Synthetic Holograms With Known Phase Truth Set a Reproducible Benchmark for Reconstruction Algorithms

Researchers at Tiangong University in Tianjin, China, have built a controlled numerical framework that generates off-axis digital holograms from three analytically specified phase targets and then reconstructs them with object-specific frequency-domain filtering followed by DCT-based global phase unwrapping. Because the object phase, carrier frequency, sampling parameters, and reconstruction ground truth are all explicitly known in advance, the framework delivers something experimental holography often cannot: a reproducible benchmark in which reconstruction error can be measured directly against the true phase rather than estimated through coupled experimental uncertainty. The study, published in PLoS One on October 22, 2026, reports that reconstruction errors remain comparatively small in smooth phase regions and increase near abrupt or high-spatial-frequency phase transitions, a pattern the authors trace to the finite bandwidth of the selected spectral windows. The three cases are presented as controlled benchmark examples for algorithm verification and baseline comparison, not as a complete substitute for experimental holographic datasets or as a full training corpus for deep-learning models.

The framework addresses a persistent bottleneck in modern holographic imaging. Holography, introduced by Gabor as a method for recording and reconstructing optical wavefront information, gained a powerful computational form when off-axis configurations used a spatial carrier frequency to separate the zero-order spectrum, the +1-order original image spectrum, and the −1-order conjugate image spectrum in the frequency domain. Digital holography subsequently became an important computational approach for quantitative phase measurement and microscopic imaging, supporting applications such as label-free microscopy and biological imaging, where phase carries information not directly available from conventional intensity images. When the recovered phase exceeds the principal interval of the arctangent function, phase unwrapping becomes a key processing step, and a range of local and global methods has been developed for digital holography. Experimental holograms are valuable for assessing practical performance, but their pixel-level phase ground truth is often difficult to obtain because acquisition conditions, optical aberrations, noise, and sample variability are coupled in the measured data. Physically defined numerical cases sidestep that coupling, which is why the authors describe them as useful reproducible benchmarks for training and quantitative evaluation in deep-learning-based holographic reconstruction.

The study uses three structurally different phase targets that together span three controlled levels of complexity. The first is a smooth Gaussian phase distribution, constructed analytically with a curvature radius chosen so the model satisfies the paraxial condition; its maximum phase dynamic range is 10 rad, corresponding to a defined height modulation, and the parameter setting is described as suppressing multiple scattering and ensuring the validity of the paraxial approximation. The second is a four-level discontinuous step model with phase values of 0.5, 1.0, 1.5, and 2.0 rad, each level occupying 128 pixels in the horizontal direction. With a 4 mm computational width and 512 samples, each pixel corresponds to approximately 7.8125 μm, so each 128-pixel step spans 1 mm in the numerical field and the phase increment between adjacent levels is 0.5 rad. The authors state that this model is intended to isolate the effect of one-dimensional phase discontinuities and is not claimed to represent the full morphology of MEMS or microfluidic specimens. The third target is a radially structured Siemens Star, simulated rather than experimentally fabricated, whose phase varies with angular position and whose highest spatial-frequency structure occurs near the center, where adjacent spokes are most closely spaced. Spoke-pattern targets are widely used to assess resolution in coherent imaging because they sample a broad range of spatial frequencies and orientations within one target, but the authors emphasize that their numerical Siemens Star provides a spatially distributed high-frequency phase structure for verification rather than an experimental resolution measurement.

