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  • Vol. 9, Iss. 3 — Mar. 6, 2014
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A phase space model of Fourier ptychographic microscopy

Roarke Horstmeyer and Changhuei Yang  »View Author Affiliations


Optics Express, Vol. 22, Issue 1, pp. 338-358 (2014)
http://dx.doi.org/10.1364/OE.22.000338


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Abstract

A new computational imaging technique, termed Fourier ptychographic microscopy (FPM), uses a sequence of low-resolution images captured under varied illumination to iteratively converge upon a high-resolution complex sample estimate. Here, we propose a mathematical model of FPM that explicitly connects its operation to conventional ptychography, a common procedure applied to electron and X-ray diffractive imaging. Our mathematical framework demonstrates that under ideal illumination conditions, conventional ptychography and FPM both produce datasets that are mathematically linked by a linear transformation. We hope this finding encourages the future cross-pollination of ideas between two otherwise unconnected experimental imaging procedures. In addition, the coherence state of the illumination source used by each imaging platform is critical to successful operation, yet currently not well understood. We apply our mathematical framework to demonstrate that partial coherence uniquely alters both conventional ptychography’s and FPM’s captured data, but up to a certain threshold can still lead to accurate resolution-enhanced imaging through appropriate computational post-processing. We verify this theoretical finding through simulation and experiment.

© 2014 Optical Society of America

1. Introduction

In ptychographic imaging, also commonly referred to as scanning diffraction microscopy, a sample is shifted across a narrow illumination beam and a series of diffraction intensity patterns are recorded. The acquired image data is then computationally processed into an improved-resolution estimate of the sample’s amplitude and phase transmittance. Ptychography’s unique procedure has recently lead to the generation of many impressive X-ray and electron microscope images that defy the conventional resolution limitations of their detectors and focusing elements [1

1. P. D. Nellist, B. C. McCallum, and J. M. Rodenburg, “Resolution beyond the ‘infromation limit’ in transmission electron microscopy,” Nature 374, 630–632 (1995). [CrossRef]

5

5. M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 437–439 (2010). [CrossRef]

]. This resolution enhancement has also spread to optical imaging [6

6. A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. 35(15), 2585–2587 (2010). [CrossRef] [PubMed]

, 7

7. A. M. Maiden, M. J. Humphry, F. Zhang, and J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A. 28(4), 604–612 (2011). [CrossRef]

], where a novel technique termed Fourier ptychographic microscopy (FPM) was recently introduced [8

8. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photon. 7, 739–745 (2013). [CrossRef]

]. Like conventional ptychography (here on abbreviated as CP), FPM also offers simultaneous resolution enhancement and sample phase recovery from a collection of images. Unlike CP, however, FPM images a sample under variable-angle illumination provided by a fixed array of light-emitting diodes (LEDs). The goal of this current work is to compare and contrast the CP and FPM procedures to bring each approach under a common mathematical framework. In doing so, we hope to encourage a cross-pollination of ideas and efforts to help both techniques progress in high-resolution complex object recovery in the optical regime.

Here, we first build upon this prior work to connect the operation of CP to its new Fourier counterpart, FPM, in the optical domain. Second, we apply our unique mathematical model to account for the effects of partially coherent illumination sources in both systems. Partial coherence plays a fundamental role both in X-ray and electron microscopy where highly coherent sources are not available, and with optical setups aimed towards speckle-free imaging using LEDs. While [9

9. J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A 339, 521–553 (1992). [CrossRef]

] also presents a theoretical model of partially coherent CP, we derive a new set of expressions for both CP and FPM that clearly establish how the finite shape of an incoherent source uniquely impacts each setup. These expressions are then verified in simulation and experiment by computationally removing the effects of partial coherence from final reconstructions. While previously considered in the context of single images [11

11. J. N. Clark, X. Huang, R. Harder, and I. K. Robinsion, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012). [CrossRef] [PubMed]

] and for CP data when the illumination’s coherence state is unknown [12

12. P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013). [CrossRef] [PubMed]

], no work has yet attempted to remove a known coherence function from a collection of ptychographic images. We aim this type of removal as a first step towards a comprehensive understanding of techniques using either coherent or incoherent active illumination to improve resolution.

However, we emphasize here that the primary aim of this work is to present an accurate physical optics-based model of FPM, connecting it to CP to clearly establish its function within a broader class of computational imaging methods. Our demonstration of coherence removal is mainly aimed as a verification of this model, but also points to several new benefits that phase space offers both techniques, which warrant future investigation.

The remainder of this paper is outlined as follows. In Section 2, we use a phase space model to demonstrate that CP and FPM datasets, to first-order, are connected by a linear canonical transform (a 90° matrix rotation). In Section 3, we use this model to visualize how parameters like illumination shape, lens geometry, and detector size impact each experimental setup. In Section 4, we incorporate the effects of partial spatial coherence into our phase space framework. First, we derive how a partially spatially coherent illumination source alters the CP and FPM datasets through a unique convolution operation. Second, we show how this convolution operation can be computationally removed to maintain data useful for resolution enhancement. Section 5 tests the comparisons developed in Sections 3–4 with a simple simulation and experiment. The partially coherent phase space model is verified, and our demonstration solidifies how deconvolution can improve the fidelity of CP and FPM reconstructions. While our phase space model is closely connected to a rich array of computational post-processing tools, we explicitly avoid their discussion until the conclusion, where we list several direct extensions that will benefit from this primarily theoretical work.

