## Noise models for low counting rate coherent diffraction imaging |

Optics Express, Vol. 20, Issue 23, pp. 25914-25934 (2012)

http://dx.doi.org/10.1364/OE.20.025914

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### Abstract

Coherent diffraction imaging (CDI) is a lens-less microscopy method that extracts the complex-valued exit field from intensity measurements alone. It is of particular importance for microscopy imaging with diffraction set-ups where high quality lenses are not available. The inversion scheme allowing the phase retrieval is based on the use of an iterative algorithm. In this work, we address the question of the choice of the iterative process in the case of data corrupted by photon or electron shot noise. Several noise models are presented and further used within two inversion strategies, the ordered subset and the scaled gradient. Based on analytical and numerical analysis together with Monte-Carlo studies, we show that any physical interpretations drawn from a CDI iterative technique require a detailed understanding of the relationship between the noise model and the used inversion method. We observe that iterative algorithms often assume implicitly a noise model. For low counting rates, each noise model behaves differently. Moreover, the used optimization strategy introduces its own artefacts. Based on this analysis, we develop a hybrid strategy which works efficiently in the absence of an informed initial guess. Our work emphasises issues which should be considered carefully when inverting experimental data.

© 2012 OSA

## 1. Introduction

3. J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy **107**, 227–231 (2007). [CrossRef]

4. D. Claus, A. M. Maiden, F. Zhang, F. G. R. Sweeney, M. Humphry, H. Schluesener, and J. M. Rodenburg, “Quantitative phase contrast optimised cancerous cell differentiation via ptychography,” Opt. Express **20**, 9911– 9918 (2012). [CrossRef] [PubMed]

5. M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S. C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical constrasts in soft X-ray ptychography,” Phys. Rev. Lett. **107**, 208101 (2011). [CrossRef] [PubMed]

6. J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett. **98**, 34801 (2007). [CrossRef]

9. P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. H. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun. **2**, 1569 (2011). [CrossRef]

10. F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B **82**, 121415 (2010). [CrossRef]

12. C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett. **108**, 73901 (2012). [CrossRef]

13. S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectroholography,” Nature **432**, 885–888 (2004). [CrossRef] [PubMed]

14. V. Chamard, J. Stangl, D. Carbone, A. Diaz, G. Chen, C. Alfonso, C. Mocuta, G. Bauer, and T. H. Metzger, “Three-dimensional x-ray Fourier transform holography: the Bragg case,” Phys. Rev. Lett. **104**, 165501 (2010). [CrossRef] [PubMed]

7. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science **321**, 379–382 (2008). [CrossRef] [PubMed]

15. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy **109**, 1256–1262 (2009). [CrossRef] [PubMed]

16. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express **16**, 7264–7276 (2008). [CrossRef] [PubMed]

16. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express **16**, 7264–7276 (2008). [CrossRef] [PubMed]

17. A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy **120**, 64–72 (2012). [CrossRef] [PubMed]

18. P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. **14**, 063004 (2012). [CrossRef]

*etc*. The successive iterations lead to a solution which is reached when the constraints resulting from the overlapping condition and the intensity measurements are satisfied simultaneously. In the presence of shot noise, such a solution does not exist as the different intensity patterns are not anymore consistent one with another. Low counting statistics are of key importance in the study for instance of radiation-sensitive objects (especially biological structures), or when the object scatters weakly, or when one attempts to obtain very high-resolution images although only few photons are scattered at the needed high angles.

*maximum likelihood*principle [19, 20]. A noise-model dependent reconstruction is thereby obtained by the minimization of the corresponding fitting functions. For this purpose, two different optimization strategies are examined, namely the ordered subset (OS) and the scaled gradient (SG). The former strategy is equivalent to the well known ptychographical iterative engine (PIE) when the additional assumption of a Gaussian noise model is considered. It has the advantage of fast convergence in the early iterations, but its final convergence is precluded by the inconsistencies in the different diffraction patterns. In contrast, the latter is slower in the early iterations, but its asymptotic convergence remains in presence of noise. For the different inversion schemes, a Monte-Carlo analysis is conducted for different noise levels, allowing a direct comparison of the solutions. The quantitative evaluation of each pair “noise model/optimization strategy” is done through quality indicators like the bias and standard deviation. Our results demonstrate the large variety of trade-offs resulting directly from the use of inversion schemes and from the implicit physical models. These are discussed in detail. The conclusions we reach have important implications for experimental applications of diffractive imaging.

## 2. Noise models for ptychographic data sets

*p*(

**) and the sample scattering function**

*r**ρ*(

**), named the object in the following. In the multiplicative approximation, the**

*r**j*-th exit-field

*ψ*(

_{j}**) is given by where**

*r**ρ*is unknown and

*p*(

_{j}**) :=**

*r**p*(

**−**

*r*

*r**) is the probe function shifted in*

_{j}

*r**. From a practical viewpoint, the reconstruction from ptychographic data requires the object and the probe to be discretised. In what follows, we denote by the object to be retrieved;*

_{j}*N*is thus the number of pixels in the object plane. This object is illuminated by a support-limited probe

*M*is the number of pixels in the camera. This vector is converted into an

*M*×

*N*matrix

*P**so that the exit field is expressed in vector form by where the index*

_{j}*j*refers to the position of the probe. The corresponding far-field

**is the discrete Fourier transform operator. Provided that the size of the camera pixel or detector is small enough, the**

*W**expected*number of photons in the

*m*-th detector reads where

*b*is the expected number of

_{m,j}*background events*and 𝒜 is the area of the detector. Since 𝒜 can be incorporated into the probe, one can set 𝒜 = 1 without loss of generality, so that

*h*= |Ψ

_{m,j}*|*

_{m,j}^{2}+

*b*is the

_{m,j}*expected*number of events for the

*m*-th measurement in the

*j*-th illumination.

