A semi-implicit relaxed Douglas-Rachford algorithm (sir-DR) for Ptychograhpy

A s e m i - i m p l i c i t r el axe d Do ug la s- Ra ch fo rd a l g o r i t h m ( s i r- DR ) fo r Pt y c ho gr ah py M I N H P H A M , 1 , * A R J U N R A N A , 2 J I A N W E I M I A O 2 , A N D S TA N L E Y O S H E R 1 1 Department of Mathematics, U niv ersity of California, Los Angeles, CA 90095, USA 2 Department of Physics & Astronomy and California NanoSyst ems Institute, Univ er sity of Calif ornia, Los Ang eles, CA 90095, USA * minhrose@mat h.ucla.edu Abstract : Alternating projection based methods, such as ePIE and rPIE, hav e been used widely in ptychograph y . Ho we ver , the y only work w ell if there are adequate measurements (diffraction patter ns); in the case of sparse data (i.e. f e wer measurements) alternating projection underper f or ms and might not ev en conv erg e. In this paper, we propose semi-implicit relax ed Douglas Rachf ord (sir -DR), an accelerated iterative method, to sol v e the classical ptyc hog raph y problem. Using both simulated and e xper imental data, w e sho w that sir -DR impro v es the con verg ence speed and the reconstruction quality relative to ePIE and rPIE. Further more, in certain cases when sparsity is high, sir -DR con ver ges while ePIE and rPIE fail. T o f acilitate others to use the algor ithm, we post the Matlab source code of sir -DR on a public website (www .ph ysics.ucla.edu/researc h/imaging/sir-DR). W e anticipate that this algorithm can be generall y applied to the pty chographic reconstruction of a wide range of samples in the ph ysical and biological sciences. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement 1. Introduction Since the first e xper imental demonstration in 1999 [1], coherent diffraction imaging (CDI) through directl y inv er ting far -field diffraction patter ns to high-resolution images has been a rapidly growing field due to its broad potential applications in the phy sical and biological sciences [2 – 5]. A fundamental problem of CDI is the phase problem, that is, the diffraction pattern measured only contains the magnitude, but the phase information is lost. In the or iginal demonstration of CDI, phase retr ie val was per f ormed b y measur ing the diffraction patter n from a finite object. If the diffraction intensity is sufficientl y ov ersampled [6], the phase inf or mation can be directly retriev ed b y using iterative algorithms [7 – 11]. Ptyc hog raph y , a po werful scanning CDI method, reliev es the finite object requirement by per f orming 2D scanning of an extended relativ e to an illumination probe and measur ing multiple diffraction patter ns with each illumination probe o ver lapped with its neighboring ones [12, 13]. The o ver lap of the illumination probes not only reduces the o versampling requirement, but also improv es the conv erg ence speed of the iterative process. By taking advantag e of ev er -improving computational po wer and advanced detectors, pty chography has been applied to study a wide range of samples using both coherent x-ray s and electrons [2, 5, 14 – 23]. More recently , a time-domain pty chograph y method was dev eloped by introducing a time-inv ar iant ov erlapping region as a constraint, allo wing the reconstruction of a time ser ies of complex e xit wa v e of dynamic processes with robust and fas t conv erg ence [24]. Algorithms f or pty chography hav e been studied exhaus tivel y in theor y and practice. The majority f ollowing non-conv e x optimization approaches [25 – 27], while a f ew