Light-Enhancement Curves (LE-curves)

Updated 4 January 2026

Light-Enhancement Curves (LE-curves) are nonlinear pixel-intensity mappings designed for adaptive low-light image enhancement via both per-pixel and global adjustments.
They integrate various learning frameworks such as CNNs, diffusion models, and reinforcement learning to estimate curve parameters efficiently while enforcing strict monotonicity and concavity.
LE-curves are seamlessly incorporated into image enhancement pipelines, significantly boosting performance metrics like PSNR and facilitating robust, real-time applications.

Light-Enhancement Curves (LE-curves) are flexible, learnable, and highly efficient nonlinear pixel-intensity mappings that underpin most state-of-the-art low-light image enhancement pipelines. Modern LE-curve methodologies allow both per-pixel and global nonlinear adjustments, often serving as the backbone for fast, high-fidelity, and real-time enhancement systems. The mathematical and algorithmic versatility of LE-curves enables their integration with deep learning, reinforcement learning, diffusion models, and Retinex-based decompositions, supporting both supervised and zero-reference/unsupervised learning objectives. These curves are engineered to be differentiable, strictly monotonic, and often iteratively composed, guaranteeing stable enhancement, color fidelity, and adaptability to variable illumination regimes.

1. Mathematical Forms and Parametric Families of LE-Curves

LE-curves span a spectrum of parametric forms, with the quadratic "Zero-DCE curve" being foundational in recent literature. The standard parametric form is: $f(x;\alpha) = x + \alpha\, x\,(1-x),\qquad x\in[0,1],\;\alpha\in[-1,1]$ This mapping is strictly increasing and differentiable on $[0,1]$ for all valid $\alpha$ . Multiple higher-order mappings are built by iterative application: $I^{(n)}(x) = I^{(n-1)}(x) + \alpha_n(x)\,I^{(n-1)}(x)\left(1 - I^{(n-1)}(x)\right)$ with typically $N=8$ recursions and spatially varying $\alpha_n(x)$ per channel (Zero-DCE (Guo et al., 2020, Li et al., 2021), KinD-LCE (Lei et al., 2022), BDCE (Huang et al., 2023), ALL-E (Li et al., 2023), ReLLIE (Zhang et al., 2021)).

CURVE (Ogino et al., 29 May 2025) employs a global cubic Bézier curve for global tone mapping: $C(q) = (1-q)^3 p_0 + 3q(1-q)^2 p_1 + 3q^2(1-q) p_2 + q^3 p_3$ with fixed endpoints and two learnable control points $(p_1, p_2)$ . SACC (Wang et al., 2022) introduces a discrete concave curve $g$ constructed through a double-summed nonnegative second-derivative, enforcing concavity and monotonicity by design.

Self-DACE (Wen et al., 2023) generalizes the quadratic using pixel-wise magnitude and pivot parameters $(\alpha,\beta)$ , combined with a sigmoid gating: $y = x + \alpha \frac{1}{\beta} S(\beta; x) x(\beta-x)$ where $S(\beta;x)$ is a sigmoid function suppressing over-enhancement for $x>\beta$ .

2. Curve Parameter Estimation: Architectures and Learning Frameworks

Parameter estimation for LE-curves is implemented through various neural architectures adapted to enhancement goals:

Convolutional Networks: DCE-Net (Zero-DCE, BDCE) is a shallow fully convolutional network outputting per-pixel, per-channel $\alpha$ maps. Zero-DCE++ uses depthwise-separable convolutions for extreme lightness and inference speed (Li et al., 2021).
Diffusion Models: BDCE models the posterior over curve parameters using a U-Net-based diffusion model on downsampled images, upsampling to full resolution for efficient HR support (Huang et al., 2023).
Reinforcement Learning: ReLLIE, CURVE, and ALL-E use RL agents. Actions correspond either to per-pixel curve parameters (A3C, as in ReLLIE, ALL-E) or global Bézier control points (CURVE, via SAC). Reward designs are driven by non-reference losses, human aesthetic estimators (NIMA in ALL-E), or CLIP-based semantic rewards (CURVE) (Ogino et al., 29 May 2025).
Retinex Fusion + Curve: KinD-LCE applies LE-curves to illumination maps extracted via Retinex decomposition and optimized jointly with reflection restoration modules (Lei et al., 2022).
Concave Curve Construction: SACC predicts a nonnegative discrete second derivative, integrating twice and normalizing, providing a fully constrained concave, monotonic curve suitable for adaptation in high-level machine vision tasks (Wang et al., 2022).

3. Training Objectives: Reference-Free, Supervised, and Self-Supervised Schemes

Zero-Reference Losses:

A representative set of unsupervised, reference-free loss functions includes:

Spatial Consistency Loss ( $L_\text{spa}$ ): Preserves local patch-wise contrast across enhancement.
Exposure Control Loss ( $L_\text{exp}$ ): Encourages local mean intensity toward a specified "well-exposed" target.
Color Constancy Loss ( $L_\text{col}/L_\text{crl}/L_\text{gcol}$ ): Favours gray-world channel balance (prevents color cast).
Illumination Smoothness/Tot. Var. Loss ( $L_\text{tv}$ ): Penalizes spatial roughness in parameter maps (or curve outputs).

Reference-based methods (KinD-LCE, BDCE) include pixel-wise MSE against ground-truth illumination or normal-light images, often augmented with structure-aware (gradient) and denoising loss terms (Lei et al., 2022, Huang et al., 2023).