The reconstruction pipeline proceeds entirely in the Fourier domain. Each object phase is encoded into a complex phase-only field and interfered numerically with a tilted reference plane wave whose carrier frequency is determined by the reference-light tilt angle, generating an off-axis hologram in which the fringe period is chosen so the three diffraction orders are completely separated in the frequency domain. A two-dimensional Fourier transform is then calculated, and the selected +1-order spectrum is isolated with a predefined window whose shape is matched to the target: the Gaussian case, whose spectrum is comparatively concentrated, uses a rectangular window, while the four-level step and Siemens Star cases, whose spectra contain broader high-frequency structure, use Gaussian soft windows whose standard deviation is set from the detected spectral extent. The authors describe this as object-specific filtering rather than an automatically optimized window-size algorithm, and they stress that the filtering is benchmark-specific, not trained or optimized from the test data. The filtered spectrum is shifted to the Fourier origin before inverse transformation, an operation that removes the carrier-frequency displacement from the selected order and allows the recovered complex field to be processed without an additional numerical carrier-removal step. The wrapped phase is calculated from the product of the reconstructed field and the conjugate reference field.

Unwrapping is handled by a DCT-based global procedure that formulates phase recovery as a Poisson problem. Horizontal and vertical wrapped-phase gradients are calculated and mapped to the principal interval using atan2(sin(x), cos(x)), the corrected gradients are combined into a divergence field, Neumann boundary conditions are imposed directly at the image boundaries, and the divergence field is transformed by the two-dimensional DCT so the Poisson equation can be solved in the transform domain before the inverse DCT yields the continuous phase estimate. The authors note that this transform-domain solution provides a deterministic global reconstruction for the noise-free benchmark cases and that no separate quality-guided or learned phase-unwrapping stage is used. The full procedure therefore consists of Fourier-domain order selection, inverse Fourier transformation, wrapped-phase extraction, gradient correction, divergence-field construction, DCT-domain Poisson solving, and removal of the residual global offset or fitted background, with no additional propagation, denoising, or learned post-processing applied. The implementation deliberately avoids an angular-spectrum propagation stage or weighted least-squares unwrapping, and the authors make no quantitative claim of nanometre-level accuracy, describing the reconstruction as a computational step rather than an experimental propagation measurement.

Simulation parameters were chosen to balance efficiency and spectral resolution while matching commercial CCD sensor characteristics. The total sampling is 512 points across a 4 mm physical hologram width, a pixel size that satisfies the Nyquist sampling theorem, with a maximum measurable spatial frequency that covers all high-frequency components of the tested phase objects and avoids spectral aliasing. Optical parameters follow conventional laboratory off-axis holography configurations, with a DC bias and modulation contrast chosen so the modulation depth of the interference fringes sits in the linear response range of a CCD device. A He-Ne laser serves as the light source, with a coherence length exceeding 20 cm that meets the coherence requirements of holographic interference. The three benchmarks were assigned different carrier and filtering settings: the smooth Gaussian case uses a 40-pixel carrier period with the rectangular +1-order passband, the four-level step case uses a 4-pixel carrier period with a Gaussian soft window, and the Siemens Star uses a 3-pixel carrier period with a Gaussian soft window. The authors state that these parameters are fixed settings of the present simulation and are not proposed as universal classification thresholds.

The quantitative results make the error structure explicit. For the Gaussian model, the measured global root mean square error is 0.000087 rad, a value the authors attribute to the smooth phase profile and the controlled, noise-free simulation; the original and reconstructed phase values cluster closely around the ideal one-to-one relationship, and the diagonal profile reproduces the broad phase maximum with errors that remain small compared with the 10-rad phase range. For the four-level step model, the global RMSE is 0.073500 rad and the correlation coefficient is 0.991417; the scatter plot forms four clusters corresponding to the prescribed phase levels, and the spatial error map identifies the phase-transition boundaries as the dominant source of deviation. For the Siemens Star model, the global RMSE is 0.038036 rad, the reconstruction retains the main radial structure, and the larger local deviations occur in high-spatial-frequency transition regions, with the error distribution following the radial sector boundaries and becoming more pronounced toward the center, where the local spatial frequency is highest. The authors report the Siemens Star correlation coefficient as a quantitative indicator of structural consistency rather than as evidence that the numerical model fully reproduces an experimental specimen.