**and the expected (noise-free) data set {**

*ρ**h*} that is at the basis of any reconstruction numerical scheme. However, when realistic data are considered, the presence of photon noise results in a substantial degradation of the measured data set

_{m,j}**:= {**

*y**y*} relative to {

_{m,j}*h*}. In order to take into account the noise issue in a ptychographic experiment, three distinct noise models are introduced in the following sections. Each of them leads to a specific criterion that links the unknown object to the measured data. We will show that these criteria are fitting functions that provide an estimate of the object further obtained

_{m,j}*via*a minimization algorithm.

### 2.1. Noise Model 𝒫: The standard photon counting model

*y*, the probability that the entire data set

_{m,j}**is collected reads For experiments performed with a single photon counting detector, like a cooled charge coupled device camera [21**

*y*21. F. Livet, F. Bley, J. Mainville, R. Caudron, S. G. J. Mochrie, E. Geissler, G. Dolino, D. Abernathy, G. Grübel, and M. Sutton, “Using direct illumination CCDs as high resolution area detector for X-ray scattering,” Nucl. Instr. Meth. A **451**, 596–609 (2000). [CrossRef]

*e.g.*the Maxipix [22

22. C. Ponchut, J. Clément, J.-M. Rigal, E. Papillon, J. Vallerga, D. LaMarra, and B. Mikulec, “Photon-counting X-ray imaging at kilohertz frame rates,” Nucl. Instrum. Meth. A **576**, 109–112 (2007). [CrossRef]

23. C. Broennimann, E. F. Eikenberry, B. Henrich, R. Horisberger, G. Huelsen, E. Pohl, B. Schmitt, C. Schulze-Briese, M. Suzuki, T. Tomizaki, H. Toyokawa, and A. Wagner, “The Pilatus 1M detector,” J. Synchrotron Rad. **13**, 120–130 (2006). [CrossRef]

25. L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag. **1**, 113–122 (1982). [CrossRef]

15. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy **109**, 1256–1262 (2009). [CrossRef] [PubMed]

28. P. Godard, M. Allain, and V. Chamard, “Imaging of highly inhomogeneous strain field in nanocrystals using x-ray Bragg ptychography: A numerical study,” Phys. Rev. B **84**, 144109 (2011). [CrossRef]

18. P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. **14**, 063004 (2012). [CrossRef]

29. J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive X-ray optics by ptychographic coherent diffractive imaging,” Opt. Express **19**, 21333–21344 (2011). [CrossRef] [PubMed]

### 2.2. Noise Model 𝒢: stabilizing the variance of the counting process

### 2.3. Noise Model 𝒬: An approximation of the counting model

*ε*is an independent centred fluctuation drawn from Gaussian random vector with variance

_{m,j}*h*. It leads to the following PDF for the data set

_{m,j}**Provided that the number of expected counts {**

*y**h*} is “large enough”, the

_{m,j}*central limit theorem*(Ref. [20], Sec. 8.47) ensures that the Gaussian PDF in Eq. (6) is a good approximation of its Poissonian counterpart given in Eq. (2). Hence, from the ptychographic image reconstruction viewpoint, Eq. (6) is a fair noise model that could be used for the design of a reconstruction algorithm. However, simpler iterative algorithms are readily obtained if the additional approximation

*h*≈

_{j,m}*y*is used for the variance so that the resulting noise model finally reads Note that data with no detected photon have been suppressed to avoid division by zero. Since the standard deviation depends on the data, this last noise model is no longer Gaussian. It is used for imaging reconstruction issues with photon noise in

_{m,j}*e.g.,*Ref. [33

33. C. A. Bouman and K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process. **5**, 480–492 (1996). [CrossRef] [PubMed]

35. M. Allain and J.-P. Roques, “High resolution techniques for gamma-ray diffuse emission: application to INTEGRAL/SPI,” Astron. Astrophys. **43**, 1175–1187 (2006). [CrossRef]

## 3. Ptychographic image reconstruction by the maximum likelihood principle

**from a noisy data set is now introduced. Following the standard statistical inference literature, the so-called maximum likelihood (ML) principle can be used to estimate the object. It derives directly from the noise model. In the case of the ptychographical reconstruction problem, the ML estimator for**

*ρ***is the quantity that maximizes (with respect to**

*ρ***) the PDF of the chosen noise model. In more formal terms, this ML estimate reads where “•” stands for 𝒫, 𝒢 or 𝒬 (**

*ρ**i.e.*, the noise model under consideration), and with the neg-loglikelihood [36

36. By definition, the likelihood is the PDF of the noise model seen as a function of the unknown parameters *ρ*. In practice, the opposite of the logarithm of the likelihood is rather considered. However, the logarithm function being a monotonic increasing function, the minimiser of the neg-loglikelihood is also the maximiser of the likelihood, *i.e.*, the ML estimator.

