Equivalent Polarization Images in Computational Imaging

• Equivalent polarization images are derived from various imaging modalities to emulate conventional polarimeter outputs using operational definitions and substitution techniques.
• Advanced computational methods, including optical multiplexing, inverse problem-solving, and deep learning, are used to reconstruct polarization state information from compressed or incomplete measurements.
• Applications in biomedical diagnostics, remote sensing, and HDR imaging are guided by performance metrics like PSNR, SSIM, and LPIPS, highlighting practical and technical efficacy.

Equivalent polarization images are image-domain quantities that are not necessarily acquired as direct polarization-channel measurements, yet are intended to serve the same role as conventional polarization-resolved outputs. In recent literature, the term covers several distinct but related constructions: four linear analyzer images $I_0, I_{45}, I_{90}, I_{135}$ reconstructed from a single lensless snapshot; retardance and orientation maps transformed from a single-state hologram into outputs equivalent to single-shot computational polarized light microscopy (SCPLM); Stokes-component images inferred from RGB alone; and polarization-camera channels interpreted as effective multi-exposure observations for HDR reconstruction (Kraicer et al., 17 Mar 2026, Liu et al., 2020, Lin et al., 19 May 2025, Ting et al., 2021).

1. Definitions and regimes of equivalence

In the cited imaging literature, equivalence is defined operationally: an output is “equivalent” when it can be used in place of directly measured polarization data for the intended imaging task. In the lensless polarization camera of Monakhova and colleagues, recovering equivalent polarization images means computationally inverting one encoded measurement into the same kind of polarization-resolved images that a conventional polarimeter would obtain by taking multiple measurements under different analyzer angles, specifically $I_0, I_{45}, I_{90}, I_{135}$ in the linear case (Kraicer et al., 17 Mar 2026). In deep learning-based holographic polarization microscopy, the equivalent outputs are retardance and orientation maps that are visually and quantitatively close to SCPLM, even though only a single-state lensfree hologram is measured (Liu et al., 2020).

A second regime of equivalence appears in sensor-free prediction. The RGB-to-polarization benchmark defines a task in which a standard RGB image is used to infer Stokes-component images that stand in for sensor-acquired polarization data, with the RGB input treated as the total intensity channel and the network predicting the remaining polarization channels (Lin et al., 19 May 2025). A third regime appears in coherent X-ray imaging, where a single linear-polarization ptychographic scan is reconstructed into two orthogonal object modes whose combinations reproduce the magnetic and electronic contrasts that would normally require polarization-resolved measurements with opposite circular helicities (Martínez et al., 2024).

A fourth regime is task-specific rather than channel-specific. In snapshot HDR from a polarization camera, the four micro-polarizer measurements are treated as if they were captured under different effective exposure times. The paper is explicit that these are only effective exposures, not literal bracketed shutter settings, but the equivalence is sufficient to support multiexposure-style HDR fusion (Ting et al., 2021).

2. Polarization representations and target image spaces

The most direct form of equivalent polarization imagery is the set of linear analyzer images

$I_0,\ I_{45},\ I_{90},\ I_{135},$

which represent scene intensity viewed through analyzers at $0^\circ,45^\circ,90^\circ,$ and $135^\circ$. In the linear-only setting, these four sub-images are sufficient for reconstruction, whereas full Stokes recovery would require six sub-images including circular polarization terms: $$\begin{cases} S_0 = I_{0} + I_{90},\quad S_1 = I_{0} - I_{90},\ S_2 = I_{45} - I_{135},\quad S_3 = I_{RCP} - I_{LCP}. \end{cases}$$ This makes the linear sub-images both an endpoint in their own right and a basis for derived polarimetric quantities (Kraicer et al., 17 Mar 2026).

Other works define the equivalent output in a different coordinate system. The RGB-to-polarization benchmark uses the Stokes vector

$$(\mathrm{S}_0,\mathrm{S}_1,\mathrm{S}_2,\mathrm{S}_3),$$

with

$$\mathrm{S}_1 = \mathrm{S}_0 \cos(2\psi)\cos(2\chi),\quad \mathrm{S}_2 = \mathrm{S}_0 \sin(2\psi)\cos(2\chi),\quad \mathrm{S}_3 = \mathrm{S}_0 \sin(2\chi),$$

and defines the normalized polarization direction on the Poincaré sphere as

$$\mathbf{S} = (\mathrm{S}_1,\mathrm{S}_2,\mathrm{S}_3)/\mathrm{S}_0.$$

Its learning target is a $9$-channel image tensor in which each of $I_0, I_{45}, I_{90}, I_{135}$0 is represented as a $I_0, I_{45}, I_{90}, I_{135}$1-channel image, so the final target is

$I_0, I_{45}, I_{90}, I_{135}$2

Here equivalence means reconstructing the polarization representation, especially the Stokes-component images, from non-polarimetric input (Lin et al., 19 May 2025).