2. Mathematically connecting conventional and Fourier ptychography

In this section, we introduce a mathematical framework to summarize the operation of both CP and FPM. We show how two otherwise unique optical setups - one capturing the diffracted light from a moving sample, and the other capturing images of a fixed sample evenly illuminated by an array of sources - create nearly identical datasets.

2.1. The conventional ptychography (CP) setup

Our first steps toward a common mathematical framework are to outline the standard elements of a CP setup, model how light passes through it, and then convert our findings into a suitable phase space representation. The basic setup, notations and derivations used here closely follow those previously employed in [9

9. J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A 339, 521–553 (1992). [CrossRef]

, 10

10. H. N. Chapman, “Phase retrieval x-ray microscopy by Wigner distribution deconvolution,” Ultramicroscopy 66, 153 (1996). [CrossRef]

]. Unlike these prior works, our final expression demonstrates a unique convolution relationship that will help us directly connect CP’s parameters with FPM’s. Furthermore, the following derivation sets the stage for simple inclusion of partial coherence effects, which are vital to our careful comparison of the two setups’ performance in Section 5. Reciprocal space coordinates will be designated with the absence of a prime, and reciprocal space functions will include a tilde (e.g., the Fourier transform of a(r′) is ã(r)). Note that here both r and r′ will have units of meters, since they represent the spatial axis of an imaging system’s two Fourier conjugate planes. A schematic diagram of a scanning CP setup containing two sets of such planes is in Fig. 1. While deviations exist, most recent ptychographic experiments generally follow Fig. 1’s optical outline. The following analysis considers a two-dimensional imaging geometry, for simplicity. Extension to three dimensions is direct.

Fig. 1 Conventional ptychography’s optical setup. A sample ψ (in green) is shifted through many positions as the intensity of the probe light it diffracts is recorded at a far-field detector. In a typical visible light setup, the lens at A(r′) is a multi-element system containing the aperture stop a(r′) at some intermediate plane, as diagrammed.

A standard CP setup first focuses light from an illumination plane I(r) onto a shifting sample and records a series of far-field diffraction patterns. We assume I(r) contains an ideal point light source that produces a quasi-monochromatic plane wave (wavelength λ) propagating parallel to the optical axis at a large distance . The case of a non-ideal point source will be considered in Section 4. At distance is an aperture plane A(r′) containing a lens of focal length f. Directly past this plane, the optical field may be described across all space simply as a(r′), the aperture transmission function.

Fig. 2 Conventional ptychography (CP) data acquisition. A chirped amplitude grating (400 μm wide, 4μm minimum pitch) serves as our sample ψ(r). It is shifted and illuminated by a probe function ã(r), here a sinc function from a rectangular-shaped focusing element. At detector plane D, the diffracted light’s intensity is recorded. (Bottom) Corresponding probe and sample Wigner functions, whose two-dimensional convolution creates CP’s data matrix m(x, r′). Note specific parameters used for this simulation are listed in Section 3.

Independent of its specific distribution, the confined beam ã(r) then interacts with a shifted sample ψ to produce an exiting optical field, S(r). We assume the effect of sample thickness upon diffraction is negligible, allowing us to define the optical field S(r) directly past the sample as a multiplication of ã(r) and the sample transmission function ψ:
S(r)=a˜(r)ψ(rx).
(2)
Here, x is the sample’s shift distance perpendicular to the optical axis. The thin object approximation holds if the maximum sample thickness t obeys t4δres2/πλ, where δres is the sampling resolution [14

14. K. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. 59(1), 1–99 (2010). [CrossRef]

]. S(r) then propagates a large distance d to far-field detector plane D, where (as a first approximation) the intensity of the Fourier transform of S is measured:
m(x,r)=|[a˜(r)ψ(rx)]|2.
(3)
Here, m(x, r′) is a two dimensional function of probe shift distance (x) and space (r′), and comprises our data matrix. In experiment, m(x, r′) is filled up, column-by-column, with discretized diffraction images captured at the detector for many shift distances x (see example in Fig. 2). For two-dimensional images, m(x, r′) is a four-dimensional function.

2.2. Phase space representation of CP

The WDF is a well-studied phase space distribution that is often used to analyze optical imaging setups [15

15. M. Testorf, B. M. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill, 2010).

17

17. R. Horstmeyer, S. B. Oh, and R. Raskar, “Iterative aperture mask design in phase space using a rank constraint,” Opt. Express 18(21), 22545–22555 (2010). [CrossRef] [PubMed]

]. Like the Fourier transform, it transfers a function of one “primal” variable r into a new space. Unlike the Fourier transform, which offers a one-to-one mapping between the primal variable r and its conjugate u (here a mapping between space and spatial frequency), this new space is two-dimensional. The WDF is a joint function of both the primal spatial variable r and the conjugate spatial frequency variable u. Although defined in a higher-dimensional space, Wψ maintains a one-to-one relationship with the complex function ψ (apart from a constant phase shift). While not always exact, it is convenient to connect the value of W (r0, u0) to the amount of optical power at point r0 propagating in direction u0. However, while the WDF is real-valued it is not necessarily non-negative, which requires this interpretation to be taken loosely.

