Direct Ptychography: Methods & Insights

• Direct ptychography is a class of reconstruction methods that directly inverts far-field diffraction data to produce a complex transmission function, capturing subtle charge redistributions at the atomic scale.
• It encompasses both analytical techniques like single-side-band and Wigner-distribution deconvolution and learned models such as PtychoNN and PtychoFormer, offering fast and dose-efficient imaging.
• The approach relies on the weak-phase approximation and known probe characteristics, making it ideal for high-resolution electron microscopy of thin samples and quantitative bonding charge analysis.

Direct ptychography denotes a family of ptychographic reconstruction strategies that seek a direct inversion from measured far-field diffraction intensities to the complex specimen transmission function, rather than relying solely on long iterative phase-retrieval loops. In the electron-microscopy formulation, a coherent probe $P(r-R_j)$ is scanned across a specimen with transmission $O(r)$, producing exit waves $\psi_j(r)=P(r-R_j)\,O(r)$ and diffraction intensities $I_j(q)=\bigl|\mathcal{F}\{\psi_j(r)\}\bigr|^2$ at each probe position. Under the weak-phase approximation, $O(r)=\exp[i\phi(r)]\approx 1+i\phi(r)$ with $\phi(r)=\sigma\int V(r,z)\,dz$, so the reconstructed phase is directly sensitive to the projected electrostatic potential, including charge redistribution from bonding. In current usage, the term encompasses non-iterative analytical inversions such as single-side-band and Wigner-distribution deconvolution, learned single-shot mappings such as PtychoNN and PtychoFormer, and in-focus electron-ptychographic workflows used for direct imaging of bonding at atomic resolution (Clark et al., 13 Mar 2025, Martinez et al., 2019).

1. Forward model and weak-phase regime

The canonical ptychographic forward model describes the object by a complex transmission function $O(r)\equiv A(r)e^{i\phi(r)}$ and the probe by a complex illumination function $P(r-R_j)$. At scan position $R_j$, the exit wave is

$\psi_j(r)=O(r)\,P(r-R_j),$

and the detector records the far-field intensity

$O(r)$0

The full measurement is therefore a 4D dataset $O(r)$1 indexed by scan position and detector coordinate. In STEM formulations this dataset is often described as momentum-resolved or 4D-STEM data, with dense sampling in both $O(r)$2 and $O(r)$3 assumed by the standard theory (Clark et al., 13 Mar 2025, Alhassan, 5 Jun 2026).

For thin weak-phase samples, the object transmission is approximated as

$O(r)$4

where

$O(r)$5

This establishes the direct link between reconstructed phase and projected electrostatic potential. In electron microscopy that phase is sensitive not only to atomic number and projected potential in the independent-atom sense, but also to bonding-induced charge redistribution. A central consequence is that phase imaging can detect subtle deviations between independent-atom and density-functional descriptions when the phase precision is sufficiently high (Martinez et al., 2019).

The weak-phase regime is also the natural setting for the principal non-iterative direct methods. In that limit the measured intensities can be linearized in the object phase, and the inversion can be expressed as a deconvolution or overlap-filtering problem involving a known or well-characterized probe. This suggests why direct ptychography is especially prominent for thin samples, known probes, and acquisition settings that preserve reciprocal-space overlap.

2. Analytical direct inversions: single-side-band and Wigner deconvolution

The two major classes of non-iterative direct reconstruction in the recent STEM literature are Single-Side-Band (SSB) ptychography and Wigner-distribution deconvolution (WDD). Both begin by Fourier transforming the measured intensity with respect to the probe position,

$O(r)$6

so that the mixed-space representation contains the overlap structure between shifted probe apertures and the object spectrum. Under the weak-phase-object approximation, SSB expresses $O(r)$7 as a sum of a central term and two first-order sidebands,