follo w con ve x relaxation [28]. In recent years, po werful ptyc hographic algor ithms ha ve been dev eloped to handle par tial coherence [29], solv e f or the probe [13, 30, 31], cor rect positioning er rors [32 – 34], reduce noise [35, 36], and deal with multiple scatter ing [37, 38]. These iterative algorithm can be g enerall y divided into three classes: i) the conjugate g radient (CG) [30, 34], ii) extended ptyc hog raph y iterative engine (ePIE) [31], and iii) difference map (DM) [13], whereas the last tw o hav e a close relationship. ePIE is an alternating projection algorithm, while DM is built on both projection and reflection which is believ ed to provide a momentum to speed up the conv erg ence. The relaxed av erage alter nating reflection (RAAR) [8] is a relaxation of DM and has been shown to be effectiv e in phase retr ie val [39]. All algor ithms e xcept ePIE tak e a global approach, i.e. using the entire collection of diffraction patterns to perform an update of the probe and object in each iteration. In contrast, ePIE goes through the measured data sequentiall y to refine the probe and object. How e v er, ePIE has a slow conv erg ence due to the step size restriction which may cause diver gence if violated. T o fix this issue, rPIE was proposed, in which regular ization is used for stability [40]. The significant results also sho w that rPIE obtains a larger field of vie w (FO V) than ePIE. In this paper, we show that DM and RAAR can also be implemented locally , similarly to ePIE. W e then apply non-conv ex optimization tools to impro v e the robustness, con v ergence and FO V of the ptyc hog raph y reconstruction. The proposed algor ithm incor porates tw o techniques. The first modifies the update of the probe and object as the algor ithm iterates via semi-implicit method or Pro ximal Mapping. The second technique is the implementation of relax ed Douglas Rachf ord, a generalized version of DM and RAAR, on the local scale. 2. The proposed algorithm Giv en N measured diffraction patter ns at N positions, the pty chographic algorithm aims to find a 2D object O and probe P that satisfy the o ver lap constraint and the Fourier magnitude constraint | F ( PO n ) | = p I n f or n = 1 , . . , N . (1) Where O n is the object at the n t h scan position. Here, w e omit the spatial variables for a simple notation and use the notations P , O n and I n f or both continuous and discrete cases. The absolute value, multiplication, division, conjugate, and square root operators are applied element-wise on P , O n and I n which represent 2D complex matr ices of the same size in the discrete case. W e can argue that O n is the object restricted to a sub-domain Ω n . The ov erlap constraint can be mathematically inter preted as O ( x + r n ) = O n ( x ) if r n + x ∈ Ω n f or n = 1 , . . . , N (2) where { r n } N n = 1 are displacement v ectors. In shor t notation, we write O n = O | Ω n to imply the object is restricted to sub domain Ω n . The equiv alent constraint in the discrete case is the agreement betw een the sub-matrix of O and O n . W e find a better representation of the problem b y introducing the exit wa v e variable Ψ = PO . By denoting the Fourier measurement constraint set T and the ov erlap object constraint set S , we ha ve T : =  Ψ = { Ψ n } N n = 1 : | F Ψ n | = p I n f or n = 1 , . . . , N  S : =  Ψ = { Ψ n } N n = 1 : ∃ P , O s.t. Ψ n = PO n f or n = 1 , . . . , N }  . Then we write the pty chography problem in a minimization fashion min Ψ i S ( Ψ ) + i T ( Ψ ) (3) where i