Coherent Diffractive Imaging Overview

• Coherent diffractive imaging is a lensless modality that reconstructs 2D or 3D sample structures from far-field diffraction patterns through computational phase retrieval.
• It leverages diverse phase-retrieval algorithms—including iterative, global optimization, and deep learning methods—to solve the phase problem and enhance image fidelity.
• Applications span nanoscale metrology, biological imaging, and crystalline defect analysis using advanced sources such as synchrotrons, XFELs, and high-harmonic lasers.

Coherent diffractive imaging (CDI) is a lens-less imaging modality in which a coherent beam is scattered by a sample and only the far-field intensity pattern is recorded; the missing phase information is then recovered computationally to reconstruct the sample’s complex object function in two or three dimensions (Ai et al., 2024). In complementary formulations, CDI recovers the complex scattering function of an object from the modulus of its Fourier transform, or reconstructs the three-dimensional morphology of isolated nanoscale objects by recording their far-field diffraction and numerically solving the phase problem (Colombo et al., 2017, Rupp et al., 2016). Across synchrotron radiation, XFELs, high harmonic generation, electrons, and optical lasers, CDI has been applied to physical and biological specimens, strained crystals, isolated nanoparticles, and dynamic processes in solution (Lo et al., 2017).

1. Foundational formulation

In the Fraunhofer regime, the measured intensity is the squared modulus of a Fourier transform. For a three-dimensional object density $\rho(\mathbf r)$, one writes

$$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$

while for a transmission object illuminated by a probe $P(\mathbf r)$ the exit wave is $\psi(\mathbf r)=P(\mathbf r)\,O(\mathbf r)$ and the detector measures

$$I(\mathbf q)=\bigl|\mathcal F\{P(\mathbf r)\,O(\mathbf r)\}\bigr|^2$$

(Ai et al., 2024, Chang et al., 2022). In quantitative phase and absorption formulations, the object transmission may be written as

$$t(x',y')=\exp\{-\mu(x',y')/2\}\exp\{i\phi(x',y')\},$$

with $\phi$ and $\mu$ encoding refraction and attenuation, respectively; after multiplication by a known mask $w_i$, the diffracted field is $b_i=F\,\mathrm{diag}(w_i)\,t$ and the measured data are intensities $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$0 (Moscoso et al., 2022).

The same formalism extends naturally to parameterized CDI. In sparsity-based CDI, an object is represented by a low-dimensional parameter vector $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$1, and one records intensity patterns

$$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$2

over detector pixel $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$3 and measurement index $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$4 (Bouchet et al., 2020). This representation is useful when the objective is not a full image reconstruction but precise metrology of sample parameters.

The forward model is not always adequately described by the weakest-scattering approximation. Standard CDI treatments often invoke linear response, elastic scattering, and first Born or Rytov approximations, but wide-angle single-shot CDI of nanoscale particles requires more accurate propagation physics (Kruse et al., 2020). For wide-angle scattering, the propagation multi-slice Fourier transform method represents the scattered field as a slice-by-slice sequence of material correction in real space and vacuum propagation in $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$5-space; benchmarked against exact Mie theory for a sphere of diameter $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$6, it gives feature error $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$7 and amplitude error $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$8 for all $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$9, $P(\mathbf r)$0, and is recommended when the refractive index departs appreciably from unity or when $P(\mathbf r)$1 (Tuemmler et al., 30 Jul 2025). This suggests that the fidelity of CDI reconstructions is governed as much by the adequacy of the scattering model as by the phase-retrieval algorithm.

2. Phase retrieval, uniqueness, and constraints

The central inverse problem in CDI is the phase problem: detectors record $P(\mathbf r)$2 and discard $P(\mathbf r)$3, so the object cannot be obtained by a direct inverse Fourier transform (Latychevskaia et al., 2011). Classical iterative schemes therefore alternate reciprocal-space modulus constraints with real-space constraints such as compact support, positivity, or known object extent. Error-reduction replaces the Fourier-domain amplitude by the measured magnitude and sets the real-space estimate to zero outside the support, while Hybrid Input–Output modifies the outside-support update with a feedback parameter $P(\mathbf r)$4 (Latychevskaia et al., 2011, Rupp et al., 2016).