*f*

_{𝒫},

*f*

_{𝒢}or

*f*

_{𝒬}; for more details concerning the ML principle the reader is referred to

*e.g.*, Ref. [20] (Chap. 18). For the noise models considered in this article, these fitting functions split as a sum over all the probe positions: where ℒ

_{•;j}is given by (up to irrelevant constant terms) where the dependencies with respect to (w.r.t.) the unknown object

**are made explicit. From these expressions, one notes that the value**

*ρ**y*= 0 leads to a contribution

_{m,j}*h*(

_{m,j}**) in the summands of both Eq. (10b) and Eq. (10c). As a result, the fitting functions ℒ**

*ρ*_{𝒫}and ℒ

_{𝒢}are equivalent w.r.t. the camera pixels that do not detect any photon. On the contrary, zero intensity camera pixels are discarded from ℒ

_{𝒬}(Eq. (10d)) which is expected consequently to lead to very noisy solutions since these pixels are legitimate constraints for the final solution (see Sec. 4.5 for an example). This problem is clearly circumvented if Eq. (10d) is modified so that the empty pixels are accounted for,

*i.e.*, The accuracy of this approximation [37] w.r.t. the Poissonian fitting function ℒ

_{𝒫}is studied in [33

33. C. A. Bouman and K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process. **5**, 480–492 (1996). [CrossRef] [PubMed]

_{𝒫}is expected to be the “best” fitting function since it is perfectly adapted to the data fluctuations. With photon noise, the ML estimator drawn from ℒ

_{𝒫}is attractive because it benefits from good asymptotic properties: for high counting rates, the ML estimator is free of systematic errors and presents the best variance estimation (Ref. [20], p. 56). For limited counting rates, however, the situation can be different and another fitting function may be more appropriate. We also stress that (by definition) the ML does not account for any additional

*a priori*constraints concerning the electronic density to be retrieved (

*e.g.*, support constraint, positivity). If the oversampling is too low and/or the number of diffraction patterns is limited (possibly equal to one), the ML may perform poorly and such additional constraints may be desirable (or even mandatory). This situation appears in support-based phase retrieval problems. However, since the present study aims at evaluating noise models for diffraction-pattern information only, the addition of object constraints has to be avoided because it may most probably blur the analysis. Hence, the ML is the appropriate tool to be considered. For sake of completeness, we also note that one can resort to the maximum

*a posteriori*principle [38, p. 183] to introduce additional constraints within a statistical framework.

*h*> 0, ∀(

_{m,j}*m*,

*j*), is mandatory in order to ensure that ℒ

_{𝒫}given in Eq. (10b) is always defined. Indeed, the same condition results in the existence of the ℒ

_{𝒫}and ℒ

_{𝒢}gradients, allowing the iterative algorithms introduced in the next section to be defined. For the sake of simplicity, we assume in the following that the assumption

*h*> 0 holds [39].

_{m,j}### 3.1. Computing the ML estimate

_{•}vanish. As a result, gradient-based algorithms are natural candidates for the optimization of the chosen likelihood. The gradient of the likelihoods given in Eq. (10) reads where the gradient for the

*j*-th probe position

*∂*

_{•;}

*is given by with “†” the conjugate-transpose operator, and where*

_{j}*corrected*far-field that depends on the chosen fitting function

*not*strictly convex, local minima may exist and can trap gradient algorithms. Moreover, it is well known that ambiguous solutions exist so that a unique ML cannot be defined for the ptychographical problem [16

16. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express **16**, 7264–7276 (2008). [CrossRef] [PubMed]

40. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A **3**, 1897–1907 (1986). [CrossRef]

### 3.2. Ordered-subset optimization strategies

*Ordered-subset*(OS) algorithms [41] [44

44. H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered-subset of projection data,” IEEE Trans. Med. Imag. **13**, 601–609 (1994). [CrossRef]

45. S. Ahn and J. A. Fessler, “Globally convergent image reconstruction for emission tomography using relaxed ordered subset algorithms,” IEEE Trans. Med. Imag. **22**, 613–626 (2003). [CrossRef]

*j*= 1 · · ·

*J*, updating consecutively the illuminated portion of the object, one full iteration

*k*→

*k*+ 1 occurs once the

*J*probes are considered. Thus, for a given initial guess

*ρ*^{(0)}, the algorithm is defined by the following updates for

*k*= 0, 1, · · · with

*D**a diagonal scaling matrix and where*

_{j}*β*> 0 is the step-length. One may note that the classical

*ptychographical iterative engine*(PIE) is a special case of this generic OS strategy. For instance, the choice (where

**is the identity matrix) is precisely [46] the version of the PIE introduced in Ref. [15**

*I*15. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy **109**, 1256–1262 (2009). [CrossRef] [PubMed]

*∂*

_{•;}

*≡*

_{j}*∂*

_{𝒫;}

*,*

_{j}*∂*

_{•;}

*≡*

_{j}*∂*

_{𝒬;}

*or*

_{j}*∂*

_{•;}

*≡*

_{j}*∂*

_{ℛ;}

*provide natural extensions of the PIE algorithm to the noise models*

_{j}*f*

_{𝒫},

*f*

_{𝒬}and

*f*

_{ℛ}, respectively; such extensions are considered in Sec. 4.5.