S ( Ψ ) and i T ( Ψ ) are the indicator functions of sets S and T respectiv ely , defined as i S ( Ψ ) = ( 0 Ψ ∈ S ∞ otherwise (4) T o solv e Eq. (3), an alter nating projection method is proposed. At each iteration, w e select a random position n and update Ψ n Ψ 0 n = Π T ( Ψ k n ) = F − 1  p I n arg (F Ψ k n )  (5) { P k + 1 , O k + 1 n } = argmin P , O n 1 2 k PO n − Ψ 0 n k 2 (6) Ψ k + 1 n = P k + 1 O k + 1 n (7) The Frobenius norm is used in this minimization problem and entire paper unless a different norm is specified. The minimization of Eq. ( 6 ) is difficult due to instability . One wa y to solv e this non-conv e x problem is to minimize each variable while fixing the other ones. O k + 1 n = argmin O n 1 2 k P k O n − Ψ 0 n k 2 = Ψ 0 n P k P k + 1 = argmin P 1 2 k PO k + 1 n − Ψ 0 n k 2 = Ψ 0 n O k + 1 n (8) This approach is unstable because of the division. A cut-off method is used to av oid the div ergence and zero-division. A modification is recommended by adding a penalizing least square er ror term (i.e. regularizer) { P k + 1 , O k + 1 n } = argmin P , O n 1 2 k PO n − Ψ 0 n k 2 + 1 2 s k P − P k k 2 + 1 2 t k O n − O k n k 2 (9) The idea of regularization appears throughout the literature such as pro ximal algor ithms [41, 42]. Eq. (9) is more reliable to sol ve than Eq. (6) but is still v er y e xpensive since the variables are coupled. P k + 1 and O k + 1 n can be sol v ed via a Backw ard-Euler sys tem der iv ed from Eq. (9). O k + 1 n = O k n − t P k + 1  P k + 1 O k + 1 n − Ψ 0 n  P k + 1 = P k − s O k + 1 n  P k + 1 O k + 1 n − Ψ 0 n ) (10) ePIE proposes a simple approximation by linear izing the system so that it can be sol ved sequentiall y . O k + 1 n = O k n − t P k  P k O k n − Ψ 0 n  P k + 1 = P k − s O k + 1 n  P k O k + 1 n − Ψ 0 n ) (11) The sys tem is solv ed b y alternating direction methods (ADM) [43]. The remaining par t is to choose appropr iate step sizes t and s to ensure stability . ePIE sugg ests t = β O / k P k k 2 max and s = β P / k O k + 1 k 2 max where β O , β P ∈ ( 0 , 1 ] are nor malized step sizes. The max matr ix nor m is the element-wise nor m, taking the maximum in absolute values of all elements in the matrix. The final v ersion of ePIE is giv en by O k + 1 n = O k n − β O P k  P k O k n − Ψ 0 n  k P k k 2 max P k + 1 = P k − β P O k + 1 n  P k O k + 1 n − Ψ 0 n ) k O k + 1 n k 2 max (12) W e will exploit the structure of Eq. (10) to give a better approximation. 2.1. A semi-implicit algorithm W e replace the minimization of Eq. (9) by two steps Step 1: O k + 1 n = argmin O n 1 2 k P k O n − Ψ 0 n k 2 + 1 2 t k O n − O k n k 2 Step 2: P k + 1 = argmin P 1 2 k PO k + 1 n − Ψ 0 n k 2 + 1 2 s k P − P k k 2 (13) This results in a better approximation to the linear ized sys tem of Eq. (11) and simpler than the Backw ard-Euler Eq. (10) O k + 1 n = O k n − t P k  P k O k + 1 n − Ψ 0 n  P k + 1 = P k − s O k + 1 n  P k + 1 O k + 1 n − Ψ 0 n ) (14) In this uncoupled system, we can derive a closed f or m solution for each sub-problem. O k + 1 n = O k n + t P k Ψ 0 n 1 + t | P k | 2 P k + 1 = p k + s O k + 1 n Ψ 0 n 1 + s | O k + 1 n | 2 (15) By choosing the step sizes s and t as in the ePIE algorithm and normalizing the parameters β O and β P , we obtain O k + 1 n = ( 1 − β O ) k P k k 2 max O k n + β O P k Ψ 0 n ( 1 − β O ) k P k k 2 max + β O | P k | 2 P k + 1 = ( 1 − β P ) k O k + 1 n k 2 max P k + β P O k + 1 n Ψ 0 n ( 1 − β P ) k O k + 1 n k 2 max + β P | O k + 1 n | 2 (16) This f ormula can be e xplained as a w eighted av erage between the previous