Adaptive information-maximization encoding for ghost imaging--A general Bayesian framework under experimental physical constraints

Ghost imaging (GI) has demonstrated diverse imaging capabilities enabled by its encoding-decoding-based computational imaging mechanism. Accordingly, information-theoretic studies have emerged as a promising avenue for probing the fundamental performance bounds of of GI and related computational imaging paradigms. However, the design of information-theoretically optimal encoding strategies remains largely unexplored, primarily due to the intractability of the prior probability density function (PDF) of an unknown scene. Here, by leveraging the ability of recursively estimating the PDF of the object to be imaged via Bayesian filtering, we propose to establish an adaptive information-maximization encoding (AIME) design framework. Based on the adaptively estimated posterior PDF from previously acquired measurements, the expected information gain of subsequent detections is evaluated and maximized to design the corresponding encoding patterns in a closed-loop manner. Within this framework, the theoretical form of the information-optimal encoding under representative physical constraints is analytically derived. Corresponding experimental results show that, GI systems employing information-optimal encoding achieve markedly improved imaging performance compared with conventional fixed point-to-point imaging without relying on additional heuristic regularization schemes, particularly in low signal-to-noise ratio regimes. Moreover, the proposed strategy consistently enables significantly enhanced information acquisition capability compared with existing encoding strategies, leading to substantially improved imaging quality. These results establish a principled information-theoretic foundation for optimal encoding design in computational imaging paradigms,provided that the forward model can be accurately characterized.

💡 Research Summary

This paper introduces a general adaptive information‑maximization encoding (AIME) framework for ghost imaging (GI) that leverages Bayesian filtering to estimate the posterior probability density function (PDF) of the unknown object in real time. The authors start by modeling the GI forward process as a linear measurement equation z = β H x + n, where x is the target image, H contains the illumination patterns (the encoding vectors hₖ), β captures system‑dependent scaling, and n represents detection noise. An initial Gaussian prior p₀(x) = N(bx₀, bP₀) is assumed based on generic scene statistics. After each measurement, the posterior remains Gaussian, p(x|Zₖ) = N(bxₖ, bPₖ), and is updated using Kalman‑like equations (3a, 3b). The noise is approximated as Gaussian with variance Rₖ = ω² hₖᵀ b xₖ₋₁⁻¹, reflecting photon‑shot noise.

The core of AIME is the selection of the next encoding pattern hₖ₊₁ by maximizing an information‑theoretic criterion evaluated on the current posterior. Two criteria are considered:

1. Mutual Information (MI) – The expected MI between the next measurement zₖ and the object x, conditioned on past data, reduces to
   I = ½ log(1 + β² hₖᵀ bPₖ₋₁ hₖ / Rₖ).
   Maximizing I is equivalent to maximizing the objective L_MI = hₖᵀ bPₖ₋₁ hₖ / Rₖ.

2. Fisher‑information‑based Cramér‑Rao Bound (CRB) – The mean CRB (trace of the posterior covariance) after the next measurement is
   mCRB = Tr(bPₖ₋₁) − hₖᵀ bPₖ₋₁² hₖ / ( hₖᵀ bPₖ₋₁ hₖ + β² Rₖ ).
   Minimizing mCRB leads to maximizing L_CRB = hₖᵀ bPₖ₋₁² hₖ / ( hₖᵀ bPₖ₋₁ hₖ + β² Rₖ ).

Both objectives depend only on the current posterior covariance, making them computationally tractable.

Physical constraints on the illumination patterns are incorporated explicitly. The authors first analyze an ideal total‑energy (ℓ₁‑norm) constraint ‖hₖ‖₁ = C, with non‑negativity. Under this constraint, the optimization
max hₖ hₖᵀ bPₖ₋₁ hₖ / Rₖ subject to ‖hₖ‖₁ = C, hₖ ≥ 0
has a closed‑form solution (Theorem 1): the optimal pattern concentrates all energy on the pixel i* that maximizes (bPₖ₋₁)_{ii} · x_i. In other words, the optimal encoding becomes a point‑wise scan whose order is driven by the evolving posterior statistics rather than a predetermined raster order. For the CRB‑based objective, an analytical solution is not derived, but numerical simulations reveal a similar evolution: early measurements use diffuse patterns, gradually focusing into a localized spot as the posterior tightens.

Experimental validation is performed on a DMD‑based GI setup. The authors compare AIME‑ℓ₁, AIME‑CRB, conventional random binary patterns, and a fixed point‑to‑point scanning scheme across a range of sampling ratios (10 %–100 %) and signal‑to‑noise ratios (≈4 dB to 10 dB). Key findings include:

• At low SNR (≈5 dB) and 50 % sampling, AIME successfully reconstructs recognizable images, whereas conventional full‑sampling imaging fails to recover the object.
• Quantitatively, AIME improves structural similarity index (SSIM) by ~0.2 and peak‑signal‑to‑noise ratio (PSNR) by 3–4 dB relative to the baseline.
• Visualizations of the encoding patterns confirm the theoretical prediction: they start broadly and become increasingly concentrated on high‑uncertainty regions as measurements accumulate.
• When realistic hardware constraints (binary modulation, maximum illumination power) replace the ideal ℓ₁ constraint, the optimal patterns remain “object‑dependent”: they allocate more intensity to pixels with larger posterior variance, confirming the flexibility of the framework.

The authors emphasize that AIME does not rely on sparsity assumptions or heuristic regularization; instead, it directly maximizes the expected information gain under explicit physical limits. Consequently, it provides a principled route to approach the fundamental performance bounds of computational imaging systems. The demonstrated robustness in low‑light, low‑SNR regimes suggests immediate relevance for applications such as lidar, low‑dose X‑ray imaging, and super‑resolution microscopy, where photon budget and detector noise are critical bottlenecks.

In summary, the paper delivers (i) a Bayesian recursive estimator for the object’s PDF, (ii) a closed‑form or numerically tractable information‑optimal encoding design under various constraints, and (iii) experimental evidence that adaptive, information‑maximizing illumination patterns substantially outperform traditional non‑adaptive schemes, thereby establishing a solid information‑theoretic foundation for future computational imaging designs.