### 3.3. Scaled-gradient optimization strategies

*ρ*^{(0)}, the following

*scaled-gradient*(SG) strategy is defined for

*k*= 0, 1, · · · where the gradient

*∂*

_{•}given by Eq. (11a) accounts for all the probes, and

**Λ**∈ ℝ

^{N×N}is a diagonal scaling matrix chosen as As underlined in Ref. [48], the iteration (15) is a natural extension of the Error Reduction algorithm to the ptychographical approach. Since

*β*and

**Λ**are not dependent on the iteration number, the condition

*ρ*^{(k)}→

*ρ*^{∞}implies ‖

*∂*

_{•}(

*ρ*^{(k)}) ‖ → 0: the convergence toward a limit point implies that this point is a local optimum of ℒ

_{•}. In practice, the step-length

*β*> 0 is adjusted in order to generate a sequence converging toward a global or (at least) a local minimum of the fitting function. To our best knowledge, no result exists that gives admissible values for

*β*ensuring the (local) convergence of Eq. (15). However, the tuning

*β*≈ 1 was found to ensure convergence in most cases investigated in the present study. Indeed, provided that

*β*is properly tuned, the SG iteration was always found to be a convergent algorithm.

*J*updates when the former performs only one. However, the SG strategy converges to a (local or global) minimum of ℒ

_{•}, even with a noisy data set. An illustration of these distinct convergence behaviors is given in Sec. 4.4.

## 4. Data inversion: a resolution *vs.* robustness trade-off

**. Because four distinct fitting functions are discussed here, it is appropriate to search for the “best” model for the reconstruction purpose. This task requires to first define how the estimators will be compared.**

*y*### 4.1. Some quality indicators

*via*two standard indicators, the

*estimation bias*and the

*estimation standard deviation*. Let 〈·〉 be the expectation (

*i.e.*, average over several realizations of the noise) operator, and

*ρ*

_{•;n}and

*ρ*

_{★;n}denote the

*n*-th component of the ML solution

*ρ*_{•}and the true object

*ρ*_{★}, respectively. Then, the estimation bias reads with

*ρ̄*

_{•;n}the

*n*-th component of the averaged solution where aims at compensating a global phase ambiguity. For complex random variables, the standard deviation of the estimation is defined by

*ρ*_{•}: whereas the bias (17) gives the systematic error, the variance (20) tells if the estimator is robust w.r.t. the noise. A third indicator is interesting to introduce: the

*mean square error*(MSE) that combines conveniently the preceding indicators. While a general closed-form expression for the bias and the standard deviation is not available, the computation of the averaged quantities given by Eq. (17) and Eq. (20) can however be achieved

*via*Monte-Carlo simulations.

### 4.2. Some implicit effects induced by the noise models

*asymptotic*case of an arbitrary large signal to noise ratio (SNR) is first investigated: since we are dealing with photon noise, this results in

*y*→

_{m,j}*h*(

_{m,j}

*ρ*_{★}). Consequently from Eq. (11), the gradient evaluated in

*ρ*_{★}vanishes whatever the noise model is. In this context, the true object

*ρ*_{★}minimizes the four fitting functions and the bias vanishes,

*i.e.*the four noise models are equivalent. Therefore, consideration of the four noise models is only relevant at low SNR. In particular, as Eq. (12) gives the following relation between the corrected exit-field drawn from the models 𝒫 and 𝒢, it shows that the contribution in the final solution of a low SNR measurement [49]

*y*∼ 1 is

_{m,j}*enhanced*with the noise model 𝒫 because its typical expected value is then

*h*< 1 ≤

_{m,j}*y*. Such measurements being spread over the borders of the intensity pattern, one expects that the noise model 𝒫 enhances the spatial resolution (

_{m,j}*i.e.*reduces the bias) w.r.t. the noise model 𝒢. However, this gain has necessarily a cost: because these low SNR measurements are plagued by large fluctuations, the model 𝒫 should also lead to larger estimation variance. The

*opposite*arguments holds for the noise model ℛ since the condition

*h*≪ 1 leads to

_{m,j}*i.e.*the model ℛ should lead to higher biases and to lower variances as compared to the noise model 𝒢. In summary, we can see that the specific behavior of each model is dominated by the set of pixels that collects the lowest number of photons.

*low-pass filtered version*of the original object (with a loss in resolution being driven by the SNR). This result should hold whatever is the considered noise model.

### 4.3. A test-chart that highlights the predicted effects

^{−1}. The ptychographical data-set is composed of a total of 81 diffraction patterns, each one of size 100 × 100 pixels. The choice of a step size of about 20 pixels in both directions leads to an overlap ratio of 65%. In addition, two SNRs are considered in these simulations: the highest SNR provides a maximum of 10

^{6}expected counts over the 2D camera; the lowest provides 10

^{3}expected photons over the camera.

### 4.4. Some issues concerning the iterative strategy

*same*fitting function. It is therefore appropriate to investigate the impact of the iterative strategy on the retrieved object. For that purpose, the inversion of a single ptychographic data set by either the OS or the SG strategy is now considered. For sake of simplicity, the fitting function ℒ

_{𝒢}is considered, but similar results are obtained with the other fitting functions.

