HMFPM: Hybrid-illumination Fourier Ptychography

• HMFPM is a microscopy framework that splits acquisition into eight sequential bright-field images for analytic aberration recovery and 12 multiplexed dark-field images for high-frequency extension.
• It achieves 1.08µm resolution over a 1.77×1.77 mm² field and offers up to 84µm digital refocusing, demonstrating high throughput and stability under severe defocus.
• The hybrid design reduces measurements to 20 (or 28 under extreme defocus), significantly speeding up reconstruction and outperforming conventional sequential methods in robustness.

Hybrid-illumination multiplexed Fourier ptychographic microscopy (HMFPM) is a Fourier ptychography framework that combines a small analytically tractable set of sequential bright-field measurements with a small set of multiplexed dark-field measurements, so that aberrations are recovered from the bright-field data and high-frequency spectrum extension is completed from the multiplexed data. In the reported implementation, HMFPM uses eight NA-matching bright-field images and typically twelve multiplexed dark-field images, achieving $1.08\,\mu\mathrm{m}$ resolution across a $1.77\times1.77\,\mathrm{mm}^2$ field of view with 20 measurements, while remaining robust under diverse aberrations and providing up to $84\,\mu\mathrm{m}$ digital refocusing capability (Zhao et al., 6 Sep 2025).

1. Foundational context

Fourier ptychographic microscopy (FPM) was introduced as a synthetic-aperture, phase-retrieval method in which a thin sample is illuminated sequentially from multiple angles, one low-resolution intensity image is captured for each angle, and the resulting shifted pupil-limited spectral patches are stitched into a high-resolution complex reconstruction. The foundational FPM prototype reported $0.78\,\mu\mathrm{m}$ resolution at $632\,\mathrm{nm}$, a field of view of $\sim 120\,\mathrm{mm}^2$, and a resolution-invariant depth of focus of $0.3\,\mathrm{mm}$, but it required 137 images per monochrome dataset and about 3 minutes of acquisition time (Zheng et al., 2014). A subsequent quantitative-phase study showed that, for thin samples, FPM phase maps could be quantitative and artifact-free, and explicitly tied phase fidelity to angular diversity, spectral overlap, and the recoverability of low-frequency phase content (Ou et al., 2013).

This baseline immediately exposed two persistent bottlenecks. The first was temporal resolution: sequential LED acquisition scales poorly to dynamic specimens, high-throughput screening, and whole-slide imaging. The second was model robustness: high-quality reconstruction depends on accurate pupil modeling, sufficient spectral overlap, and well-conditioned phase retrieval. HMFPM is best understood as a response to both issues at once. Rather than treating acceleration and aberration correction as separate engineering problems, it restructures the acquisition so that the portion of the data most suitable for analytic inversion is retained sequentially, while the portion that mainly contributes high-frequency support is compressed by multiplexing.

2. Hybrid illumination architecture

The defining architectural choice in HMFPM is the split between bright-field and dark-field acquisition. The method first acquires eight sequential NA-matching bright-field images. These are not compressed away, because they are used to reconstruct a bright-field spectrum and to extract the pupil aberration analytically. It then acquires a much smaller set of multiplexed dark-field images, typically 12, each produced by turning on several dark-field LEDs simultaneously. The resulting 20-measurement schedule is therefore neither conventional sequential FPM nor fully multiplexed FPM; it is explicitly hybrid in both illumination design and algorithmic role assignment (Zhao et al., 6 Sep 2025).

The dark-field LEDs are arranged in three concentric rings extending the illumination NA to about three times the objective NA. Within each ring, LEDs are partitioned into groups according to a predefined multiplexing factor, with LEDs in each group chosen at equal angular intervals and activated alternately. Acquisition proceeds radially outward from the inner ring to the outer ring, so that the recovered spectrum expands gradually from lower to higher spatial frequencies. In the simulation configuration reported for 12 dark-field measurements, the ring angles were $16^\circ$, $20^\circ$, and $23^\circ$, with four LEDs at equal angular intervals per ring and simultaneous activation of 4, 6, and 6 LEDs in the inner, middle, and outer rings, respectively. Under severe defocus, the method increases the total number of measurements to 28 by reducing multiplexing density (Zhao et al., 6 Sep 2025).