Oversampling is the traditional route to uniqueness. In conventional CDI, these algorithms normally require $P(\mathbf r)$5 to converge reliably to a unique solution, and Bragg CDI conventionally demands $P(\mathbf r)$6 in all three reciprocal-space directions (Latychevskaia et al., 2011, Ulvestad et al., 2016). That requirement is not absolute, however. Holographic coherent diffraction imaging combines inline holography with CDI: the hologram supplies an approximate diffraction phase

$P(\mathbf r)$7

which is combined with the exact diffraction magnitude $P(\mathbf r)$8 to initialize reconstruction. In that formulation, the condition of oversampling the diffraction patterns can be relaxed, and the phase problem can be solved in a fast and unambiguous manner (Latychevskaia et al., 2011).

Several CDI variants replace or supplement support constraints by other sources of diversity. Randomly coded masks provide known multiplicative modulations between sample and detector, enabling phase retrieval with prior information about the masks rather than typical object-domain constraints; in the reported experiment, the results indicate that with enough masks, in this case 3 or 4, the diffraction phases converge reliably, implying stability and uniqueness of the retrieved solution (Seaberg et al., 2015). A sparsity-based phaseless algorithm treats the forward model as a combination of coherent and incoherent waves, uses several detectors and masks, and solves a sequence of $P(\mathbf r)$9 problems with a “Noise Collector”; it guarantees exact recovery if the image is sparse for a given basis and has computational cost linear in the number of pixels of the image (Moscoso et al., 2022).

Global optimization has also been introduced into phase retrieval. Memetic Phase Retrieval combines deterministic ER/HIO updates with stochastic genetic operators including rigged-roulette selection and differential crossover in Fourier space (Colombo et al., 2017). In coherent electron diffraction of $\psi(\mathbf r)=P(\mathbf r)\,O(\mathbf r)$0, after about 200 generations with $\psi(\mathbf r)=P(\mathbf r)\,O(\mathbf r)$1, $\psi(\mathbf r)=P(\mathbf r)\,O(\mathbf r)$2, $\psi(\mathbf r)=P(\mathbf r)\,O(\mathbf r)$3, $\psi(\mathbf r)=P(\mathbf r)\,O(\mathbf r)$4, $\psi(\mathbf r)=P(\mathbf r)\,O(\mathbf r)$5, and $\psi(\mathbf r)=P(\mathbf r)\,O(\mathbf r)$6, the method recovered the projected atomic potential and the weak O-column contrast, with measured intensity ratios $\psi(\mathbf r)=P(\mathbf r)\,O(\mathbf r)$7 and $\psi(\mathbf r)=P(\mathbf r)\,O(\mathbf r)$8 (Colombo et al., 2017).

3. Acquisition geometries and experimental realizations

CDI includes multiple acquisition geometries that encode different redundancies. Single-shot CDI records one diffraction pattern from an isolated object; ptychography generalizes CDI to extended samples by raster scanning across a localized coherent beam and using overlap redundancy to reconstruct both object and probe (Tadesse et al., 2018). Bragg CDI records a stack of two-dimensional diffraction patterns around a Bragg peak as the sample is rotated, thereby reconstructing morphology and strain-sensitive phase (Ulvestad et al., 2016). In situ CDI introduces a time-invariant overlapping region as a real-space constraint common to all frames in a time series (Lo et al., 2017).

