Lensless Compressive Imaging

Updated 1 April 2026

Lensless Compressive Imaging is a computational imaging technique that replaces traditional lenses with coded masks to encode scenes, exploiting compressive sensing theory.
It leverages structured sensing matrices and advanced reconstruction algorithms to achieve high SNR and enable modalities like high-speed video and endoscopy.
The approach offers benefits such as broad wavelength compatibility and compact design while facing trade-offs in acquisition speed and computational cost.

Lensless Compressive Imaging (LCI) is an approach to computational imaging that entirely eliminates refractive or diffractive optics, substituting the lens element with a coded mask or programmable aperture. This paradigm exploits compressive sensing (CS) theory by encoding the high-dimensional scene onto a lower-dimensional measurement space through global multiplexing on the sensor. Subsequent reconstruction algorithms, leveraging sparsity priors or learned image models, aim to recover high-fidelity images, videos, or task-specific features from sub-Nyquist measurements. LCI architectures are characterized by their minimal optical stack, broad wavelength compatibility, and the ability to integrate customized signal processing directly into the acquisition pipeline.

1. Architectural Principles and Forward Model

The canonical LCI system consists of a two-dimensional array of aperture elements (mask or programmable assembly) and a single non-imaging photodetector or a bare sensor array; no lens is present (Huang et al., 2013, Yuan et al., 2015). Each measurement corresponds to a modulation of the scene via a mask pattern, producing an integrated scalar (for single-pixel detectors) or a globally coded sensor response (for sensor arrays or diffuser-based cameras). The measurement model is universally linear:

$y = A\,x + n$

where $x\in\mathbb{R}^{N}$ represents the pixelized scene, $y\in\mathbb{R}^{M}$ the vector of measurements (with $M\ll N$ in the compressive regime), $A\in\mathbb{R}^{M\times N}$ the system or sensing matrix encoding the optical forward operator, and $n$ measurement noise.

Programmable aperture single-pixel systems: The aperture assembly is typically a binary or grayscale mask (e.g., LCD, DMD, or MEMS array) (Huang et al., 2013), producing scalar bucket measurements.
Diffuser/mask-on-sensor architectures: A scattering element (random phase diffuser, random multi-focal lenslet, coded amplitude mask) is placed directly above a pixelated sensor, so each scene point is mapped to a spatially-extended, often pseudo-random point-spread function (PSF) (Kabuli et al., 4 Feb 2026, Monakhova et al., 2021).

In both cases, CS recovery is viable when $A$ satisfies appropriate incoherence conditions with respect to the scene’s sparsifying basis.

2. Sensing Matrix Design and Optical Encoding

Optical encoding in LCI is designed to maximize information throughput under physical and computational constraints.

Hadamard/Bernoulli random masks: Hardware-implemented by mapping binary or $\pm1$ matrix entries to aperture transmittance; Hadamard-based matrices provide favorable mutual coherence and allow fast transforms (Huang et al., 2013).
Pseudo-random or physically random masks: Stochastic SLM designs (e.g., random particle distributions) realize a random sampling operator in modalities where digital SLMs are infeasible (Schaake et al., 2018).
Diffuser/lenslet-based encoding: Masks with engineered PSFs, such as random diffusers or random multi-focal lenslet (RML) masks, spatially mix scene points over the sensor in a manner optimal for CS (Kabuli et al., 4 Feb 2026, Kabuli et al., 24 Jan 2025). The choice of PSF directly controls the “multiplexing” or coupling among scene elements.

Recent advancements in mask fabrication (precision random lenslet arrays (Kabuli et al., 4 Feb 2026)) allow more nuanced trade-offs between information transfer (measured via modulation transfer function, mutual information) and system invertibility.

3. Measurement Noise, Statistical Limits, and SNR Scaling

A central question in LCI is the sensitivity to measurement noise and how system SNR scales with resolution, noise sources, and pixel count (Jiang et al., 2014, Jiang et al., 2014).

Noise model: Typically, both photon (shot) noise (modeled as Poisson) and additive electronic (Gaussian) noise are present.
SNR invariance: For properly designed sensing matrices (e.g., modified Hadamard), the total output SNR is asymptotically independent of image resolution:

$\mathrm{SNR}_{\rm LCI} \ge \frac{X^0}{\sqrt{2X^0 + 4\sigma^2}}$

where $X^0$ is the scene brightness and $x\in\mathbb{R}^{N}$ 0 the variance of additive noise (Jiang et al., 2014). By contrast, conventional pinhole or lens-based imagery SNR decays as $x\in\mathbb{R}^{N}$ 1 for large $x\in\mathbb{R}^{N}$ 2 due to the uncorrelated accumulation of additive noise.

Multiplexing and estimation limits: Estimation-theoretic analyses show that for dense objects, increased spatial multiplexing in the optical encoder degrades Cramér-Rao bound (CRB) performance, while sparse targets are robust to higher degrees of multiplexing (Kabuli et al., 24 Jan 2025). Mask optimization thus depends on anticipated object structure and noise regime.

4. Reconstruction Algorithms and Computational Strategies

LCI inverts the ill-posed measurement model via regularized optimization, leveraging structured signal priors.