*i.e.*, corrupted by photon noise) however, the gradient norm starts to decrease before it reaches a stagnation, such that the convergence does not occur. Furthermore, Fig. 2(b, c) shows that the OS strategy should be stopped early in the iteration process [50] in order to pick the best solution w.r.t. the

*relative error*(in the object plane) defined by where

*ρ*_{★}is the true object and where

_{•}, even with a noisy data set. This minimum defines an estimate which is a

*global*trade-off over the set of inconsistent diffraction patterns, leading to a lower relative error than the best relative error reached by the OS strategy (see Fig. 2(c) and the reconstructions shown in Fig. 2(d, e, f)).

### 4.5. Monte-Carlo analysis

#### Investigation of the figure of merit for each noise model

*intrinsic*merit of each noise-model, the impact of the minimization method has to be as low as possible. In other words, the minima of ℒ

_{•}can be only compared w.r.t. the noise models if one ensures that the reconstruction quality is not affected by the way the data are handled along the iterative process. Therefore, the use of the (convergent) SG strategy is mandatory, with the additional condition of an initialization as close as possible to the minima. For that purpose, the true solution is chosen as initial guess,

*i.e.,*

*ρ*^{(0)}=

*ρ*_{★}. The algorithms are stopped when the norm of the gradient reaches a conveniently small value.

**̄**

*ρ*_{•}(both modulus and phase) given in Eq. (18) and the standard deviation given in Eq. (20) are provided in Fig. 3 and Fig. 4, respectively. The predicted cut-off frequency effect is clearly visible in Fig. 3: for ℒ

_{ℛ}, the edges of the modulus are smoothed and the phase is damped whereas they remain much more resolved (undamped) for ℒ

_{𝒫}. For this low SNR, the modulus is contaminated by fluctuations that come from the object phase function. The relative amplitude of these fluctuations is 8, 10, 17 and 10 % for ℒ

_{𝒫}, ℒ

_{𝒢}, ℒ

_{𝒬}and ℒ

_{ℛ}, respectively. They reduce when the SNR increases and they become negligible (around 1%) for 10

^{6}photons. As explained in Sec. 4.2, such artifacts appear because the retrieved object is essentially a low-pass filtered version of the original object (see Fig. 5). In the case of the present object, it results from a mixing between the real and imaginary parts of the object, leading to the observation of a phase-like structure in the modulus component.

_{ℛ}has the highest robustness w.r.t. the photon noise whereas ℒ

_{𝒫}has the lowest. For all the fitting functions, the standard deviation grows with the collected number of photons, which is a standard result when one deals with photon noise (Ref. [51], p. 181). As expected, the fitting function ℒ

_{𝒢}reaches a tradeoff w.r.t. these two fitting functions, as expected from Sec. 4.2. Quantitatively, the numerical evaluation of the quality indicators defined in Sec. 4.1 are reported in Table 1. For both SNRs, although the variance is lower with ℒ

_{𝒢}or ℒ

_{ℛ}, one notes that ℒ

_{𝒫}gives however the best results w.r.t. the MSE and the error in the object plane. The fitting function ℒ

_{𝒬}being much less robust to the noise than the other three fitting functions (see Sec. 3), it is not considered as a valuable alternative for CDI in the low counting rate regime.

#### Starting from a coarse initial guess

*a priori*object information can be used, resulting in the choice of a free-space estimate for the initial guess. The quality indicators achieved with 10

^{3}photons are reported in the Table 2; as the OS strategy does not lead to converging iterations (see Sec. 3.2), the iteration that gives the best (

*i.e.*, the smallest) error in the object plane is selected for each data set.

_{𝒫}offers the lowest bias but the worst variance while the fitting function ℒ

_{ℛ}has the opposite characteristics. Moreover, every criteria is improved when using the SG strategy, the gain being more clearly evidenced with the noise model 𝒫. For the sake of completeness, the difference-map (DM) iteration [52

52. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A **20**, 40–55 (2003). [CrossRef]

53. P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy **109**, 338–343 (2009). [CrossRef] [PubMed]

_{𝒢}perform equivalently [54]. In Fig. 6, the results of the SG strategy obtained from a single noisy data set is shown for the various fitting functions; these illustrate how a typical reconstruction looks like for each noise model when the SG strategy is used. Finally, the algorithms presented in this work have been tested on several object classes: phase objects, absorption objects, objects with low or high contrasts,

*etc*. It is always the case that the Poissonian noise model 𝒫 presents the least systematic errors, whereas the noise model ℛ is the most robust, the Gaussian model 𝒢 reaching a trade-off between the two others. It is also observed that the differences between all these algorithms tend to vanish when the SNR increases.