update O k n and Ψ k n P k . The object update is similar to the rPIE algor ithm when rewriting it as O k + 1 n = O k + β O P k  Ψ 0 n − Ψ k n  ( 1 − β O ) k P k k 2 max + β O | P k | 2 (17) The difference is rPIE does not hav e the parameter β O in front of the fraction. i.e. rPIE has a larg er step size than sir-DR. This helps con ver ge faster but might also get trapped in local minima. The regular ization (weighted a verag e) in sir -DR is more mathematicall y cor rect and enhances the algor ithm’ s stability . In the ne xt section, w e apply the Douglas-Rachf ord algorithm to sol v e f or the exit wa v e Ψ . 2.2. The relaxed Douglas-Rachf ord algor ithm The Douglas-Rachf ord algorithm was or iginall y proposed to sol ve the heat conduction problem [44], which represents a composite minimization problem min Ψ f ( Ψ ) + g ( Ψ ) (18) The iteration consists of Ψ k + 1 = Ψ k + prox t f  2 prox t g ( Ψ k ) − Ψ k  − prox t g ( Ψ k ) (19) Ov er the past decades, this accelerated conv ex optimization algorithm has been exhaus tivel y studied in both theor y and practice with many applications [45 – 50].Here we apply the algorithm to the pty chographic phase retr ie val. Note that the Douglas-Rachf ord algor ithm reduces to Difference Map (DM) when f = i T and g = i S are characteristic functions of constraint sets T and S , respectiv ely Ψ k + 1 = Ψ k + Π T  2 Π S ( Ψ k ) − Ψ k  − Π S ( Ψ k ) (20) W e realize that the reflection operator 2 Π S − I helps to accelerate the conv erg ence in the con ve x case and escape local minima in the non-con ve x case. How ev er this momentum, caused b y reflection, might be too large and can lead to ov er -fitting. Theref ore, we relax the reflection by introducing the relaxation parameter σ ∈ [ 0 , 1 ] Ψ k + 1 = prox t f  ( 1 + σ ) prox t g ( Ψ k ) − σ Ψ k  + σ  Ψ k − prox t g ( Ψ k )  (21) Since the e xperimental measurements are contaminated by noise, a direct projection of measure- ment constraint is not an appropr iate approach. W e thus relax the Fourier magnitude constraint b y a least square penalty min Ψ n N Õ n = 1 k | F Ψ n | − p I n k 2 + i S ( Ψ n ) (22) Recall that pro x t f ( Ψ ) has a closed form solution pro x t f ( Ψ k ) = argmin Ψ 1 2 k | F Ψ | − p I n k 2 + 1 2 t k Ψ − Ψ k k 2 = Ψ k + t F − 1 h √ I n arg (F Ψ ) i 1 + t = ( 1 − τ ) Ψ k + τ F − 1 h p I n arg (F Ψ ) i (23) where τ = t /( 1 + t ) ∈ ( 0 , 1 ) e xclusiv ely is the nor malized step size. Combining this result with DM, we obtain Ψ k + 1 = ( 1 − τ )  ( 1 + σ ) π S ( Ψ k ) − σ Ψ k  + τ Π T  ( 1 + σ ) π S ( Ψ k ) − σ Ψ k  + σ  Ψ k − Π S ( Ψ k )  = τ  σ Ψ k + Π T  ( 1 + σ ) π S ( Ψ k ) − σ Ψ k   +  1 − τ ( 1 + σ )  Π S ( Ψ k ) (24) When we let β = 1 − τ and σ = 1 , the update reduces to RAAR. Theref ore, w e sho w that relaxed Douglas-Rachf ord is a generalized v ersion of RAAR. W e now mo ve to our main algor ithm. 2.3. The sir-DR algorithm In combination of the semi-implicit algor ithm and relaxed Douglas Rachf ord algor ithm, we propose the sir -DR algorithm, sho wn in Fig. 1. In this algorithm, we only apply the semi-implicit method on O k n while P k can be integrated with the Forward Euler (g radient descent) method. τ ∈ [ 0 , 1 ] is chosen to be small. In most of our experiments, we select τ ≈ 0 . 1 , while the choice of σ depends on the specific problem. In man y cases, σ = 1 works very well (full reflection). But in some specific cases, large σ might cause diver gence or small reco vered FO V . W e decrease σ in these cases, f or e xample σ = 0 . 5 . W e choose β O = 0 . 