$O(r)$8

where $O(r)$9 and $\psi_j(r)=P(r-R_j)\,O(r)$0. In a focused-probe STEM with a circular aperture of radius $\psi_j(r)=P(r-R_j)\,O(r)$1, the relevant information lies in the “double-overlap” regions of the three overlapping discs. For each $\psi_j(r)=P(r-R_j)\,O(r)$2 with $\psi_j(r)=P(r-R_j)\,O(r)$3, SSB extracts the phase-object spectrum by integrating $\psi_j(r)=P(r-R_j)\,O(r)$4 over the appropriate overlap region $\psi_j(r)=P(r-R_j)\,O(r)$5, then recovers $\psi_j(r)=P(r-R_j)\,O(r)$6 by inverse Fourier transform. The method is non-iterative, single pass through $\psi_j(r)=P(r-R_j)\,O(r)$7, and highly noise-rejecting because all $\psi_j(r)=P(r-R_j)\,O(r)$8 are discarded. Its non-super-resolution transfer is limited to $\psi_j(r)=P(r-R_j)\,O(r)$9, corresponding to twice conventional BF-STEM (Clark et al., 13 Mar 2025).

WDD reformulates the same data in terms of Wigner distributions. After computing

$I_j(q)=\bigl|\mathcal{F}\{\psi_j(r)\}\bigr|^2$0

one obtains

$I_j(q)=\bigl|\mathcal{F}\{\psi_j(r)\}\bigr|^2$1

with probe and object Wigner distributions

$I_j(q)=\bigl|\mathcal{F}\{\psi_j(r)\}\bigr|^2$2

$I_j(q)=\bigl|\mathcal{F}\{\psi_j(r)\}\bigr|^2$3

A Wiener-filter deconvolution then gives

$I_j(q)=\bigl|\mathcal{F}\{\psi_j(r)\}\bigr|^2$4

where $I_j(q)=\bigl|\mathcal{F}\{\psi_j(r)\}\bigr|^2$5 is a small regularizer to avoid division by zero. A further Fourier transform yields

$I_j(q)=\bigl|\mathcal{F}\{\psi_j(r)\}\bigr|^2$6

from which the object spectrum and finally $I_j(q)=\bigl|\mathcal{F}\{\psi_j(r)\}\bigr|^2$7 are recovered. WDD remains one-shot but requires knowledge of the probe or its Wigner distribution, and its regularizer trades resolution against noise (Clark et al., 13 Mar 2025, Alhassan, 5 Jun 2026).

The contrast with iterative approaches is structural. Direct methods have algorithmic cost $I_j(q)=\bigl|\mathcal{F}\{\psi_j(r)\}\bigr|^2$8 deconv, are one-shot, and assume a weak object and known probe. Iterative methods such as PIE, ePIE, Difference-Map, RAAR, or maximum-likelihood formulations have cost $I_j(q)=\bigl|\mathcal{F}\{\psi_j(r)\}\bigr|^2$9, require $O(r)=\exp[i\phi(r)]\approx 1+i\phi(r)$0–$O(r)=\exp[i\phi(r)]\approx 1+i\phi(r)$1 iterations in typical summaries, can update both object and probe, and can handle stronger scattering and more flexible forward models, but may stagnate or fit noise. WDD super-resolution can in principle recover up to $O(r)=\exp[i\phi(r)]\approx 1+i\phi(r)$2 by judicious “compress-to-2D” choices, but only at extremely high dose $O(r)=\exp[i\phi(r)]\approx 1+i\phi(r)$3 (Clark et al., 13 Mar 2025).

3. Atomic-resolution direct electron ptychography of bonding charge

A benchmark experimental realization of electron ptychography as a bond-sensitive phase microscope was demonstrated on single-layer boron nitride. The implementation used a JEOL ARM200CF at 80 kV with a 31.5 mrad convergence semi-angle, a probe scan of $O(r)=\exp[i\phi(r)]\approx 1+i\phi(r)$4 positions over a $O(r)=\exp[i\phi(r)]\approx 1+i\phi(r)$5 nm field of view corresponding to pixel size $O(r)=\exp[i\phi(r)]\approx 1+i\phi(r)$6 Å, and a pnCCD (4D-Canvas system) recording $O(r)=\exp[i\phi(r)]\approx 1+i\phi(r)$7 pixels per diffraction pattern at 4000 frames/s. The dose per frame was $O(r)=\exp[i\phi(r)]\approx 1+i\phi(r)$8, and the total dose after nonrigid alignment of 16 repeats was $O(r)=\exp[i\phi(r)]\approx 1+i\phi(r)$9 (Martinez et al., 2019).