#### The minimization of ℒ_{𝒫}: a hybrid optimization strategy

_{𝒫}is the fitting function that undergoes the strongest degradation if the initialization is far from the final solution. On the contrary, the minimization of ℒ

_{ℛ}is very robust w.r.t. the starting point. Moreover, the OS and the SG strategies are mostly equivalent for that fitting function. Hence, it is appropriate to search for a hybrid strategy that profits from both fitting functions. Therefore, we propose to use the OS strategy starting with the fitting function ℒ

_{ℛ}in order to get quickly to a first estimate which is subsequently introduced as an initial guess for the further minimization of the fitting function ℒ

_{𝒫}. As an example, one can perform 1000 OS iterations with ℒ

_{ℛ}followed by 1000 SG iterations with the fitting function ℒ

_{𝒫}. The quality indicators obtained with this strategy are shown in Table 2. One notes that these indicators are improved: they reach values similar to the ones obtained with the true object (see Table 1). Figure 7 also shows the reconstructed phase obtained by either the “hybrid” strategy or by the SG strategy with the true object as an initial guess. These phases are similar, showing that the hybrid strategy is a valuable technique for the optimization of the Poissonian fitting function.

## 5. Conclusion

*vs.*robustness trade-off. The merit of each noise model may be user and/or object dependent and, from an experimental perspective, the impact of the intensity fluctuations w.r.t. the noise model has to be tested on numerical samples prior to the inversion. An efficient strategy to circumvent the problem in the case of experimental intensity analysis consists in building a set of data for a model sample, designed as close as possible to the available experimental data set (Fourier space resolution, number of probe positions, SNR,

*etc*.). This numerical data set can then be used to test the different noise model approaches and emphasizes the possible reconstruction artifacts.

## A. The variance stabilization transform

*h*around

*μ*in the first order is where

*R*stands for higher order terms. One then has where 〈(

*y*−

*μ*)

*h*′(

*μ*)〉 = 0 is used in the last line. Neglecting all the contributions from the terms higher than the first order gives: Thus, within the first order approximation, the variance of

*h*(

*y*) is independent of

*μ*if a function

*h*is exhibited such that

*h*′(

*x*)|

_{x=μ}=

*b/f*(

*μ*) for a constant

*b*in ℝ. The obvious candidate

*h*(

*x*) =

*bx/f*(

*μ*) is of no interest, being a linear function. A suitable choice for the Poissonian case in which

*b*= 1/4. This is the variance stabilization used in the Section 2.2. Anscombe, in [55], showed that the function

*h*(

*x*) = (

*x*+ 3/8)

^{1/2}has a better variance-stabilization capability than the square-root transform.

## Acknowledgments

## References and links

1. | K. A. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. |

2. | J. M. Rodenburg, “Ptychography and related diffracted imaging methods,” in “ |

3. | J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy |

4. | D. Claus, A. M. Maiden, F. Zhang, F. G. R. Sweeney, M. Humphry, H. Schluesener, and J. M. Rodenburg, “Quantitative phase contrast optimised cancerous cell differentiation via ptychography,” Opt. Express |

5. | M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S. C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical constrasts in soft X-ray ptychography,” Phys. Rev. Lett. |

6. | J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett. |

7. | P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science |

8. | M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature |

9. | P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. H. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun. |

10. | F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B |

11. | M. J. Humphry, B. Kraus, A. C. Hurst, A. M. Maiden, and J. M. Rodenburg, “Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging,” Nat. Commun. |

12. | C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett. |

13. | S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectroholography,” Nature |

14. | V. Chamard, J. Stangl, D. Carbone, A. Diaz, G. Chen, C. Alfonso, C. Mocuta, G. Bauer, and T. H. Metzger, “Three-dimensional x-ray Fourier transform holography: the Bragg case,” Phys. Rev. Lett. |

15. | A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy |

16. | M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express |

17. | A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy |

18. | P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. |

19. | R. A. Fisher, |

20. | M. G. Kendall and A. Stuart, |

21. | F. Livet, F. Bley, J. Mainville, R. Caudron, S. G. J. Mochrie, E. Geissler, G. Dolino, D. Abernathy, G. Grübel, and M. Sutton, “Using direct illumination CCDs as high resolution area detector for X-ray scattering,” Nucl. Instr. Meth. A |

22. | C. Ponchut, J. Clément, J.-M. Rigal, E. Papillon, J. Vallerga, D. LaMarra, and B. Mikulec, “Photon-counting X-ray imaging at kilohertz frame rates,” Nucl. Instrum. Meth. A |

23. | C. Broennimann, E. F. Eikenberry, B. Henrich, R. Horisberger, G. Huelsen, E. Pohl, B. Schmitt, C. Schulze-Briese, M. Suzuki, T. Tomizaki, H. Toyokawa, and A. Wagner, “The Pilatus 1M detector,” J. Synchrotron Rad. |

24. | K. Lange and R. Carson, “EM reconstruction algorithm for emission and transmission tomography,” IEEE Trans. Med. Imag. |

25. | L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag. |

26. | L. B. Lucy, “An iterative technique for the rectification of observed distribution,” New Astron. Rev. |

27. | G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Cryst. |

28. | P. Godard, M. Allain, and V. Chamard, “Imaging of highly inhomogeneous strain field in nanocrystals using x-ray Bragg ptychography: A numerical study,” Phys. Rev. B |

29. | J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive X-ray optics by ptychographic coherent diffractive imaging,” Opt. Express |

30. | Provided that the fluctuations in one measurement are accurately described by a Poisson PDF, then this PDF is defined by a single (positive) parameter that is the mean and the variance. |

31. | J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. |

32. | M. F. Freeman and J. W. Tuckey, “Transformations related to the angular and the square root,” Ann. Math. Statist. |