This acquisition logic belongs to a broader lineage of hybrid source-coded FPM. A direct precursor is source-coded FPM for live-cell imaging, which used four DPC bright-field half-circle images and 17 multiplexed dark-field images, reducing acquisition from 173 sequential images to 21 while preserving $1.77\times1.77\,\mathrm{mm}^2$0 NA reconstruction over a $1.77\times1.77\,\mathrm{mm}^2$1 objective field (Tian et al., 2015). HMFPM differs in that its bright-field stage is not merely an initialization heuristic; it is the locus of analytic aberration extraction, and that analytic stage is what stabilizes the subsequent multiplexed dark-field inversion.

3. Image formation and inverse formulation

HMFPM inherits the standard single-LED FPM forward model. For the $1.77\times1.77\,\mathrm{mm}^2$2-th illumination angle, with object spectrum $1.77\times1.77\,\mathrm{mm}^2$3 and pupil

$1.77\times1.77\,\mathrm{mm}^2$4

the recorded intensity is

$1.77\times1.77\,\mathrm{mm}^2$5

When several LEDs are active simultaneously, HMFPM uses the usual mutually incoherent multiplexing model,

$1.77\times1.77\,\mathrm{mm}^2$6

so each multiplexed frame is an incoherent sum of coherent single-LED images (Zhao et al., 6 Sep 2025).

The analytic bright-field stage is built around eight NA-matching measurements. For each such image, HMFPM defines a shifted modulated field $1.77\times1.77\,\mathrm{mm}^2$7 whose magnitude is $1.77\times1.77\,\mathrm{mm}^2$8, and then introduces the supporting function

$1.77\times1.77\,\mathrm{mm}^2$9

with $84\,\mu\mathrm{m}$0. Kramers–Kronig relations are then used to reconstruct the corresponding modulated spectra. The crucial aberration step follows from overlap phase cancellation. For two NA-matching illuminations $84\,\mu\mathrm{m}$1 and $84\,\mu\mathrm{m}$2, on the overlap region $84\,\mu\mathrm{m}$3,

$84\,\mu\mathrm{m}$4

The sample phase cancels, leaving a relation that depends only on the pupil phase. By aggregating such equations across all overlaps, HMFPM solves for the aberration phase linearly and then stitches an aberration-corrected bright-field spectrum $84\,\mu\mathrm{m}$5 (Zhao et al., 6 Sep 2025).

The dark-field stage is formulated as a constrained multiplexed optimization problem. With $84\,\mu\mathrm{m}$6 fixed on its known support $84\,\mu\mathrm{m}$7 and the recovered pupil $84\,\mu\mathrm{m}$8 held fixed, HMFPM minimizes

$84\,\mu\mathrm{m}$9

subject to

$0.78\,\mu\mathrm{m}$0

This is a narrower inverse problem than blind multiplexed FPM, because the low-frequency spectrum and the pupil are already known.

4. Reconstruction algorithm and aberration robustness

The reconstruction is deliberately split into an analytic stage and an optimization stage. In the bright-field stage, HMFPM reconstructs the modulated spectra from NA-matching images using Kramers–Kronig relations, estimates the pupil from overlap-induced phase differences, removes the aberration phase, and stitches an aberration-free bright-field spectrum. In the multiplexed dark-field stage, it initializes

$0.78\,\mu\mathrm{m}$1

and updates only the unknown high-frequency spectrum (Zhao et al., 6 Sep 2025).