9 in most cases. β P is chosen to be large at the beginning ( β P = 1 ) and decreases as a function of iteration. This adaptive step-size has been introduced as a strategy for noise-robust Fourier ptyc hog raph y [51]. Algorithm 1 sir -DR algorithm Input : N measurements { I n } N n = 1 , number of iterations K , parameters σ , τ , β O , β P Initialize : O 0 , P 0 , { Z 0 n } N n = 1 . f or k = 1 , . . . , K do randomly pick the n t h diffraction patter n, e xtract O k n = O k | Ω n update Ψ k + 1 n Ψ S = O k n P k Z S = F Ψ S ˆ Z = ( 1 + σ ) Z S − σ Z k n Z T = ( 1 − τ ) √ I n arg ˆ Z + τ ˆ Z Z k + 1 n = Z T + σ ( Z k n − Z S ) Ψ k + 1 n = F − 1 Z k + 1 n update O k + 1 n , P k + 1 O k + 1 n = ( 1 − β O ) k P k k 2 max O k n + β O P k Ψ k + 1 n ( 1 − β O ) k P k k 2 max + β O | P k | 2 P k + 1 = P k − β P O k + 1 n  P k O k + 1 n − Ψ k + 1 n ) k O k + 1 n k 2 max update O k + 1 O k + 1 | Ω n = O k + 1 n end for Output : O N , P N O k P k extract O k n Ψ S = O k n P k n Z S = F Ψ S ˆ Z T =(1 + σ ) Z S − σ Z k n Z T = (1 − τ ) √ I n arg( ˆ Z ) + τ ˆ Z { Z k n } N n = 1 Z k +1 n = Z T + σ ( Z k n − Z S ) Ψ k + 1 n = F − 1 Z k + 1 n update O k +1 n update P k +1 n update O k +1 P O Fig. 1: Flo w char t of the sir -DR algorithm a b Fig. 2: A simulated complex object with the amplitude being a camera man image (a) and the phase being a pepper image (b). 3. Experimental results 3.1. Reconstruction from simulated data T o e xamine the sir -DR algorithm, we simulate a comple x object of 128 × 128 pixels with a cameraman and a pepper images representing the amplitude and the phase, respectiv ely (Fig. 2).The circular aper ture is chosen as probe with a radius of 50 pixels. W e raster scan the aper ture o ver the object with a step size of 35 pixels, resulting in 4x4 scan positions. The ov erlap is theref ore 56.4%, the approximate low er limit for ePIE to work. Poisson noise is added to the diffraction patter ns with a flux of 10 8 photons per scan position. W e use R no is e to quantify the relativ e er ror with respect to the noise-free diffraction patter ns R no is e = 1 N N Õ n = 1 k | F ( P 0 O 0 n ) | − √ I n k 1 , 1 k √ I n k 1 , 1 . (25) where P 0 and O 0 is the noise-free model and the L 1 , 1 matrix nor m represents the sum of all elements in absolute value of the matr ix. The abov e flux results in R no is e = 3 . 73% . F ig. 3 show s that three algorithms (ePIE, rPIE and sir -DR) all successfully reconstr uct the object in the case where the o ver lap between adjacent positions is high and the noise lev el is low . As a baseline comparison, Fig. 3 sho ws that all three algorithms cor rectl y reconstruct the object in the ideal case when the ov erlap betw een adjacent positions is high and the noise lev el is lo w . Ne xt, we apply the three algor ithms to the reconstr uction of sparse data, which is centrally important to reducing computation time, data storag e requirements and incident dose to the sample. W e increase the scan step size to 50 pixels while keeping the same field of vie w , which reduces the number of diffraction patterns to 3 × 3 . Consequently , the o ver lap is reduced to 39.1%. Not only is the ov erlap betw een adjacent positions low , but the total number of measurements is also small, creating a challenging data set for conv entional ptyc hog raphic algor ithms. Fig. 4 sho w that sir -DR can work w ell with sparse data, while ePIE and rPIE f ail to reconstr uct the object faithfull y . 