33. | C. A. Bouman and K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process. |

34. | L. Bouchet, “A comparative-study of deconvolution methods for gamma-ray spectra,” Astron. Astrophys. |

35. | M. Allain and J.-P. Roques, “High resolution techniques for gamma-ray diffuse emission: application to INTEGRAL/SPI,” Astron. Astrophys. |

36. | By definition, the likelihood is the PDF of the noise model seen as a function of the unknown parameters |

37. | Following [33], it is shown that a second order Taylor expansion around y of the Poissonian fitting function ℒ_{m,j}_{𝒫} leads to (10e). |

38. | M. Bertero and P. Boccacci, |

39. | Since the condition b > 0 [cf. Eq. (1)], an arbitrary small background component can be introduced, hence allowing all the fitting functions and gradients to be well defined._{m,j} |

40. | J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A |

41. | In the optimization literature, OS algorithms are also known as |

42. | D. P. Bertsekas, |

43. | Y. Censor, D. Gordon, and R. Gordon, “BICAV: a block-iterative parallel algorithm for sparse systems with pixel-related weighting,” IEEE Trans. Med. Imag. |

44. | H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered-subset of projection data,” IEEE Trans. Med. Imag. |

45. | S. Ahn and J. A. Fessler, “Globally convergent image reconstruction for emission tomography using relaxed ordered subset algorithms,” IEEE Trans. Med. Imag. |

46. | The original version of the PIE introduced by Rodenburg and Faulkner in [31] considers another definition for |

47. | A. M. Maiden, Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. |

48. | C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv:optics (2011). |

49. | Since the three fitting functions ℒ |

50. | This non-monotonic behaviour of the relative error is standard when inverse problems ( |

51. | Ph. Réfrégier, |

52. | V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A |

53. | P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy |

54. | From [52, p. 339], one notes that the constraint defined by the data set in the DM strategy takes the form of Eq. (12b), suggesting that the data fluctuations are described by the Gaussian model defined in Sec. 2.2. |

55. | J. F. Anscombe, “The transformation of Poisson, binomial and negative-binomial data,” Biometrika |

**OCIS Codes**

(030.4280) Coherence and statistical optics : Noise in imaging systems

(100.5070) Image processing : Phase retrieval

(110.0180) Imaging systems : Microscopy

**ToC Category:**

Imaging Systems

**Virtual Issues**

Vol. 7, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

Pierre Godard, Marc Allain, Virginie Chamard, and John Rodenburg, "Noise models for low counting rate coherent diffraction imaging," Opt. Express **20**, 25914-25934 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-23-25914

