Compressive Cryo-EM Sensing

• Compressive cryo-EM sensing is an approach that leverages compressive sensing theory and targeted acquisition to significantly reduce measurements while maintaining high structural fidelity.
• It utilizes both pixel-space and Fourier-space measurement models with innovative masking strategies to enhance imaging speed and lower electron doses.
• Advanced reconstruction techniques, including sparse priors, dictionary learning, and generative diffusion models, enable robust image recovery and improved data throughput.

Compressive cryo-EM sensing encompasses algorithmic and system innovations that substantially reduce the number of measurements required in cryogenic electron microscopy (cryo-EM) while preserving structural fidelity. By leveraging compressive sensing (CS) theory, targeted acquisition strategies, and advanced priors—including sparse and generative models—these approaches enable lower electron doses, accelerated imaging speed, and increased data throughput, addressing limitations posed by sample damage and detector bandwidth. Below is an in-depth overview and technical synthesis of methodologies and results in compressive cryo-EM sensing, integrating recent advances in both 3D cryo FIB-SEM and 2D/3D single-particle frameworks (Nicholls et al., 2022, Shabeeb et al., 17 Nov 2025).

1. Measurement Models for Compressive Acquisition

Compressive cryo-EM replaces exhaustive, rasterized data collection with acquisition of a reduced set of measurements $m \ll n$, where $n$ is the signal or image dimension. The process can be described abstractly as:

$y = \mathcal{A}(x^*) + \eta,$

where $x^* \in \mathbb{R}^n$ is the unknown high-resolution image, $\mathcal{A}$ is the sensing operator determined by the acquisition protocol, $y \in \mathbb{R}^m$ is the measurement vector, and $\eta \sim \mathcal{N}(0, \sigma^2 I)$ is additive noise.

Two principal classes of measurement models are employed:

• Pixel-space compressive acquisition: In imaging modalities such as cryo FIB-SEM and pixel-masked single-particle cryo-EM, binary masks $B_i$ are applied in the real space, often with subsequent spatial binning (sum pooling over $K \times K$ windows), yielding $m = b\cdot (n/K^2)$ measurements for $b$ masks and compression factor $C = n/m = K^2 / b$ (Shabeeb et al., 17 Nov 2025, Nicholls et al., 2022).
• Fourier-space compressive acquisition: Subsets of spatial frequencies are selected via binary or structured masks $B$ in the Fourier domain. For example, in programmable back-focal-plane mask setups,

$$\mathcal{A}_{\text{Fourier}}(x) = \mathcal{F}^{-1}(B \odot \mathcal{F} x),$$

capturing exactly $m = n/C$ Fourier coefficients (Shabeeb et al., 17 Nov 2025).

• Layered compressive sampling for 3D FIB-SEM: In slice-based volumetric imaging, each layer $x_\ell$ (vectorized 2D slice) is sampled at $M \ll \bar{N}$ probe locations via mask operator $P_{\Omega_\ell}$, i.e., $y_\ell^{cs} = P_{\Omega_\ell} x_\ell + n_\ell^{cs}$ (Nicholls et al., 2022).

Empirically, both random and data-tailored deterministic sampling strategies are supported, with mask design critically impacting reconstruction quality and dose reduction.

2. Targeted and Randomized Sampling Strategies

Sampling strategies govern the selection of mask coefficients or spatial pixels, balancing between randomization to ensure incoherence and data-driven targeting for information efficiency.

• Targeted Sampling (TS) in Cryo FIB-SEM: Each slice $\ell$ allocates $M_t = \lfloor \rho M \rfloor$ "targeted" samples, drawn according to a probability mass function $p_\ell$ informed by the previous layer's reconstruction $\hat{x}_{\ell-1}$, and $M_r = M - M_t$ random samples uniformly (Nicholls et al., 2022).
  • TS-intensity: $$p_\ell(q) = \hat{x}_{\ell-1, q} / \| \hat{x}_{\ell-1} \|_1$$
  • TS-gradient: $$p_\ell(q) = \operatorname{Grad}(\hat{x}_{\ell-1})_q / \| \operatorname{Grad}(\hat{x}_{\ell-1}) \|_1$$

The mask for layer $\ell$ is $\Omega_\ell =$ targeted $\cup$ random.

• Masking in cryoSENSE:
  • Pixel masks: Generated via coded apertures plus detector binning; higher compression is achievable.
  • Fourier masks: Uniformly random, low-frequency-biased (concentric rings), or random spokes in reciprocal space. Fourier-space masking is used predominantly for moderate compression and is preferred when edge preservation is critical (Shabeeb et al., 17 Nov 2025).

A practical implication is that the optimal ratio $\rho$ and mask type depend on anticipated image heterogeneity, the degree of compression, and downstream reconstruction priors.

3. Reconstruction Algorithms and Priors

Recovering high-fidelity images from undersampled data necessitates powerful priors:

• Blind Bayesian Dictionary Learning via BPFA (cryogenic FIB-SEM):
  • Slices are decomposed into overlapping patches $x_{\ell, i}$, each modeled as $x_{\ell, i} = D_\ell \alpha_{\ell, i}$, where $D$ is a learned dictionary and $\alpha_{\ell, i}$ sparse coefficients with BPFA priors. Latent variables (dictionary atoms, coefficients, sparsity masks) and hyperparameters (noise, dictionary size) are inferred via stochastic EM (sampling in E-step, M-step by likelihood maximization).
  • Reconstruction is patch-wise, averaged to form $\hat{x}_\ell$ (Nicholls et al., 2022).
• Sparse Prior Reconstruction (cryoSENSE):
  • The image $x$ is assumed sparse in a fixed basis $\Psi$ (DCT, wavelet, or finite differences for TV). Reconstruction solves

  $$\hat{x} = \arg\min_x \| \mathcal{A}(x) - y \|_2^2 + \lambda \| \Psi x \|_1$$

  via iterative soft-thresholding (ISTA) or its accelerated variant FISTA (Shabeeb et al., 17 Nov 2025).