The dominant error mechanism is associated with rapid phase variation and finite spectral support. Abrupt spatial changes require broader frequency content than smooth regions, so boundary-localized error is consistent with the finite bandwidth of the selected spectral window, an interpretation the authors limit to the present numerical configuration and describe as not a measurement of an experimental optical transfer function. The error maps show that the largest deviations are spatially organized rather than randomly distributed, with the step boundaries and the Siemens Star spoke transitions accounting for a substantial fraction of the local error, while the error histograms are centered near zero with widths that differ between the smooth and high-frequency targets. In the step case, the horizontal and vertical center-line errors are directionally different because the step boundaries are oriented along one image direction and the carrier is also oriented along the x direction; the authors treat this asymmetry as a property of the simulated configuration rather than a universal feature of step-phase holography. The reported numerical results characterize this specific implementation and parameter set and should not be interpreted as direct evidence of experimental real-time performance, optical resolution, or robustness to acquisition noise.

The limitations are stated as deliberately as the strengths. The simulations are noise-free, phase-only, and two-dimensional, and they intentionally omit amplitude-phase coupling, speckle, sensor nonuniformity, optical aberration, multiple scattering, and complex three-dimensional specimen structure; these omissions allow the effect of phase structure and spectral filtering to be isolated, but they mean the results are controlled numerical evidence rather than a claim that the synthetic cases reproduce the full complexity of experimental digital holography. No AFM-based calibration is performed, no experimental measurement of detector modulation transfer function or optical aberration is carried out, and the three cases are benchmark examples rather than a statistically representative dataset, so they cannot support claims about the prevalence of particular phase distributions in industrial or biomedical imaging. Because the simulations are noise-free, the results should not be interpreted as evidence of noise-suppression performance; noise robustness remains a separate experimental question. The principal strength is the explicit analytical ground truth, which permits reconstruction errors to be separated from experimental uncertainty, and the three structurally different targets provide a controlled comparison of smooth, discontinuous, and radially varying phase content.

Looking forward, the authors outline a systematic extension path. Future simulations should introduce amplitude modulation, detector noise, speckle, optical aberration, illumination nonuniformity, and multiple depth planes, followed by validation against experimentally recorded holograms. Before the simulations are used for deep-learning-based phase reconstruction, larger and more diverse datasets should be developed, with independent training, validation, and test sets and explicit reporting of dataset size and variation, since a practical deep-learning dataset would require substantially greater diversity and experimentally realistic variability. The numerical model could also be extended toward experimentally validated three-dimensional objects and, where appropriate, multi-wavelength or color holography. To support reproduction, the minimal anonymized dataset has been deposited in the Zenodo repository and is publicly available, containing the numerical data required to reproduce the phase-reconstruction results for the Gaussian, four-level step, and Siemens star phase models, together with the corresponding source code, parameter metadata, reconstructed phase data, and error data. The work was supported by the National Natural Science Foundation of China under Grant No. 62305246, and the authors have declared no competing interests. Within its stated scope, the framework offers a reproducible baseline for algorithm verification and for subsequent comparison with more diverse experimental or learning-based datasets.

Subject of Research: A reproducible numerical benchmark for off-axis digital hologram generation and phase reconstruction with analytically known phase ground truth

Article Title: Controlled numerical generation and reconstruction of off-axis digital holograms with object-specific frequency-domain filtering

Article References: Lu, J., Shi, X., Sun, J., Ren, S., & Zeng, Y. (2026). Controlled numerical generation and reconstruction of off-axis digital holograms with object-specific frequency-domain filtering. PLOS One, 21(10), e0360375. https://doi.org/10.1371/journal.pone.0360375

Image Credits: AI Generated

DOI: 10.1371/journal.pone.0360375

Keywords: digital holography, off-axis holography, quantitative phase imaging, phase unwrapping, frequency-domain filtering, DCT-based Poisson solver, Siemens Star, Gaussian phase model, four-level step phase, reconstruction benchmark, deep-learning training data, RMSE evaluation