The goal of ptychography’s many post-processing algorithms is to recover the complex sample function ψ, which has a one-to-one relationship with Wψ, from its recorded dataset m. This goal is computationally related to deconvolving the effect of the aperture a, described by Wã, from m(x, r′) in Eq. (5). Deconvolution is often indirectly achieved through a phase retrieval algorithm [18

18. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004). [CrossRef] [PubMed]

]. Before proceeding, it is worth mentioning several challenging features exhibited by the above CP arrangement when considered in an optical microscopy context: its low collection efficiency of lensless detection hampers signal-to-noise, scanning of the sample requires mechanical motion that introduces instabilities during detection, and the large extent of the probe across space is challenging to accurately characterize, although several recently developed algorithms now account for this [19

19. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1562 (2009). [CrossRef] [PubMed]

,20

20. 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]

]. As we show next, the recently proposed FPM technique in [8

8. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photon. 7, 739–745 (2013). [CrossRef]

] also recovers a complex sample ψ via deconvolution of a high-dimensional dataset, but is able to circumvent the above list of limitations.

2.3. Mathematical representation of Fourier ptychographic microscopy (FPM)

FPM also acquires a sequence of images that are compiled into a data matrix (here labeled mF) but does so using the unique optical setup in Fig. 3. Two primary experimental differences set FPM apart from the CP setup outlined above: an array of n LEDs now occupy the illumination plane I(r), and the locations of the sample and aperture planes are effectively switched. Instead of recording the diffraction pattern from a small illuminated sample region, FPM images the entire sample under illumination from different directions.

Fig. 3 Fourier ptychographic microscopy’s (FPM) optical setup. An LED array replaces CP’s single illumination source in Fig. 1, and planes S(r′) and A(r) have switched places along the optical axis. Each LED sequentially illuminates the sample from a different angle.

Again, we begin by assuming each LED in the array occupying the illumination plane I(r) emits a quasi-monochromatic and spatially coherent field at wavelength λ (partially coherent illumination is included in Section 4). Each LED sequentially illuminates the entire sample plane S(r′) a distance away with an angled plane wave. Next, the illuminated sample is imaged by a lens of focal length f located at aperture plane A(r). In practice, the employed lens is a microscope objective (MO), although in principle, any style of image-forming optic will result in a similar analysis. At detector plane D(r′), a pixel array samples the image intensity at spacing δx = λw/2f (to avoid aliasing issues). From Eq. (1), we note that the optical field at A(r) is proportional to the Fourier transform of the field both at the sample plane S(r′) and the image plane D(r′), a feature that distinguishes FPM from CP and lends to its name.

Again applying the thin object approximation, the optical field S(r′) directly past the sample plane may be written as a multiplication between the incident field and the sample transmission function ψ as,
S(r)=ψ(r)ejkxr.
(7)
Here, x represents the sine of the angle at which the plane wave generated by the ith LED, located a distance hi away from the optical axis, travels: x=hi/hi2+2, with the distance between the LED array and the sample. As with CP, x is again connected to an illumination shift distance. Since we here define this shift distance at the illumination plane instead of the sample plane, x now becomes a variable modifying the sample’s spatial frequency. The optical field S(r′) continues to propagate to aperture plane A(r), mathematically represented through the scaled Fourier transform in Eq. (1). The field is attenuated at A(r) by the aperture transmittance function a(r) (i.e., the shape of the MO pupil plane), creating the optical field,
[S(r)]a(r)=S˜(r)a(r)=ψ˜(rx)a(r).
(8)
Again, we’ve neglected coordinate scaling factors for clarity (see Appendix A). Finally, this attenuated field propagates to image plane D(r′), represented through a scaled Fourier transform. At D(r′), the digital pixel array detects the field’s intensity mF(x, r′):
mF(x,r)=|[ψ˜(rx)a(r)]|2
(9)

Fig. 4 FPM data acquisition diagram. (Top) The same grating sample ψ(r) used in Fig. 2 is sequentially illuminated by tilted plane waves, adding a different linear phase ∝ x to each image (tilted green line). At plane A(r), the aperture a(r) limits the extent of the field before the sample is imaged to detector plane D(r′) at low resolution. (Bottom) Corresponding WDF’s and their convolution, representing FPM’s data matrix. Color maps here follow those included in Fig. 2.

3. Visualizing connections between both ptychographic domains

Fig. 5 The experimental factors influencing CP and FPM data matrices. (top) Geometrical factors define the data matrix scaling and sampling, while (bottom) parameters specific to the focusing/imaging lens define data matrix blurring for both setups.

The simulations presented in Fig. 2, Fig. 4 and Section 5 use a fixed set of example setup parameters to ensure the CP and FPM setups data matrices only vary by a rotation. For CP, we assume a lens (diameter w = 37.5 mm, focal length f = 105 mm) creates a sinc of estimated width 18 μm (peak-to-zero) at the sample from an LED located = 300 mm away. The sample plane contains a grating with 4 μm minimum feature size that is shifted in 4 μm steps. In Fig. 2 and Fig. 4, the simulated grating is 0.4 mm wide, while in Section 5 it is 1.33 mm wide. We assume a 4 mm-wide detector containing 4 μm pixels with full factor captures its diffraction pattern, which approximately requires d = 30 mm, assuming free space propagation. For FPM, we assume a similar lens (with parameters w = 37.5 mm, do = 300 mm and di = 105 mm) images the sample onto an identical detector. FPM’s LED array is fixed at a distance l = 100 mm and illuminates the same sample. The array extends across a total distance h = 24 mm perpendicular to the optical axis, yielding a 240 μm pitch for Fig. 2 and Fig. 4. One important parameter still missing from the above analysis is the light’s coherence state, connected to the active area of each optical source. We will now extend our phase space model to account for this critical effect.