In holographic polarization microscopy, the equivalent outputs are not analyzer-angle images or Stokes images but birefringence descriptors: retardance and orientation. These are pseudo-colored using the same convention as compensated polarized light microscopy, producing images visually equivalent to polarization microscopy outputs (Liu et al., 2020). Guided lensless polarization imaging uses a tensorial representation

$I_0, I_{45}, I_{90}, I_{135}$3

where each $I_0, I_{45}, I_{90}, I_{135}$4 is the intensity that would be seen after an ideal polarizer at angle $I_0, I_{45}, I_{90}, I_{135}$5, for either grayscale or RGB reconstruction and for either three or four polarization angles (Kraicer et al., 28 Mar 2026).

3. Inverse formation from single, compressed, or incomplete measurements

A central technical route to equivalent polarization images is optical multiplexing followed by inverse reconstruction. In the lensless polarization camera, a diffuser acts as the main lensless encoder and a striped polarization mask at the sensor plane provides modality multiplexing. For each color channel $I_0, I_{45}, I_{90}, I_{135}$6, the measurement model is

$I_0, I_{45}, I_{90}, I_{135}$7

with $I_0, I_{45}, I_{90}, I_{135}$8, where $I_0, I_{45}, I_{90}, I_{135}$9 are the unknown polarization-resolved scene images, $I_0,\ I_{45},\ I_{90},\ I_{135},$0 is the diffuser PSF, and $I_0,\ I_{45},\ I_{90},\ I_{135},$1 selects the sensor pixels belonging to each polarization stripe. The compact form is

$I_0,\ I_{45},\ I_{90},\ I_{135},$2

Reconstruction is posed as a regularized least-squares problem with a weighted anisotropic TV prior and solved with ADMM; the regularizer is applied across spatial, color, and polarization dimensions, with weaker regularization along the polarization axis to preserve inter-channel differences (Kraicer et al., 17 Mar 2026).

The same forward model underlies RGB-guided lensless polarization imaging, but the reconstruction is explicitly two-stage. Stage I performs physics-based inversion using FISTA as the main solver and ADMM as an alternative baseline, with a weighted 3DTV prior and a non-negativity constraint. Stage II uses a refinement network adapted from SwinFuSR, with separate branches for the initial polarization reconstruction and the aligned RGB guidance image, fused through Attention-guided Cross-domain Fusion blocks that alternate self-attention and cross-attention. In the four-angle RGB setting, the final output has $I_0,\ I_{45},\ I_{90},\ I_{135},$3 channels, corresponding to $I_0,\ I_{45},\ I_{90},\ I_{135},$4 polarization states across $I_0,\ I_{45},\ I_{90},\ I_{135},$5 colors. On PIP, the color model reaches about $I_0,\ I_{45},\ I_{90},\ I_{135},$6 PSNR, $I_0,\ I_{45},\ I_{90},\ I_{135},$7 SSIM, and $I_0,\ I_{45},\ I_{90},\ I_{135},$8 LPIPS with FISTA input, while plain FISTA is far worse (Kraicer et al., 28 Mar 2026).

Equivalent polarization recovery can also arise from coherent redundancy rather than spatial masking. In multimodal ptychography, a linearly polarized incident beam is decomposed into two orthogonal circular eigenmodes of a magnetic sample, and the measured diffraction intensity becomes

$I_0,\ I_{45},\ I_{90},\ I_{135},$9

Because the two modes are orthogonal, they do not interfere. Ptychographic overlap redundancy allows reconstruction of both modes from a single dataset, thereby recovering magnetic and electronic image components that are equivalent to what conventional XMCD ptychography obtains from opposite circular polarizations (Martínez et al., 2024).

A related but older formulation appears in array imaging of small polarizable scatterers. There the measured data are the coherency matrix, or equivalently the Stokes parameters, rather than the full complex vector field. A preprocessing map applied to the coherency matrix partially recovers the projected ideal response, and electromagnetic Kirchhoff migration of the preprocessed data yields images that are asymptotically identical, at high frequency, to those obtained from the ideal projected data. In the Fraunhofer regime, the cross-range resolution is $0^\circ,45^\circ,90^\circ,$0 and the range resolution is $0^\circ,45^\circ,90^\circ,$1 (Bardsley et al., 2018).