3.2. Reconstruction from experimental data 3.2.1. Optical laser data As an initial test of sir -DR with e xper imental data, w e collect diffraction patter ns from an USAF resolution patter n using a green laser with a wa velength of 543 nm. The incident illumination is created by a 15 µ m diameter pinhole. The pinhole is placed approximatel y 6 mm in front of the sample, creating a illumination wa v efront on the sample plane that can be approximated a b c d e f Fig. 3: The reconstructions of ePIE, rPIE and sir -DR of a complex object consisting of 128 × 128 pix els, a scan step size of 35 pixels and 4 × 4 diffraction patterns. Poisson noise was added to the diffraction patter ns with R no is e = 3 . 73% . T op ro w (a-c) show s the amplitude and bottom row (d-f ) show s the phase of ePIE, rPIE and sir -DR reconstructions, respectiv ely . b y Fresnel propagation. The detector is positioned 26 cm downs tream of the sample to collect far -field diffraction patter ns. W e raster scan across the sample with a step size of 50 µ m and acquire 169 diffraction patter ns. W e per f orm a sparsity test b y randomly choosing 85 diffraction patterns (50% density) and run ePIE, rPIE, and sir -DR on this subset with 300 iterations. If w e assume the probe diameter is to where the intensity falls to 10% of the maximum, then the the ov erlaps are 73% and 46.4% f or the full and sparsity sets respectiv ely . F ig. 5 sho ws that rPIE and sir -DR obtain a larg er FO V than ePIE as both use regular ization. Further more, sir -DR remo ves noise more effectivel y and obtains a flatter background than ePIE and rPIE. W e monitor the R -factor (relative er ror) to quantify the reconstruction, defined as R F = 1 N N Õ n = 1 k | F ( PO n ) | − √ I n k 1 , 1 k √ I n k 1 , 1 (26) R F is 16.94%, 13.95% and 13.28% f or the ePIE, rPIE and sir-DR reconstr uctions, respectivel y . 3.2.2. Synchrotron radiation data T o demonstrate the applicability of sir -DR to synchrotron radiation data, w e reconstr uct a pty chographic data set collected from the Adv anced Light Sources [16]. In this e xper iment, 710 e V soft x-ra ys are f ocused onto a sample using a zone plate and the far -field diffraction patterns are collected by a detector . A 2D scan consists of 7,500 positions, which span approximatel y 10 × 4 µ m . The sample is a por tion of a HeLa cell labeled with nanopar ticles, which is suppor ted on a g raphene-o xide lay er . Fig. 6 show s the ePIE reconstruction of the whole FO V of the sample. T o compare the three algorithms, w e choose a subdomain of a 4 × 4 µ m region, consisting of 2,450 diffraction patterns. With the same assumption, the ov erlap is computed to be 79.5%. a b c d e f Fig. 4: Pty chographic reconstructions of sparse data by ePIE, rPIE and sir -DR. The data consist of 3 × 3 diffraction patter ns with a scan step of 50 pixels. Poisson noise was added to the diffraction patterns with R no is e = 3 . 73% . T op row (a-c) show s the amplitude and bottom row (d-f ) show s the phase of ePIE, rPIE and sir -DR reconstructions, respectiv ely . For this sparse data, ePIE and rPIE fail to conv erg e no matter how many iterations are used, but sir -DR conv erg es to an image of good quality . a b c Fig. 5: The ePIE (a), rPIE (b) and sir -DR (c) reconstructions of a sparse data with 300 iterations, where sir -DR obtains a better quality reconstruction than ePIE and rPIE. Both sir -DR and rPIE produce a larg er FO V than ePIE. Scale bar 200 µ m Fig. 7 sho w the ePIE, rPIE, and sir -DR reconstructions, respectivel y . With 300 iterations, all three algor ithms conv erge to images with good quality . When