4. A complete statistical model with partially coherent light

Here, we use our phase space model to show that in either optical setup, partially coherent LED illumination does not limit the ability to recover an exact sample amplitude and phase estimate. We conclude that while partial coherence impacts CP and FPM performance differently, it remains a mathematical separable expression that can be removed by computational post-processing. Section 5 applies our model to remove known coherence blur from both CP and FPM data for the first time, and Section 6 considers future extensions to build upon this initial demonstration.

4.1. Partially coherent source description

To accurately model experimentally realistic optical sources, we must introduce a statistical measure of spatial coherence into our phase space descriptions of CP in Eq. (5) and FPM in Eq. (11). We achieve this by treating the optical source’s emitted field U(r, t) as a temporally stationary stochastic process and examining its correlation across space and time: 〈U(r1, t1)U*(r2, t2)〉 = Γ̃(r1, r2, τ). Here, Γ̃ is the light’s mutual coherence, τ = t2t1 is a constant time difference, and the expectation value is performed over time. From the Weiner-Khinchine theorem, the cross-spectral density (CSD) of this stochastic process is defined as Γ(r1, r2, ω) = ∫ Γ̃(r1, r2, τ)ejωτ. The spectral density C(r, ω) = Γ(r, r, ω) represents the intensity of light at location r at a certain frequency ω. We will assume our illumination sources are fully spatially incoherent within their photon-generating area, leading to a CSD function at source plane I,
ΓI(r1,r2,ω)=γ2C(r1,ω)δ(r1r2),
(13)
where C represents the geometric shape of the source intensity for each frequency ω (typically a circ-function in two dimensions), γ is its spatial coherence cross section and δ is a Dirac delta function. For the remainder of this section, we will drop spectral dependance on ω for simplicity, assuming a notch filter is used in experiment to effectively isolate a narrow spectrum from the source. Although not detailed here, effects of a spectrally broad (i.e., temporally incoherent) source are an important consideration and may be included through incoherent superposition of the following equations. The Van Cittert-Zernike theorem relates Eq. (13)’s CSD of the source ΓI in to the CSD a distance z away, Γz:
Γz(Δr)=ejkq2zC(r)ejkz(rΔr)drC˜(r),
(14)
where a constant multiplier is neglected for simplicity, Δr′ = r′1r′2 and q=r12r22. Assuming (r12r22)/λz1 allows us to neglect the phase factor up front. With this assumption, we arrive at an approximate scaled Fourier relationship between the shape of an incoherent illumination source, C, and the CSD function Γz at any subsequent plane a large distance z from this source.

4.2. CP with partially coherent light

Fig. 6 Partially coherent light manifests itself as an additional convolution along the data matrix scan dimension x for both (a) CP and (b) FPM. The convolution is one-dimensional, as indicated by the vertical bar. With matrices rotated by 90° with respect to one another, this convolution will mix the data from each respective setup in a unique manner. For this simulation, we used the same setup parameters as for Fig. 2 and Fig. 4, but assumed each illumination source C(x) (i.e., LED) is a rectangle 200 μm in diameter.

4.3. FPM with partially coherent light

5. Case study: CP and FPM under partially coherent illumination

To briefly demonstrate the validity of our phase space model, we now attempt to measure and remove the effects of partial coherence in example CP and FPM data matrices, both in simulation and experiment. This exercise allows us to check the accuracy of our final statistical descriptions in Eq. (17) and Eq. (21). In addition, this demonstration also offers the following three primary insights. First, FPM setups that currently rely upon partially coherent LED arrays may improve the fidelity of their reconstructions by adopting this coherence removal procedure, as our tests establish. Second, the only currently demonstrated procedure that accounts for partial coherence within CP data does so without knowledge of the illumination coherence function, C(p) [12

12. P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013). [CrossRef] [PubMed]

]. The proposed coherence removal algorithm takes into account a-priori knowledge of C(p), offering a more robust procedure when an estimate of the illumination source’s shape is available. Third, our experiment tracks the slow degradation of phase imaging performance as a function of decreasing source coherence. To the best of our knowledge, it is still not currently well-understood why phase acquisition is possible yet noisy with low-coherence illumination, and our findings may generalize to benefit this area of investigation.

5.1. Simulation

Fig. 7 Simulation of partially coherent effects produce blurred (a) CP and (b) FPM data matrices of an example grating. A Wiener filter can approximately recover the coherent data matrix for each setup, from which an accurate sample reconstruction is direct. (c) Reconstruction error as a function of LED diameter (i.e., blur kernel width) increases for both CP and FPM, although FPM’s error is consistently lower. (d) The chirped grating sample and its coherent CP data matrix, for comparison.

The example blurred CP and FPM data matrix inputs in the left of Fig. 7(a)–(b) assume quasi-monochromatic illumination from sources with 100 μm-diameter active area (0.11° angular extent). The associated Wiener deconvolution outputs are shown directly to the right. Gaussian noise (normalized variance of 10−3) was added to the data before deconvolution. Noise variance and source size were assumed as prior knowledge to assure optimal filter performance. Figure 7(c) plots the average root-mean-squared error (RMSE) of recovered data matrices as a function of source diameter after Wiener deconvolution. Each point in this plot is an average over 10 experiments with noise variances ranging evenly from 10−2 to 10−4. The linear process of recovering a sample estimate from its coherent data matrix ensures sample reconstruction RMSE will follow a similar curve. CP and FPM setups that do not create a fully sampled data matrix (i.e., that under-sample along x) still benefit from a similar deconvolution approach. While beyond the scope of this work, we have successfully applied a blind deconvolution algorithm to under-sampled CP and FPM data matrices to achieve nearly equivalent coherence removal performance.