Classical convex solvers:
- ℓ₁-sparsity or total variation (TV) minimization is standard for promoting compressibility (Yuan et al., 2015). For TV, the problem is:
$x\in\mathbb{R}^{N}$ 3

with $x\in\mathbb{R}^{N}$ 4 a gradient or sparsifying transform. - Algorithms such as ISTA/FISTA, ADMM, and TwIST are widely used; convergence is typically declared when relative solution change drops below $x\in\mathbb{R}^{N}$ 5.
Patch-based local sparsity and denoising:
- SLOPE introduces local transform domain sparsity by patch extraction, DCT transform, soft-thresholding, and aggregation (Yuan et al., 2015).
Untrained deep priors:
- Deep image prior frameworks treat the reconstruction as an optimization over a convolutional generator’s weights, constrained only by the measurement consistency ( $x\in\mathbb{R}^{N}$ 6), effecting powerful implicit regularization (Monakhova et al., 2021). No external training data is required.
Task-driven and hybrid priors:
- Generative models (e.g., diffusion, learned Wiener filters) can be employed for denoising and high-frequency restoration, especially in low-light settings (Liu et al., 7 Jan 2025).
- For specific tasks (e.g., edge detection), the measurement matrix is preconditioned by the inverse of a filtering operator to recover the task output directly, bypassing post-processing (Zheng et al., 2023).

5. Extensions: Temporal, Multispectral, Edge, and Endoscopic Imaging

The flexibility of LCI enables diverse imaging modes by enriching the system model and reconstruction.

Temporal multiplexing and high-speed video:
- Rolling-shutter sensors distribute measurements over time, permitting temporal encoding of videos into single still captures (Antipa et al., 2019). The measurement operator is block diagonal, aligning individual sensor regions to distinct time frames. With appropriate priors (3DTV, learned models), video volumes of up to 140 frames at >4.5 kHz are reconstructed from a single capture.
Edge and feature-driven LCI:
- By absorbing the inverse of an edge filter directly into the forward model ( $x\in\mathbb{R}^{N}$ 7), systems can directly reconstruct spatiotemporal edge maps sequence-wise from a single data cube, without post-processing (Zheng et al., 2023).
Multispectral, polarization, multi-view:
- Coded spectral or polarization masks enable snapshot spectral or polarimetric imaging (Monakhova et al., 2021, Jiang et al., 2013).
- Multi-view extension is realized by having multiple detectors behind the aperture; joint sparse recovery fuses the correlated views for improved SNR or super-resolution (Jiang et al., 2013).
Lensless compressive endoscopy:
- Speckle illumination patterns generated from multicore fibers, combined with TV-regularized CS inversion, enable cellular-scale imaging at compression rates well below unity (Guérit et al., 2018, Guérit et al., 2021). Innovations such as partial speckle scanning (multiple shifts per SLM pattern, exploiting “memory effect”) accelerate acquisition while preserving RIP conditions (Guérit et al., 2021).

6. Physical Implementation, Mask Design, and Information Metrics

Recent work emphasizes co-optimization of mask design, hardware implementation, and quantification of information throughput.

Mask fabrication and PSF engineering:
- Random multi-focal lenslet (RML) masks with controlled multiplexing extend the trade-off space between diffusivity and invertibility, exhibiting higher modulation transfer function (MTF) and mutual information than diffusers under equivalent noise/quantization (Kabuli et al., 4 Feb 2026).
Information-theoretic metrics:
- Measurement mutual information quantifies average scene-to-measurement dependency, found to be robust against quantization noise for optimized RML masks, unlike high-multiplexing diffusers (Kabuli et al., 4 Feb 2026).
Coded illumination:
- Orthogonal block/“shifting dot” patterns (illumination-side) dramatically improve the conditioning of the joint forward operator, yielding 10–15 dB PSNR improvements in experimental prototypes and facilitating closed-form separable recovery (Zheng et al., 2021).

Encoding type	MTF (high f)	Mutual Info (bits @σ=5)	Task-specific?
Lens (reference)	maximal	7.0	No
RML mask	high (>diff)	4.0	No
Diffuser	lowest	1.9	No
A' $x\in\mathbb{R}^{N}$ 8	-	-	Yes (edges)

7. Limitations, Trade-offs, and Practical Considerations

Notable limitations and trade-offs in LCI include:

Acquisition speed: Pattern switching rates (LCD, DMD) and integration times often limit temporal resolution in single-pixel geometries (Yuan et al., 2015). New paradigms such as time-tagged ultrafast detection or rolling-shutter video address this for dynamic scenes (Satat et al., 2016, Antipa et al., 2019).
Reconstruction cost: Convex and deep learning-based solvers require significant compute; untrained networks offer superior performance but demand high memory and prolonged optimization (Monakhova et al., 2021).
Calibration and stability: Mask or PSF calibration must remain valid; the angular memory effect or physical perturbations in the mask/sensor can degrade model accuracy (Zheng et al., 2023).
Scene-dependent optimality: The optimal degree of optical multiplexing is contingent on anticipated sparsity and noise regime. For dense scenes and read-noise dominance, opt for low-multiplexing encoders; for sparse scenes under shot-noise, aggressive multiplexing is permissible (Kabuli et al., 24 Jan 2025).
Throughput and SNR: While LCI enables high-SNR scaling in the large- $x\in\mathbb{R}^{N}$ 9 regime, precise light budgets and photon statistics must be managed in low-light or highly compressed scenarios (Jiang et al., 2014, Liu et al., 7 Jan 2025).

In summary, lensless compressive imaging is a highly flexible computational imaging architecture that eliminates lenses, utilizes spatially varying mask codes, and exploits compressive sensing with advanced measurement and reconstruction paradigms. Ongoing advances in mask design, information-theoretic system optimization, and deep learning-driven inverse methods are closing the performance gap to conventional optics while enabling new applications in scientific imaging, biomedical microscopy, single-shot video, and ultra-compact deployable sensors (Yuan et al., 2015, Kabuli et al., 4 Feb 2026, Kabuli et al., 24 Jan 2025, Zheng et al., 2023, Zheng et al., 2021, Monakhova et al., 2021).