Laboratory-scale implementations are a major branch of CDI development. Wide-angle single-shot CDI of individual superfluid helium nanodroplets has been demonstrated using high-harmonic-generation-driven extreme-ultraviolet pulses from a table-top femtosecond laser system (Rupp et al., 2016). In that setup, the maximum recorded spatial frequency was $\psi(\mathbf r)=P(\mathbf r)\,O(\mathbf r)$9 for $$I(\mathbf q)=\bigl|\mathcal F\{P(\mathbf r)\,O(\mathbf r)\}\bigr|^2$$0, giving nominal resolution $$I(\mathbf q)=\bigl|\mathcal F\{P(\mathbf r)\,O(\mathbf r)\}\bigr|^2$$1; among $$I(\mathbf q)=\bigl|\mathcal F\{P(\mathbf r)\,O(\mathbf r)\}\bigr|^2$$2 shots, about 2300 bright patterns were recorded, of which $$I(\mathbf q)=\bigl|\mathcal F\{P(\mathbf r)\,O(\mathbf r)\}\bigr|^2$$3 were concentric rings, $$I(\mathbf q)=\bigl|\mathcal F\{P(\mathbf r)\,O(\mathbf r)\}\bigr|^2$$4 elliptical rings, and $$I(\mathbf q)=\bigl|\mathcal F\{P(\mathbf r)\,O(\mathbf r)\}\bigr|^2$$5 streak patterns (Rupp et al., 2016). Table-top XUV ptychography has also resolved features as small as $$I(\mathbf q)=\bigl|\mathcal F\{P(\mathbf r)\,O(\mathbf r)\}\bigr|^2$$6 $$I(\mathbf q)=\bigl|\mathcal F\{P(\mathbf r)\,O(\mathbf r)\}\bigr|^2$$7 and introduced a Rayleigh-type criterion as a direct and unambiguous resolution metric for high-resolution table-top setup (Tadesse et al., 2018).

Electron CDI has become another experimentally mature branch. Deep learning coherent electron diffractive imaging was experimentally demonstrated on twisted hexagonal boron nitride, monolayer graphene, and a Au nanoparticle, with Fourier ring correlation between CNN and ptychographic images giving $$I(\mathbf q)=\bigl|\mathcal F\{P(\mathbf r)\,O(\mathbf r)\}\bigr|^2$$8 $$I(\mathbf q)=\bigl|\mathcal F\{P(\mathbf r)\,O(\mathbf r)\}\bigr|^2$$9 criterion$$t(x',y')=\exp\{-\mu(x',y')/2\}\exp\{i\phi(x',y')\},$$0 and $$t(x',y')=\exp\{-\mu(x',y')/2\}\exp\{i\phi(x',y')\},$$1 $$t(x',y')=\exp\{-\mu(x',y')/2\}\exp\{i\phi(x',y')\},$$2 for twisted hBN, and $$t(x',y')=\exp\{-\mu(x',y')/2\}\exp\{i\phi(x',y')\},$$3 $$t(x',y')=\exp\{-\mu(x',y')/2\}\exp\{i\phi(x',y')\},$$4 and $$t(x',y')=\exp\{-\mu(x',y')/2\}\exp\{i\phi(x',y')\},$$5 $$t(x',y')=\exp\{-\mu(x',y')/2\}\exp\{i\phi(x',y')\},$$6 for a Au nanoparticle (Chang et al., 2022).

Crystalline-defect imaging is a distinct CDI use case. In coherent X-ray diffraction from bulk silicon, a dislocation line centered in the beam produces destructive interference at exact Bragg position, splitting the Bragg peak into two lobes and suppressing $$t(x',y')=\exp\{-\mu(x',y')/2\}\exp\{i\phi(x',y')\},$$7; by raster-scanning a coherent beam, a two-dimensional map of $$t(x',y')=\exp\{-\mu(x',y')/2\}\exp\{i\phi(x',y')\},$$8 directly traces out the dislocation loop with $$t(x',y')=\exp\{-\mu(x',y')/2\}\exp\{i\phi(x',y')\},$$9beam-size $\phi$0 resolution (Jacques et al., 2010). This establishes CDI not only as a morphology-reconstruction method, but also as a phase-sensitive probe of embedded lattice defects.