5.2. Experiment

To experimentally verify the findings of the simulation in Fig. 7, we constructed a simplified FPM setup with an illumination system to scan along one dimension. Experimental parameters closely match the parameters used in simulation (see Section 3). Our experimental setup exhibits two primary differences from the diagram in Fig. 3. First, a single LED on a motorized linear stage (Newport ESP301) was used instead of a fixed LED array at illumination plane I to facilitate easy variation of LED coherence area. This variation was achieved by placing pinholes of different diameter (100 μm–1000 μm) directly in front of the active area of a 532 nm central-wavelength diode. Note that while sufficient for the current experiment, a mechanical stage setup offers resolutions that are generally inferior to LED array-based FPM, since mechanical motion introduces the same inaccuracies limiting CP’s achievable resolution. Second, an f = 50 mm, w = 50 mm collection lens was inserted 50 mm in front of the LED source to assure uniform illumination of the sample. We experimentally determined this lens has minimal effect on the coherence area at the detector plane. Our imaging setup used a f = 105 mm, w = 37.5 mm compound lens (Nikon Micro-Nikkor f/2.8G) positioned do = 300 mm from the sample that imaged onto a 4.54μm pixel CMOS array (Prosilica-GX 1920).

Figure 8 displays an example set of simulated and experimental data matrices of the same chirped grating sample in Fig. 7 under three different illumination coherence states. Each data matrix was compiled by scanning the LED-pinhole unit at 250μm steps across 25 mm, for a total of 100 samples along x. This sampling rate is approximately 4–5 times higher than prior demonstrations of FPM [8

8. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photon. 7, 739–745 (2013). [CrossRef]

,26

26. X. Ou, R. Horstmeyer, G. Zheng, and C. Yang, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(2), 4845–4848 (2013). [CrossRef] [PubMed]

], which is not significant enough to alter any of our experimental conclusions. At each step along x, we capture an image of the linear grating and select a single row of the CMOS detector array to form data matrix column x. Each image’s maximum pixel value is scaled to 1 (i.e., each data matrix column in Fig. 8 is normalized to it’s maximum value), which enhances the appearance of noise in low-intensity areas but aids with visualization of coherence effects. The wiggling effect observable within the experimental data matrix (i.e., shifting of the grating image as a function of illumination angle) has two primary causes. First, shifting at the image plane may occur for samples not in ideal focus, which our detector’s slight undersampling prevents an exact verification of. Second, the grating’s finite thickness (3 mm) does not accurately match the thin object approximation from Section 2, leading to an unaccounted for phase modification that manifests itself as this irregular artifact.

Fig. 8 (a) Simulated and (b) experimental FPM data matrices with varying degrees of partially coherent illumination. The experimental sample closely matches the distribution of ψ(r) in Fig. 7(d). C at top indicates the LED active area diameter used in each experiment.

Figure 8 highlights three important effects of illumination coherence on FPM’s data. First, the striped “diffraction cone” within each matrix mF(x, r′) broadens along the x-dimension when using a larger-diameter source, as the convolution relationship in Eq. (21) predicts. Conceptually, an increasingly incoherent source will extend the lens’s coherent spatial frequency cutoff at NA to its incoherent spatial frequency cutoff at 2NA, hence broadening what is captured along x. This slight improvement in spatial resolution is also present (although difficult to discern) within each individual image along the r′-dimension. Second, Eq. (21)’s convolution also predicts features along x to blur with increased incoherence, which is clearly observed at the edge of the diffraction cone. As just noted, this blurring does not impact the spatial resolution of each image, but instead causes images captured by adjacent LEDs to become increasingly similar, and thus harder to accurately extract sample phase from. Finally, incoherent illumination still allows the FPM setup to acquire high-frequency sample information that otherwise would not be captured by a conventional imaging setup. This is indicated by the dark “tails” at the bottom of each data matrix, which represent high-frequency grating information that is diffracted into the imaging lens from an off-axis LED, otherwise cutoff from a single image. The density of this high-frequency information tail decreases with increasingly incoherent illumination. However, it is still clearly present with a low-coherence source, thus allowing computational improvement of a reconstructed image’s resolution beyond the conventional imaging lens NA cutoff. This information-preserving feature of ptychography in the presence of incoherent light is a very powerful tool that has yet to be studied in full, and is the main conclusion of this experiment.

6. Conclusion and future work

To briefly summarize, we first derived a linear relationship connecting the data matrices captured by conventional and Fourier ptychography. We then demonstrated that partial coherence alters different features of each setup’s data matrix, although effectively blurring both. Simulation and experiment verified the successful removal of such partial coherence artifacts for both setups, although removal from FPM’s data set is expected to yield lower error for most sparse biological samples of interest. Besides this ancillary benefit, the FPM setup requires no moving components, which thus suggests it may be capable of greater stability with respect to CP.