4. Learned transformations and estimation from non-polarimetric inputs

A second major route to equivalent polarization images is learned transformation from measurements that do not explicitly contain the full set of required polarization channels. Deep learning-based holographic polarization microscopy begins with a lensfree holographic microscope modified by the addition of one polarizer/analyzer pair. Raw holograms are recorded at $0^\circ,45^\circ,90^\circ,$2 sample-to-sensor heights for multi-height phase recovery and $0^\circ,45^\circ,90^\circ,$3 lateral positions for pixel super-resolution. After multi-height phase recovery, pixel super-resolution, back propagation, and autofocusing, the reconstructed hologram is normalized so that background amplitude is $0^\circ,45^\circ,90^\circ,$4 and background phase is $0^\circ,45^\circ,90^\circ,$5, and its amplitude and phase are fed into a GAN-based network trained against SCPLM targets. The network outputs retardance and orientation channels. On $0^\circ,45^\circ,90^\circ,$6 birefringent objects, the reported overall object-wise averaged absolute error is $0^\circ,45^\circ,90^\circ,$7 rad for retardance and $0^\circ,45^\circ,90^\circ,$8 rad for orientation, with a field of view greater than $0^\circ,45^\circ,90^\circ,$9 (Liu et al., 2020).

The RGB-to-polarization benchmark formalizes an even stronger inference problem: direct estimation of polarization from a single RGB image. Using the Jeon et al. spectral and polarization real-world dataset, the benchmark trains on the first $135^\circ$0 aligned RGB–Stokes pairs and tests on the last $135^\circ$1, after resizing all images to $135^\circ$2 and normalizing them to $135^\circ$3. The models are evaluated with PSNR, SSIM, and LPIPS on $135^\circ$4, individually and averaged overall. The main result is that restoration-style direct reconstruction is stronger than diffusion-based generative synthesis for fidelity. MAE is the strongest method overall, with average PSNR $135^\circ$5, average SSIM $135^\circ$6, and average LPIPS $135^\circ$7. Uformer achieves the best average LPIPS at $135^\circ$8. Among generative methods, Img2ImgTurbo is the strongest, while WDiff is the weakest with average PSNR $135^\circ$9, average SSIM $$\begin{cases} S_0 = I_{0} + I_{90},\quad S_1 = I_{0} - I_{90},\ S_2 = I_{45} - I_{135},\quad S_3 = I_{RCP} - I_{LCP}. \end{cases}$$0, and average LPIPS $$\begin{cases} S_0 = I_{0} + I_{90},\quad S_1 = I_{0} - I_{90},\ S_2 = I_{45} - I_{135},\quad S_3 = I_{RCP} - I_{LCP}. \end{cases}$$1 (Lin et al., 19 May 2025).

These results establish that equivalent polarization images can be generated without explicit multi-state polarimetric capture, but they also delimit the present ceiling. The benchmark states that even the best methods still do not fully recover the true polarization cues, and the holographic microscopy paper shows that performance depends on combining morphology with holographic amplitude and phase rather than on morphology alone (Lin et al., 19 May 2025, Liu et al., 2020).

5. Task-specific reinterpretations and application domains

Equivalent polarization images are not always used to replicate a conventional polarimeter channel-by-channel; in some cases they provide an alternate representation with the same downstream utility. The HDR reconstruction work based on a polarization camera treats the four on-chip micro-polarizer images as a set captured under different effective exposure times. Starting from

$$\begin{cases} S_0 = I_{0} + I_{90},\quad S_1 = I_{0} - I_{90},\ S_2 = I_{45} - I_{135},\quad S_3 = I_{RCP} - I_{LCP}. \end{cases}$$2

the paper rewrites the effect as

$$\begin{cases} S_0 = I_{0} + I_{90},\quad S_1 = I_{0} - I_{90},\ S_2 = I_{45} - I_{135},\quad S_3 = I_{RCP} - I_{LCP}. \end{cases}$$3

with special cases for the four fixed angles $$\begin{cases} S_0 = I_{0} + I_{90},\quad S_1 = I_{0} - I_{90},\ S_2 = I_{45} - I_{135},\quad S_3 = I_{RCP} - I_{LCP}. \end{cases}$$4. The authors are explicit that this is not identical to conventional exposure bracketing: the variation is pixel-specific, depends on the local polarization state, and weakens when the light is nearly unpolarized. Nonetheless, the equivalence is strong enough to justify a Debevec-style fusion into an HDR-like input and an $$\begin{cases} S_0 = I_{0} + I_{90},\quad S_1 = I_{0} - I_{90},\ S_2 = I_{45} - I_{135},\quad S_3 = I_{RCP} - I_{LCP}. \end{cases}$$5-layer autoencoder for snapshot HDR recovery (Ting et al., 2021).