Sort: Year | Journal | Reset

### References

- K. A. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys.59, 1–100 (2010). [CrossRef]
- J. M. Rodenburg, “Ptychography and related diffracted imaging methods,” in “Advances in Imaging and Electron Physics,” 150, P. W. Hawkesed. (Elsevier, 2008), 87–184. [CrossRef]
- J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy107, 227–231 (2007). [CrossRef]
- D. Claus, A. M. Maiden, F. Zhang, F. G. R. Sweeney, M. Humphry, H. Schluesener, and J. M. Rodenburg, “Quantitative phase contrast optimised cancerous cell differentiation via ptychography,” Opt. Express20, 9911– 9918 (2012). [CrossRef] [PubMed]
- M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S. C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical constrasts in soft X-ray ptychography,” Phys. Rev. Lett.107, 208101 (2011). [CrossRef] [PubMed]
- J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett.98, 34801 (2007). [CrossRef]
- P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science321, 379–382 (2008). [CrossRef] [PubMed]
- M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature467, 436–439 (2010). [CrossRef] [PubMed]
- P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. H. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun.2, 1569 (2011). [CrossRef]
- F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B82, 121415 (2010). [CrossRef]
- M. J. Humphry, B. Kraus, A. C. Hurst, A. M. Maiden, and J. M. Rodenburg, “Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging,” Nat. Commun.3, 730 (2012). [CrossRef] [PubMed]
- C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett.108, 73901 (2012). [CrossRef]
- S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectroholography,” Nature432, 885–888 (2004). [CrossRef] [PubMed]
- V. Chamard, J. Stangl, D. Carbone, A. Diaz, G. Chen, C. Alfonso, C. Mocuta, G. Bauer, and T. H. Metzger, “Three-dimensional x-ray Fourier transform holography: the Bragg case,” Phys. Rev. Lett.104, 165501 (2010). [CrossRef] [PubMed]
- A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy109, 1256–1262 (2009). [CrossRef] [PubMed]
- M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express16, 7264–7276 (2008). [CrossRef] [PubMed]
- A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy120, 64–72 (2012). [CrossRef] [PubMed]
- P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys.14, 063004 (2012). [CrossRef]
- R. A. Fisher, Statistical Methods and Scientific Inference (Oliver & Boyd, 1956).
- M. G. Kendall and A. Stuart, The advanced theory of statistics2a (Griffin, 1963).
- F. Livet, F. Bley, J. Mainville, R. Caudron, S. G. J. Mochrie, E. Geissler, G. Dolino, D. Abernathy, G. Grübel, and M. Sutton, “Using direct illumination CCDs as high resolution area detector for X-ray scattering,” Nucl. Instr. Meth. A451, 596–609 (2000). [CrossRef]
- C. Ponchut, J. Clément, J.-M. Rigal, E. Papillon, J. Vallerga, D. LaMarra, and B. Mikulec, “Photon-counting X-ray imaging at kilohertz frame rates,” Nucl. Instrum. Meth. A576, 109–112 (2007). [CrossRef]
- C. Broennimann, E. F. Eikenberry, B. Henrich, R. Horisberger, G. Huelsen, E. Pohl, B. Schmitt, C. Schulze-Briese, M. Suzuki, T. Tomizaki, H. Toyokawa, and A. Wagner, “The Pilatus 1M detector,” J. Synchrotron Rad.13, 120–130 (2006). [CrossRef]
- K. Lange and R. Carson, “EM reconstruction algorithm for emission and transmission tomography,” IEEE Trans. Med. Imag.8, 306–316 (1984).
- L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag.1, 113–122 (1982). [CrossRef]
- L. B. Lucy, “An iterative technique for the rectification of observed distribution,” New Astron. Rev.79, 745–754 (1974).
- G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Cryst.A63, 36–42 (2007).
- P. Godard, M. Allain, and V. Chamard, “Imaging of highly inhomogeneous strain field in nanocrystals using x-ray Bragg ptychography: A numerical study,” Phys. Rev. B84, 144109 (2011). [CrossRef]
- J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive X-ray optics by ptychographic coherent diffractive imaging,” Opt. Express19, 21333–21344 (2011). [CrossRef] [PubMed]
- Provided that the fluctuations in one measurement are accurately described by a Poisson PDF, then this PDF is defined by a single (positive) parameter that is the mean and the variance.
- J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett.85, 4795–4797 (2004). [CrossRef]
- M. F. Freeman and J. W. Tuckey, “Transformations related to the angular and the square root,” Ann. Math. Statist.21, 607–611 (1950). [CrossRef]
- C. A. Bouman and K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process.5, 480–492 (1996). [CrossRef] [PubMed]
- L. Bouchet, “A comparative-study of deconvolution methods for gamma-ray spectra,” Astron. Astrophys.113, 167–183 (1995).
- M. Allain and J.-P. Roques, “High resolution techniques for gamma-ray diffuse emission: application to INTEGRAL/SPI,” Astron. Astrophys.43, 1175–1187 (2006). [CrossRef]
- By definition, the likelihood is the PDF of the noise model seen as a function of the unknown parameters ρ. In practice, the opposite of the logarithm of the likelihood is rather considered. However, the logarithm function being a monotonic increasing function, the minimiser of the neg-loglikelihood is also the maximiser of the likelihood, i.e., the ML estimator.
- Following [33], it is shown that a second order Taylor expansion around hm,j = ym,j of the Poissonian fitting function ℒ𝒫 leads to (10e).
- M. Bertero and P. Boccacci, Introduction to inverse problems in imaging (Institute of Physics Publishing, 1998). [CrossRef]
- Since the condition hm,j > 0 is enforced if bm,j > 0 [cf. Eq. (1)], an arbitrary small background component can be introduced, hence allowing all the fitting functions and gradients to be well defined.
- J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A3, 1897–1907 (1986). [CrossRef]
- In the optimization literature, OS algorithms are also known as incremental gradient methods or block iterative methods, see for instance [42, Sec. 1.5.2 ] or [43] for details.
- D. P. Bertsekas, Nonlinear programming, 2nd ed. (Athena Scientific, 1999).
- Y. Censor, D. Gordon, and R. Gordon, “BICAV: a block-iterative parallel algorithm for sparse systems with pixel-related weighting,” IEEE Trans. Med. Imag.20, 1050–1060 (2001). [CrossRef]
- H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered-subset of projection data,” IEEE Trans. Med. Imag.13, 601–609 (1994). [CrossRef]
- S. Ahn and J. A. Fessler, “Globally convergent image reconstruction for emission tomography using relaxed ordered subset algorithms,” IEEE Trans. Med. Imag.22, 613–626 (2003). [CrossRef]
- The original version of the PIE introduced by Rodenburg and Faulkner in [31] considers another definition for Dj.
- A. M. Maiden, Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am.29, 1606–1614 (2012). [CrossRef]
- C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv:optics (2011).
- Since the three fitting functions ℒ𝒫, ℒ𝒢 and ℒℛ are equivalent w.r.t. a nil data, only the data such that ym,j ≠ 0 should be considered in order to discriminate the noise-models.
- This non-monotonic behaviour of the relative error is standard when inverse problems (e.g., image restoration or tomographic reconstruction) are solved with gradient optimization technics, see for instance [38, Chap. 6].
- Ph. Réfrégier, Noise Theory and Application to Physics: From Fluctuation to Information (Springer, 2004).
- V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A20, 40–55 (2003). [CrossRef]
- P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 338–343 (2009). [CrossRef] [PubMed]
- From [52, p. 339], one notes that the constraint defined by the data set in the DM strategy takes the form of Eq. (12b), suggesting that the data fluctuations are described by the Gaussian model defined in Sec. 2.2.
- J. F. Anscombe, “The transformation of Poisson, binomial and negative-binomial data,” Biometrika35, 246–254 (1948).

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