In the future, our derived phase space model may help advance ptychography’s development with several useful hardware modifications. First, following concepts well-known in linear filter design, the convolution relationships in Eq. (17) and Eq. (21) indicate that careful modification of each data matrix blur kernel can greatly reduce sample recovery error. CP’s conventional sinc probe and FPM’s typical circular aperture both include many transfer function zeros, which are computationally impossible to invert. Apodization of the probe and aperture with a designed mask can improve this inversion, offering increased solution stability, independent of recovery algorithm specifics. Apodization of the incoherent illumination source’s finite shape, C(p), will also improve removal of partial coherence effects. Second, Eq. (12) suggests that alternative optical setups can capture the data matrix m under different linear transformations (e.g., a matrix rotation that is not 90°, or another isomorphic transform besides rotation). These alternatives to CP and FPM will most likely offer application-specific advantages. For example, one could imagine both shifting the sample across a limited range and using a small number of illumination sources to increase collection efficiency. This specific joint CP-FPM setup may benefit applications only tolerating minimal movement, but many other hybrid designs may be easily imagined to fulfill niche design constraints. Finally, we minimally considered the computational post-processing aspect of ptychography in our analysis. As recently demonstrated, phase space offers a rich array of image reconstruction tools [22

22. C. Teale, D. Adams, M. Murnane, H. Kapteyn, and D. J. Kane, “Imaging by integrating stitched spectrograms,” Opt. Express 21(6), 6783–6793 (2012). [CrossRef]

]. Working within a high-dimensional space like the WDF’s is required when including partial coherence, so our model will most immediately impact ptychographic algorithms that must account for the effects of large, high-throughput sources. Furthermore, our demonstration of a linear mapping between CP and FPM assures that any future computational developments may jointly benefit both setups. For example, we now know FPM can immediately benefit from recent CP algorithms like ePIE [19

19. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1562 (2009). [CrossRef] [PubMed]

], annealing [20

20. 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]

], and other procedures accounting for partial coherence [12

12. P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013). [CrossRef] [PubMed]

]. Such sharing between two previously disconnected research areas is the most immediate impact our phase space model, which we believe offers a solid foundation for many future insights to expand upon as ptychography continues to evolve.

Appendix A: Phase space expressions with scaling factors included

Re-working CP’s data matrix to include coordinate scaling reveals two primary effects. First, propagation from the lens to the sample includes a λf scaling factor [13

13. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

], where λ is wavelength and f the lens focal length. Second, propagation from the sample to the detector includes a similar scaling factor by λd, with d the detector distance. A scaled version of Eq. (3) is thus,
m(x,r)=|r,r/λd[a˜(r/λf)ψ(rx)]|2,
(22)
where the subscripts indicate the original and transformed coordinates used within the Fourier transform exponent. This can be rewritten in integral form as,
m(x,r)=a˜(r1λf)a˜*(r2λf)ψ(r1x)ψ*(r2x)exp[jkdr(r1r2)]dr1dr2
(23)
From here, a scaled Wigner convolution relationship is found as,
m(x,r)=Wψλf(rxλf,u)Wa˜(r,λfdru)drdu.
(24)
where the ψλf subscript indicates the coordinate system of Wψ is multiplicatively scaled by a constant λf factor. Pre-integral multiplicative constants are omitted for clarity. Equation (24) includes three primary effects of scaling. First is the λf scaling factor along Wψ’s spatial variable r, which also necessarily requires the phase space function’s spatial frequency variable u to be contracted by the same proportion before computing the convolution. Second, the resulting data matrix’s r′ coordinate is scaled by a λf/d factor, and third its x coordinate by 1/λf.

Scaling effects can similarly be incorporated into FPM’s data matrix Eq. (10) as,
mF(x,r)=ψ˜(r1λdoxλ)ψ˜*(r2λdoxλ)a(r1)a*(r2)exp(jkrλdi(r1r2))dr1dr2.
(25)
Here, do is the distance from the sample to the lens and di is the distance from the lens to the detector (Fig. 3). Straightforward manipulations following Appendix B’s steps lead to,
mF(x,r)=Wψλdo(udox,r)Wa˜(u,rλdordi)dudr.
(26)
where ψλdo here indicates Wψ is fully scaled by a constant factor 1/λdo. Again, three main differences are apparent comparing the above to the FPM convolution expression in Eq. (11): r′ is scaled by λdo/di, x is scaled by do, and Wψ’s joint coordinates are scaled by 1/λdo before convolution. Similar manipulations yield scaling factors for data matrices containing the effects of partially coherent illumination.