4. Estimation theory, optimization, and computational acceleration

A notable development in CDI is the explicit use of estimation theory. Under Poisson shot noise, the Fisher information matrix for parametrized CDI is

$\phi$1

and the covariance of any unbiased estimator obeys $\phi$2 (Bouchet et al., 2020). Bouchet et al. used the worst-case standard error

$\phi$3

as a scalar objective and optimized illumination parameters with Adam in TensorFlow v2. In their four-probe scheme, a conventional ptychography-like grid yielded $\phi$4, optimized probes achieved $\phi$5, and a single-shot optimized zone plate achieved $\phi$6 under the same total photon budget (Bouchet et al., 2020). Monte-Carlo ML estimation on $\phi$7 Poisson-noisy patterns confirmed that the RMS errors saturate the CRLB (Bouchet et al., 2020).

Computation has also shifted from purely iterative reconstruction toward learned inversion and real-time pipelines. Deep learning CDI used a U-Net–style encoder–decoder with residual blocks and skip connections, trained only on simulated diffraction data from random phase objects, and achieved phase inference in milliseconds on a GPU plus a few gradient-descent iterations for stitching (Chang et al., 2022). Real-time three-dimensional CDI has been pursued with the carousel phase retrieval algorithm, which reformulates 3D reconstruction as a set of coupled 2D reconstructions linked through the Fourier slice theorem; on a desktop with a 16-core AMD R9-5950X CPU and NVIDIA RTX 3080 Ti GPU, CPRA was reported to be $\phi$8–$\phi$9 faster on CPU and $\mu$0–$\mu$1 faster on GPU versus conventional Shrinkwrap, with a Staphylococcus aureus cell reconstructed at sub-$\mu$2 resolution $\mu$3 (Ai et al., 2024).

Throughput reduction is an independent computational theme. Axial ptychographic CDI replaces the conventional 2D raster scan with a 1D axial scan, reducing the number of required diffraction patterns by approximately an order of magnitude while maintaining high-fidelity reconstruction (You et al., 21 Nov 2025). In the reported dual-color wavefront-retrieval experiment, the measured axial chromatic shift was $\mu$4, compared with a theory value of $\mu$5 for a UV fused silica lens (You et al., 21 Nov 2025). A related super-resolution variant, tilted-incidence multi-rotation-angle fusion ptychography, uses tilted incidence and multiple sample rotations to extend Fourier coverage; in simulation at $\mu$6 and $\mu$7, the four-view union reached $\mu$8, a factor $\mu$9 beyond the Abbe limit (Youyang et al., 6 Apr 2025).

5. Resolution, dose, and benchmarking

CDI is often characterized as offering resolution not limited by lens aberrations, but practical resolution is determined by detector extent, coherence, source bandwidth, and dose (Colombo et al., 2017). In ptychographic tabletop XUV CDI at $w_i$0, the measured half-pitch resolutions were $w_i$1 vertically and $w_i$2 horizontally; the paper attributes the horizontal limit to astigmatism, temporal coherence, and waveguiding in the absorber (Tadesse et al., 2018). In single-shot HHG CDI of helium nanodroplets, wide-angle scattering encoded enough tomographic information that bent crescent streaks were uniquely reproduced by prolate “pill” shapes tilted out of the plane, whereas oblate “wheel” shapes could not reproduce two-sided bending (Rupp et al., 2016).

Benchmarking in CDI increasingly uses transfer-function criteria rather than visual inspection alone. Fourier ring correlation was used to compare CNN and ePIE reconstructions in electron CDI (Chang et al., 2022); Phase-Retrieval Transfer Function and Fourier Shell Correlation were used to evaluate CPRA (Ai et al., 2024); and phase retrieval transfer functions were used in random-mask CDI, where PRTF approached unity as the number of masks increased and near-perfect phase retrieval was observed by $w_i$3 masks (Seaberg et al., 2015).

Dose efficiency has become a central evaluation axis. In situ CDI was introduced as a general real-time method for dynamic processes in solution, and numerical simulations further indicated that it can potentially reduce the radiation dose by more than an order of magnitude relative to conventional CDI (Lo et al., 2017). A later analysis of computational microscopy beyond perfect lenses reported that the combination of in situ CDI and ptychography can reduce the dose by two orders of magnitude over ptychography, and that at low doses in situ CDI can achieve higher resolution than perfect lenses with the point spread function as a delta function (Lu et al., 2023). In the HeLa-cell model used there, LoCDI achieved $w_i$4 at $w_i$5, while standalone ptychography gave $w_i$6 at the same fluence (Lu et al., 2023). This suggests that, under dose-limited conditions, the relevant comparison is not aberration-free optics versus lensless imaging in the abstract, but the signal-to-noise transfer of specific measurement designs.

