust to noise, which is demonstrated empirically.
The structure of this paper is as follows. To begin with, the PR prior work is reviewed in Section 2. Then, in Section 3, we formulate our new PR problem and introduce our alternative iterative algorithm in detail. Section 4 presents our experimental simulations. Finally, concluding remarks and directions for future research are presented in Section 5.
2 Related work
2.1 The alternative projection strategy
2.2 ADMM for the PR optimization problem
The above problem is a non-convex problem, Yang et. al. [14] suggested ADMM technique to solve the problem and obtained a better reconstruction. Yang’s algorithm suffers from the limit of recovering the image that is sparse in spatial domain; when it comes to recover the images that are non-sparse in spatial domain, it fails to reconstruct. However, most of the natural images are non-sparse in spatial domain; Yang’s algorithm cannot enable to retrieve the phase of these images. Moreover, the previous alternative projection algorithms of utilizing sparsity prior suffer from either the relatively low reconstruction quality at low oversampled factors or being sensitive to noise. To address these issues, we propose a framework of utilizing TIHP tight frame, and experimental results indicate its efficiency for natural images.
3 The proposed approach
3.1 Problem formulation
Where x ∈ ℝ ^{ N } is the interest signal, W represents the TIHP tight frame admits W ^{ T } W = I (here I is an identify matrix), and λ is the regular parameter. The first term of problem (9) is the data consistency, and the second term represents sparse representation regularization term in the TIHP tight frame. For the sparse representation regularization term, both the l _{1} norm and l _{0} norm can be considered to promote sparsity. In the simulations section (Subsection 4.1), we demonstrate that the reconstruction obtained by l _{1} norm is better than utilizing l _{0} norm in terms of reconstruction quality. The third term is an indicator function that can combine some additional constraints, such as support or non-negative constraint, into PR process.
3.2 The proposed phase retrieval method
- 1.
x subproblem
- 2.
z subproblem
- 3.
y subproblem
Where Ρ _{ S }(•) represents the projection operator onto the constraint set S.
Complete description of our proposed algorithm
Input: the Fourier magnitude b; |
Initialization: t = 1, initial estimated image x ^{(0)}, error tolerance τ > 0, parameters λ, γ, ρ _{1}, ρ _{2} |
Repeat |
Update image x ^{(t)} by (14); |
Update z ^{(t)}, y ^{(t)} by (20) and (22), respectively; |
Update the scaled dual variables \( {\mathbf{u}}_1^{(t)} \), \( {\mathbf{u}}_2^{(t)} \) by (23); |
t = t + 1; |
Until maximum iteration number is reached or res ≤ τ |
Output: final estimated image |
4 Experimental simulations
4.1 Parameter setup
To demonstrate the effectiveness of the sparsity in our algorithm, we compared our algorithm with the alternative projection algorithm without sparsity constraint, and the HIO algorithm is chosen as the benchmark algorithm under the noise-free case. The non-negative constraint S _{+} is utilized in the HIO algorithm, and the HIO MATLAB codes can be downloaded from https://github.com/leeneil/ghio-matlab. We also incorporated the state-of-the-art sparsity-based algorithms such as the Max-K algorithm [8] and RAAR framework with shearlet sparsity method [9] for comparison. The Max-K algorithm can be regarded as a parameterized relaxation with respect to RAAR [8] of utilizing K sparse constraint; therefore, we chose the RAAR method with sparsity constraint in the TIHP tight frame for comparison. Due to the l _{0} norm that is incorporated to promote sparsity, we termed this algorithm as RAAR-l _{0} algorithm, which is also introduced in [8]. For RAAR-l _{0} algorithm, the sparsity level K is set to 0.4 N, where N is the total number of the measurements, and β = 0.99. Moreover, the proposed RAAR-based algorithm in [9] is selected for comparison. We utilize the sparsity in TIHP tight frame instead of shearlet sparsity for reconstruction from the far-field data. The l _{1} norm is utilized in [9] to promote sparsity; thus, the corresponding algorithm is called RAAR-l _{1} algorithm. It is difficult to give the theoretical guarantee for the choice of the parameters for PR algorithms. In general, these parameters are tuned heuristically. To give a better performance for the RAAR-l _{1} algorithm, we suggest a rule of updating thresholding ε empirically: ε = C _{1} + C _{2}/t (here C _{1} and C _{2} are some constants that need to be tuned empirically). The thresholding ε of the RAAR-l _{1} algorithm is decreasing in dependence of the iteration number t, which gives a promising result that has been demonstrated in [9]. Each parameter in the RAAR-l _{1} algorithm was evaluated by varying one parameter at a time while keeping the rest fixed based on the principle of obtaining the higher peak signal to noise ratio (PSNR). We tried several choices of β, C _{1}, and C _{2} for this algorithm at oversampled factor 1.58, and experimental results show that β = 0.99, ε = 1.5 + 8/t are suitable parameter choices.