Appendix B: The Wigner representation of the FPM data matrix

Our goal in this derivation is to transform Eq. (10) into an expression separable in ψ and ã, for direct comparison with Eq. (5). This can be achieved by taking advantage of the Wigner distribution function (WDF). As noted in Section 2, the WDF is a convenient tool to achieve variable separation. The WDFs describing ψ and a are obtained by first transforming (r1, r2) to center-difference coordinates (r, y), using r1 = r + y/2 and r2 = ry/2:
mF(x,r)=ψ˜(r+y2x)ψ˜*(ry2x)a(r+y2)a*(ry2)×exp(jkry)dydr,
(27)
where in the exponent we use the fact that r1r2 = y. Following the definition of the WDF in Eq. (6), we can define the WDF of the Fourier transform of our sample function ψ̃ as,
Wψ˜(r,u)=ψ˜(r+y2)ψ˜*(ry2)exp(jkyu)dy.
(28)
Applying an inverse Fourier transform to both sides of Eq. (28) yields,
ψ˜(r+y2)ψ˜*(ry2)=k2πWψ˜(r,u)exp(jkyu)du
(29)
The Wigner distribution of the aperture function a(r), Wa(r, u), will take a similar form as Eq. (28). As we will see next, it is more useful to express the WDF of the aperture with a shifted spatial frequency term, Wa(r, r′u):
Wa(r,ru)=a(r+y2)a*(ry2)exp(jky(ru))dy
(30)
The inverse Fourier transform of Eq. (30) yields,
a(r+y2)a*(ry2)=k2πWa(r,ru)exp(jky(ru))d(ru)
(31)
Inserting Eq. (29) and Eq. (31) into Eq. (27) and noting all terms in the exponent cancel produces a near-final FPM data matrix expression:
mF(x,r)=Wψ˜(rx,u)Wa(r,ru)drdu,
(32)
where the pre-integral multiplier is omitted for clarity. To fully connect FPM’s data matrix with CP’s in Eq. (5), we can take advantage of a convenient property of the Wigner distribution. As a function of both space and spatial frequency, it is clear that Wψ(r, u) must contain the same information as when it is applied to the sample’s Fourier transform, Wψ̃ (r, u). The two Wigner functions are connected by,
Wψ˜(r,u)=Wψ(u,r).
(33)
The Wigner distribution of the sample’s Fourier transform ψ̃ is given by the Wigner distribution of the sample ψ in Eq. (28) but rotated 90° [15

15. M. Testorf, B. M. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill, 2010).

]. Applying this property to both WDFs in Eq. (32), without swapping the dummy convolution variables, produces an expression directly comparable with CP’s Eq. (5):
mF(x,r)=Wψ(ux,r)Wa˜(u,rr)dudr,
(34)
which is in Eq. (11). Two manipulations applied to Eq. (34) lead to the 90° rotation relationship between the CP and FPM data matrices in Eq. (12). First, both the data matrix on the left of Eq. (34) and the two Wigner functions on the right of Eq. (34) must be rotated by swapping the order of their variables (i.e., mF(x, r′) becomes mF(r′, x)). Second, the dummy integration variables x and r′ must switch from one Wigner function to the other (i.e., x goes to Wã and r′ goes to Wψ). Comparing the result of these operations with Eq. (5) leads to Eq. (12)’s linear transform.

Appendix C: Conventional ptychography with partially coherent source, derivation

The influence of partially coherent light on conventional ptychography is derived by first inserting the CSD function of light at the sample plane (Eq. (15)) into our data matrix m(x, r′) in Eq. (16) to produce,
m(x,r)=C(p)a˜(r1p)a˜*(r2p)ψ(r1x)ψ*(r2x)×exp[jkr(r1r2)]dr1dr2dp.
(35)
Next, we perform the variable substitution r1 = r + y/2 and r2 = ry/2 to create,
m(x,r)=C(p)a˜(r+y2p)a˜*(ry2p)ψ(r+y2x)ψ*(ry2x)×exp[jkry]dydrdp.
(36)
Following the same steps as Eq. (28)Eq. (29), we may replace the ψ(r+y2x)ψ*(ry2x) term with its WDF to yield,
m(x,r)=k2πC(p)a˜(r+y2p)a˜*(ry2p)Wψ(rx,u)×exp[jky(ur)]dydrdpdu
(37)
Likewise, a similar WDF relationship may be constructed for the aperture:
a˜(r+y2p)a˜*(ry2p)=k2πWa˜(rp,ru)exp(jky(ru))d(ru)
(38)
Inserting Eq. (38) into Eq. (37) and neglecting constant multipliers leads to,
m(x,r)=C(p)Wa˜(rp,ru)Wψ(rx,u)×exp(jky(ur+ru))dydrdpdud(ru).
(39)
Noting all terms in the exponent cancel and the dy and d(−r′u) integrals drop to leave,
m(x,r)=C(p)Wa˜(rp,ru)Wψ(rx,u)drdudp,
(40)
the final expression in Eq. (17). The finite extent of the incoherent source C(p) alters our original expression for CP’s data matrix through a convolution along the x-dimension of the data matrix, similar to FPM as derived in Appendix D.

Appendix D: FPM with partially coherent sources, derivation

Acknowledgments

The authors acknowledge funding support from the National Institutes of Health (grant no. 1DP2OD007307-01) and Clearbridge Biophotonics Pte Ltd., Singapore (Agency Award no. Clearbridge 1). The authors would like to thank Mark Harfouche, Benjamin Judkewitz and Xiaoze Ou for helpful feedback during manuscript preparation.

References and links

1.

P. D. Nellist, B. C. McCallum, and J. M. Rodenburg, “Resolution beyond the ‘infromation limit’ in transmission electron microscopy,” Nature 374, 630–632 (1995). [CrossRef]

2.

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]

3.

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,” PRL 98, 034801 (2007). [CrossRef]

4.

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

5.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 437–439 (2010). [CrossRef]

6.

A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. 35(15), 2585–2587 (2010). [CrossRef] [PubMed]

7.

A. M. Maiden, M. J. Humphry, F. Zhang, and J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A. 28(4), 604–612 (2011). [CrossRef]

8.

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photon. 7, 739–745 (2013). [CrossRef]

9.

J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A 339, 521–553 (1992). [CrossRef]

10.

H. N. Chapman, “Phase retrieval x-ray microscopy by Wigner distribution deconvolution,” Ultramicroscopy 66, 153 (1996). [CrossRef]

11.