The reconstruction workflow combined ptychographic phase retrieval with post-collection aberration correction and simultaneous Z-contrast imaging. Residual aberrations were modeled in Fourier space by

$\phi(r)=\sigma\int V(r,z)\,dz$0

with

$\phi(r)=\sigma\int V(r,z)\,dz$1

where the coefficients were fitted by minimizing phase variance or by cross-correlation with a reference region, and the conjugate phase factor was then applied to correct the retrieved probe and final object phase. In parallel, a high-angle ADF detector recorded the in-focus Z-contrast image. Because ADF intensity $\phi(r)=\sigma\int V(r,z)\,dz$2 is insensitive to bonding, it provided unambiguous B-versus-N sublattice identification and prevented mis-assignment of bright and dark spots in the phase map (Martinez et al., 2019).

The reported phase sensitivity after aberration correction was on the order of $\phi(r)=\sigma\int V(r,z)\,dz$3 rad r.m.s. noise. To quantify the bonding signal, Martinez et al. defined the Integrated Squared-Phase Cross-Section (ISPCS) per atom by squaring the phase image, partitioning the image into Voronoi cells around each atomic site, and integrating $\phi(r)=\sigma\int V(r,z)\,dz$4 inside each cell. The measured difference,

$\phi(r)=\sigma\int V(r,z)\,dz$5

was statistically significant and opposite in sign to the independent-atom model, which predicted $\phi(r)=\sigma\int V(r,z)\,dz$6. DFT-based projected potentials from CASTEP plane-wave and WIEN2k LAPW+lo with PBE, LDA, rSCAN, and PBE0 functionals yielded $\phi(r)=\sigma\int V(r,z)\,dz$7 in the range $\phi(r)=\sigma\int V(r,z)\,dz$8–$\phi(r)=\sigma\int V(r,z)\,dz$9, much closer to experiment. The final phase image showed a slightly smaller phase-shift on N sites than B sites, exactly as expected when electrons are transferred from B$O(r)\equiv A(r)e^{i\phi(r)}$0N, and mapped the spatial extent of the charge redistribution around each bond at sub-Å resolution. The study characterized this as the first direct real-space imaging of bonding charges by electron microscopy at true atomic resolution (Martinez et al., 2019).

A plausible implication is that, when phase sensitivity, aberration calibration, and chemically specific lattice labeling are all available in a single dataset, direct electron ptychography can serve as a quantitative probe of chemically induced changes in the projected potential rather than merely of projected atomic columns.

4. Sampling, focus, thickness, and upsampling

The principal sampling constraint in direct ptychography is set by the scan step. For a regular grid of spacing $O(r)\equiv A(r)e^{i\phi(r)}$1, Nyquist sampling requires

$O(r)\equiv A(r)e^{i\phi(r)}$2

with $O(r)\equiv A(r)e^{i\phi(r)}$3 determined by the convergence semi-angle $O(r)\equiv A(r)e^{i\phi(r)}$4 and wavelength $O(r)\equiv A(r)e^{i\phi(r)}$5. The nominal resolution is

$O(r)\equiv A(r)e^{i\phi(r)}$6

and in practice direct ptychography often requires $O(r)\equiv A(r)e^{i\phi(r)}$7 to fully sample the transfer of information. Dose and coherence also enter explicitly: the phase precision obeys $O(r)\equiv A(r)e^{i\phi(r)}$8 per pixel under Poisson noise, or roughly $O(r)\equiv A(r)e^{i\phi(r)}$9, and direct SSB rejects out-of-band noise so that at low dose $P(r-R_j)$0–$P(r-R_j)$1 it can outperform iterative methods (Clark et al., 13 Mar 2025).

