Infrared and Laser Engineering, Volume. 51, Issue 11, 20220402(2022)
Phase retrieval algorithms: principles, developments and applications (invited)
Figures & Tables(15)
Fig. 2. Flow diagrams and schematic diagrams of single-intensity alternating projection iterative algorithms. (a) Flow diagram of GS algorithm; (b) Flow diagram of HIO algorithms; (c1)-(c3) The relationship between the input of the next iteration and the output of the current iteration for ER, HIO, CHIO respectively
Fig. 3. Schematic diagrams of radial multi-intensity phase retrieval. (a) Schematic diagram of ptychography; (b) Flow diagram of ePIE algorithm; (c) Principles and experimental setup of FPM[95]
Fig. 4. Systematic setup and imaging result of TIE[126]. (a)-(b) TIE systematic setup based on 4
Fig. 6. Performance comparison of phase retrieval algorithms based on optimization theory. (a1)-(a2) Relative error curves of phase retrieval for PhaseLift and PhaseCut under Gaussian noise and image scan-lines[35]; (b1) Relative error curves of spectral method and truncated spectral method for the phase retrieval of 1D Gaussian signal[37]; (b2) Empirical success rate curves of WF and TWF for the phase retrieval of real-valued signal[37]; (c1)-(c2) Empirical success rate curves of algorithms based on intensity/amplitude loss function for the phase retrieval of real-valued and complex-valued signals[38]
Fig. 7. Comparison between GESPAR, SDP and Sparse-Fienup algorithm[58]. (a1)-(a2) Comparison of algorithm complexity for the phase recovery of signals with different vector lengths; (b) Comparison curves of phase recovery successful rate
Fig. 8. Phase retrieval based on deep learning. (a) Physical model of deep learning solving phase retrieval problems; (b) Single-frame lensless phase retrieval using deep learning[59]; (c) Fast FPM imaging with a few images using deep learning[61]; (d) Basic framework of DnCNN in prDeep algorithm[62]
Fig. 9. Several typical application scenarios of phase retrieval. (a) X-ray diffraction imaging of crystal microstructure; (b) Optical encryption system based on phase retrieval with a 4
Table 1. Classification of phase retrieval algorithms
View tableView in ArticleTable 1. Classification of phase retrieval algorithms
Algorithm Pros Cons Alternating projections[ 29 ,54 ]Simple implementation Wide applicability Time-consuming Low precision Non- prior TIE[ 55 ]Deterministic solution Simple computation Require no phase unwrapping and complex optical systems Harsh preconditions Lower precision compared with interferometry Limited applications KK relations[ 56 ]Reconstruction with high quality and high speed Require less data collection Require high precision of NA matching SDP[ 34 -35 ]Guarantee for convergence and reconstruction accuracy Not applicable to large-scale signals Non-convex optimization[ 36 -38 ]Struggle with parameter optimization Rely on initial estimate Sparsity- based General[ 57 -58 ]Reconstruction with high quality and stability Low computational cost Applicable to objects with sparsity only Deep learning[ 59 -62 ]Powerful expressive ability Lack of interpretability Unable to deal with multiple models and applications
Table 2. Uniqueness of phase retrieval problem[66]
View tableView in ArticleTable 2. Uniqueness of phase retrieval problem[66]
Item Content Fourier measurements 1D No uniqueness ≥2D Uniqueness for real nonreducible signals with a twice oversampling rate General measurements Real signal Satisfying the complement property is necessary 2N–1 random measurements guarantee uniqueness with high probability Real signal (noisy) Stable uniqueness requires nlogn measurements Complex signal 4n–4 generic measurements are sufficient
Table 3. Flow chart for Misell algorithm
View tableView in ArticleTable 3. Flow chart for Misell algorithm
Algorithm 1 ：MisellInput： Intensity measurements in the frequency domain without modulation ${\left| {{F_0}} \right|^2}$ and phase modulation factors${t_m}$ with their corresponding intensity measurements${\left| {{F_m}} \right|^2}\left( {1 \leqslant m \leqslant n} \right)$ . Output：Spatial complex distribution f.S1. Select the initial Guess of spatial complex distribution without modulation f0 at random. S2. Add factor t1 to the phase of f0 , and obtain the spatial image after the first phase modulation f1. S3. Fourier transform f1 to obtain the complex distribution in the frequency domain G1. S4. Replace the amplitude of G1 with the measurement ${\left| {{F_1}} \right|^{}}$ , and obtain the new complex distribution in the frequency domain$G_1'$ .S5. Inverse Fourier transform$G_1'$ and substract the phase factor t1 , and obtain the new spatial image without modulation$f_1'$ . S6. Add the factor$ {t}_{2} $ to the phase of$f_1'$ , and obtain the spatial image after the second phase modulation f2 . S7. Repeat S2-S6 to cover all modulated images. S8. Repeat S2-S7 until the convergence conditions are satisfied, and output the final result.