For each multiplexed measurement, the predicted field for LED $0.78\,\mu\mathrm{m}$2 is

$0.78\,\mu\mathrm{m}$3

The spatial-domain projection enforcing the measured multiplexed intensity is

$0.78\,\mu\mathrm{m}$4

After Fourier transformation of $0.78\,\mu\mathrm{m}$5, the unknown spectrum outside the bright-field support is updated in closed form,

$0.78\,\mu\mathrm{m}$6

The paper emphasizes that this update requires no relaxation factor, in contrast to many conventional multiplexed FPM solvers (Zhao et al., 6 Sep 2025).

This analytic–optimization partition is the reason HMFPM is markedly more robust to aberration than earlier multiplexed approaches. General multiplexed FPM can be solved with amplitude-based gradient methods such as Accelerated Wirtinger Flow, which provides an analytical step size and a stationary-point convergence guarantee for arbitrary multiplexing patterns, but it still operates as a fully optimization-driven inversion over multiplexed measurements (Bostan et al., 2018). HMFPM instead removes the pupil-estimation burden from the multiplexed stage, fixes the low-frequency spectrum, and optimizes only the remaining dark-field extension. The practical consequence is improved stability under defocus, arbitrary pupil phase, field-dependent aberration, and scanning-induced focus shifts.

5. Performance and demonstrated applications

In simulation, HMFPM maintained $0.78\,\mu\mathrm{m}$7 resolution under no aberration, moderate arbitrary aberration, and large arbitrary aberration with 20 measurements, while APIC achieved comparable quality with 96 measurements and MFPM and SFPM degraded strongly under moderate aberration and failed under large aberration. Under severe $0.78\,\mu\mathrm{m}$8 defocus-dominant aberration, HMFPM preserved target resolution by reducing multiplexing density and using 28 measurements, whereas MFPM and SFPM failed even with increased measurement counts. On natural-image simulations, HMFPM achieved amplitude and phase SSIM values above 0.89 under difficult aberration cases, again at 20 measurements and with quality comparable to APIC (Zhao et al., 6 Sep 2025).

Experimentally, the method resolved a Siemens star to $0.78\,\mu\mathrm{m}$9, close to the reported theoretical $632\,\mathrm{nm}$0 resolution corresponding to $632\,\mathrm{nm}$1 NA. It retained this performance while digitally refocusing through defocus up to $632\,\mathrm{nm}$2, using 20 measurements in focus, 24 at intermediate defocus, and 28 at the largest defocus. On a $632\,\mathrm{nm}$3-pixel patch, HMFPM reconstruction required about $632\,\mathrm{nm}$4, compared with approximately $632\,\mathrm{nm}$5 for APIC on the reported CPU–GPU platform, a 72-fold speed improvement; for $632\,\mathrm{nm}$6-pixel patches, HMFPM required $632\,\mathrm{nm}$7, while APIC exceeded the reported GPU memory limit (Zhao et al., 6 Sep 2025).

The biomedical demonstrations target pathology rather than only resolution targets. HMFPM reconstructed phase from an unstained human non-small-cell lung cancer pathology slide using 20 measurements. Field-dependent aberrations were handled patchwise, with 256-pixel patches providing local pupil estimates across the $632\,\mathrm{nm}$8 field. In whole-slide imaging, the method also corrected scanning-induced aberrations arising from slide tilt and sample unevenness, so that separated scan positions with different focus states could be digitally refocused without manual adjustment. Within the reported study, this combination of few measurements, local aberration correction, and large-field pathology reconstruction is the main practical justification for the method’s designation as high-throughput and aberration-free (Zhao et al., 6 Sep 2025).

6. Relation to adjacent multiplexed and hybrid FPM variants

HMFPM sits within a longer development of multiplexed and hybrid computational illumination. State-multiplexed Fourier ptychography established the general mixed-state model in which a single camera frame represents an incoherent sum of several coherent illumination states, and showed both angular and color multiplexing with explicit computational demultiplexing (Dong et al., 2014). Source-coded FPM for live-cell imaging then introduced an explicitly hybrid bright-field/dark-field schedule—four DPC bright-field images plus 17 multiplexed dark-field images—to reduce 173 sequential measurements to 21 while preserving high synthetic NA and quantitative phase (Tian et al., 2015). HMFPM is closest to this second lineage, but replaces DPC initialization and blind multiplexed inversion with NA-matching analytic reconstruction and analytic aberration extraction.