Recent work has addressed the scan-sampling bottleneck by relating parallax imaging to direct ptychography. In this formulation, direct ptychography under the weak-phase-object approximation is written as a non-iterative deconvolution with a contrast transfer function $P(r-R_j)$2 and scan-demodulated Fourier stack $P(r-R_j)$3,

$P(r-R_j)$4

Parallax imaging then appears as a quadratic Taylor truncation of the direct-ptychography overlap kernel. This leads to an upsampling strategy in which the virtual bright-field stack is upsampled on a finer scan grid and combined with the overlap filter,

$P(r-R_j)$5

followed by inverse filtering with $P(r-R_j)$6 or a Tikhonov-regularized variant. Analytical and experimental results showed $P(r-R_j)$7–$P(r-R_j)$8 scan upsampling, with upsampling factors $P(r-R_j)$9 recovering the analytical CTF out to $R_j$0 when the diversity condition $R_j$1 was satisfied. For beam-sensitive MOF and virus-like-particle samples at dose $R_j$2 and strong defocus, parallax and upsampled direct ptychography were visually indistinguishable, whereas for high-fluence, high-aberration imaging the direct method retained superior zero-crossing recovery (Varnavides et al., 24 Jul 2025).

Thickness and focus introduce a separate limitation. For moderately thick samples, contrast reversals can occur in direct electron ptychography when the projected potential becomes strong. Central focusing addresses this by placing the probe focal plane at the sample mid-plane. In the slice-wise formulation, the integrated double-aperture term acquires a sinc envelope and a defocus phase factor, and when the focal plane is at the center,

$R_j$3

The phase contributions from slices above and below the central plane therefore cancel, suppressing defocus-induced phase errors and often removing contrast reversals when dynamical scattering is not overly strong. The same analysis gives $R_j$4 and the rule of thumb

$R_j$5

Reported examples include SrTiO$R_j$6 up to about 28 nm at 200 kV and 20 mrad, where central focusing gave nearly reversal-free images, and GaN, where a 10–17 nm thickness range still exhibited residual reversals even at mid-focus, signaling the onset of dynamical effects (Gao et al., 2023).

5. Learned direct ptychography

Data-driven direct ptychography replaces analytical inversion by an end-to-end approximation of the inverse map from diffraction intensities to amplitude and phase. PtychoNN is a fully convolutional network that predicts real-space structure and phase at each scan point solely from the corresponding far-field diffraction data. Its architecture is a U-net-style encoder-decoder with two parallel decoders, one for amplitude and one for phase; the encoder uses four hierarchical blocks of $R_j$7 convolutions with ReLU and $R_j$8 max-pooling, while the decoder uses nearest-neighbor up-sampling, skip connections, and final $R_j$9 convolutions. The input is a single-shot far-field intensity patch, normalized per pattern, and the output is a complex transmission map represented as two real channels. Training used an experimental tungsten test pattern on a $\psi_j(r)=O(r)\,P(r-R_j),$0 grid with 50% overlap and 30 nm step at beamline 26-ID, APS, with ground truth from 400 ePIE iterations in Ptycholib. The supervised loss was

$\psi_j(r)=O(r)\,P(r-R_j),$1

with $\psi_j(r)=O(r)\,P(r-R_j),$2 (Cherukara et al., 2020).

PtychoNN converged to MAE $\psi_j(r)=O(r)\,P(r-R_j),$3 for amplitude and $\psi_j(r)=O(r)\,P(r-R_j),$4 rad for phase relative to ePIE ground truth. On the test set, network-versus-ePIE errors were reported as $\psi_j(r)=O(r)\,P(r-R_j),$5 and $\psi_j(r)=O(r)\,P(r-R_j),$6 rad, with Fourier ring correlation showing less than 10 nm difference in resolution. Inference cost was approximately 1 ms per point on a V100 GPU, versus approximately 300 ms per single-point ePIE reconstruction for 400 iterations, yielding an overall speedup of approximately $\psi_j(r)=O(r)\,P(r-R_j),$7 and a full $\psi_j(r)=O(r)\,P(r-R_j),$8 scan time of less than 1 s versus more than 1 min. The same network remained visually and quantitatively accurate when the scan step was increased from 30 nm to 150 nm, corresponding to $\psi_j(r)=O(r)\,P(r-R_j),$9 fewer scan points and overlap below 5%, with no retraining needed and concomitant $O(r)$00 dose and acquisition-time savings (Cherukara et al., 2020).