Table 4. Flow chart for KKSAI technique
View tableView in ArticleTable 4. Flow chart for KKSAI technique
Algorithm 2 ：KKSAIInput：Intensity measurements Ii (x, y) at each sub-aperture and reference plane wave ri (x, y). Their definitions are given respectively as: $ I_{i}(x, y)=\left|\mathcal{F}^{-1}\left\{S_{i}(u, v)\right\}\right|^{2}$ $v_{1}(x, y)={\rm{e}}^{-{y\left(\omega_{1}-x+y_{1}-y\right)} }$ where $ S_{i}(u, v)=S(u, v) D\left(u-u_{i}, v-v_{i}\right)$ denotes the subregion spectrum at the imaging plane cropped by the sub-aperture at position (ui, vi). Output: Pupil-limited object function S(u, v).S1. For the starting intensity I1, introduce a Hilbert kernel $ H_{1}(u, v)=-j \operatorname{sgn}\left(v_{1}\right) \cdot \operatorname{sgn}(v)$ and define an intermediate variable as:$ X_{1}=\ln \left\{\left[\tilde{s}_{1}(x, y)+r_{1}(x, y)\right] / r_{1}(x, y)\right\}$ where $ \tilde{s}_{1}(x, y)$ denotes the shifted scattered field exiting from the object plane. S2. Impose KK relations to calculate the real part and imaginary part of the intermediate variable:$\operatorname{Re}\left\{X_{1}\right\}=\dfrac{1}{2} \ln \left[I_{1}(x, y) /\left|r_{1}(x, y)\right|^{2}\right]$ $ \operatorname{Im}\left\{X_{1}\right\}=\mathcal{F}^{-1}\left\{\mathcal{F}\left\{\operatorname{Re}\left\{X_{1}\right\}\right\} \cdot H_{1}(u, v)\right\}$ S3. Obtain the shifted form of the subregion: $S_{1}\left(u+u_{1}, v+v_{1}\right)=F\left\{{\rm{e}}^{\operatorname{Re}\left\{X_{1}\right\}+j \ln \left\{X_{1}\right\} } \cdot v_{1}(x, y)\right\}$ move it back to the correct position to get Si (u, v). S4. Repeat above operations for other subregions. S5. Subsequently stitch all subregional complex fields into a full pupil plane spectrum S(u, v): $S(u, v)=\displaystyle\sum_{i=1}^{4} S_{i}(u, v) /\left[\displaystyle\sum_{i=1}^{4} D\left(u-u_{i}, v-v_{i}\right)+\varepsilon\right]$ where $\varepsilon$ is a small constant to keep numerical stability in the zero-valued region.
Table 5. Flow chart for TWF algorithm
View tableView in ArticleTable 5. Flow chart for TWF algorithm
Algorithm 3 ：Truncated Wirtinger Flow (TWF)Input：Intensity measurements yi and sampling vectors αi; trimming thresholds αy and truncation criteria sets ε1, ε2. Output：Iteration value zT. S1. Initialize estimated value ${{\textit{z}}_0} = \lambda \tilde {\textit{z}}$ ，where λ is defined by Eq. (23) and$\tilde {\textit{z}}$ is the leading eigenvector of the following matrix:$Y = \dfrac{1}{m}\displaystyle\sum\limits_{i = 1}^m { {y_i} } {a_i} a_i^{\rm{H}}{1_{\left\{ {\left| { {y_i} } \right| \leqslant \alpha _y^2\left( {\dfrac{1}{m}\displaystyle\sum\nolimits_{i = 1}^m { {y_i} } } \right)} \right\} } }$ S2. Impose iterations on the initial estimated value by gradient update, and the update formula is: ${ {\textit{z} }_{k + 1} } = { {\textit{z} }_k} + \dfrac{ {2{\mu _k} } }{m}\displaystyle\sum\limits_{i = 1}^m {\dfrac{ { {y_i} - \left| {a_i^{\rm{H}}{ {\textit{z} }_k} } \right|} }{ {a_i^{\rm{H}}{ {\textit{z} }_k} } } } {\text{ } }{a_i}{1_{\varepsilon _1^i \cap \varepsilon _2^i} }$ S3. Output the final result after completing iterations of predetermined number.
Table 6. Comparison of runtime performance[58]
View tableView in ArticleTable 6. Comparison of runtime performance[58]
SDP/s Sparse-Fienup/s GESPAR/s s=3 1.32 0.09 0.04 s=5 1.78 0.12 0.05 s=8 3.85 0.50 0.06
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Aiye Wang, An Pan, Caiwen Ma, Baoli Yao. Phase retrieval algorithms: principles, developments and applications (invited)[J]. Infrared and Laser Engineering, 2022, 51(11): 20220402
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Received: Jun. 13, 2022
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Published Online: Feb. 9, 2023
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