• Generative Diffusion Prior (cryoSENSE):

  • A denoising diffusion probabilistic model (DDPM) is trained to capture the natural image manifold of protein cryo-EM micrographs. Guided reverse SDE integration is fused with measurement consistency, using Nesterov-style corrections to enforce $\mathcal{A}(x)$ alignment with $y$. This approach excels at higher compression regimes, particularly when pixel-space masking discards much detail (Shabeeb et al., 17 Nov 2025).

A plausible implication is that generative priors might outperform sparse priors, especially as compression increases and measurement information becomes less redundant.

4. Implementation and Computational Performance

• GPU acceleration (cryo FIB-SEM): CUDA/C++ implementations support real-time reconstruction. Slice recovery times are approximately $7.3 \pm 0.6$ s for a $1280 \times 960$ FIB-SEM slice, compared to a 3-minute physical slicing step, enabling on-the-fly mask design and active experiment adaptation (Nicholls et al., 2022).
• cryosense throughput: The method accommodates compression factors up to $C\approx2.5$–$2.7$, with high-fidelity reconstructions in both pixel and Fourier domains. The measurement and reconstruction pipeline is hardware-coordinated: pixel-space masks are implemented as coded apertures and binning on direct detectors; Fourier masks rely on phase plates or diffractive elements (Shabeeb et al., 17 Nov 2025).

The computational cost scales with dictionary or model size, patch number (in BPFA), or number of iterations (in ISTA/FISTA/DDPM schemes), motivating efficient implementation and, where needed, distributed inference.

5. Quantitative Performance and Empirical Validation

Performance is quantitated in terms of SSIM, PSNR, MSE, LPIPS, 3D structural fidelity (FSC, VC), and biological downstream compatibility.

Modality	Max. Compression	Metric Gains (vs baseline)*	Dose/Throughput
Cryo FIB-SEM (BPFA+TS)	20× dose reduc.	+0.75 SSIM (TS-intensity, 5% sample)	16% faster at 10%
cryoSENSE (pixel, DCT/WT)	$2.5\times$	SSIM $0.75-0.92$, LPIPS $0.05-0.15$	Up to $2.7\times$
cryoSENSE (pixel, diffusion)	$2.7\times$	LPIPS $0.12-0.17$, VC $0.97$
cryoSENSE (Fourier, DCT)	$2.5\times$	SSIM $0.66-0.92$, LPIPS $0.05-0.14$
Biological Model Fitting	$\approx1.3\times$	RMSD $2.07-2.34$ Å

* Relative to uniform density sampling or non-compressive raster scan; see experimental details in (Nicholls et al., 2022, Shabeeb et al., 17 Nov 2025).

Key findings include:

• Up to $20\times$ electron dose reduction (cryo FIB-SEM) with visually and metrically faithful reconstructions at 5% sampling (Nicholls et al., 2022).
• At 10% sampling: time per volumetric acquisition declines from 12.4 h to 10.5 h (16% speed-up) (Nicholls et al., 2022).
• cryoSENSE achieves $2.5\times$–$2.7\times$ higher throughput while retaining original 3D volume resolution (e.g., 6.65 Å at $C=2.5$, VC=0.98), with minimal penalty in downstream tasks such as conformational analysis and atomic model building (Shabeeb et al., 17 Nov 2025).
• Metric superiority of TS over uniform random sampling, and generative priors over hand-crafted ones at high compression, is consistently reported.

6. Extensions, Limitations, and Outlook

Compressive cryo-EM sensing strategies are extensible beyond the studied modalities:

• Generalization: Techniques can be adapted to cryo-STEM, ptychography, cryo-TEM tilt series, and 4D-STEM whenever adjacent views exhibit high similarity or lie on a low-dimensional manifold (Nicholls et al., 2022).
• Limitations: Efficacy assumes minimal drift/deformation between layers; high drift may limit utility of guided masks. BPFA and diffusion model scaling warrants attention for large fields of view or 3D joint inference. Current noise models are Gaussian; real detectors may require Poisson or heteroskedastic models for optimal performance (Nicholls et al., 2022, Shabeeb et al., 17 Nov 2025).
• Practical recommendations:
  • Calibrate targeted sampling fraction $\rho$ to specimen heterogeneity: smaller $\rho$ for variable samples.
  • For compression $C\leq2$ with Fourier masking and moderate information loss, employ sparse DCT/WT priors and ISTA; for $C>2$ and pixel-space masking, use diffusion priors with measurement guidance (Shabeeb et al., 17 Nov 2025).
  • Explore hybrid 3D sampling combining in-plane and depth-wise adaptive patterns.
  • Investigate alternative priors such as deep generative models if BPFA becomes a computational bottleneck (Nicholls et al., 2022).

A plausible implication is that future directions include integrating adaptive mask optimization, deeper learning-based priors, and bespoke noise models to further push the limits of dose and throughput constraints while maintaining or enhancing structure preservation in cryo-EM modalities.