6. Dynamic, polychromatic, and nonlinear extensions

A major extension of CDI is reconstruction across time. Chrono CDI incorporates the entire measurement time series into phase retrieval by adding a nearest-neighbor temporal coupling term

$w_i$7

with fixed initial and final states measured at full oversampling (Ulvestad et al., 2016). In simulation and experiment, stable reconstructions were demonstrated down to $w_i$8–$w_i$9, corresponding to $b_i=F\,\mathrm{diag}(w_i)\,t$0–$b_i=F\,\mathrm{diag}(w_i)\,t$1 lower oversampling than the conventional $b_i=F\,\mathrm{diag}(w_i)\,t$2, and therefore the same factor improvement in temporal resolution (Ulvestad et al., 2016). In situ CDI reached $b_i=F\,\mathrm{diag}(w_i)\,t$3 spatial resolution at $b_i=F\,\mathrm{diag}(w_i)\,t$4 temporal resolution in simulation and experimentally reconstructed processes including Pb dendrite growth and live U-87 MG glioblastoma cell fusion (Lo et al., 2017).

Spectral generalizations relax the quasi-monochromatic assumption. Multi-wavelength CDI treats the measured diffraction as an incoherent sum of $b_i=F\,\mathrm{diag}(w_i)\,t$5 wavelength-dependent exit-wave intensities,

$b_i=F\,\mathrm{diag}(w_i)\,t$6

and reconstructs the separate wavelength channels by alternating scaled support constraints, spectral-norm constraints, and a joint modulus projection (Malm et al., 2020). The associated polychromatic constraint ratio,

$b_i=F\,\mathrm{diag}(w_i)\,t$7

must exceed unity, with the practical limit $b_i=F\,\mathrm{diag}(w_i)\,t$8 (Malm et al., 2020). In the element-specific example using the 11th and 13th harmonics from a HHG source, the retrieved Al-thickness map matched the nominal $b_i=F\,\mathrm{diag}(w_i)\,t$9 with rms error $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$00, and Si inclusions of $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$01 were visible with $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$02 thickness accuracy (Malm et al., 2020). Broadband coherent diffraction for single-shot attosecond imaging approaches the same problem from the opposite direction: it numerically monochromatizes a broadband diffraction pattern by regularized inversion of a spectrum-dependent matrix, with visible-light and hard-X-ray demonstrations at $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$03 and $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$04, respectively (Huijts et al., 2019).

Strong-field and nonlinear regimes challenge the standard linear-response assumptions of CDI. Quantum coherent diffractive imaging replaces the classical susceptibility model by a Maxwell–density-matrix formalism and predicts that, for the He $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$05 transition at $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$06 with $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$07, intensities above a few $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$08–$$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$09 drive saturation, stimulated emission, and Mollow-type sidebands, leading to substantial departures from linear-response scattering (Kruse et al., 2020). Non-linear X-ray coherent diffractive imaging isolates the nonlinear diffraction component from the much stronger linear scattering by exploiting mutual incoherence between different wavelengths; in the ferroelectric sum-frequency-generation example, the retrieved nonlinear phase map distinguishes $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$10 versus $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$11 domain polarity, and the noise analysis states that because nonlinear conversion efficiencies are low $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$12, averaging $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$13 shots is needed to attain $$I(\mathbf q)=|E(\mathbf q)|^2=|\mathcal F\{\rho(\mathbf r)\}(\mathbf q)|^2,$$14 over the full detector (Sarkar et al., 17 Dec 2025). These developments indicate that CDI is no longer confined to monochromatic, linear, and static scattering, but is evolving into a broader framework for phase-sensitive inference across spectral, temporal, and nonlinear dimensions.