J. N. Clark, X. Huang, R. Harder, and I. K. Robinsion, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012). [CrossRef] [PubMed]

12.

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013). [CrossRef] [PubMed]

13.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

14.

K. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. 59(1), 1–99 (2010). [CrossRef]

15.

M. Testorf, B. M. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill, 2010).

16.

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” JOSA A 3(8), 1227–1238 (1986). [CrossRef]

17.

R. Horstmeyer, S. B. Oh, and R. Raskar, “Iterative aperture mask design in phase space using a rank constraint,” Opt. Express 18(21), 22545–22555 (2010). [CrossRef] [PubMed]

18.

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004). [CrossRef] [PubMed]

19.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1562 (2009). [CrossRef] [PubMed]

20.

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]

21.

M. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008). [CrossRef]

22.

C. Teale, D. Adams, M. Murnane, H. Kapteyn, and D. J. Kane, “Imaging by integrating stitched spectrograms,” Opt. Express 21(6), 6783–6793 (2012). [CrossRef]

23.

G. Zheng, X. Ou, R. Horstmeyer, and C. Yang, “Characterization of spatially varying aberrations for wide field-of-view microscopy,” Opt. Express 21(13), 15131–15143 (2013). [CrossRef] [PubMed]

24.

D. Brady, Optical Imaging and Spectroscopy (John Wiley & Sons, 2009). [CrossRef]

25.

R. G. Brown and P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering (John Wiley & Sons, 1996).

26.

X. Ou, R. Horstmeyer, G. Zheng, and C. Yang, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(2), 4845–4848 (2013). [CrossRef] [PubMed]

OCIS Codes
(110.4980) Imaging systems : Partial coherence in imaging
(110.1758) Imaging systems : Computational imaging
(080.5084) Geometric optics : Phase space methods of analysis
(070.7425) Fourier optics and signal processing : Quasi-probability distribution functions

ToC Category:
Imaging Systems

History
Original Manuscript: October 22, 2013
Revised Manuscript: December 16, 2013
Manuscript Accepted: December 17, 2013
Published: January 2, 2014

Virtual Issues
Vol. 9, Iss. 3 Virtual Journal for Biomedical Optics

Citation
Roarke Horstmeyer and Changhuei Yang, "A phase space model of Fourier ptychographic microscopy," Opt. Express 22, 338-358 (2014)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-22-1-338


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References

  1. P. D. Nellist, B. C. McCallum, J. M. Rodenburg, “Resolution beyond the ‘infromation limit’ in transmission electron microscopy,” Nature 374, 630–632 (1995). [CrossRef]
  2. F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010). [CrossRef]
  3. J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, I. Johnson, “Hard-X-ray lensless imaging of extended objects,” PRL 98, 034801 (2007). [CrossRef]
  4. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, F. Pheiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008). [CrossRef] [PubMed]
  5. M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, F. Pheiffer, “Ptychographic X-ray computed tomography at the nanoscale,” Nature 467, 437–439 (2010). [CrossRef]
  6. A. M. Maiden, J. M. Rodenburg, M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. 35(15), 2585–2587 (2010). [CrossRef] [PubMed]
  7. A. M. Maiden, M. J. Humphry, F. Zhang, J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A. 28(4), 604–612 (2011). [CrossRef]
  8. G. Zheng, R. Horstmeyer, C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photon. 7, 739–745 (2013). [CrossRef]
  9. J. M. Rodenburg, R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A 339, 521–553 (1992). [CrossRef]
  10. H. N. Chapman, “Phase retrieval x-ray microscopy by Wigner distribution deconvolution,” Ultramicroscopy 66, 153 (1996). [CrossRef]
  11. J. N. Clark, X. Huang, R. Harder, I. K. Robinsion, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012). [CrossRef] [PubMed]
  12. P. Thibault, A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013). [CrossRef] [PubMed]
  13. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  14. K. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. 59(1), 1–99 (2010). [CrossRef]
  15. M. Testorf, B. M. Hennelly, J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill, 2010).
  16. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” JOSA A 3(8), 1227–1238 (1986). [CrossRef]
  17. R. Horstmeyer, S. B. Oh, R. Raskar, “Iterative aperture mask design in phase space using a rank constraint,” Opt. Express 18(21), 22545–22555 (2010). [CrossRef] [PubMed]
  18. H. M. L. Faulkner, J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004). [CrossRef] [PubMed]
  19. A. M. Maiden, J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1562 (2009). [CrossRef] [PubMed]
  20. A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012). [CrossRef] [PubMed]
  21. M. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008). [CrossRef]
  22. C. Teale, D. Adams, M. Murnane, H. Kapteyn, D. J. Kane, “Imaging by integrating stitched spectrograms,” Opt. Express 21(6), 6783–6793 (2012). [CrossRef]
  23. G. Zheng, X. Ou, R. Horstmeyer, C. Yang, “Characterization of spatially varying aberrations for wide field-of-view microscopy,” Opt. Express 21(13), 15131–15143 (2013). [CrossRef] [PubMed]
  24. D. Brady, Optical Imaging and Spectroscopy (John Wiley & Sons, 2009). [CrossRef]
  25. R. G. Brown, P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering (John Wiley & Sons, 1996).
  26. X. Ou, R. Horstmeyer, G. Zheng, C. Yang, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(2), 4845–4848 (2013). [CrossRef] [PubMed]

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