PtychoFormer extends the learned direct paradigm with a hierarchical transformer-based single-shot phase-retrieval model. It groups up to 9 diffraction frames in a local $O(r)$01 neighborhood into a 9-channel input, uses a four-stage MiT encoder with channel depths $O(r)$02, and reconstructs amplitude and phase through a convolutional decoder. Overlapping local reconstructions are stitched by cropping 10% borders and applying feathered edge tapering. On a 3,100-sample synthetic test set at 20 px offset, the reported metrics were amplitude $O(r)$03, $O(r)$04, and phase $O(r)$05, $O(r)$06, outperforming previous deep-learning approaches by 25–60% in NRMSE. One-shot inference plus stitching took 0.14 s on GPU, while ePIE on an $O(r)$07 scan required $O(r)$08 s per iteration for 800–1500 iterations, or about 5–8.5 min, so the model was 2,100–3,600$O(r)$09 faster than ePIE. Its hybrid extension, ePF, uses the transformer reconstruction as the initial guess for ePIE, reduces global-phase-shift ambiguity because the model predicts absolute $O(r)$10, and required approximately 100 fewer iterations than ePIE to reach SSE tolerance, with final NRMSE further improved by 47–73%. The reported limitation is that these results were obtained on synthetic data and that real experiments require careful phase calibration (Nakahata et al., 2024).

These learned approaches enlarge the meaning of “direct ptychography.” In the analytical literature, “direct” usually denotes one-shot inversion under a weak-object model; in the deep-learning literature, it denotes a single forward pass through a trained inverse model. The shared feature is the replacement of long, sample-specific iterative optimization by a precomputed inversion rule.

6. Capabilities, limitations, and relation to iterative ptychography

Direct ptychography is used because ptychographic phase retrieval can be highly dose efficient, can reveal light elements next to heavy atoms, can map functional properties, and can achieve image resolutions far beyond the conventional resolution limit defined by the imaging aperture size. In the non-iterative SSB/WDD setting, the principal advantages are one-shot reconstruction, fast computation, a known transfer mechanism, and direct access to phase and amplitude from 4D-STEM data. The electron-bonding study further showed that in-focus STEM ptychography combined with aberration-free reconstruction and simultaneous Z-contrast labeling yields a quantitative bond-sensitive phase microscope, with proposed applicability to charge transfers at defects, grain boundaries, hetero-interfaces, ferroelectric domains, and catalytic centers (Clark et al., 13 Mar 2025, Martinez et al., 2019).

The limitations are equally well defined. SSB demands a weak-phase object, well-characterized probe, and nonvanishing overlap integral over the spatial frequencies of interest; full WDD relaxes some of those restrictions but still assumes single scattering and a known probe Wigner distribution. Both analytical direct methods depend on dense sampling and sufficient reciprocal-space overlap, and multiple or dynamical scattering in thick crystals is not handled by simple WDD or SSB. For direct learned models, generalization is best when the object class and probe shape resemble the training distribution; a significant change in sample type or probe profile may require retraining or fine-tuning, and completely new materials not represented in training may yield biased reconstructions (Alhassan, 5 Jun 2026, Cherukara et al., 2020).

A common misconception is that “direct” and “iterative” are mutually exclusive categories of ptychography. The recent literature instead treats them as complementary regimes. Analytical direct methods are non-iterative by construction; deep-learning direct methods are single-shot predictors; and hybrid pipelines explicitly use a direct reconstruction as initialization for an iterative refinement, as in ePF. This suggests a continuum rather than a dichotomy: known-probe weak-object data favor direct inversion, strong-scattering or probe-unknown cases favor iterative refinement, and hybrid strategies combine data-driven speed with physics-based correction (Nakahata et al., 2024).

A second misconception is that direct ptychography is only a computational shortcut. The atomic-resolution hBN result indicates that, with sufficient phase sensitivity and calibration, direct electron ptychography is also a metrological tool for measuring subtle projected-potential changes associated with bonding. Conversely, the central-focusing and sampling studies show that this metrological role depends critically on the validity of the weak-phase model, on focus placement, on scan Nyquist conditions, and on whether higher-order aberration or dynamical-scattering effects are negligible or explicitly accounted for (Gao et al., 2023, Varnavides et al., 24 Jul 2025).