PEAR-FINCH: Post-Engineered Axial Resolution in FINCH

• The paper introduces a novel method that engineers the axial resolution post-recording by synthesizing a library of FINCH holograms.
• It employs a two-step reconstruction process—numerical back-propagation and deconvolution with synthetic point spread holograms—to achieve an extended depth-of-focus and improved SNR.
• Applications in biomedical imaging, 3D holography, and fluorescence microscopy demonstrate the method's potential, despite challenges in acquisition speed and data storage.

Post-Engineering of Axial Resolution in FINCH (PEAR-FINCH) is a digital holographic imaging methodology that enables post-recording control over the depth-of-focus (DOF) in Fresnel Incoherent Correlation Holography (FINCH). Unlike conventional FINCH, where the depth response is constrained by the physical and optical configuration set at acquisition, PEAR-FINCH “engineers” new axial characteristics after data collection by synthesizing information from a library of distinct FINCH holograms. This approach offers extended DOF and improved signal-to-noise ratio (SNR), with promising applications in biomedical imaging, holography, and fluorescence microscopy (Gopinath et al., 30 Sep 2025).

1. Conceptual Overview

FINCH is a self-interference-based incoherent digital holography technique in which light from an object point is split, modulated via two lenses with different focal lengths, and recombined to form a hologram. At least three phase-shifted holograms are synthesized into a complex hologram free of twin image and bias noise. FINCH inherently offers a longer DOF and higher lateral resolution than direct incoherent imaging but its axial resolving capability is fixed by the hardware at the time of acquisition.

PEAR-FINCH introduces a post-recording degree of freedom to axial response engineering by first recording a library of FINCH holograms, each acquired with a distinct modulating diffractive lens focal length on the spatial light modulator (SLM). Each entry in this library has a unique axial focus and DOF. Selected holograms are then numerically combined post-acquisition—producing a synthetic object hologram (SOH) with a tailored axial profile. The process achieves new DOF characteristics not attainable with standard FINCH (Gopinath et al., 30 Sep 2025).

2. Library Construction and Synthetic Hologram Generation

The PEAR-FINCH workflow begins by varying the focal length of a diffractive lens displayed on the SLM during multiple acquisitions. For each configuration (indexed by $m$), a set of at least three phase-shifted holograms is recorded (e.g., phase shifts of $0$, $2\pi/3$, $4\pi/3$) and synthesized into a complex hologram $H_{OH_{z(m)}}$.

A “synthetic object hologram” (SOH) is generated as the sum of these complex holograms:

$HSOH = \sum_{m} H_{OH_{z(m)}}$

A library of corresponding point spread holograms (PSH) for each focus configuration is similarly recorded. The “synthetic point spread hologram” (SPSH) is:

$HSPSH = \sum_{m} H_{PH_{z(m)}}$

This construction enables the axial response of the system to be engineered post-recording via the choice of hologram entries included in the sum (Gopinath et al., 30 Sep 2025).

3. Two-Step Axial Reconstruction Process

PEAR-FINCH employs a two-step image reconstruction procedure:

A. Numerical Back-Propagation.

The SOH is numerically back-propagated to the desired reconstruction plane using the Fresnel operator (multiplication by a quadratic phase function $Q(1/z_r)$):

$I_{R1} = |HSOH * Q(1/z_r)|$

This yields an image where the depth-of-focus is extended due to the superposition of multiple depth responses—sharp reconstruction is achieved over a larger axial range but includes increased out-of-focus background.

B. Deconvolution with Synthetic PSH.

To suppress out-of-focus contributions and enhance SNR, the numerically back-propagated SOH is deconvolved with the SPSH. The paper references using the Lucy–Richardson–Rosen algorithm (LRRA), nonlinear reconstruction, or incoherent nonlinear deconvolution methods for this step. The final image can be written formally as:

$I_{R2} = |(HSOH * Q(1/z_r)) + (HSPSH * Q(1/z_r))|$

where “$*$” denotes convolution. Cross-correlation with the synthetic PSH further isolates in-focus features (Gopinath et al., 30 Sep 2025).

4. Experimental Demonstration and Performance

Experimental validation in (Gopinath et al., 30 Sep 2025) uses multi-plane objects (e.g., test target digits) positioned at various distances from the reference plane. Three main comparisons were performed:

• Direct Imaging (DI): Only objects at the focal plane are sharply resolved.
• Standard FINCH: Multiple axial planes can be reconstructed, but with a limited DOF when objects are displaced far from the plane of focus.
• PEAR-FINCH: By synthesizing SOH using three focal lengths (e.g., 9, 10, 11 cm), both near and far objects (e.g., at ±3 mm and ±12 mm from reference) are reconstructed in focus.

Quantitative measures, including the Structural Similarity Index Measure (SSIM) and image entropy, demonstrate better structural fidelity and SNR for PEAR-FINCH versus DI and conventional FINCH (Gopinath et al., 30 Sep 2025).

5. Distinction from Existing DOF Engineering Methods

PEAR-FINCH differs fundamentally from classical axial engineering in FINCH and direct imaging. In traditional FINCH and related PSF/DOF engineering approaches, the depth response is set by the physical PSF at acquisition, which cannot be tuned post-recording. In contrast, PEAR-FINCH decouples capture and DOF selection by leveraging a stored hologram library, permitting synthesis of arbitrarily tailored axial profiles after the fact.

A related, though distinct, methodology is the coded-aperture FINCH framework, where computational methods such as the Transport of Amplitude into Phase–Gerchberg Saxton Algorithm (TAP-GSA) and Lucy–Richardson–Rosen Algorithm (LRRA) are deployed for single-shot super-resolution and axial enhancement (Arockiaraj et al., 2023). However, such approaches engineer the axial response through inverse problem-solving and deconvolution, potentially in real time, whereas PEAR-FINCH emphasizes flexibility in post-acquisition synthesis of new DOF profiles.

6. Applications in Biomedical and 3D Imaging

The extended DOF and improved SNR achieved with PEAR-FINCH have significant implications for:

• Biomedical Imaging and Microscopy: Imaging biological tissues or organisms with features distributed across multiple axial planes, where simultaneous sharp imaging at all depths is advantageous.
• Holographic 3D Imaging: Flexible post-facto refocusing, “all-in-focus” reconstruction, and extended range 3D imaging without repeated acquisition.
• Fluorescence Microscopy: Enhanced 3D structural analysis by engineering the axial response independently from the lateral resolution.

PEAR-FINCH’s post-acquisition tunability is particularly beneficial where sample stability, acquisition time, or re-imaging is constrained (Gopinath et al., 30 Sep 2025).

7. Current Limitations and Prospects

Key limitations noted for PEAR-FINCH include the requirement for multiple (≥3 per configuration) phase-shifted camera shots and the practical storage and processing burdens associated with maintaining a hologram library. The authors indicate future research directions will prioritize:

• Reducing the number of required shots, ideally approaching single-shot DOF tuning.
• Extension to more complex or dynamic scenes, enhancing the versatility for in vivo or real-time applications.
• Continued refinement of post-processing and deconvolution algorithms to streamline computational loads and maximize image fidelity.

These avenues suggest ongoing optimization of PEAR-FINCH toward fast, flexible, and practical 3D imaging with engineered axial responses (Gopinath et al., 30 Sep 2025).