Self-supervised adaptation (SACC) penalizes loss on high-level proxy tasks such as rotated jigsaw puzzles, aligning low-light enhanced features with those of normal-light images, while enforcing concavity and monotonicity on the learned curve (Wang et al., 2022).

Reinforcement learning pipelines define reward as the negative sum of these losses (ReLLIE, ALL-E), or more recently, as the improvement in CLIP-based or aesthetic quality scores (CURVE, ALL-E) (Li et al., 2023, Ogino et al., 29 May 2025).

4. Integration with Image Enhancement Pipelines

LE-curve evaluation and application protocols generally proceed as follows:

Per-pixel or global parameter estimation: Input image $\to$ curve-parameter map(s), either per-pixel/channel or global/global + patch.
Iterative curve application: Nonlinear mapping is composed over $N$ steps (commonly 6--8).
Fusion with denoising (optional): As LE-curve-based brightening amplifies noise, specialized denoisers are either interleaved per step (BDCE) or integrated as a subsequent stage (Self-DACE, ReLLIE) (Huang et al., 2023, Zhang et al., 2021, Wen et al., 2023).
Retinex decompositions: LE-curves are applied to illumination maps with reflectance fusion for maintaining spatial and chromatic fidelity (KinD-LCE) (Lei et al., 2022).

Example pseudocode (Zero-DCE/BDCE-type):

I_e = I_0
for n in range(N):
    I_e = I_e + alpha_n * I_e * (1 - I_e)
    # Optionally, denoise I_e at each step

Self-DACE/BDCE pipelines interleave denoising after each curve application.

5. Impact on Low-Light Enhancement and High-Level Vision

LE-curve methods achieve state-of-the-art efficiency and enhancement quality across standard benchmarks. Notable empirical findings include:

Zero-DCE and KinD-LCE achieve PSNR gains of 2--3 dB by curve recursion and illumination-map integration (Li et al., 2021, Lei et al., 2022).
CURVE and ALL-E outperform deep U-Net and GAN baselines in human preference and no-reference metrics by leveraging flexible global curves and aesthetic/CLIP-based rewards (Ogino et al., 29 May 2025, Li et al., 2023).
BDCE outperforms prior curve- and learning-based models on PSNR/SSIM for high-resolution inputs due to the diffusion-enhanced parameter estimation and denoising (Huang et al., 2023).
SACC provides robust gains (mAP, EPE) on downstream tasks—classification, detection, flow—by self-supervised feature alignment via constrained concave curves (Wang et al., 2022).
All curve-based methods are highly parameter efficient: Zero-DCE++ has $10^4$ parameters, KinD-LCE $\sim$ 10k--100k, Self-DACE $\sim$ 70k, and BDCE achieves full HD/4K scalability.

6. Architectural Constraints and Theoretical Considerations

Strict monotonicity and concavity constraints are often imposed:

Monotonicity: Enforced by analytic curve forms $f(x;\alpha)$ with bounded parameter ranges, or implicitly by second difference integration (SACC) (Wang et al., 2022).
Concavity: Empirically justified by the prevalence of concave response functions in cameras, enforced by nonnegative second-derivative parametrizations (Wang et al., 2022).
Smoothness: Illumination/parameter TV loss avoids checkerboard or “halo” artifacts at boundaries.

These constraints ensure that local details and global structure are preserved while avoiding unnatural intensity inversions. Ablations confirm their necessity for stable training and high downstream task performance.

Method	Curve Parametrization	Parameterization	Key Constraints
Zero-DCE/BDCE	Iterated quadratic	$\alpha(x), N$	Monotonic, $[-1,1]$
KinD-LCE	Iterated quadratic (illum. map)	$\alpha_i(x), N$	Monotonic, TV
CURVE	Cubic Bézier (global)	$p_1,p_2$	Endpoint fixed, smooth
SACC	Discrete, concave curve (pixel-wise)	$a_k \ge 0$ , double-sum	Monotonic, concave
Self-DACE	AAC: parametric w/ gating	$\alpha,\beta$	Monotonic, local pivot
ALL-E, ReLLIE	Iterated quadratic	$A^t(x)$ , RL	Monotonic, RL-stable

7. Extensions, Limitations, and Future Directions

LE-curves have demonstrated strong generalizability in both supervised and unsupervised/zero-reference regimes. Their modularity facilitates plug-and-play integration into larger enhancement, detection, and recognition frameworks. However, several limitations persist:

Extreme low-light noise can overwhelm non-denoising pipelines; hybrid denoise-enhance approaches (BDCE, Self-DACE) address this at increased complexity (Huang et al., 2023, Wen et al., 2023).
LE-curve parameter estimation is sensitive to the spatial/temporal statistics of input lighting.
Fixed global "well-exposed" targets or global curves may be suboptimal in severe HDR or scene-dependent contexts (Wang et al., 2022).

Emerging trends include fully differentiable and self-supervised enhancement pipelines using feature-level proxies (rotated jigsaw, CLIP/NIMA rewards), advanced global curve families (Bézier, S-shaped), and hybrid diffusion or RL-based parameter inference strategies offering both interpretability and robust automation across imaging scenarios.

Key References: (Guo et al., 2020, Li et al., 2021, Zhang et al., 2021, Lei et al., 2022, Wang et al., 2022, Li et al., 2023, Wen et al., 2023, Huang et al., 2023, Ogino et al., 29 May 2025)