Here, T _{ K }(•) represents the operator that retains the K largest coefficients, and B(•) = max(min(•, 255), 0) denotes as the pixel intensities constraint. The projection P _{S}(•) represents projection onto the support constraint. Differing from the RAAR-l _{0} algorithm, the RAAR-l _{1}algorithm incorporates \( {P}_{\mathrm{S}}^{\varepsilon}\left(\bullet \right)=B\left\{{P}_{\mathrm{S}}\left[{\mathbf{W}}^T{T}_{\varepsilon}\left(\mathbf{W}\left(\bullet \right)\right)\right]\right\} \) into (4). The maximums of iterations for all algorithms in the experiments are set to 3000.
Note that an initial guess is important for PR, we utilize x ^{(0)} = P _{ S } P _{ Μ }(v) as the initial guess for all algorithms; here, v is a random image. We tune the parameters of the proposed two algorithms finely, heuristically, we set the dyadic scales 7, and ρ _{1} = ρ _{2} = 0.01, γ = 0.5, τ = 0.01 for the two algorithms. For the parameter λ, we set PR-TIHP-l _{0} and PR-TIHP-l _{1} to 0.1 and 0.05, respectively.
4.2 Phase retrieval from noise-free oversampled diffraction pattern
We performed the proposed algorithm and the three benchmark algorithms for various images at oversampled factor 1.58. In this simulation, the non-negative support region is simply the window with size 324 × 324 in the center of the 512 × 512 padded image.
Time (s) for phase retrieval of our PR-TIHP-l _{1} algorithm and the benchmark algorithms
Testing image | HIO | RAAR-l _{0} | RAAR-l _{1} | PR-TIHP-l _{1} |
---|---|---|---|---|
Lena | 269.00 | 3403.06 | 1559.63 | 797.27 |
Fruits | 268.52 | 3423.63 | 1578.53 | 159.94 |
Average | 268.76 | 3413.34 | 1569.08 | 478.60 |
4.3 Phase retrieval from noisy oversampled diffraction pattern
We simulated a noisy diffraction pattern from the image “Lena” at oversampled factor 1.74. Under the noisy data case, we incorporate the oversampling smoothness (OSS) algorithm [20], which produces consistently better reconstruction than the HIO algorithm under the noisy scenario, instead of the HIO algorithm for comparison. The OSS code can be downloaded from the author’s homepage and the maximum iterations for OSS we set is also 3000. Note that the random phase without support constraint is suitable for OSS, we utilized its initial method for initial guess of OSS; moreover, the same initialization is as described in Section 4.1 for the other algorithms. We added the random noise n on the true oversampled diffraction pattern to generate the noisy measurement data b _{ noise } = b + n.. The noise n is scaled so that the R _{noise} defined by R _{noise} = ||b − b _{ noise }||_{1}/||b||_{1} is ranging from 5–20 %. For each noise level, we performed 20 independent runs and calculated the R _{real} [20]: R _{ real } = ||x ^{(t)} − x||_{1}/||x||_{1} to evaluate the reconstruction quality.
5 Conclusions
In this paper, we have introduced a framework for PR based on translation invariant Haar pyramid. Our main idea is to formulate sparse representation regularization term of utilizing TIHP tight frame for PR. We incorporated the formulated regularization term into the PR problem, which yields a new non-convex optimization problem. ADMM technology was utilized for solving the resulting problem, and a satisfied solution is obtained. We demonstrated the l _{1} norm that promoted sparsity can obtain better reconstruction than l _{0} norm for our approach heuristically. Moreover, experimental simulations showed that our proposed approach considerably outperforms the previous PR algorithms in terms of reconstruction quality at low oversampled factors. The Gaussian noise with various noise levels was added on the true oversampled diffraction pattern to evaluate the reconstruction quality of our algorithm showing robust to noise. In this paper, TIHP tight frame is chosen for our approach to retrieval phases as well as recover images. Exploiting finer tight frame to improve the reconstruction quality is our future work.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant 61471313 and by the Natural Science Foundation of Hebei Province under Grant F2014203076.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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