reducing the number of iterations to 100, w e observe that sir -DR conv erg es f aster and reconstruct a larger FO V than ePIE. The individual nanopar ticles, which serve as a resolution benchmark, are better resolv ed in the sir -DR reconstruction than the ePIE and rPIE ones. Fur thermore, the reconstruction by ePIE contains artifacts as a faint square g rid, which is remo ved b y rPIE and sir-DR. a b c Fig. 6: The ePIE reconstruction of a por tion of a HeLa cell labeled with nanopar ticles after 300 iterations. The data consist of 7,500 diffraction patterns and co vers a 9 . 71 × 3 . 70 µ m region. (a) The full FO V of the sample. (b) Magnified view of a region ( 3 . 70 × 3 . 70 µ m ) in (a). (c) Magnified vie w of a region ( 1 . 66 × 1 . 66 µ m ) in (b). The scale bars are 1000 nm , 500 nm and 200 nm respectivel y . W e ne xt perform a sparsity test by randoml y picking 980 out of 2,450 diffraction patterns, i.e. a reduction of data by 60%. The cor responding o v erlap of the sparsity set is 50.8%. Fig. 8 show s the reconstructions by ePIE, rPIE, and sir -DR with 300 iterations. Both the ePIE and rPIE reconstructions exhibit noticeable degradation. In particular, the nanopar ticles are not well resol ved. But the sir -DR reconstruction has no noticeable ar tif acts noise and the individual nanopar ticles are clearl y visible. The quality of sir -DR reconstruction with 60% data reduction is still comparable to that of ePIE using all the diffraction patter ns a b c d e f g h i j k l Fig. 7: The ePIE (a), rPIE (b) and sir -DR (c) reconstructions of a 3 . 70 × 3 . 70 µ m region of the HeLa cell after 300 iterations with R F = 16 . 48% , 16 . 50% and 14 . 40% , respectivel y . (d-f ) Magnified regions ( 1 . 66 × 1 . 66 µ m ) in (a-c), respectivel y . (g-i) ePIE, rPIE and sir -DR reconstr uctions after 100 with R F = 18 . 69% , 16 . 60% and 14 . 92% , respectiv ely . (j-l) Magnified regions in (g-i), respectiv ely . Among the three algor ithms, sir -DR not onl y con ver g es the fas test, but also produces the best reconstruction. Scale bar 500 n m and 200 nm respectivel y . ePIE rPIE sir -DR a b c d e f Fig. 8: (a-c) The ePIE, rPIE, and sir -DR reconstructions of a sparse data set after 300 iterations with R F = 19 . 52% , 16 . 58% and 14 . 26% , respectivel y . T o create the sparse data, we randoml y pick 980 out of 2,450 diffraction patter ns from the HeLa sample. (d-f ) Magnified regions in (a-c), respectivel y . While the quality of the ePIE and rPIE reconstruction is deg raded, sir -DR reproduces a good quality imag e with more distinguishable features. 4. Conclusion In this w ork, we hav e dev eloped a fast and robust ptyc hographic algor ithm, ter med sir -DR. The algorithm relaxes Douglas-Rachf ord to improv e robustness and applies a semi-implicit scheme (semi-Backw ard Euler) to solv e f or the object and to e xpand the reconstructed FO V . Using both simulated and experimental data, we hav e demonstrated that sir -DR outper f or ms ePIE and rPIE with sparse data. Being able to obtain good pty chographic reconstructions from sparse measurements, sir -DR can reduce the computation time, data storage requirement and radiation dose to the sample. Ackno wledgments This work was suppor ted by STR OBE: A National Science Foundation Science & T echnology Center (DMR -1548924). J.M. also ackno w ledges the suppor t by the Office of Basic Energy Sciences of the US DOE (DE-SC0010378). References 1. J. Miao, P . Charalambous, J. Kirz, and D. 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