Metasurface image processing introduces another form of equivalence: the same spatial image can yield different or invariant edge maps depending on polarization-resolved transfer functions. The metasurface acts on the s- and p-polarized components through a $$\begin{cases} S_0 = I_{0} + I_{90},\quad S_1 = I_{0} - I_{90},\ S_2 = I_{45} - I_{135},\quad S_3 = I_{RCP} - I_{LCP}. \end{cases}$$6 transfer matrix

$$\begin{cases} S_0 = I_{0} + I_{90},\quad S_1 = I_{0} - I_{90},\ S_2 = I_{45} - I_{135},\quad S_3 = I_{RCP} - I_{LCP}. \end{cases}$$7

For a polarization-independent Laplacian differentiator, the desired behavior is

$$\begin{cases} S_0 = I_{0} + I_{90},\quad S_1 = I_{0} - I_{90},\ S_2 = I_{45} - I_{135},\quad S_3 = I_{RCP} - I_{LCP}. \end{cases}$$8

whereas a strongly asymmetric design suppresses one polarization channel. The first regime supports unpolarized or arbitrarily polarized imaging with similar output; the second enables direction-dependent edge detection whose output depends on the input polarization state (Cotrufo et al., 2023).

The application space attached to equivalent polarization images is correspondingly broad. The lensless polarization camera explicitly targets polarization-based strain analysis and polarization-aware machine vision (Kraicer et al., 17 Mar 2026). Guided lensless polarization imaging is positioned for biomedical diagnostics, autonomous driving, and remote sensing (Kraicer et al., 28 Mar 2026). Holographic polarization microscopy is demonstrated on clinically relevant MSU and TCA crystals, with MSU specifically identified as the diagnostic signature of gout (Liu et al., 2020).

6. Conditions of validity, ambiguities, and limitations

The equivalence claim is consistently contingent on model assumptions. In the basic lensless polarization camera, successful recovery assumes a largely shift-invariant diffuser response, an accurately known mask response, well-characterized polarization-mask transmission, and a static scene during the snapshot. The authors emphasize that reconstruction quality is sensitive to model mismatch, especially the finite gap between mask and sensor, fabrication imperfections, stripe boundary effects, and cross-talk between adjacent polarization stripes. In matched forward-model simulations, recovery is high-quality, but real prototype performance degrades relative to simulation, and controlled mismatch experiments show that even moderate deviations from the assumed mask reduce PSNR and SSIM noticeably (Kraicer et al., 17 Mar 2026).

The RGB-guided extension mitigates some of these deficiencies but introduces new dependencies. Its reported robustness to unseen PSFs and residual lensless/RGB misalignment is improved by translation augmentation during training, and fine-tuning with only $$\begin{cases} S_0 = I_{0} + I_{90},\quad S_1 = I_{0} - I_{90},\ S_2 = I_{45} - I_{135},\quad S_3 = I_{RCP} - I_{LCP}. \end{cases}$$9 target-domain pairs further improves results on UPLight and ZJU-RGB-P. At the same time, the method requires a registered auxiliary RGB camera, so the gain in fidelity comes with an additional sensing modality and a calibration problem of its own (Kraicer et al., 28 Mar 2026).

Learned sensor-free estimation has its own limitations. The RGB-to-polarization benchmark identifies limited dataset diversity, lack of robustness evaluation under adverse conditions, insufficient fine-grained polarization fidelity, and the fact that existing image pre-trained models are not optimized for polarization structure. It also raises downstream risk, suggesting confidence-map prediction alongside polarization estimates. In holographic microscopy, the network generalizes to TCA crystals and corn starch, but performance improves after transfer learning on TCA, indicating that cross-material generalization is present but incomplete (Lin et al., 19 May 2025, Liu et al., 2020).

Single-measurement coherent methods also involve structural ambiguities. In multimodal ptychography, the orthogonal modes must be mutually incoherent in the measured intensity; the reconstruction has a unitary ambiguity, so the algorithm does not inherently know which mode is “charge” and which is “magnetic”; good initialization is needed; and more general anisotropic media may require more than two modes. The authors therefore do not present it as a universal polarization analyzer for arbitrary materials (Martínez et al., 2024). In coherency-based array imaging, the preprocessing recovers the transverse $$(\mathrm{S}_0,\mathrm{S}_1,\mathrm{S}_2,\mathrm{S}_3),$$0 projected response rather than the full $$(\mathrm{S}_0,\mathrm{S}_1,\mathrm{S}_2,\mathrm{S}_3),$$1 field, although the resulting Kirchhoff images have the same asymptotic location resolution as those from the ideal projected data (Bardsley et al., 2018).

Taken together, these works define equivalent polarization images not as a single representation but as a family of measurement surrogates: analyzer-angle intensities, Stokes components, birefringence descriptors, magnetic and electronic mode images, or effective exposure channels. Their common feature is substitution under a specified forward model or learned mapping. The substitution can be exact only under matched assumptions; outside those assumptions, the quality of equivalence is limited by calibration error, inverse-problem ill-posedness, training-data coverage, or physical model ambiguity.