Another parallel branch optimized illumination patterns rather than hard-coding them. Single-shot learned multiplexed FPM jointly optimized one grayscale LED pattern over 69 central bright-field LEDs and a CNN to reconstruct high-resolution complex images from a single captured frame, but it remained sample-specific and did not implement hybrid bright-field/dark-field multiplexing in the usual sense (Cheng et al., 2018). Data-driven design for FPM learned task-specific source patterns and showed that bright-field and dark-field illumination should often be designed separately, with asymmetric bright-field patterns favored for phase and symmetric bright-field patterns for amplitude (Kellman et al., 2019). Adaptive coded illumination based on a physical neural network went further by learning sample-specific binary equal-brightness multiplexed patterns and directly capturing them in hardware, often producing patterns that separated bright-field and dark-field information rather than mixing them indiscriminately (Sun et al., 2023). Compared with these learned methods, HMFPM is not sample-specific and does not rely on a training prior; its distinguishing feature is that hybrid illumination is chosen to preserve an analytic route to pupil recovery.

High-brightness sequential platforms provide a different point of comparison. Laser-illuminated FPM using Galvo-steered plane-wave angles achieved 96 raw images in $632\,\mathrm{nm}$9 and explicitly identified the camera, rather than illumination power, as the acquisition bottleneck. That result strengthens the rationale for multiplexing and hybrid coding once bright illumination hardware is available (Chung et al., 2016). HMFPM belongs to that same throughput-oriented regime, but uses hybrid illumination to reduce frame count while retaining robust aberration correction.

7. Practical constraints, failure modes, and complementary extensions

Several system-level considerations remain important for HMFPM even though they are not unique to it. First, vignetting can invalidate the standard space-invariant shifted-pupil model near the bright-field/dark-field transition zone. In low-NA, large-FOV FPM, such transition LEDs can produce “semi-bright and semi-dark” images that are poorly described by the usual model; field segmentation and omission of imperfect transition-zone images were proposed as countermeasures (Pan et al., 2017). For HMFPM, this implies that hybrid pattern design should avoid or explicitly model transition-zone LEDs rather than treat them as ordinary bright-field or dark-field states.

Second, multiplexed reconstruction remains sensitive to illumination geometry. A physics-based defocusing strategy for LED-array calibration can recover global lateral offset, rotation angle, and LED-board distance from sequential defocused bright-field images, thereby providing corrected illumination wavevectors before FPM reconstruction (Zheng et al., 2021). This is especially pertinent when multiplexing reduces measurement redundancy: geometric miscalibration is then harder to absorb by iterative correction.

Third, dark-field data remain vulnerable to stray light, underexposure, overexposure, and nonuniform background. A systematic preprocessing pipeline for FPM proposed sparse handling of invalid pixels, weighted dark-frame subtraction, thresholding, and masked amplitude updates to suppress stray-light-corrupted regions, with the explicit conclusion that improved inter-measurement consistency can outweigh modest signal loss (Zhang et al., 2017). Those considerations transfer directly to HMFPM, whose multiplexed dark-field measurements still depend on accurate low-SNR intensity modeling.

Finally, HMFPM addresses acquisition efficiency and aberration correction, but not the separate problem of missing low-spatial-frequency phase in practical FPM. A later hybrid method combining FPM with one additional on-axis defocused image and transport of intensity equation (TIE) reconstruction restored low-frequency phase by using TIE for the broad phase background and FPM for high-frequency detail (Rogalski et al., 30 May 2025). A plausible implication is that a future system could combine HMFPM’s hybrid bright-field/dark-field multiplexing with a TIE side channel, thereby addressing both acquisition efficiency and low-frequency phase observability; that combined architecture is suggested by the literature but not demonstrated in the cited works.